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"Por favor, poderia me dizer que caminho devo seguir agora? Isso depende bastante de at onde voc quer chegar." Lewis Carrol - Alice no Pa s das Maravilhas Atrav s dos s culos a Matem tica tem sido a mais poderosa e efetiva ferramenta para a compreens o das leis que regem a Natureza e o Universo. Os t picos introdut rios que apresentamos neste livro originaram-se, inicialmente, dos problemas pr ticos que surgiram no dia a dia e que continuaram impulsionados pela curiosidade humana de entender e explicar os fen nemos que regem a natureza. Historicamente, o C lculo Diferencial e Integral de uma vari vel estuda dois tipos de problemas: os associados no o de derivada, antigamente chamados de tang ncias e os problemas de integra o, antigamente chamados de quadraturas. Os relativos deriva o envolvem varia es ou mudan as, como por exemplo, a extens o de uma epidemia, os comportamentos econ micos ou a propaga o de poluentes na atmosfera, dentre outros. Como exemplos de problemas relacionados integra o destacam-se o c lculo da reas de regi es delimitadas por curvas, do volume de s lidos e do trabalho realizado por uma part cula. Grande parte do C lculo Diferencial e Integral foi desenvolvida no s culo XVIII por Isaac Newton para estudar problemas de F sica e Astronomia. Aproximadamente na mesma poca, Gottfried Wilhelm Leibniz, independentemente de Newton, tamb m desenvolveu consider vel parte do assunto. Devemos a Newton e Leibniz o estabelecimento da estreita rela o entre derivada e integral por meio de um teorema fundamental. As nota es sugeridas por Leibniz s o as universalmente usadas. O principal objetivo do livro foi apresentar os primeiros passos do C lculo Diferencial e Integral de uma vari vel com simplicidade, atrav s de exemplos, mas sem descuidar do aspecto formal da disciplina, dando nfase interpreta o geom trica e intuitiva dos conte dos. O livro inclui a maioria da teoria b sica, assim como exemplos aplicados e problemas. As provas muito t cnicas ou os teoremas mais sofisticados que n o foram provados no ap ndice, foram ilustrados atrav s de exemplos, aplica es e indica es bibliogr ficas adequadas e est o incluidos como refer ncia ou leitura adicional para os leitores interessados. Os conceitos centrais do C lculo Diferencial e Integral de uma vari vel s o relativamente profundos e n o se espera que possam ser assimilados de uma s

ii vez. Neste n vel, o importante que o leitor desenvolva a habilidade de calcular e adquira a compreens o intuitiva dos problemas. As express es do tipo " facil ver"ou semelhantes, que aparecem no texto, n o devem ser encaradas de forma literal e tem o prop sito de dar um aviso ao leitor de que naquele lugar a apresenta o resumida e os detalhes, perfeitamente acess veis, dever o ser preenchidos. Esperamos que o livro permita ao leitor um acesso r pido e agrad vel ao C lculo Diferencial e Integral de uma vari vel. N o podemos deixar de recomendar aos alunos a utiliza o, criteriosa, dos softwares de C lculo existente no mercado, pois eles s o um complemento til ao aprendizado da disciplina. Desejamos agradecer aos nossos colegas do Departamento de An lise e do IME-UERJ que, de algum modo, nos motivaram e deram condi es para escrever estas notas e Sra. Sonia M. Alves pela digita o. O texto foi digitado utlizando Amstex e a maioria dos desenhos foram feitos utilizando o software Mathematica. Certamente, todos os erros s o exclusivamente de responsabilidade dos autores.

Mauricio A. Vilches- Maria Luiza Correa Rio de Janeiro

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Copyright by Mauricio A. Vilches Todos os direitos reservados Proibida a reprodu o parcial ou total

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Conte do

1 Introdu o 1.1 Desiguldades . . . . . . . . . . . . . . . . . . . . . 1.2 Intervalos . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Valor Absoluto . . . . . . . . . . . . . . . . . . . . . 1.4 Dist ncia . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Plano Coordenado . . . . . . . . . . . . . . . . . . 1.6 Equa o da Reta . . . . . . . . . . . . . . . . . . . . 1.6.1 Equa o Geral da Reta . . . . . . . . . . . . 1.6.2 Equa o Reduzida da Reta . . . . . . . . . 1.6.3 Paralelismo e Perpendicularismo de Retas 1.7 Equa o das C nicas . . . . . . . . . . . . . . . . . 1.8 Trigonometria . . . . . . . . . . . . . . . . . . . . . 1.9 Exerc cios . . . . . . . . . . . . . . . . . . . . . . . . 2 Fun es de uma Vari vel Real 2.1 Defini es e Exemplos . . . . . . . . . . . . . . 2.2 Gr ficos de Fun es . . . . . . . . . . . . . . . . 2.3 Exemplos de Fun es . . . . . . . . . . . . . . . 2.3.1 Fun o Modular ou Valor Absoluto . . 2.3.2 Fun es Polinomiais . . . . . . . . . . . 2.3.3 Fun es Pares e mpares . . . . . . . . . 2.4 Interse o de Gr ficos . . . . . . . . . . . . . . 2.5 gebra de Fun es . . . . . . . . . . . . . . . . 2.5.1 Fun es Racionais . . . . . . . . . . . . 2.6 Composta de Fun es . . . . . . . . . . . . . . 2.7 Inversa de uma Fun o . . . . . . . . . . . . . . 2.7.1 M todo para Determinar a Inversa . . . 2.8 Fun o Exponencial . . . . . . . . . . . . . . . . 2.8.1 Crescimento e Decaimento Exponencial 2.8.2 Fun o Log stica . . . . . . . . . . . . . 2.9 Fun o Logar tmica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4 5 5 8 8 9 10 11 15 20 25 25 33 39 39 40 47 49 51 53 53 57 58 62 65 66 67

