maple 7 na resolução de cálculos algébricos

CURSO DE INFORMAacuteTICA NA EDUCACcedilAtildeO

USO DE SOFTWARE MATEMAacuteTICO

INTRODUCcedilAtildeO DO MAPLE

NA RESOLUCcedilAtildeO DE CAacuteLCULOS ALGEacuteBRICOS

Marcus Antonio de Oliveira Santos

Porto Velho ndash RO

2014

Sumaacuterio

O que eacute MAPLE helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 4

Onde surgiu helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 4

A folha de trabalho 5

Ajuda 5

Operaccedilotildees aritmeacuteticas 6

Alguns operadores helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 6

Nuacutemeros 6

Representaccedilatildeo interna de uma expressatildeo helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 7

Resultado de uma operaccedilatildeo 8

Atribuiccedilatildeo de valores helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 10

Expressotildees algeacutebricas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 10

Reduccedilatildeo de Polinocircmios helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 11

Multiplicaccedilatildeo de Monocircmio por Monocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 12

Multiplicaccedilatildeo de Polinocircmio por Monocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 12

Multiplicaccedilatildeo de Polinocircmio por Polinocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 13

Divisatildeo de Monocircmio por Monocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 14

Divisatildeo de Polinocircmio por Monocircmio 14

Divisatildeo de Polinocircmio por Polinocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 15

Adiccedilatildeo algeacutebrica de Polinocircmios helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 16

Produtos Notaacuteveis helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 17

Quadrado da diferenccedila helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 18

Quadrado da soma helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 18

Fraccedilotildees algeacutebricas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 19

Equaccedilatildeo fracionaacuteria helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 20

Somatoacuterio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 22

Sistemas de equaccedilotildees de 1deg grau com duas incoacutegnitas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 23

Matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 26

Matriz transposta 26

Adiccedilatildeo de matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 27

Subtraccedilatildeo de matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 28

Multiplicaccedilatildeo de um nuacutemero real por matriz helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 28

Multiplicaccedilatildeo de matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 28

Determinantes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 29

Matriz inversa helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 31

Equaccedilatildeo do 2deg grau helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 33

Graacutefico da funccedilatildeo polinomial de 1deg grau helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 34

Graacutefico da funccedilatildeo polinomial de 2deg grau (Funccedilatildeo Quadraacutetica) helliphelliphelliphelliphelliphelliphelliphelliphellip 36

Fatorial 38

Nuacutemeros Binomiais 39

Combinaccedilatildeo simples 39

Arranjo simples 40

Equaccedilotildees irracionais 41

Conversotildees 42

Arcos e Radianos 42

Relaccedilotildees trigonomeacutetricas 43

Escalas termomeacutetricas 44

Nuacutemeros complexos 45

Limites 46

Lista de funccedilotildees 47

Resoluccedilatildeo de problemas com a inserccedilatildeo de dados 47

O que eacute Winplot 50

Abrindo o Winplot e construindo graacuteficos 50

Abrindo o Winplot 50

Criando o graacutefico da funccedilatildeo de 1ordm grau f(x) = 2x + 5 50

Criando o graacutefico da funccedilatildeo de 2ordm grau f(x) = x2 52

Referecircncias bibliograacuteficas 56

Respostas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 57

O que eacute MAPLE

MAPLE

(Manipulation Algebraic Polinomial Linear Equation)

O MAPLE eacute um programa de computador produzido por Waterloo Maple Inc Eacute um tipo de

sistema conhecido como ldquosistema de manipulaccedilatildeo algeacutebricardquo ou ldquosistema matemaacutetico

computacionalrdquo O MAPLE possui quatro aspectos gerais que satildeo Aspectos algeacutebricos

Aspectos numeacutericos Aspectos graacuteficos e Aspectos de programaccedilatildeo Todos estes aspectos

estatildeo integrados formando um corpo uacutenico Tais sistemas satildeo uacuteteis na resoluccedilatildeo faacutecil e precisa

de uma diversidade de problemas em matemaacutetica Nos dias de hoje estes programas tecircm se

mostrado instrumentos poderosos para o trabalho de cientistas engenheiros e matemaacuteticos

MAPLE tambeacutem eacute um sistema de aacutelgebra computacional comercial de uso geneacuterico Constitui

um ambiente informaacutetico para a computaccedilatildeo de expressotildees algeacutebricas e siacutembolos permitindo

o desenho de graacuteficos a duas ou trecircs dimensotildees O seu desenvolvimento comeccedilou em

1981pelo Grupo de Computaccedilatildeo Simboacutelica na Universidade de Waterloo em Waterloo no

Canadaacute proviacutencia de Ontaacuterio

Desde 1988 o MAPLE tem sido desenvolvido e comercializado pela Maplesoft uma

companhia canadense tambeacutem baseada em Waterloo Ontaacuterio A versatildeo atual eacute MAPLE 110

Onde surgiu

O primeiro conceito de Bordo surgiu de uma reuniatildeo em novembro de 1980 na Universidade

de Waterloo Pesquisadores na universidade pretendiam comprar um computador poderoso o

suficiente para executar Macsyma Em vez disso ficou decidido que iriam desenvolver seu

proacuteprio sistema de aacutelgebra computacional que seria capaz de rodar em computadores com

preccedilos mais razoaacuteveis Assim o projeto comeccedilou com o objetivo de criar um sistema de

aacutelgebra simboacutelica acessiacutevel aos investigadores e estudantes

O desenvolvimento inicial de Bordo procedeu muito rapidamente com a primeira versatildeo

limitada aparecendo em Dezembro de 1980 MAPLE foi demonstrada primeiro em

conferecircncias no iniacutecio de 1982

Ateacute ao final de 1983 mais de 50 universidades tinham coacutepias de Bordo instalados em suas

maacutequinas Devido ao grande nuacutemero de pedidos de licenciamento e de apoio em 1984 o

grupo de investigaccedilatildeo se organizou com WATCOM Products Inc para licenciar e distribuir

MAPLE

Em 1988 devido agraves crescentes solicitaccedilotildees de apoio Waterloo MAPLE Inc Foi fundada A

meta inicial da empresa era de gerenciar a distribuiccedilatildeo do software Eventualmente a empresa

evoluiu para ter um departamento de I amp D em que grande parte do desenvolvimento da

MAPLE eacute feito hoje O desenvolvimento significativo de Bordo continua nos laboratoacuterios da

universidade de investigaccedilatildeo incluindo o Laboratoacuterio de Computaccedilatildeo Simboacutelica da

Universidade de Waterloo o Ontaacuterio Research Center for Computer Aacutelgebra da Universidade

de Western Ontaacuterio e laboratoacuterios de outras universidades em todo o mundo

A folha de trabalho

Nos microcomputadores com o MAPLE e instalado a folha de trabalho ou worksheet eacute

disparada clicando-se no iacutecone do programa Ela eacute o principal meio para gravar e ler os

trabalhos desenvolvidos no MAPLE A worksheet eacute um caderno virtual de anotaccedilotildees de

caacutelculos A vantagem do caderno virtual eacute que qualquer coisa jaacute escrita pode ser modificada

sem necessidade de fazer outras alteraccedilotildees pois o trabalho se ajusta automaticamente agraves

mudanccedilas A worksheet tem quatro tipos de linhas 1 linhas de entrada de comandos que

usam a cor vermelha e satildeo precedidas pelo sinal de pronto ldquogtrdquo 2 linhas de saiacuteda dos

comandos na cor azul 3 linhas de texto na cor preta e 4 linhas de graacutefico Algumas dessas

linhas podem ser convertidas umas nas outras Mais detalhes sobre as worksheets podem ser

encontrados no New Users Tour depois de clicar em Help

Ajuda

O MAPLE possui um sistema de ajuda interativo chamado help on line Para pedir ajuda

sobre uma funccedilatildeo ou qualquer assunto pertinente deve-se preceder a funccedilatildeo ou o assunto com

um sinal de interrogaccedilatildeo Por exemplo

gt maple

A paacutegina de ajuda conteacutem vaacuterias partes 1 descriccedilatildeo da sintaxe 2 descriccedilatildeo detalhada 3

exemplos e 4 palavras chaves relacionadas ao assunto A partir das palavras chaves pode-se

ldquonavegarrdquo pelo sistema de ajuda ateacute que a informaccedilatildeo desejada seja encontrada Outra

modalidade de ajuda consiste na procura por assunto No caso do MAPLE for Windows deve-

se clicar com o mouse no Help e depois Introduction (primeiro de cima para baixo) A nova

janela teraacute uma seccedilatildeo cinza na parte superior Clique em Mathematics por exemplo Na

segunda coluna procure por um sub-toacutepico e assim por diante Os usuaacuterios mais persistentes

podem procurar ajuda clicando em Topic Search ou Full Text Search depois de ter

clicado em Help

Apresentaremos uma introduccedilatildeo aos comandos baacutesicos para utilizar o programa MAPLE e

seus aplicativos graacuteficos no ensino fundamental e meacutedio

Operaccedilotildees Aritmeacuteticas

O MAPLE usa as operaccedilotildees na seguinte ordem

1deg potenciaccedilatildeo ( ^ ou )

2deg multiplicaccedilatildeo ( ) e divisatildeo ( )

3deg adiccedilatildeo ( + ) e subtraccedilatildeo ( )

Alguns Operadores

sqrt (lsquorsquo) raiz quadrada

gt sensor (soacute podemos digitar apoacutes o sensor)

resolve mas natildeo mostra

daacute a resposta e mostra na tela

= atribui nome a nuacutemero ou expressatildeo

gt restart limpa a memoacuteria

gt simplify (lsquorsquo) simplifica uma expressatildeo

gt expand (lsquorsquo) expande uma expressatildeo

gt solve (lsquorsquo) resolve uma equaccedilatildeo

Sum mostra o somatoacuterio

sum resolve o somatoacuterio

Nuacutemeros

O MAPLE trabalha com os nuacutemeros de maneira exata Nenhuma aproximaccedilatildeo eacute feita

gt (253+59)^3

314432000729

Podemos ver que o resultado eacute um nuacutemero racional Para obter uma aproximaccedilatildeo decimal

devemos usar o comando evalf (evaluate in floating point)

gt evalf() 4313196159

O sinal de percentagem guarda o valor do uacuteltimo resultado calculado O uacuteltimo resultado

tem 10 diacutegitos (valor default) Podemos calcular com mais diacutegitos significativos por exemplo

com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690

A medida que formos explanando o assunto mostraremos outras maneiras de determinar a

quantidade de diacutegitos e outra maneira de ter resultados com casas decimais eacute entrar pelo

menos com um nuacutemero com casa decimal lembrando que no MAPLE o ponto substitui a

viacutergula

Representaccedilatildeo Interna de uma Expressatildeo

Descrevendo em detalhes a representaccedilatildeo interna de expressotildees no MAPLE

a) 1234 minusminusminus+ xxxx

gt poly=x^4+x^3-x^2-x-1

= poly + minus minus minus x4 x3 x2 x 1

b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)

