Matemática - Apostila UERJ Cálculo - Volume 01, Notas de estudo de Matemática

Baixe Matemática - Apostila UERJ Cálculo - Volume 01 e outras Notas de estudo em PDF para Matemática, somente na Docsity! CÁLCULO: VOLUME I MAURICIO A. VILCHES MARIA L. CORREA i "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 continuaram impulsionados pela curiosidade humana de entender e explicar os fenônemos que 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 chamados de tangências e os problemas de integração, antigamente chamados de qua- draturas. Os relativos à derivação envolvem variações ou mudanças, como por exemplo, a extensão de uma epidemia, os comportamentos econômicos ou a propagação de poluentes na atmosfera, dentre outros. Como exemplos de problemas relacionados à integração destacam-se o cálculo da áreas de re- giões delimitadas por curvas, do volume de sólidos e do trabalho realizado por 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, independentemente de Newton, também desenvolveu considerável parte do assunto. Devemos a Newton e Leibniz o estabelecimento da estreita relação entre derivada e inte- gral por meio de um teorema fundamental. As notações sugeridas por Leibniz são 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 e problemas. As provas muito técnicas ou os teoremas mais sofisticados que não foram provados no apêndice, foram ilustrados através de exemplos, apli- cações e indicações bibliográficas adequadas e estão incluidos como referência ou 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 iv Conteúdo 1 Introdução 1 1.1 Desiguldades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Intervalos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Valor Absoluto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Distância . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Plano Coordenado . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Equação da Reta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.1 Equação Geral da Reta . . . . . . . . . . . . . . . . . . . . 8 1.6.2 Equação Reduzida da Reta . . . . . . . . . . . . . . . . . 9 1.6.3 Paralelismo e Perpendicularismo de Retas . . . . . . . . 10 1.7 Equação das Cônicas . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 Trigonometria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.9 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Funções de uma Variável Real 25 2.1 Definições e Exemplos . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Gráficos de Funções . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Exemplos de Funções . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.1 Função Modular ou Valor Absoluto . . . . . . . . . . . . 39 2.3.2 Funções Polinomiais . . . . . . . . . . . . . . . . . . . . . 40 2.3.3 Funções Pares e Ímpares . . . . . . . . . . . . . . . . . . . 47 2.4 Interseção de Gráficos . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5 Ágebra de Funções . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5.1 Funções Racionais . . . . . . . . . . . . . . . . . . . . . . 53 2.6 Composta de Funções . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7 Inversa de uma Função . . . . . . . . . . . . . . . . . . . . . . . . 57 2.7.1 Método para Determinar a Inversa . . . . . . . . . . . . . 58 2.8 Função Exponencial . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.8.1 Crescimento e Decaimento Exponencial . . . . . . . . . . 65 2.8.2 Função Logística . . . . . . . . . . . . . . . . . . . . . . . 66 2.9 Função Logarítmica . . . . . . . . . . . . . . . . . . . . . . . . . . 67 v vi CONTEÚDO 2.9.1 Desintegração Radioativa . . . . . . . . . . . . . . . . . . 69 2.10 Funções Trigonométricas . . . . . . . . . . . . . . . . . . . . . . . 72 2.10.1 Função Seno e Co-seno . . . . . . . . . . . . . . . . . . . . 72 2.10.2 Função Tangente e Secante . . . . . . . . . . . . . . . . . 73 2.10.3 Função Co-tangente e Co-secante . . . . . . . . . . . . . . 74 2.11 Funções Trigonométricas Inversas . . . . . . . . . . . . . . . . . 80 2.11.1 Função Arco seno . . . . . . . . . . . . . . . . . . . . . . . 80 2.11.2 Função Arco co-seno . . . . . . . . . . . . . . . . . . . . . 81 2.11.3 Função Arco tangente . . . . . . . . . . . . . . . . . . . . 82 2.11.4 Função Arco secante, co-tangente e co-secante . . . . . . 83 2.12 Funções Hiperbólicas . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.13 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3 Limite e Continuidade de uma Função 95 3.1 Limites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.1 Definições e Exemplos . . . . . . . . . . . . . . . . . . . . 95 3.1.2 Limites Laterais . . . . . . . . . . . . . . . . . . . . . . . . 102 3.1.3 Limites no Infinito . . . . . . . . . . . . . . . . . . . . . . 106 3.1.4 Limites Infinitos . . . . . . . . . . . . . . . . . . . . . . . . 109 3.1.5 Símbolos de Inderminação . . . . . . . . . . . . . . . . . 113 3.1.6 Limites Fundamentais . . . . . . . . . . . . . . . . . . . . 114 3.1.7 Assíntotas . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2 Continuidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.2.1 Teorema do Valor Intermediário . . . . . . . . . . . . . . 126 3.3 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4 Derivada 139 4.1 Reta Tangente . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2 Funções Deriváveis . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.3 Regras de Derivação . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.4 Derivada da Função Composta . . . . . . . . . . . . . . . . . . . 151 4.4.1 Derivada da Função Exponencial . . . . . . . . . . . . . . 154 4.4.2 Derivada da Função Logarítmica . . . . . . . . . . . . . . 156 4.4.3 Derivada das Funções Trigonométricas . . . . . . . . . . 159 4.4.4 Derivada das Funções Trigonométricas Inversas . . . . . 161 4.4.5 Derivada das Funções Hiperbólicas . . . . . . . . . . . . 162 4.5 Derivação Implícita . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.5.1 Cálculo da Derivada de uma Função Implícita . . . . . . 165 4.6 Famílias de Curvas Ortogonais . . . . . . . . . . . . . . . . . . . 170 4.7 Derivadas de Ordem Superior . . . . . . . . . . . . . . . . . . . . 172 4.8 Aproximação Linear . . . . . . . . . . . . . . . . . . . . . . . . . 175
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