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vi 2.9.1 Desintegra o Radioativa . . . . . . . . . . . . Fun es Trigonom tricas . . . . . . . . . . . . . . . . . 2.10.1 Fun o Seno e Co-seno . . . . . . . . . . . . . . 2.10.2 Fun o Tangente e Secante . . . . . . . . . . . 2.10.3 Fun o Co-tangente e Co-secante . . . . . . . . Fun es Trigonom tricas Inversas . . . . . . . . . . . 2.11.1 Fun o Arco seno . . . . . . . . . . . . . . . . . 2.11.2 Fun o Arco co-seno . . . . . . . . . . . . . . . 2.11.3 Fun o Arco tangente . . . . . . . . . . . . . . 2.11.4 Fun o Arco secante, co-tangente e co-secante Fun es Hiperb licas . . . . . . . . . . . . . . . . . . . Exerc cios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTE DO

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2.10

2.11

2.12 2.13

3 Limite e Continuidade de uma Fun o 3.1 Limites . . . . . . . . . . . . . . . . . . . 3.1.1 Defini es e Exemplos . . . . . . 3.1.2 Limites Laterais . . . . . . . . . . 3.1.3 Limites no Infinito . . . . . . . . 3.1.4 Limites Infinitos . . . . . . . . . . 3.1.5 S mbolos de Indermina o . . . 3.1.6 Limites Fundamentais . . . . . . 3.1.7 Ass ntotas . . . . . . . . . . . . . 3.2 Continuidade . . . . . . . . . . . . . . . 3.2.1 Teorema do Valor Intermedi rio 3.3 Exerc cios . . . . . . . . . . . . . . . . . .

4 Derivada 4.1 Reta Tangente . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fun es Deriv veis . . . . . . . . . . . . . . . . . . . . . 4.3 Regras de Deriva o . . . . . . . . . . . . . . . . . . . . 4.4 Derivada da Fun o Composta . . . . . . . . . . . . . . 4.4.1 Derivada da Fun o Exponencial . . . . . . . . . 4.4.2 Derivada da Fun o Logar tmica . . . . . . . . . 4.4.3 Derivada das Fun es Trigonom tricas . . . . . 4.4.4 Derivada das Fun es Trigonom tricas Inversas 4.4.5 Derivada das Fun es Hiperb licas . . . . . . . 4.5 Deriva o Impl cita . . . . . . . . . . . . . . . . . . . . . 4.5.1 C lculo da Derivada de uma Fun o Impl cita . 4.6 Fam lias de Curvas Ortogonais . . . . . . . . . . . . . . 4.7 Derivadas de Ordem Superior . . . . . . . . . . . . . . . 4.8 Aproxima o Linear . . . . . . . . . . . . . . . . . . . .

CONTE DO

4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 Velocidade e Acelera o . . . . . . . . A Derivada como Taxa de Varia o . . Exerc cios I . . . . . . . . . . . . . . . . Varia o de Fun es . . . . . . . . . . Fun es Mon tonas . . . . . . . . . . . Determina o de M ximos e M nimos Concavidade e Pontos de Inflex o . . Esboco do Gr fico de Fun es . . . . . Problemas de Otimiza o . . . . . . . Teorema de L'H pital . . . . . . . . . . Exerc cios II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 178 181 186 194 200 204 209 213 221 231 239 246 246 250 250 253 254 257 262 271 272 272 273 276 282 282 288 290 297 300 300 303 319 321 327 330 334 337

5 Integra o Indefinida 5.1 Defini es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 M todos de Integra o . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 M todo de Substitui o . . . . . . . . . . . . . . . . . . . 5.2.2 Integrais que Envolvem Produtos e Pot ncias de Fun es Trigonom tricas . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Integra o Por Partes . . . . . . . . . . . . . . . . . . . . . 5.2.4 Substitui o Trigonom trica . . . . . . . . . . . . . . . . . 5.2.5 Integra o de Fun es Racionais . . . . . . . . . . . . . . 5.2.6 Tangente do ngulo M dio . . . . . . . . . . . . . . . . . 5.3 Aplica es da Integral Indefinida . . . . . . . . . . . . . . . . . . 5.3.1 Obten o de Fam lias de Curvas . . . . . . . . . . . . . . 5.3.2 Outras Aplica es . . . . . . . . . . . . . . . . . . . . . . 5.4 Exerc cios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Integra o Definida 6.1 Introdu o . . . . . . . . . . . . . . . . . . 6.2 Defini o e C lculo da Integral Definida . 6.2.1 Teorema Fundamental do C lculo 6.3 Contru o de Primitivas . . . . . . . . . . 6.4 Aplica es da Integral Definida . . . . . . 6.4.1 Acelera o, velocidade e posi o . 6.5 C lculo de reas . . . . . . . . . . . . . . 6.6 Volume de S lidos de Revolu o . . . . . 6.6.1 C lculo do Volume do S lidos . . 6.6.2 Outros Eixos de Revolu o . . . . 6.6.3 M todos das Arruelas . . . . . . . 6.7 Comprimento de Arco . . . . . . . . . . . 6.8 Logaritmo Natural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii 6.8.1 Logaritmo Natural como rea . . . . . . . . 6.9 Trabalho . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Integrais Impr prias . . . . . . . . . . . . . . . . . . 6.10.1 Integrais Definidas em Intervalos Ilimitados 6.10.2 Integrais de Fun es Descont nuas . . . . . . 6.11 Exerc cios . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exerc cios Resolvidos 8 Ap ndice 9 Respostas 10 Bibliografia . . . . . . . . . . . .

CONTE DO

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