= a minus x2 3 + x 1

d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5

f) 332 γβα∆ +minus=

gt Delta=(alpha^2-beta^3+gamma^3)

= ∆ minus + α2 β3 γ3

g) 46 102103 sdot+sdot minus

gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2

OBS natildeo esqueccedila de apertar ENTER apoacutes digitar ( ) para confirmar

Resultado de uma Operaccedilatildeo

Uma vez descrita em detalhes uma expressatildeo no MAPLE colocado () apoacutes a descriccedilatildeo e

pressionado ENTER O programa pode

I Mostrar o resultado da operaccedilatildeo

a) x4log 3

2

1 =

gt log[12](root[3](4))

-23

b) Qual eacute o nuacutemero x expresso por 9850318 ++

gt sqrt(18)+3sqrt(50)+sqrt(98)

25 2

c) Determine o valor da cossec 6

π

gt csc(Pi6)

2

d) Calcule o valor da expressatildeo 0

0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2

II Mostrar a expressatildeo (operaccedilatildeo) e esperar que vocecirc use um dos operadores para obter o

resultado esperado

a) Resolva a equaccedilatildeo 2x1x 48 ++ =

gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1

b) Resolva a equaccedilatildeo 81

13x =

gt 3^x=181

= 3x 181

gt solve()

minus( )ln 81( )ln 3

gt simplify()

-4

c) Encontrar o valor decimal da expressatildeo 3

1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626

d) Racionalize o denominador da expressatildeo 53

4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5

Atribuiccedilatildeo de Valores

Lembre-se para nunca atribuir valor a uma letra porque todas as vezes que ela aparecer o

valor seraacute usado na operaccedilatildeo

Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12

Quando isso acontecer digite a palavra restart para limpar a memoacuteria

gt restart

Expressotildees Algeacutebricas

Uma expressatildeo matemaacutetica que apresenta nuacutemeros e letras ou somente letras eacute denominada

expressatildeo algeacutebrica ou literal As letras que normalmente representam nuacutemeros reais satildeo

chamadas variaacuteveis Quando a expressatildeo algeacutebrica natildeo conteacutem variaacutevel ou variaacuteveis no

denominador eacute chamada expressatildeo algeacutebrica inteira quando conteacutem eacute chamada expressatildeo

algeacutebrica fracionaacuteria A seguir mostraremos as diversas operaccedilotildees com monocircmios e

polinocircmios e os respectivos operadores que satildeo utilizados no MAPLE para resolvecirc-los

Reduccedilatildeo de Polinocircmios

Numa expressatildeo algeacutebrica observando que o polinocircmio possui termos ou monocircmios

semelhantes podemos tornar mais simples a expressatildeo adicionando esses termos somando

algebricamente os coeficientes numeacutericos e mantendo a parte literal Natildeo esqueccedila que entre o

coeficiente numeacuterico e a parte literal existe o sinal de multiplicaccedilatildeo

Efetue xxxxx 9853 +minusminus+

gt 3x+5x-x-8x+9x

8 x

Resolva 2535132 222 minus+++minusminus+minus xxxxxx

gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1

Reduza os seguintes monocircmios utilizando o programa MAPLE

1) 4a ndash 5b ndash b + 2a ndash a

2) 222 y5yy3yy minus+minus+

3) 6tt5

45tt

2

1 22 minusminusminus

4) m4m6mm2mm 2323 minusminusminus++

5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste

Usando o Word com ajuda do equation editor para digitar e o MAPLE para resolver

responda as questotildees

1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6

zxy 323232 minusminusminusminusminus

Multiplicaccedilatildeo de Monocircmio por Monocircmio

Para multiplicar dois ou mais monocircmios multiplicamos os coeficientes numeacutericos entre si e

multiplicamos as partes literais entre si lembrando a propriedade das potecircncias

0a com aaa nmnm ne=sdot +

Resolva ( ) ( )3223 52 zxyzyx minus+

Resoluccedilatildeo no MAPLE

gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4

Multiplicaccedilatildeo de Polinocircmio por Monocircmio

A multiplicaccedilatildeo de um monocircmio por um polinocircmio eacute feita multiplicando-se o monocircmio por

cada termo do polinocircmio

Resolva ( ) ( )232 24103 xyxyxyyx minusminus+minus


gt a=(-3x^2y+10xy^3-4xy)

= a minus + minus 3 x2 y 10x y3 4 x y

gt b=(-2xy^2)

= b minus2 x y2

gt ab

minus2 ( )minus + minus 3 x2 y 10x y3 4 x y x y2

gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3

Resolva ( ) ( )2322 2453 axaxxaxa minusminus+


gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)

= d minus2 ( ) + minus 3 a2 x 5 a2 x3 4 a x a x2

gt expand(d)

minus minus + 6 a3 x3 10a3 x5 8 a2 x3

Multiplicaccedilatildeo de Polinocircmio por Polinocircmio

A multiplicaccedilatildeo de um polinocircmio por outro polinocircmio eacute feita multiplicando-se cada termo (ou

monocircmio) de um deles pelos termos (ou monocircmios) do outro e reduzindo-se os termos

semelhantes (se houver)

Calcule ( ) ( )314 minus+ xx

Resoluccedilatildeo algeacutebrica

( ) ( )

3114

3124

311344

314

2

2

minusminusminus+minus

sdotminussdot+sdotminussdotminussdot+

xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)

minus minus 4 x2 11x 3

Atividade 2

Usando o programa MAPLE calcule

1) ( ) ( )2ax5abx3 minusminus

2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23

4) ( ) ( )1x12x3x2 minus+minus

5) ( ) ( )12a3a4a 32 +minusminus

6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+

10) ( ) ( )12ββ3β5β 22 minusminus+minus

Teste

3) ( ) ( )34x2434x15x18x 23 minusminusminus+

4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +

Divisatildeo de Monocircmio por Monocircmio

Para dividir dois ou mais monocircmios dividimos os coeficientes numeacutericos entre si e dividimos

as partes literais entre si recordando a seguinte propriedade das potencias

0a com aaa nmnm ne= minus

Resolva ( ) ( )cabcba 223 312minus


gt (-12a^3b^2c)(3ab^2c)

minus4 a2

Divisatildeo de polinocircmio por monocircmio

Efetuamos a divisatildeo de um polinocircmio por um monocircmio natildeo-nulo fazendo a divisatildeo de cada

termo do polinocircmio pelo monocircmio

Dividir ( ) ( )4ab4abb28ab16a 32224 +minus


gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)

minus + 4 a3 b 7 a b b2

Divisatildeo de Polinocircmio por Polinocircmio

Para dividir polinocircmio pelo polinocircmio adotamos um procedimento anaacutelogo ao algoritmo

usado na aritmeacutetica Observe a resoluccedilatildeo algeacutebrica

Resolva ( ) ( )372158247 23 minusminus+minus xxxx


7x3 ndash 24x2 + 58x ndash 21 7x ndash 3

ndash 7x3 + 3x2 x2 ndash 3x + 7

0 ndash 21x2 + 58x ndash 21

21x2 ndash 9x x2 (7x ndash 3) = 7x3 ndash3x

0 49x ndash 21 3x (7x ndash 3) = 21x2 ndash 9x

ndash 49x + 21 7 ( 7x ndash 3) = 49x ndash 21

0 0

Resoluccedilatildeo no MAPLE (o Maple soacute resolve de forma clara se o resto for zero)

gt c=(7x^3-24x^2+58x-21)(7x-3)

= c minus + minus 7 x3 24x2 58x 21

minus 7 x 3

gt simplify(c)

minus + x2 3 x 7

Resolva ( ) ( )132562 22345 ++++minus++ xxxxxxx


gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3

Resolva as divisotildees utilizando o MAPLE

1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide

3) ( ) ( )4ab4abb28ab12a 32224 divide+minus

4)

divide

minus 234 y3

2y

6

1y

3

2

5) ( ) ( )1x3x34x12x5x6x 2234 +minusdivide+minus+minus

6) ( ) ( )2x6x3xx 23 minusdivide+minusminus

7) ( ) ( )5x157x2x2 +divideminus+

8) ( ) ( )1x3x5x 223 minusdivideminus+

Teste

7) Determine a soma dos polinocircmios 10x2x2x5xxx 24235 minus+minus+minus e 306x6x3 +minus A

seguir divida essa soma por 62xx 2 +minus e encontre o valor numeacuterico do resultado para

x = ndash 2

Adiccedilatildeo algeacutebrica de polinocircmios

A soma de dois ou mais polinocircmios eacute o polinocircmio cujos coeficientes satildeo obtidos adicionando-

se os coeficientes dos termos que apresentam o mesmo grau De forma anaacuteloga a diferenccedila de

dois polinocircmios eacute o polinocircmio cujos coeficientes satildeo obtidos subtraindo-se numa certa

ordem os coeficientes dos termos que apresentam o mesmo grau

Calcular ( ) ( )yxyxxyyx 72652 ++minusminusminus+

1 Resoluccedilatildeo algeacutebrica

( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652

minusminusminusminusminus++minusminus+minus+

++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)

minus minus 3 x 2 y 8 x y

2 Dados 32ce563b734a 2323 ++=minus+=+minus+= xxxxxxx determinar a + b + c

b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c

+ minus minus 3 x3 4 x 8 x2

gt a+c

+ minus + x3 5 x2 x 10

Produtos notaacuteveis

No caacutelculo algeacutebrico os produtos que aparecem com muita frequumlecircncia satildeo chamados de

produtos notaacuteveis Entre eles temos o produto da soma pela diferenccedila de dois termos o

quadrado da soma de dois termos e o quadrado da diferenccedila de dois termos

Quadrado da diferenccedila

Calcule ( )2yx minus


( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus

+minusminussdot+sdotminussdotminussdot

minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4

1) Dados P1 = a + b + c P2 = a ndash b + c e P3 = a + b ndash c Determinar P1 + P2 + P3 P1 +

