Matemática - Apostila UERJ Cálculo - Volume 01

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"Por favor, poderia me dizer que caminho devo seguir agora?Isso depende bastante de até onde você quer chegar."Lewis Carrol - Alice no País das Maravilhas

Através dos séculos a Matemática tem sido a mais poderosa e efetiva ferra-menta para a compreensão das leis que regem a Natureza e o Universo.Os tópicos introdutórios que apresentamos neste livro originaram-se, inicial-mente, dos problemas práticos que surgiram no dia a dia e que continuaramimpulsionados pela curiosidade humana de entender e explicar os fenônemosque regem a natureza.Historicamente, o Cálculo Diferencial e Integral de uma variável estuda dois ti-pos de problemas: os associados à noção de derivada, antigamente chamadosde tangências e os problemas de integração, antigamente chamados de qua-draturas. Os relativos à derivação envolvem variações ou mudanças, comopor exemplo, a extensão de uma epidemia, os comportamentos econômicosou a propagação de poluentes na atmosfera, dentre outros. Como exemplosde problemas relacionados à integração destacam-se o cálculo da áreas de re-giões delimitadas por curvas, do volume de sólidos e do trabalho realizadopor uma partícula.Grande parte do Cálculo Diferencial e Integral foi desenvolvida no século XVI-II por Isaac Newton para estudar problemas de Física e Astronomia. Aproxi-madamente na mesma época, Gottfried Wilhelm Leibniz, independentementede Newton, também desenvolveu considerável parte do assunto. Devemos aNewton e Leibniz o estabelecimento da estreita relação entre derivada e inte-gral por meio de um teorema fundamental. As notações sugeridas por Leibnizsão as universalmente usadas.O principal objetivo do livro foi apresentar os primeiros passos do Cálculo Di-ferencial e Integral de uma variável com simplicidade, através de exemplos,mas sem descuidar do aspecto formal da disciplina, dando ênfase à interpre-tação geométrica e intuitiva dos conteúdos.O livro inclui a maioria da teoria básica, assim como exemplos aplicados eproblemas. As provas muito técnicas ou os teoremas mais sofisticados quenão foram provados no apêndice, foram ilustrados através de exemplos, apli-cações e indicações bibliográficas adequadas e estão incluidos como referênciaou leitura adicional para os leitores interessados.Os conceitos centrais do Cálculo Diferencial e Integral de uma variável são re-lativamente profundos e não se espera que possam ser assimilados de uma só

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vez. Neste nível, o importante é que o leitor desenvolva a habilidade de calcu-lar e adquira a compreensão intuitiva dos problemas. As expressões do tipo "éfacil ver"ou semelhantes, que aparecem no texto, não devem ser encaradas deforma literal e tem o propósito de dar um aviso ao leitor de que naquele lugara apresentação é resumida e os detalhes, perfeitamente acessíveis, deverão serpreenchidos.Esperamos que o livro permita ao leitor um acesso rápido e agradável ao Cál-culo Diferencial e Integral de uma variável.Não podemos deixar de recomendar aos alunos a utilização, criteriosa, dossoftwares de Cálculo existente no mercado, pois eles são um complemento útilao aprendizado da disciplina.Desejamos agradecer aos nossos colegas do Departamento de Análise e doIME-UERJ que, de algum modo, nos motivaram e deram condições para es-crever estas notas e à Sra. Sonia M. Alves pela digitação. O texto foi digitadoutlizando Amstex e a maioria dos desenhos foram feitos utilizando o softwareMathematica. Certamente, todos os erros são exclusivamente de responsabili-dade dos autores.

