Calculus I

Preface......Page 4 Outline......Page 5 Introduction......Page 11 Review : Functions......Page 13 Review : Inverse Functions......Page 23 Review : Trig Functions......Page 30 Review : Solving Trig Equations......Page 37 Review : Solving Trig Equations with Calculators, Part I......Page 46 Review : Solving Trig Equations with Calculators, Part II......Page 57 Review : Exponential Functions......Page 62 Review : Logarithm Functions......Page 65 Review : Exponential and Logarithm Equations......Page 71 Review : Common Graphs......Page 77 Introduction......Page 89 Rates of Change and Tangent Lines......Page 91 The Limit......Page 100 One-Sided Limits......Page 110 Limit Properties......Page 116 Computing Limits......Page 122 Infinite Limits......Page 130 Limits At Infinity, Part I......Page 140 Limits At Infinity, Part II......Page 149 Continuity......Page 159 The Definition of the Limit......Page 166 Introduction......Page 181 The Definition of the Derivative......Page 183 Interpretations of the Derivative......Page 189 Differentiation Formulas......Page 198 Product and Quotient Rule......Page 206 Derivatives of Trig Functions......Page 212 Derivatives of Exponential and Logarithm Functions......Page 223 Derivatives of Inverse Trig Functions......Page 228 Derivatives of Hyperbolic Functions......Page 234 Implicit Differentiation......Page 246 Related Rates......Page 255 Higher Order Derivatives......Page 269 Logarithmic Differentiation......Page 274 Introduction......Page 277 Rates of Change......Page 279 Critical Points......Page 282 Minimum and Maximum Values......Page 288 Finding Absolute Extrema......Page 296 The Shape of a Graph, Part I......Page 302 The Shape of a Graph, Part II......Page 311 The Mean Value Theorem......Page 320 Optimization......Page 327 More Optimization Problems......Page 341 Indeterminate Forms and L’Hospital’s Rule......Page 356 Linear Approximations......Page 362 Differentials......Page 365 Newton’s Method......Page 368 Business Applications......Page 373 Introduction......Page 379 Indefinite Integrals......Page 380 Computing Indefinite Integrals......Page 386 Substitution Rule for Indefinite Integrals......Page 396 More Substitution Rule......Page 409 Area Problem......Page 422 The Definition of the Definite Integral......Page 432 Computing Definite Integrals......Page 442 Substitution Rule for Definite Integrals......Page 454 Introduction......Page 465 Average Function Value......Page 466 Area Between Curves......Page 469 Volumes of Solids of Revolution / Method of Rings......Page 480 Volumes of Solids of Revolution / Method of Cylinders......Page 490 More Volume Problems......Page 498 Work......Page 509 Introduction......Page 513 Proof of Various Limit Properties......Page 514 Proof of Various Derivative Facts/Formulas/Properties......Page 530 Proof of Trig Limits......Page 543 Proofs of Derivative Applications Facts/Formulas......Page 548 Proof of Various Integral Facts/Formulas/Properties......Page 559 Area and Volume Formulas......Page 571 Types of Infinity......Page 575 Summation Notation......Page 579 Constants of Integration......Page 581