P2 ndash P3 P1 ndash P2 + P3 e P1 ndash P2 ndash P3

2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +

Fraccedilotildees algeacutebricas

Um quociente de dois polinocircmios indicado na forma fracionaacuteria na qual uma ou mais

variaacuteveis aparecem no denominador chama-se fraccedilatildeo algeacutebrica O denominador de uma

fraccedilatildeo algeacutebrica deve representar sempre um nuacutemero real diferente de zero pois natildeo tem

sentido dividir por zero

Simplificar a fraccedilatildeo algeacutebrica

a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=

+sdotminusminussdotminus=

minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)

= b minus minus 3 a 4

minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1

minus 2 minus a 2 + 1 a

gt simplify(c)

+ 1 a + a 4

Equaccedilatildeo fracionaacuteria

Uma equaccedilatildeo se diz fracionaacuteria quando tem pelo menos uma incoacutegnita no denominador A

resoluccedilatildeo de uma equaccedilatildeo fracionaacuteria eacute feita de maneira semelhante agrave resoluccedilatildeo de uma

equaccedilatildeo Apenas devemos excluir do conjunto universo da equaccedilatildeo fracionaacuteria os valores da

incoacutegnita que anulam o denominador de cada um dos termos da equaccedilatildeo

Resolva as seguintes equaccedilotildees fracionaacuterias

a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3

=there4minusminus=rArrminus=minusrArr

minus=minusrArr=+rArr=+rArr

rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5

Simplifique as fraccedilotildees algeacutebricas e resolva as equaccedilotildees fracionaacuterias

1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste

8) Encontre o valor de x da equaccedilatildeo algeacutebrica usando o MAPLE

( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio

Para abreviar expressotildees que envolvam adiccedilatildeo utiliza-se o somatoacuterio representado por

sum=

n

piix

Lecirc-se somatoacuterio dos termos xi com i variando de p ateacute n

O siacutembolo Σ (letra grega sigma) substitui a palavra soma p e n satildeo nuacutemeros naturais e p le n

Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2

=there4sdotsdotrArrrArr+++rArr

+sdot++sdot++sdot++sdotrArr+sum=

n

ii


gt n=sqrt(Sum(2i+3i=25))

= n sum = i 2

5

( ) + 2 i 3

gt sqrt(sum(2i+3i=25))

2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n

nnS entatildeo quanto vale S


gt S=Sum(3(n^2+n+1)n=14)Sum(nn=111)

= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2

Sistemas de equaccedilotildees de 1deg grau com duas incoacutegnitas

Quando escrevemos duas equaccedilotildees de 1ordm grau com duas incoacutegnitas ligadas pelo conectivo e

estamos escrevendo um sistema de duas equaccedilotildees de 1ordm grau com duas incoacutegnitas (no caso x

e y) A soluccedilatildeo de um sistema de duas equaccedilotildees de 1ordm grau com duas incoacutegnitas x e y eacute um

par ordenado (x y) que eacute soluccedilatildeo tanto da primeira equaccedilatildeo como da segunda Para resolver

um sistema de equaccedilatildeo no MAPLE devemos acessar os comandos que estatildeo no pacote Linalg

que deve ser carregado com o comando with Apoacutes aberto soacute podemos trabalhar com caacutelculos

que envolvam este pacote caso contraacuterio ele se encerra automaticamente

gt restart gt with(linalg)

1) Sabendo que em um terreno haacute galinhas e coelhos num total de 23 animais e 82 peacutes

Determine a quantidade de galinhas e a quantidade de coelhos

Resoluccedilatildeo algeacutebrica pelo meacutetodo da substituiccedilatildeo

=+=+

824y2x

23y x temos

coelhosy e galinhas x Sendo

5x

1823x

y23x

23yx

=rArrthere4minus=rArr

minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e

galinhas 5 haacute Logo


gt with(linalg)

Warning the protected names norm and trace have been

redefined and unprotected

gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)

gt soluccedilatildeo=(backsub(a))

= soluccedilatildeo [ ]5 18

2) Determinar a soluccedilatildeo (x y) do sistema

=+=+

2932

12

yx

yx

Resoluccedilatildeo algeacutebrica pelo meacutetodo da adiccedilatildeo

( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx

A soluccedilatildeo eacute ( 75)


gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5

gt soluccedilatildeo=(backsub(g))


3) Determinar a soluccedilatildeo (xy) do sistema

=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383

gt soluccedilatildeo=(backsub(h))


Atividade 6

Determinar a soluccedilatildeo (xy) dos sistemas

1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes

Matriz eacute uma tabela de nuacutemeros formada por m linhas e n colunas Dizemos que essa matriz

tem ordem m X n (lecirc-se m por n) sendo m ge 1 e n ge 1 Geralmente dispomos os elementos de

uma matriz entre parecircnteses ou entre colchetes

Um modo simplificado de fazer a representaccedilatildeo eacute

[ ] ji Nn m sendo n x m aA isin=

onde

jia elemento da matriz sendo os iacutendices i e j indicadores da posiccedilatildeo do elemento na matriz

O iacutendice i indica a linha 1 le i le m

O iacutendice j indica a coluna 1 le j le n

1) Escreva a matriz ( )3x3jiaA = em que j2ia 2

ji minus=

Para resoluccedilatildeo no MAPLE usamos o pacote Linalg

gt restart

gt with(linalg)

gt matrix(33(ij)-gti^2-2j)

-1 -3 -52 0 -27 5 3

Matriz transposta

Dada uma matriz A de ordem m X n chama-se matriz transposta de A indicada por At a

matriz cuja ordem eacute n X m sendo as suas linhas ordenadamente iguais agraves colunas da matriz

A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6

Adiccedilatildeo de matrizes

A soma de duas matrizes A e B de mesma ordem eacute a matriz tambeacutem de mesma ordem obtida

com a adiccedilatildeo dos elementos de mesma posiccedilatildeo das matrizes A e B

Lembrando que A + B existe se e somente se A e B forem do mesmo tipo

1) Calcule A + B sendo

minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1

Subtraccedilatildeo de matrizes

A diferenccedila entre duas matrizes A e B de mesma ordem eacute a matriz obtida pela adiccedilatildeo da

matriz A com a oposta da matriz B ou seja

A ndash B = A + (ndash B)

1) Calcule A ndash B das matrizes acima


gt evalm(A-B)

3 23 2

-2 3

Multiplicaccedilatildeo de um nuacutemero real por uma matriz

O produto de um nuacutemero real x por uma matriz A eacute obtido pela multiplicaccedilatildeo de cada

elemento da matriz A por esse nuacutemero real x

1) Calcule 3B da matriz acima


( ) ( )

( )

minus

minusminus=

minussdotsdotminussdot

sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3

Multiplicaccedilatildeo de Matrizes

O produto das matrizes A = (aij)m x p e B = (bij)p x n eacute a matriz C = (cij)m x n em que cada

elemento cij eacute obtido por meio da soma dos produtos dos elementos correspondentes da i-

eacutesima linha de A pelos elementos da j-eacutesima coluna de B

Da definiccedilatildeo temos que a matriz produto A middot B soacute existe se o nuacutemero de colunas de A for

igual ao nuacutemero de linhas de B

1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D

Resoluccedilatildeo algeacutebrica (linha vezes coluna)

=

sdot+sdotsdot+sdotsdot+sdot


rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2

gt D=matrix(22[4768])

= d

4 76 8

gt evalm(CD)

Error (in evalmevaluate) use the amp operator for

matrixvector multiplication

gt evalm(CampD)

12 2110 1532 51

Determinantes

Determinante de uma matriz quadrada eacute um nuacutemero real que associamos a essa matriz

segundo algumas regras Sendo a matriz A seu determinante eacute indicado como det A

1) Calcule det D


10det4232678486

74minus=there4minusrArrsdotminussdotrArr

D


gt det(D)

-10

2) Calcule o valor dos seguintes determinantes

8

3

14

πsen

a)minus


gt restart

gt with(linalg)

gt a=matrix(22[sin(Pi4)3-18])

= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus


gt b=matrix(22[log(8)log(2)34-1])

= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa

Consideremos A uma matriz quadrada de ordem n dizemos que Andash1 eacute a matriz inversa de A

se e somente se A middot Andash1 = In e Andash1 middot A = In onde A eacute a matriz Andash1 eacute a matriz inversa da

matriz A In eacute a matriz identidade de mesma ordem da matriz A

1) Sendo

=

3

8

2

5A determine sua inversa se existir


Impondo a condiccedilatildeo de que A middot Arsquo = In e determinamos Arsquo

=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a

Da igualdade de matrizes temos

=+=+

032

185

ca

ca

=+=+

132

085

db

db

Resolvendo os sistemas pelo meacutetodo da adiccedilatildeo vem

220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca

Substituindo o valor obtido para c em uma das equaccedilotildees do sistema temos

35

15161511651285185 minus=rArr

minus=rArrminus=rArr=+rArr=sdot+rArr=+ aaaaaca

550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db

Substituindo o valor obtido para d em uma das equaccedilotildees do sistema temos

85

4004050)5(85085 =rArr=rArr=minusrArr=minussdot+rArr=+ bbbbdb

Assim Arsquo =

d

b

c

a que eacute a inversa de A e pode ser representada por

minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)

gt A=matrix(22[5823])

= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule

a) A + B b) A ndash B c) B ndash A

2 Dadas as matrizes A =

minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule

a) 4A b) 9B c) 5A ndash 3B d) A2

5minus


7

3

0

2 e B =

minus5

4

2

1 calcule

a) A middot B b) B middot A c) A3 d) B3 ndash 4B

4 Calcule o valor dos seguintes determinantes

a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus

6 Calcule se existir 1A minus em cada caso

=

4

2

3

1A a)

minus=

3

1

2

0A b)

Equaccedilatildeo do 2deg grau

Denomina-se equaccedilatildeo de 2ordm grau na incoacutegnita x toda equaccedilatildeo da forma ax2 + bx + c = 0 onde

a b c satildeo nuacutemeros reais e a ne 0 No MAPLE resolvemos usando o operador ldquosolverdquo

1) Determine em R o conjunto-verdade e as raiacutezes da equaccedilatildeo 011336 2 =+minus mm


1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆

sdotsdotminusminus=∆minus=∆ acb

9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2

=there4dividedivide=

sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19

2) Resolva 024142 2 =+minus xx


gt solve(2x^2-14x+24=0)

3 4

3) Resolva 01072 =+minus xx


gt solve(x^2-7x+10=0)

