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