Mauricio A. Vilches- Maria Luiza CorreaRio de Janeiro

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Copyright by Mauricio A. VilchesTodos os direitos reservados

Proibida a reprodução parcial ou total

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Conteúdo

1 Introdução 11.1 Desiguldades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Intervalos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Valor Absoluto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Distância . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Plano Coordenado . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Equação da Reta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6.1 Equação Geral da Reta . . . . . . . . . . . . . . . . . . . . 81.6.2 Equação Reduzida da Reta . . . . . . . . . . . . . . . . . 91.6.3 Paralelismo e Perpendicularismo de Retas . . . . . . . . 10

1.7 Equação das Cônicas . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Trigonometria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.9 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Funções de uma Variável Real 252.1 Definições e Exemplos . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Gráficos de Funções . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Exemplos de Funções . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1 Função Modular ou Valor Absoluto . . . . . . . . . . . . 392.3.2 Funções Polinomiais . . . . . . . . . . . . . . . . . . . . . 402.3.3 Funções Pares e Ímpares . . . . . . . . . . . . . . . . . . . 47

2.4 Interseção de Gráficos . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Ágebra de Funções . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.1 Funções Racionais . . . . . . . . . . . . . . . . . . . . . . 532.6 Composta de Funções . . . . . . . . . . . . . . . . . . . . . . . . 532.7 Inversa de uma Função . . . . . . . . . . . . . . . . . . . . . . . . 57

2.7.1 Método para Determinar a Inversa . . . . . . . . . . . . . 582.8 Função Exponencial . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.8.1 Crescimento e Decaimento Exponencial . . . . . . . . . . 652.8.2 Função Logística . . . . . . . . . . . . . . . . . . . . . . . 66

2.9 Função Logarítmica . . . . . . . . . . . . . . . . . . . . . . . . . . 67

v

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2.9.1 Desintegração Radioativa . . . . . . . . . . . . . . . . . . 692.10 Funções Trigonométricas . . . . . . . . . . . . . . . . . . . . . . . 72

2.10.1 Função Seno e Co-seno . . . . . . . . . . . . . . . . . . . . 722.10.2 Função Tangente e Secante . . . . . . . . . . . . . . . . . 732.10.3 Função Co-tangente e Co-secante . . . . . . . . . . . . . . 74

2.11 Funções Trigonométricas Inversas . . . . . . . . . . . . . . . . . 802.11.1 Função Arco seno . . . . . . . . . . . . . . . . . . . . . . . 802.11.2 Função Arco co-seno . . . . . . . . . . . . . . . . . . . . . 812.11.3 Função Arco tangente . . . . . . . . . . . . . . . . . . . . 822.11.4 Função Arco secante, co-tangente e co-secante . . . . . . 83

2.12 Funções Hiperbólicas . . . . . . . . . . . . . . . . . . . . . . . . . 852.13 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3 Limite e Continuidade de uma Função 953.1 Limites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.1.1 Definições e Exemplos . . . . . . . . . . . . . . . . . . . . 953.1.2 Limites Laterais . . . . . . . . . . . . . . . . . . . . . . . . 1023.1.3 Limites no Infinito . . . . . . . . . . . . . . . . . . . . . . 1063.1.4 Limites Infinitos . . . . . . . . . . . . . . . . . . . . . . . . 1093.1.5 Símbolos de Inderminação . . . . . . . . . . . . . . . . . 1133.1.6 Limites Fundamentais . . . . . . . . . . . . . . . . . . . . 1143.1.7 Assíntotas . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.2 Continuidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.2.1 Teorema do Valor Intermediário . . . . . . . . . . . . . . 126

3.3 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4 Derivada 1394.1 Reta Tangente . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.2 Funções Deriváveis . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.3 Regras de Derivação . . . . . . . . . . . . . . . . . . . . . . . . . 1494.4 Derivada da Função Composta . . . . . . . . . . . . . . . . . . . 151

4.4.1 Derivada da Função Exponencial . . . . . . . . . . . . . . 1544.4.2 Derivada da Função Logarítmica . . . . . . . . . . . . . . 1564.4.3 Derivada das Funções Trigonométricas . . . . . . . . . . 1594.4.4 Derivada das Funções Trigonométricas Inversas . . . . . 1614.4.5 Derivada das Funções Hiperbólicas . . . . . . . . . . . . 162

4.5 Derivação Implícita . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.5.1 Cálculo da Derivada de uma Função Implícita . . . . . . 165

4.6 Famílias de Curvas Ortogonais . . . . . . . . . . . . . . . . . . . 1704.7 Derivadas de Ordem Superior . . . . . . . . . . . . . . . . . . . . 1724.8 Aproximação Linear . . . . . . . . . . . . . . . . . . . . . . . . . 175