2 5

4) Resolva 22 4)5(2)1(7 xxxx +minus=minus


gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5

Graacutefico da Funccedilatildeo Polinomial de 1deg grau

A representaccedilatildeo graacutefica de uma funccedilatildeo do 1ordm grau y = ax + b (a ne 0) eacute uma reta natildeo-paralela

aos eixos Ox ou Oy sendo raiz ou zero da funccedilatildeo a abscissa do ponto onde a reta intercepta o

eixo Ox

A construccedilatildeo do graacutefico de uma funccedilatildeo do 1ordm grau y = 2x pode ser feita atribuindo-se alguns

valores reais a x e obtendo-se valores de y correspondentes organizando-os em uma tabela e

em seguida localizando no plano cartesiano os pontos (xy) e traccedilando a reta que passa por

eles

No MAPLE usamos os comandos de repeticcedilatildeo (for ndash from ndash to ndash do ndash od) para atribuir os

valores de x e encontrar os valores de y

1 Construir no plano cartesiano o graacutefico da funccedilatildeo y = 2x no intervalo de ndash2 ateacute 2

1deg Descobrindo o valor de y

gt for x from -2 to 2 do 2x od

-4

-2

0

2

4

2deg Usamos o comando ldquoplotrdquo para fazer o graacutefico e para colocarmos nome ou tiacutetulo a

esse graacutefico usaremos o acento grave O tiacutetulo deve ficar entre os acentos

gt restart

gt plot(2xx=-22title=`funccedilatildeo polinomial`)

2 Representar no plano cartesiano o graacutefico da funccedilatildeo y = 2x ndash3 no intervalo de ndash1 ateacute 2

1deg Como representar a funccedilatildeo

gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2

2ordm Como encontrar o valor de y da funccedilatildeo

gt for x from -1 to 2 do 2x-3 od

-5

-3

-1

1

3deg Como fazer o graacutefico e dar um nome ao graacutefico

gt restart

gt restartplot(2x-3x=-12title=`funccedilatildeo polinomial `)

Graacutefico da Funccedilatildeo Polinomial de 2deg grau (Funccedilatildeo Quadraacutetica)

Em um sistema cartesiano ortogonal o graacutefico de uma funccedilatildeo quadraacutetica eacute representado por

uma curva agrave qual damos o nome de paraacutebola

1 Construir no plano cartesiano o graacutefico da funccedilatildeo y = x2 ndash 4 no intervalo de ndash3 ateacute 3

1deg Como representar a funccedilatildeo no MAPLE

gt (x^2-4x=-33)

minus x2 4 = x -3 3

2deg Como encontrar o valor de y da funccedilatildeo

gt for x from -3 to 3 do x^2-4 od

5

0

-3

-4

-3

0

5

3deg Como fazer o graacutefico

gt restart

gt plot(x^2-4x=-33)

2 Traccedilar o graacutefico da funccedilatildeo f(x) = cos 2x no intervalo [0 2π] na cor marrom com 8 de

grossura e com o seu nome no titulo

gt restart gtplot(cos(2x)x=02Picolor=brownthickness=8tit le=`Cristina`)

Testes

9) Encontre o valor de y e faccedila o graacutefico da funccedilatildeo ( ) 1xxf 2 minus= para x = ndash 2 ateacute 3

10) Faccedila o graacutefico da funccedilatildeo f(x) = cos x para x = 0 ateacute 2π

11) Faccedila o graacutefico da funccedilatildeo f(x) = 3sen x para x = 0 ateacute 2π na cor verde com 10 de

grossura e com seu nome no tiacutetulo

Fatorial

Sendo n um nuacutemero natural maior que 1 definimos como fatorial de n (n) o nuacutemero

( )( )( ) 123n2n1nnn sdotsdotsdotminusminusminussdot= K

Lemos n como n fatorial Exemplo

7201234566 =sdotsdotsdotsdotsdot=bull

A digitaccedilatildeo e resoluccedilatildeo no MAPLE natildeo sofre nenhuma modificaccedilatildeo

gt 6

720

1 Calcule o valor de 8

11

Resoluccedilatildeo no Maple

gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1

Nuacutemeros Binomiais

Dados os nuacutemeros naturais n e p com n ge p o nuacutemero

p

n eacute chamado de nuacutemero binomial n

sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10

Combinaccedilatildeo simples

Sendo A = naaaa 321 um conjunto de n elementos e p um numero natural menor ou

igual a n chamamos de combinaccedilatildeo simples dos n elementos de A tomados p a p todo

subconjunto de A com p elementos que eacute representado por pnC No MAPLE entraremos no

combinat numcomb choose Dentro do programa utilizaremos o operador numcomb onde

informaremos os elementos do conjunto A e a quantidade de p elementos que queremos em

cada subconjunto C Para ver esses subconjuntos utilizaremos o operador choose

1 Quantos subconjuntos tem a combinaccedilatildeo simples dos elementos de A = 2 3 4 6

tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr

sdotsdotsdotsdotsdotsdot

rArrminus

=rArrminus

=


gt with(combinatnumbcomb)

[ ]numbcomb

gt numbcomb([2346]3)

4

2 Quais satildeo esses subconjuntos


gt with(combinatchoose)

[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples


igual a n chamamos de arranjo simples dos n elementos tomados p a p todo agrupamento

ordenado com p elementos distintos que se forma com os n elementos de A De modo geral eacute

representado por pnA No MAPLE entraremos no combinat numbperm permute Dentro do

programa utilizaremos o operador numbperm onde informaremos os elementos do conjunto

A e a quantidade de elementos de cada subconjunto C para vermos a quantidade de

subconjuntos de p elementos Para ver esses subconjuntos utilizaremos o operador permute

1 Determinar todos os subconjuntos de trecircs elementos do conjunto A = 2 3 4 6


gt with(combinatpermute)

[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]

2 Quantos subconjuntos formaram


241

1234

3)(4

4A

p)(n

nA 34pn =sdotsdotsdot

rArrminus

=rArrminus

=


gt with(combinatnumbperm)

[ ]numbperm

gt numbperm([2346]3)

24

Equaccedilotildees Irracionais

As equaccedilotildees irracionais satildeo as que apresentam incoacutegnita no radicando e a resoluccedilatildeo eacute feita

elevando-se os dois membros da equaccedilatildeo a uma potecircncia conveniente para transformaacute-la em

uma equaccedilatildeo racional Eacute necessaacuterio verificar os resultados obtidos

1 Resolver a equaccedilatildeo xx minus=minus 256 e verificar o resultado


( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22

=∆rArrthere4minussdotsdotminusrArrminus=∆minus===

=minus+

=minus++minus

+minus=minusminus=minus

acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx

minus=there4minus=minusminus=

=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx

Verificando o resultado para x = 1

11111215626 =there4=rArrminus=sdotminusrArrminus=minus xx

Verificando o resultado para x = ndash 2

( ) ( ) 4441622256256 =there4=rArrminusminus=minusminusrArrminus=minus xx

Logo tanto o nuacutemero 1 e o nuacutemero ndash 2 verificam a equaccedilatildeo irracional dada


gt restart

gt a=sqrt(6-5x)=2-x

= a = minus 6 5x minus 2 x

gt solve(a)

-2 1

2 Quais os valores reais de x para os quais a expressatildeo 1662 +minus xx eacute igual a 22

gt d=sqrt(x^2-6x+16)=2sqrt(2)

= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2

3 Resolva no conjunto real a equaccedilatildeo irracional 6115 +=minus+ xx

gt a=sqrt(5x+1)-1=sqrt(x+6)

= a = minus + 5 x 1 1 + x 6

gt solve(a)

3

4 Resolva a equaccedilatildeo 1211 +=minus++ xxx

gt b=sqrt(x+1)+sqrt(x-1)=sqrt(2x+1)

= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5

5 Resolva a equaccedilatildeo 2132

132 =minusminusminus+

xx

xx

gt c=(sqrt(2x)+sqrt(3x-1))(sqrt(2x)-sqrt(3x-1))= 2

= c = + 2 x minus 3 x 1

minus 2 x minus 3 x 12

gt solve(c)

925

Conversotildees

Arcos e Radianos

Arco de uma circunferecircncia eacute cada uma das partes em que uma circunferecircncia fica dividida

por dois de seus pontos Assim sendo A e B dois pontos quaisquer de uma circunferecircncia

eles a dividem em duas partes

Grau eacute o arco unitaacuterio equivalente a 1360 da circunferecircncia que o conteacutem

Radiano eacute o arco cujo comprimento eacute igual ao comprimento do raio da circunferecircncia que o

conteacutem

No Maple usaremos os operadores convert degrees e radians

1 Determine em radianos a medida do arco 60deg


180deg mdashmdashmdashmdashmdash π

60deg mdashmdashmdashmdashmdash x rad3

πx

180

π60x =there4rArr

degsdotdeg=rArr


gt convert(60degreesradians)

13

π

2 Calcule em graus a medida do arco de 4

3π rad


deg=there4rArrdegsdot

rArr= 135x4

1803

4

3πx


gt convert(3Pi4degrees)

135degrees

3 Calcule em radianos a medida do arco de 330deg


gt convert(330unitsdegreesradians)

116

π

Relaccedilotildees trigonomeacutetricas

As principais relaccedilotildees trigonomeacutetricas satildeo seno (sin) cosseno (cos) tangente (tan)

cotangente (cot) secante (sec) e cossecante (csc) No Maple soacute podemos encontrar os valores

das mesmas se estiverem em radiano

1 Determine secante cossecante e cotangente do arco de 30deg


gt convert(30degreesradians) 16

π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++


gt cos(Pi3)+cos(2Pi)+cos(2Pi3) 1

Escalas Termomeacutetricas

Uma escala termomeacutetrica corresponde a um conjunto de valores numeacutericos onde cada desses

valores estaacute associado a uma temperatura Na escala Celsius o intervalo vai de 0degC a 100degC e

na Kelvin de 273K a 373K ou seja eacute dividida em 100 partes iguais Na escala Fahrenheit o

intervalo vai de 32degF a 212degF ou seja 180 partes iguais

1 A temperatura normal do corpo humano eacute de 36degC Qual eacute essa temperatura expressa nas

escalas Fahrenheit e Kelvin

Resoluccedilatildeo algeacutebrica (escala Fahrenheit)

deg=rArrthere4+=rArrsdot=minusrArr

minus= 896328645

36932

9

32

5FFF

FC


gt convert(36temperaturedegcdegF)

967838000

Resoluccedilatildeo algeacutebrica (escala Kelvin)

30927336362735

273

5=rArrthere4+=rArr=minusrArr

minus= KKKKC


gt convert(36temperatureCelsiuskelvin)

618320

gt evalf()

3091500000

Nuacutemeros Complexos

O conjunto C dos nuacutemeros complexos eacute aquele formado pelos nuacutemeros que podem ser

expressos na forma z = a + bi em que a isinR b isinR e i = 1minus A forma z = a + bi eacute

denominada forma algeacutebrica de um nuacutemero complexo em que a eacute a parte real e b a parte

imaginaacuteria No MAPLE usaremos o I como operador no lugar do i

1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I

2 Calcule (5 + i) + (3 + i) ndash (4 ndash i)


gt (5+I)+(3+I)-(4-I) + 4 3 I

3 Calcule (5 ndash i) (4 + 2i)


gt (5-I)(4+2I)

+ 22 6 I

4 Determine o moacutedulo e o argumento dos nuacutemeros complexos representados no plano

Argand-Gauss (Aqui usaremos o operador arctan da inversa trigonomeacutetrica e o polar para

converter os pontos para coordenadas polar)


0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ

eacute 5i12z de modulo O22

=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen

eacute 5i12z complexo nuacutemero do argumento O


gt convert(12+5Ipolar)

polar 13

arctan

512

gt convert(arctan(512)degrees)

180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite

Dizemos que o limite de uma funccedilatildeo f(x) quando x tende a a eacute o nuacutemero real L se e

somente se os nuacutemeros reais da imagem f(x) permanecerem proacuteximos de L para os infinitos

valores de x proacuteximos de a indica-se

Lf(x)lim =rarrax

No MAPLE utilizamos o operador limit se a primeira letra estiver maiuacutescula o programa

mostra a expressatildeo se estiver minuacutescula mostra o resultado Que pode ser obtido tambeacutem

usando o operador value (lsquorsquo)

1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1

minus + x3 x2 1 minus 1 2x

gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr


gt Limit((x-1)(sqrt(x)-1)x=4)

lim rarr x 4

minus x 1

minus x 1

gt value() 3

Lista de funccedilotildees

Estes foram os principais operadores que o MAPLE dispotildee para trabalhar com operaccedilotildees que

vai do ensino fundamental ateacute o meacutedio Eacute certo que o programa dispotildee de centenas de outros

operadores Caso o estudante tenha duacutevidas sobre como saber se ele resolve este ou aquele

caacutelculo ele deve lembrar da ajuda ou help que o programa dispotildee ou mesmo buscar as listas

de funccedilotildees dos pacotes dando o comando gt com o nome do que se quer buscar Por exemplo

gt linalg

gt exp

Resoluccedilatildeo de problemas com a inserccedilatildeo de dados

Os quatro aspectos gerais que o MAPLE possui estatildeo integrados formando um corpo uacutenico

Tais sistemas satildeo uacuteteis na resoluccedilatildeo faacutecil e precisa de uma diversidade de problemas em

matemaacutetica e fiacutesica O sistema permite atribuir nome a um nuacutemero ou expressatildeo bem como

representar uma expressatildeo e ter resultado para a mesma Acompanhe a resoluccedilatildeo dos dois

problemas abaixo

1 Um disco gira a 410 rpm Sabendo que o diacircmetro do disco eacute igual a 36 cm calcule a

velocidade angular e a velocidade escalar de um ponto da sua periferia

Obs para resoluccedilatildeo do mesmo usaremos as expressotildees de T (periacuteodo) de ω (velocidade

angular) de v (velocidade escalar) e de R (raio) onde

2

dR =

f

t=T T

2πω = Rω sdot=v

O problema forneceu o valor da frequumlecircncia (f = 410 rpm) do tempo (t = 1 min) que seraacute

transformado em segundos e do diacircmetro (d = 36 cm) Veja a resoluccedilatildeo algeacutebrica e como

inserir esses dados no programa e ter os resultados de forma mais precisa


Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410

60TT =rArrthere4=rArr=

f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π

ω =rArrthere4sdotsdot=rArrsdot=rArr=

msπ2463π738

183

41Rω =rArrthere4=rArrsdot=rArrsdot= vvvv

π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π

gt velocidadeangular=evalf() = velocidadeangular 4293509962

gt v=omegaR = v 246π

gt velocidadeescalar=evalf() = velocidadeescalar 7728317929

2 Dois espelhos planos formam entre si um certo acircngulo Calcule esse acircngulo sabendo que

reduzindo-o 20deg o nuacutemero de imagens produzidas pelo sistema de um dado objeto eacute

aumentado de 9

Obs Aqui montando a associaccedilatildeo angular de espelhos e substituindo uma expressatildeo na

outra obtemos a seguinte equaccedilatildeo

120α

36091

α

360 1

20α

3609N 1

α

360N minus

minus=+minusthere4minus

minus=+rArrminus=

Feito isso veja como inserir essa equaccedilatildeo no programa e ter o resultado


gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20

Veja que no primeiro problema os dados foram digitados diretamente para o programa com os

seus siacutembolos originais utilizando o operador ldquo=rdquo que serviu para atribuir o nome a cada

valor apoacutes isso digitamos as expressotildees para encontrar o valor do raio R do periacuteodo T da

velocidade angular ω e da velocidade escalar v utilizando o mesmo operador no lugar do ldquo=rdquo

Todos encerrados com ldquordquo operador que serve para mostrar e resolver o que se estar

digitando E para ter um resultado mais preciso da velocidade angular e da velocidade escalar

na forma decimal usamos o operador ldquoevalf()rdquo coisa que muitas vezes nas aulas de Fiacutesica

Matemaacutetica e em muitos livros didaacuteticos jaacute natildeo se usa mais o valor de Pi vez que se tem que

adequar muitos caacutelculos a capacidade de aprendizagem do aluno

No segundo problema primeiro montamos a equaccedilatildeo de associaccedilatildeo de espelhos depois

digitamos essa equaccedilatildeo no programa e para ter o resultado utilizamos o operador

ldquosolve()rdquo usado para responder equaccedilotildees do 2ordm grau e fraccedilotildees algeacutebricas que no caso vale

a resposta positiva O operador ldquo() rdquo eacute para se referir ao uacuteltimo caacutelculo executado

Aleacutem desse existe diversos outros softwares matemaacuteticos capazes de tornar a vida do

estudante cada vez mais praacutetica e melhor soacute depende do interesse do mesmo veja por

exemplo o uso do WINPLOT programa que usado junto com o MAPLE ajuda na elaboraccedilatildeo

e feitura de graacuteficos

O que eacute Winplot

O Winplot eacute um programa de domiacutenio puacuteblico produzido por Richard Parris da Phillips

Exeter Academy (httpmathexeteredurparris) Eacute utilizado para o traccedilado de funccedilotildees e

equaccedilotildees no plano e no espaccedilo Utiliza pouco espaccedilo no disco e dispotildee de vaacuterios recursos

Abrindo o Winplot e construindo graacuteficos

Abrindo o Winplot

Para abrir o Winplotexe clique duas vezes no iacutecone

Feito isso apareceraacute a Tela Principal com a Barra de Tiacutetulo que mostra o nome do

programa e os bototildees Maximizar Minimizar e Fechar e a Barra de Menus que apresenta

os menus do programa Janela e Ajuda

Clique em Janela onde apareceraacute 2ndashdim F2 e 3ndashdim F3 ou seja graacutefico em duas dimensotildees e

graacutefico em trecircs dimensotildees

Clique em 2ndashdim F2 onde apareceraacute a janela semnome1wp2 com os seguintes menus

Arquivo Equaccedilatildeo Ver Mouse Um Dois Anim e Outros

Criando o graacutefico da funccedilatildeo de 1ordm grau f(x) = 2x +5

Clique a opccedilatildeo Equaccedilatildeo e surgiraacute uma coluna com diversos menus Escolha a opccedilatildeo

1ExplicitaF1 e apareceraacute a caixa que teraacute o local para digitar a funccedilatildeo f(x)

Veja que nessa caixa vocecirc pode digitar a sua funccedilatildeo 2x + 1 ou 2x +1 travar o intervalo que

deseja por exemplo x miacuten -5 e x maacutex 5 escolher a cor e a espessura da linha do graacutefico

Feito tudo isso eacute soacute clicar em ok e apareceraacute o graacutefico da funccedilatildeo digitada

Veja que no primeiro momento aparece apenas o graacutefico sem mostrar de forma clara quais

foram os pontos pelo quais passou para vermos isso devemos clicar em Ver e escolher na

coluna que aparece a opccedilatildeo Enquadrar janela feita a escolha o graacutefico aparece mais

completo Para colocar grades no seu graacutefico vocecirc clica novamente em Ver escolhe na

coluna a opccedilatildeo Grade clique nessa opccedilatildeo para aparecer a caixa grade

Marque nessa caixa eixos ambos marcas setas roacutetulos e veja se decimais estaacute em 0 e 0

marque tambeacutem escala sobre eixos e no campo grade marque retangular apoacutes isso escolha

pontilhado Feito tudo isso clique em aplicar

Para enquadrar melhor o graacutefico que acabou de fazer vocecirc pode usar a tecla Page Up para

aumentar o zoom ou a tecla Page Down para diminuir bem como usar as teclas de seta do

teclado para mover o graacutefico para enquadraacute-lo da melhor forma que escolher

Concluiacutedo tudo isso vocecirc pode copiar esse graacutefico e colar no trabalho que desejar para isso

basta ir no menu Arquivo e escolher na coluna que aparece a opccedilatildeo Copiar e levar para o

trabalho que estaacute fazendo colar e ajustar do jeito que quiser

minus12minus11minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 1 2 3 4 5 6 7 8 9 10 11 12

minus6minus5minus4minus3minus2minus1

123456789

10111213141516

x

y

Veja como podemos ver todos os pontos por onde passa nosso graacutefico bem como em que

intervalo se encontra

Criando o graacutefico da funccedilatildeo de 2ordm grau f(x) = x2

Para criar um graacutefico de uma funccedilatildeo do segundo grau devemos seguir os mesmos passos que

fizemos para criar o graacutefico da funccedilatildeo do primeiro grau o uacutenico detalhe que devemos nos

preocupar satildeo com os operadores matemaacuteticos + - ^ sqrt (adiccedilatildeo subtraccedilatildeo

multiplicaccedilatildeo divisatildeo potenciaccedilatildeo e radiciaccedilatildeo respectivamente)