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4.9 Velocidade e Aceleração . . . . . . . . . . . . . . . . . . . . . . . 1784.10 A Derivada como Taxa de Variação . . . . . . . . . . . . . . . . . 1814.11 Exercícios I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.12 Variação de Funções . . . . . . . . . . . . . . . . . . . . . . . . . 1944.13 Funções Monótonas . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.14 Determinação de Máximos e Mínimos . . . . . . . . . . . . . . . 2044.15 Concavidade e Pontos de Inflexão . . . . . . . . . . . . . . . . . 2094.16 Esboco do Gráfico de Funções . . . . . . . . . . . . . . . . . . . . 2134.17 Problemas de Otimização . . . . . . . . . . . . . . . . . . . . . . 2214.18 Teorema de L’Hôpital . . . . . . . . . . . . . . . . . . . . . . . . . 2314.19 Exercícios II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

5 Integração Indefinida 2465.1 Definições . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.2 Métodos de Integração . . . . . . . . . . . . . . . . . . . . . . . . 250

5.2.1 Método de Substituição . . . . . . . . . . . . . . . . . . . 2505.2.2 Integrais que Envolvem Produtos e Potências de Funções

Trigonométricas . . . . . . . . . . . . . . . . . . . . . . . . 2535.2.3 Integração Por Partes . . . . . . . . . . . . . . . . . . . . . 2545.2.4 Substituição Trigonométrica . . . . . . . . . . . . . . . . . 2575.2.5 Integração de Funções Racionais . . . . . . . . . . . . . . 2625.2.6 Tangente do Ângulo Médio . . . . . . . . . . . . . . . . . 271

5.3 Aplicações da Integral Indefinida . . . . . . . . . . . . . . . . . . 2725.3.1 Obtenção de Famílias de Curvas . . . . . . . . . . . . . . 2725.3.2 Outras Aplicações . . . . . . . . . . . . . . . . . . . . . . 273

5.4 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

6 Integração Definida 2826.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2826.2 Definição e Cálculo da Integral Definida . . . . . . . . . . . . . . 288

6.2.1 Teorema Fundamental do Cálculo . . . . . . . . . . . . . 2906.3 Contrução de Primitivas . . . . . . . . . . . . . . . . . . . . . . . 2976.4 Aplicações da Integral Definida . . . . . . . . . . . . . . . . . . . 300

6.4.1 Aceleração, velocidade e posição . . . . . . . . . . . . . . 3006.5 Cálculo de Áreas . . . . . . . . . . . . . . . . . . . . . . . . . . . 3036.6 Volume de Sólidos de Revolução . . . . . . . . . . . . . . . . . . 319

6.6.1 Cálculo do Volume do Sólidos . . . . . . . . . . . . . . . 3216.6.2 Outros Eixos de Revolução . . . . . . . . . . . . . . . . . 3276.6.3 Métodos das Arruelas . . . . . . . . . . . . . . . . . . . . 330

6.7 Comprimento de Arco . . . . . . . . . . . . . . . . . . . . . . . . 3346.8 Logaritmo Natural . . . . . . . . . . . . . . . . . . . . . . . . . . 337

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6.8.1 Logaritmo Natural como Área . . . . . . . . . . . . . . . 3386.9 Trabalho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396.10 Integrais Impróprias . . . . . . . . . . . . . . . . . . . . . . . . . 341

6.10.1 Integrais Definidas em Intervalos Ilimitados . . . . . . . 3416.10.2 Integrais de Funções Descontínuas . . . . . . . . . . . . . 348

6.11 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7 Exercícios Resolvidos 365

8 Apéndice 401

9 Respostas 411

10 Bibliografia 429

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Page 13: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 14: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 15: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 16: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 17: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 18: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 28: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 29: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 30: Matemática - Apostila UERJ Cálculo - Volume 01

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Page 31: Matemática - Apostila UERJ Cálculo - Volume 01

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k ��� � o:W kvmwo8p `ebcd� g S k ��� � o�knm ½ � o�­¤k ����� oMknm�o� Õ

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