Ex

f(x) = x2 harr f(x) = x^2

f(x) = ndash2x2 ndash 5x + 7 harr f(x) = -2x^2-5x+7

Seguido todos os passos e respeitado os operadores matemaacuteticos o graacutefico da nossa funccedilatildeo do

2ordm grau f(x) = x2 no intervalo de ndash5 ateacute 5 ficaraacute assim

minus15minus14minus13minus12minus11minus10minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16minus2minus1

123456789

10111213141516171819202122232425

x

y

Clicando em Um no menu e escolhendo Zeros na coluna que aparece na janela com o

graacutefico surgiraacute uma seta indicando o um dos pontos onde o graacutefico intercepta o eixo do x

Clicando em proacuteximo da janela zeros veremos e teremos marcada a segunda raiz e do lado

esquerdo do botatildeo proacuteximo aparece o ponto ou valor da raiz marcada no graacutefico agrave medida

que clicamos os valores se alternam se forem diferentes

Fora esses recursos que satildeo utilizados para trabalhos escolares do niacutevel fundamental e meacutedio

o Winplot oferece dezenas de outros recursos para fazer graacuteficos inclusive graacutefico em terceira

dimensatildeo muito utilizado em feira de ciecircncias trabalhos de poacutes-graduaccedilatildeo mestrado e

doutorado O programa estaacute disponiacutevel gratuitamente na internet inclusive com diversos

trabalhos acadecircmicos ou natildeo de como utilizaacute-lo

Veja como fica no Winplot o graacutefico da funccedilatildeo polinomial de 1ordm grau y = 2x ndash 3 no intervalo

de ndash1 ateacute 2 que foi feito no MAPLE na paacutegina 36

minus3 minus2 minus1 1 2 3

minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3

Veja como fica no Winplot o graacutefico da funccedilatildeo polinomial de 2ordm grau y = x2 ndash 4 no intervalo


minus5 minus4 minus3 minus2 minus1 1 2 3 4 5

minus4

minus3

minus2

minus1

1

2

3

4

5

x

y

No Winplot vemos claramente nos graacuteficos onde estatildeo localizados os pares ordenados por

exemplo no primeiro graacutefico temos os pares (ndash1 ndash5) (0 ndash3) (1 ndash1) (2 1) no segundo

tambeacutem podemos localizar seus pares ordenados sem muita dificuldade

Veja como fica no Winplot o graacutefico da funccedilatildeo f(x) = cos 2x no intervalo [0 2π] que foi

feito no MAPLE na paacutegina 37

1 2 3 4 5 6

minus2

minus1

1

2

x

yy = cos(2x) 00 lt= x lt= 2pi

Aqui podemos ver claramente a oscilaccedilatildeo entre 1 e ndash1 Caso ocorram duacutevidas de como digitar

uma funccedilatildeo eacute soacute clicar em Equaccedilatildeo e escolher Biblioteca

Referecircncias bibliograacuteficas

ATIVIDADES USANDO O WINPLOT 2ndashdim em Portuguecircs Texto extraiacutedo na Internet via

httpmathexeteredurparris Arquivo capturado em 26 de outubro de 2010

AacuteVILA Geraldo Severo de Souza Introduccedilatildeo ao caacutelculo Rio de Janeiro LTC 1998

BONJORNO Regina Azenha et al Fiacutesica completa 2ordf ed Satildeo Paulo FTD 2001 vu

BRETON Philippe Histoacuteria da informaacutetica Satildeo Paulo Universidade Estadual Paulista

1998 Cap 1-5

BUTTS Thomas Colocando problemas adequadamente In NCTM Problem solving in

School Mathematics 1980 p 23-26

CALCULATING machines [online] Texto extraiacutedo na Internet via

wwwwebcomcomcalcCalcMachhtml Arquivo capturado em 09 de setembro de 2010

COSTA Adeilton Fernandes da Uso de softwares matemaacuteticos Porto Velho

UNIRRIOMAR 2005

DrsquoAMBROSIO Beatriz S Como ensinar Matemaacutetica hoje Temas amp Debates n 2 p 15-

19 1989

DANTE Luiz R Didaacutetica da resoluccedilatildeo de problemas de Matemaacutetica Satildeo Paulo Aacutetica

1989

ESTEPHAN Violeta Maria PERACCHI Edilaine do Pilar Fernandes e TOSATTO Claacuteudia

Miriam Matemaacutetica ideacuteias amp relaccedilotildees 1ordf ed Curitiba Positivo 2002

EXPLORANDO20WINPLOT20-20VOL201 Texto extraiacutedo na Internet via


GIOVANNI Joseacute Ruy BONJORNO Joseacute Roberto Matemaacutetica uma nova abordagem Satildeo

Paulo FTD 2001 3v

GIOVANNI Joseacute Ruy CASTRUCCI Benedito GIOVANI Jr Joseacute Ruy A conquista da

matemaacutetica ndash renovada 6ordm ao 9ordm ano Satildeo Paulo FTD 2009

GIOVANNI Joseacute R GIOVANNI JR Joseacute R Matemaacutetica Pensar e Descobrir Satildeo Paulo

FTD 1996

LUZ Antocircnio Maacuteximo Ribeiro da amp LUZ Beatriz Alvarenga Aacutelvares Fiacutesica 1ordf edSatildeo

Paulo Scipione 2007 3v

SAMPAIO Joseacute Luiz amp CALCcedilADA Caio Seacutergio Universo da fiacutesica 2ordf ed Satildeo Paulo

Atual 2005 3v

SILVA Claudio Xavier da amp FILHO Benigno Barreto Matemaacutetica aula por aula 2ordf ed Satildeo

Paulo FTD 2005 v 3

Respostas das atividades

Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus

4) 23 2m5m minusminus

5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus

4) 13x5x3x 23 minus+minus

5) 48aa14a3a 235 minus++minus

6) 128a15a2 minusminus

7) 23a14a15a 23 +minusminus

8) 2bc2ac2abcba 222 +++++

9) 313α10α 2 minusminus

10) 35β11β5β 34 minusminusminus

Atividade 3

1) x3

2minus

2) 32yx

3

3) 23 b7abb3a +minus

4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4

1) 3a + b +c a ndash b + 3c a + 3b ndash c ndash a +

b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)

minusminusminusminus

=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =

5) 2a) minus= 90b) = 2c) minus=

6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+

4) 22 10β11α16α minusminus

5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++

7) valor da divisatildeo 5xx 3 +minus valor

numeacuterico ndash 1

8) O valor de x = AMO ndash TE

Sumaacuterio

O que eacute MAPLE helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 4

Onde surgiu helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 4

A folha de trabalho 5

Ajuda 5

Operaccedilotildees aritmeacuteticas 6

Alguns operadores helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 6

Nuacutemeros 6

Representaccedilatildeo interna de uma expressatildeo helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 7

Resultado de uma operaccedilatildeo 8

Atribuiccedilatildeo de valores helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 10

Expressotildees algeacutebricas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 10

Reduccedilatildeo de Polinocircmios helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 11

Multiplicaccedilatildeo de Monocircmio por Monocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 12

Multiplicaccedilatildeo de Polinocircmio por Monocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 12

Multiplicaccedilatildeo de Polinocircmio por Polinocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 13

Divisatildeo de Monocircmio por Monocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 14

Divisatildeo de Polinocircmio por Monocircmio 14

Divisatildeo de Polinocircmio por Polinocircmio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 15

Adiccedilatildeo algeacutebrica de Polinocircmios helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 16

Produtos Notaacuteveis helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 17

Quadrado da diferenccedila helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 18

Quadrado da soma helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 18

Fraccedilotildees algeacutebricas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 19

Equaccedilatildeo fracionaacuteria helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 20

Somatoacuterio helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 22

Sistemas de equaccedilotildees de 1deg grau com duas incoacutegnitas helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 23

Matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 26

Matriz transposta 26

Adiccedilatildeo de matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 27

Subtraccedilatildeo de matrizes helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 28

Multiplicaccedilatildeo de um nuacutemero real por matriz helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip 28







Fatorial 38



Arranjo simples 40


Conversotildees 42

Arcos e Radianos 42




Limites 46










O que eacute MAPLE

MAPLE
















Onde surgiu













MAPLE








A folha de trabalho











Ajuda




gt maple









clicado em Help








Alguns Operadores












Nuacutemeros


gt (253+59)^3

314432000729



gt evalf() 4313196159



com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690




viacutergula






b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)


d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++










Fatorial 38



Arranjo simples 40


Conversotildees 42

Arcos e Radianos 42




Limites 46










O que eacute MAPLE

MAPLE
















Onde surgiu













MAPLE








A folha de trabalho











Ajuda




gt maple









clicado em Help








Alguns Operadores












Nuacutemeros


gt (253+59)^3

314432000729



gt evalf() 4313196159



com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690




viacutergula






b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)


d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




O que eacute MAPLE

MAPLE
















Onde surgiu













MAPLE








A folha de trabalho











Ajuda




gt maple









clicado em Help








Alguns Operadores












Nuacutemeros


gt (253+59)^3

314432000729



gt evalf() 4313196159



com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690




viacutergula






b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)


d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++







MAPLE








A folha de trabalho











Ajuda




gt maple









clicado em Help








Alguns Operadores












Nuacutemeros


gt (253+59)^3

314432000729



gt evalf() 4313196159



com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690




viacutergula






b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)


d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++









clicado em Help








Alguns Operadores












Nuacutemeros


gt (253+59)^3

314432000729



gt evalf() 4313196159



com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690




viacutergula






b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)


d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




314432000729



gt evalf() 4313196159



com 30 diacutegitos

gt evalf[30]() 431319615912208504801097393690




viacutergula






b) ( )22 1+yx

gt x(y^2+1)^2

x ( ) + y2 12

c) 1

32

+minus

x

x

gt a=(x^2-3)(x+1)


d) 33 +x

gt sqrt(x^3+3)

+ x3 3

gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt root[2](x^3+3)

+ x3 3

gt root(x^3+3 2)

+ x3 3

e) 5 2 2x minus

gt root[5](x^2-2)

( ) minus x2 2( )1 5





gt 3exp(-6)+2exp(4)

+ 3 eeee( )-6

2 eeee4

h) 16 log2

gt log[2](16)

( )ln 16( )ln 2

gt log(16)log(2)

( )ln 16( )ln 2






a) x4log 3

2

1 =


-23



25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++






25 2


π

gt csc(Pi6)

2


0

24

12

12

minus

+

gt (2^0+12)(14-2^0)

-2


resultado esperado


gt 8^(x+1)=4^(x+2)

= 8( ) + x 1

4( ) + x 2

gt simplify()

1


13x =

gt 3^x=181

= 3x 181

gt solve()


gt simplify()

-4


1

gt a=1sqrt(3)

= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




= a13

3

gt evalf(a)

5773502693

gt evalf(a5)

57736

gt evalf(a15)

577350269189626


4

minus

gt 4(3-sqrt(5))

41

minus 3 5

gt rationalize()

+ 3 5




Ex

gt x=3

gt 2x+x^2

15

gt 5x-x

12


gt restart














gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++













gt 3x+5x-x-8x+9x

8 x


gt 2x^2-3x+1-5x^2-x+3+5x+x^2-2

minus + + 2 x2 x 2

Atividade 1




3) 6tt5

45tt

2



5) x6

7x

3

2 minus

6) 7

y3x

5

4y

3

2x minus+minus

7) 3

2x5x

+minus

8) 22

x5

y4xx)(xy

5

2x minusminus+

Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




Teste



1) 4

yx

3

y5x

6

yx

2

yx

4

y7x 3244324432

+minus+minus

2) 2

zxy

18

zxy

6








gt (2x^3y^2z)(-5xy^2z^3)

minus10x4 y4 z4






gt a=(-3x^2y+10xy^3-4xy)


gt b=(-2xy^2)

= b minus2 x y2

gt ab


gt expand(ab)

minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




minus + 6 x3 y3 20x2 y5 8 x2 y3



gt restart

gt d=(3a^2x+5a^2x^3-4ax)(-2ax^2)


gt expand(d)








( ) ( )

3114

3124

311344

314

2

2



xx

xxx

xxxx

xx


gt b=(4x+1)(x-3)

= b ( ) + 4 x 1 ( ) minus x 3

gt expand(b)


Atividade 2



2)

minus

+ ab5

2cba

2

1 22

3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




3)

minus+minus x2

13x

3

2x

3

1x

2

1 23



6) ( ) ( )65a23a minus+

7) ( ) ( )14a3a25a 2 +minus+

8) ( ) ( )cbacba ++++

9) ( ) ( )32α15α minus+


Teste


4) ( ) ( )5β2α2β3α minus+

5) ( ) ( )γβαγβα ++++

6) ( ) 25x +







gt (-12a^3b^2c)(3ab^2c)

minus4 a2






gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





gt x=(16a^4b^2-28a^2b^2+4ab^3)(4ab)

= x14

minus + 16a4 b2 28a2 b2 4 a b3

a b

gt simplify(x)

b ( ) minus + 4 a3 7 a b

gt expand(x)













0 0


gt c=(7x^3-24x^2+58x-21)(7x-3)


minus 7 x 3

gt simplify(c)

minus + x2 3 x 7



gt d=(2x^5+6x^4+x^3-x^2+5x+2)(x^2+3x+1)

= d + + minus + + 2 x5 6 x4 x3 x2 5 x 2

+ + x2 3 x 1

gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt simplify(d)

minus + 2 x3 x 2

Calcular

minus

+ 22

4

3

2

1yxy


gt ((12)xy^2)((-34)y^2)

minus23

x

Atividade 3


1)

minusdivide

+ 22 y4

3xy

2

1

2) ( ) ( )5523 y5xy15x divide


4)

divide

minus 234 y3

2y

6

1y

3

2





Teste



x = ndash 2








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++








( ) ( )

xyyx

xyxyyyxx

yxyxxyyx

yxyxxyyx

823

26752

72652

72652


++minusminusminus+


gt (2x+5y-6xy)-(-x+2xy+7y)



b ndash c e a + c


gt a=(x^3+4x^2-3x+7)

= a + minus + x3 4 x2 3 x 7

gt b=(3x^3+6x-5)

= b + minus 3 x3 6 x 5

gt c=(x^2+2x+3)

= c + + x2 2 x 3

gt a+b+c

+ + + 4 x3 5 x2 5 x 5

gt b-c


gt a+c

+ minus + x3 5 x2 x 10








( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++







( )( ) ( )

22

22

2

2 yxyx

yxyxyx

yyxyyxxx

yxyx

yx

+minus


minussdotminusminus


gt c=(x-y)^2

= c ( ) minus x y 2

gt expand(c)

minus + x2 2 x y y2

Quadrado da soma

Calcule 2

3

2y

+


gt d=(y+23)^2

= d

+ y

23

2

gt expand(d)

+ + y2 43

y49

Atividade 4



2) ( )335yminus

3) 3

23yx3

2

4) 222

5

yx

minus

5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




5) ( )2205ab

6) ( )23 5βα +

7) 2

2n5

3m

minus

8) ( )3y2x +







a) 22

2

ba

aba

minus+


gt g=(a^2+ab)(a^2-b^2)

= g + a2 a b

minus a2 b2

gt simplify(g)

minusa

minus b a

b) 16

1682

2

minus+minus

x

xx


( ) ( )( ) ( ) 4

4

44

44

16

1682

2

+minus=


minus+minus

x

x

xx

xx

x

xx


gt a=(x^2-8x+16)(x^2-16)

= a minus + x2 8 x 16

minus x2 16

gt simplify(a)

minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




minus x 4 + x 4

c) 4

1

16

432 minus

minusminusminus

aa

a


gt b=((3a-4)(a^2-16))-1(a-4)


minus a2 16

1 minus a 4

gt simplify(b)

21

+ a 4

d)

a

a

+minusminus

1

22

1


gt c=1(2-((a-2)(1+a)))

= c1


gt simplify(c)

+ 1 a + a 4







a) 12

111

4

3 =+x


62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




62

12122

121191112912

11

12

12

12

9

1212

111

4

3



rArr=+

xxx

xxxxx

x

xx

x

xeacutecmmoquex


gt d=(34)+(1x)=(1112)

= d = + 34

1x

1112

gt expand(d)

= + 34

1x

1112

gt solve(d)

6

b) 3

3

9

52 +

minus=minus xx


gt e=((5)(x^2-9))=((-3)(x+3))

= e = 51

minus x2 9minus3

1 + x 3

gt solve(e)

43

Atividade 5


1) x1

x1

x1

x1

x1

4x2

2

minus++

+minusminus

minus

2) ( ) ( ) 1313x22x5 =minusminus+

3) 20

3y

4

3

5

2y =minus

4) 2

7

3

1x

4

3x =minusminus+

5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




5) 2x

3

3x

2x +=minus

Teste


( )

( ) ( )BOCx

CTE

B

AM

BOCxB

xBCAM

+minus=

++

Somatoacuterio


sum=

n

piix



Determinar ( )sum=

+=5

2

32i

in


( ) ( ) ( ) ( ) ( )

10252240131197

35234233232232

2

5

2



n

ii



= n sum = i 2

5

( ) + 2 i 3


2 10

Determinar ( )sum=

+3

1

32i

i


gt restart

gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt n=Sum(2i+3i=13)

= n sum = i 1

3

( ) + 2 i 3

gt sum(2i+3i=13)

21

Se ( )

sum

sum

=

=

++=

11

1

4

1

2 13

n

n

n




= S

sum = n 1

4

( ) + + 3 n2 3 n 3

sum = n 1

11

n

gt sum(3(n^2+n+1)n=14)sum(nn=111)

2













=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





=+=+

824y2x

23y x temos


5x

1823x

y23x

23yx


minus==+

18y2

36y

362y

46824y2y

824y2y46

824yy)2(23

824y2x

=rArrthere4=

=minus=+minus

=+minus=+minus

=+

coelhos 18 e



gt with(linalg)



gt a=matrix(23[11232482])

= a

1 1 232 4 82

gt a=gausselim(a)




=+=+

2932

12

yx

yx


( )( )

7

2932

3633

12932

312

=minus=minusminus

=+

minussdot=+sdot=+

x

yx

yx

yx

yx

5

712

127

12

=minus=

=+=+

y

y

y

yx



gt with(linalg)

gt g=matrix(23[11122329])

= g

1 1 122 3 29

gt g=gausselim(g)

= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




= g

1 1 120 1 5




=+minus=minus1252

123

yx

yx


gt with(linalg)

gt h=matrix(23[3-2-12512])

= h

3 -2 -12 5 12

gt h=gausselim(h)

= h

3 -2 -1

0193

383



Atividade 6


1)

=+=minus

415y2x

163y3x

2)

=minus=+

1y2x

5yx

3)

minus=+minus=+minus

145y7x

35y6x

4)

=+

=+

110y4

x

1103

yx

5)

=minus

=+

10yx

1103

y

4

x

Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




Matrizes






onde





ji minus=


gt restart

gt with(linalg)


-1 -3 -52 0 -27 5 3

Matriz transposta



A

1) Determinar Mt de

minus=

654

731M


gt M=matrix(23[137-456])

= M

1 3 7-4 5 6

gt transpose(M)

1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




1 -43 57 6






minus=

2

4

1

0

3

2

A

minus

minusminus=

1

2

3

2

0

1

B


( ) ( )

( )

minus=

minus++

minus+minus

++

minus+=

minus

minusminus+

minus

1

6

4

2

3

1

12

24

31

20

03

12

1

2

3

2

0

1

2

4

1

0

3

2


gt restart

gt with(linalg)



gt A=matrix(32[2-13402])

= a

2 -13 40 2

gt B=matrix(32[-1-3022-1])

= b

-1 -30 22 -1

gt evalm(A+B)

1 -43 62 1







gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++










gt evalm(A-B)

3 23 2

-2 3






( ) ( )

( )

minus

minusminus=


sdotsdotminussdot

rArr

minus

minusminus

3

6

9

6

0

3

13

23

33

23

03

13

1

2

3

2

0

1

3


gt evalm(3B)

-3 -90 66 -3







1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++






1) Calcule CD sendo

=2

1

0

5

1

3

C e

=

86

74D


=



rArr

lowast

51

15

21

32

10

12

8275

8171

8073

6245

6141

6043

8

7

2

1

5

16

403


gt C=matrix(32[301152])

= c

3 01 15 2


= d

4 76 8

gt evalm(CD)



gt evalm(CampD)

12 2110 1532 51

Determinantes



1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




1) Calcule det D


10det4232678486


D


gt det(D)

-10


8

3

14

πsen

a)minus


gt restart

gt with(linalg)


= a

12

2 3

-1 8

gt det(a)

+ 4 2 3

1

3

4

8logb)

2

minus



= b

( )ln 8( )ln 2

3

4 -1

gt det(b)

minus + ( )ln 8 12 ( )ln 2

( )ln 2

gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt simplify(b)

3 34 -1

gt det(simplify(b))

-15

Matriz inversa




1) Sendo

=

3

8

2




=

++

++

rArr

=

lowast

1

0

0

1

32

85

32

85

1

0

0

1

3

8

2

5

db

db

ca

ca

d

b

c

a


=+=+

032

185

ca

ca

=+=+

132

085

db

db


220

01510

21610

)5(032

)2(185

=rArrminus=minus

=+minus=minusminus

rArr

sdot=+minussdot=+

cc

ca

ca

ca

ca


35

15161511651285185 minus=rArr


550

11510

01610

)5(132

)2(085

minus=rArr=minus

=+=minusminus

rArr

sdot=+minussdot=+

dd

db

db

db

db


85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





85


Assim Arsquo =

d

b

c


minusminus

=minus

5

8

2

31A


gt restart

gt with(linalg)


= A

5 82 3

gt inverse(A)

-3 82 -5

Atividade 7

1 Sendo A =

3

2

1

0

4

1 e B =

minus1

1

2

0

4

3 calcule



minus 2

4

0

3

1

2 e B =

minus2

1

4

2

1

3 calcule


5minus


7

3

0

2 e B =

minus5

4

2

1 calcule



a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




a)

minus

minus 1

2

1

2

1

3

5

3

4

b)

01

2

3

1

1

2

0

1

2

3πcos

log12

πsen

minus

minus

minus


a)

5

32

6

2 b)

273

1log

813

πtg

3

c) 1

2πcos

3

10

2minus


=

4

2

3

1A a)

minus=

3

1

2

0A b)






1

13

36

=minus=

=

c

b

a

( )

5

25

144169

136413

42

=∆

=∆minus=∆


9

1

8

8

72

8

362

513

4

1

18

18

72

18

362

513

2


sdotminus=


sdot+=

sdot∆plusmnminus=

xx

xx

a

bx

Logo

=

9

1

4

1S


gt solve(36m^2-13m+1=0)

14

19



gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt solve(2x^2-14x+24=0)

3 4




2 5



gt solve(7x(x-1)=2(x^2-5)+4x^2)

2 5




eixo Ox




eles






-4

-2

0

2

4



gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt restart




gt restart

gt(y=2x-3 x=-12)

= y minus 2 x 3 = x -1 2



-5

-3

-1

1


gt restart







gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++









gt (x^2-4x=-33)

minus x2 4 = x -3 3



5

0

-3

-4

-3

0

5


gt restart

gt plot(x^2-4x=-33)




Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++







Testes





Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




Fatorial






gt 6

720


11


gt 118

990


710

sdot


gt (107)(2512)

744

3 Simplificando ( )

( )2n

n1n

+++

obtemos


( )( )

( )( ) ( )

( )( ) ( ) ( ) ( ) 1n

1

1n2n

2n

n1n2n

11nn

n1n2n

nn1n

2n

n1n

+=

+++=

++++=

++++=

+++


gt b=((n+1)+n)(n+2)

= b + ( ) + n 1 n

( ) + n 2

gt simplify(b)

1 + n 1



p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++






p


sobre p

( )pnp

n

p

nC pn minus

=

=

Exemplo

( ) 1023

5

252

5

2

5==

minus=

bull


gt binomial(52)

10









tomados 3 a 3


46

24

1123

1234

3)(43

4C

p)(np

nC 34pn =rArr


rArrminus

=rArrminus

=



[ ]numbcomb


4




[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





[ ]choose

gt choose([2346]3)

[ ] [ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 6 [ ] 3 4 6

Arranjo simples











[ ]permute

gt permute([2346]3)

[ ] 2 3 4 [ ] 2 3 6 [ ] 2 4 3 [ ] 2 4 6 [ ] 2 6 3 [ ] 2 6 4 [ ] 3 2 4 [ ] 3 2 6 [ ] 3 4 2 [[ ] 3 4 6 [ ] 3 6 2 [ ] 3 6 4 [ ] 4 2 3 [ ] 4 2 6 [ ] 4 3 2 [ ] 4 3 6 [ ] 4 6 2 [ ] 4 6 3 [ ] 6 2 3 [ ] 6 2 4 [ ] 6 3 2 [ ] 6 3 4 [ ] 6 4 2 [ ] 6 4 3 ]



241

1234

3)(4

4A

p)(n


rArrminus

=rArrminus

=



[ ]numbperm


24







( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++










( ) ( )

( ) 921414

211

02

06454

4456

256

22

2

2

2

22


=minus+

=minus++minus


acb

cba

xx

xxx

xxx

xx

2

31

12

91

2

plusmnminusrArr

sdotplusmnminus

rArr∆plusmnminus=

a

bx


=there4=+minus=

22

4

2

31

12

2

2

31

xx

xx







gt restart

gt a=sqrt(6-5x)=2-x


gt solve(a)

-2 1



= d = minus + x2 6 x 16 2 2

gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




gt solve(d)

4 2



= a = minus + 5 x 1 1 + x 6

gt solve(a)

3



= b = + + x 1 minus x 1 + 2 x 1

gt solve(b)

12

5



xx

xx


= c = + 2 x minus 3 x 1


gt solve(c)

925

Conversotildees

Arcos e Radianos






conteacutem






πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++









πx

180

π60x =there4rArr

degsdotdeg=rArr



13

π


3π rad



rArr= 135x4

1803

4

3πx



135degrees




116

π








π

gt sec(Pi6)

23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




23

3

gt csc(Pi6) 2

gt cot(Pi6) 3


2πcoscos2π

3

πcos ++












minus= 896328645

36932

9

32

5FFF

FC



967838000


30927336362735

273


minus= KKKKC



618320

gt evalf()

3091500000






1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++









1 Calcule (2 + 3i) +(6 + 4i)


gt (2+3I)+(6+4I) + 8 7 I



gt (5+I)+(3+I)-(4-I) + 4 3 I



gt (5-I)(4+2I)

+ 22 6 I





0 12

5 P(12 5)

ρ

θ

13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




13ρ

25144ρ

512ρ


=+=

+=

+=

deg==

==

==

+=

619864952212

5arctanθ

13

12

ρ

aθ cos

13

5

ρ

bθsen




polar 13

arctan

512


180

arctan

512

degrees

π

gt evalf()

2261986495degrees

Limite




Lf(x)lim =rarrax




1 Calcular 2x1

1xxlim

23

1 minus+minus

rarrx


gt Limit((x^3-x^2+1)(1-2x)x=1)

lim rarr x 1


gt limit((x^3-x^2+1)(1-2x)x=1) -1

2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




2 Calcular 1x

1xlim

4x minusminus

rarr



lim rarr x 4

minus x 1

minus x 1

gt value() 3







gt linalg

gt exp






problemas abaixo





2

dR =

f

t=T T

2πω = Rω sdot=v





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





Dados

f = 410 rpm

t = 1 min = 60 s

d = 36 cm

R =

ω =

v =

cm18R236

R2d

R =rArrthere4=rArr=

s41

6T

410


f

t

rads3

41πω

641

1π2

ω

416π2

ωT2π


msπ2463π738

183


π


gt f=410 = f 410

gt t=60 = t 60

gt d=36 = d 36

gt R=d2 = R 18

gt T=tf

= T641

gt omega=2PiT

= ω413

π






aumentado de 9



120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




120α

36091

α

360 1

20α

3609N 1

α

360N minus


minus=+rArrminus=



gt 360alpha-1+9=360(alpha-20)-1

= + 3601α

8 minus 3601

minus α 201

gt solve() 40 -20























Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++









Abrindo o Winplot































123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++























123456789

10111213141516

x

y








Ex






123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





123456789

10111213141516171819202122232425

x

y














minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





minus5

minus4

minus3

minus2

minus1

1

x

yy = 2x-3




minus4

minus3

minus2

minus1

1

2

3

4

5

x

y






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++






1 2 3 4 5 6

minus2

minus1

1

2

x










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++










1998 Cap 1-5






UNIRRIOMAR 2005


19 1989


1989






Paulo FTD 2001 3v




FTD 1996




Atual 2005 3v


Paulo FTD 2005 v 3


Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++





Atividade 1

1) 5a ndash 6b

2) 2y5y2 +

3) 11tt10

3 2 minusminus


5) x2

1minus

6) y35

33x

3

11 minus

7) 3

2x

3

14 minus

8) 5

y2x

5

3x 22

minusminus

Atividade 2

1) 42bx10a

2) cba5

1 33minus

3) x2

3x

3

1x

6

1x

4

1 234 minus+minus





8) 2bc2ac2abcba 222 +++++



Atividade 3

1) x3

2minus

2) 32yx

3


4) y4

1y2 minus

5) 3x2x2 +minus

6) 3xx 2 minusminus

7) 2x ndash 3

8) 5x +1 r = 5x ndash 2

Atividade 4


b + c

2) 9125yminus

3) 69yx27

8

4) 25

yx 44

5) 42b025a

6) 236 25ββ10αα ++

7) 422 n25

9mn

5

6m +minus

8) 3223 y6xyy12x8x +++

Atividade 5

1) x1

4x

minus

2) ndash1

3) 3

4) ndash29

5) ndash3

Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




Atividade 6

1)

3

13

3

29

2) [ ]32

3) [17 21]

4) [80 90]

5)

7

1290

7

1360

Atividade 7

1)

=

2

3

3

0

8

4a)

minusminus

=4

1

1

0

0

2b)

minusminus

=4

1

1

0

0

2c)

2)

minus=

8

16

0

12

4

8a)

minus=

18

9

36

18

9

27b)

minusminus=

4

17

12

9

8

19c)


=5

10

02

15

2

55

d)

3)

=

35

23

14

4a)

minus=

41

25

4

2b)

=

343

201

0

8c)

=

177

100

50

27d)

4) 62a) minus= 2

1b) =


6)

minus

minus=minus

2

11

2

32

A a) 1

minus=minus

02

1

12

3A b) 1

Testes

1) 4432 yx6

13yx

6

13 minus

2) 18

7z

18

7xy 32

+minus

3) 726x181x6x72x 234 ++minus+


5) α2 + β2 + γ2 + 2αβ + 2αγ + 2βγ

6) 2510xx 2 ++




maple 7 na resolução de cálculos algébricos

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