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Table of Contents
Preface …………………………………………………………………………………………………………………………. iii
Outline …………………………………………………………………………………………………………………………. iv
Review…………………………………………………………………………………………………………………………… 2
Introduction ……………………………………………………………………………………………………………………………. 2
Review : Functions ………………………………………………………………………………………………………………….. 4
Review : Inverse Functions …………………………………………………………………………………………………….. 10
Review : Trig Functions …………………………………………………………………………………………………………. 17
Review : Solving Trig Equations ……………………………………………………………………………………………… 24
Review : Solving Trig Equations with Calculators, Part I ………………………………………………………….. 33
Review : Solving Trig Equations with Calculators, Part II …………………………………………………………. 44
Review : Exponential Functions ……………………………………………………………………………………………… 49
Review : Logarithm Functions ………………………………………………………………………………………………… 52
Review : Exponential and Logarithm Equations ………………………………………………………………………. 58
Review : Common Graphs ………………………………………………………………………………………………………. 64
Limits ………………………………………………………………………………………………………………………….. 76
Introduction ………………………………………………………………………………………………………………………….. 76
Rates of Change and Tangent Lines ………………………………………………………………………………………… 78
The Limit ………………………………………………………………………………………………………………………………. 87
One‐Sided Limits …………………………………………………………………………………………………………………… 97
Limit Properties …………………………………………………………………………………………………………………….103
Computing Limits ………………………………………………………………………………………………………………….109
Infinite Limits ……………………………………………………………………………………………………………………….117
Limits At Infinity, Part I ………………………………………………………………………………………………………….126
Limits At Infinity, Part II ………………………………………………………………………………………………………..135
Continuity ……………………………………………………………………………………………………………………………..144
The Definition of the Limit ……………………………………………………………………………………………………..151
Derivatives …………………………………………………………………………………………………………………. 166
Introduction ………………………………………………………………………………………………………………………….166
The Definition of the Derivative ……………………………………………………………………………………………..168
Interpretations of the Derivative ……………………………………………………………………………………………174
Differentiation Formulas ……………………………………………………………………………………………………….179
Product and Quotient Rule …………………………………………………………………………………………………….187
Derivatives of Trig Functions …………………………………………………………………………………………………193
Derivatives of Exponential and Logarithm Functions ………………………………………………………………204
Derivatives of Inverse Trig Functions ……………………………………………………………………………………..209
Derivatives of Hyperbolic Functions ……………………………………………………………………………………….215
Chain Rule …………………………………………………………………………………………………………………………….217
Implicit Differentiation ………………………………………………………………………………………………………….227
Related Rates ………………………………………………………………………………………………………………………..236
Higher Order Derivatives ……………………………………………………………………………………………………….250
Logarithmic Differentiation ……………………………………………………………………………………………………255
Applications of Derivatives …………………………………………………………………………………………. 258
Introduction ………………………………………………………………………………………………………………………….258
Rates of Change……………………………………………………………………………………………………………………..260
Critical Points ………………………………………………………………………………………………………………………..263
Minimum and Maximum Values ……………………………………………………………………………………………..269
Finding Absolute Extrema ……………………………………………………………………………………………………..277
The Shape of a Graph, Part I ……………………………………………………………………………………………………283
The Shape of a Graph, Part II ………………………………………………………………………………………………….292
The Mean Value Theorem ………………………………………………………………………………………………………301
Optimization …………………………………………………………………………………………………………………………308
More Optimization Problems …………………………………………………………………………………………………322
Calculus I
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Indeterminate Forms and L’Hospital’s Rule …………………………………………………………………………….336
Linear Approximations ………………………………………………………………………………………………………….342
Differentials ………………………………………………………………………………………………………………………….345
Newton’s Method …………………………………………………………………………………………………………………..348
Business Applications ……………………………………………………………………………………………………………353
Integrals …………………………………………………………………………………………………………………….. 359
Introduction ………………………………………………………………………………………………………………………….359
Indefinite Integrals ………………………………………………………………………………………………………………..360
Computing Indefinite Integrals ………………………………………………………………………………………………366
Substitution Rule for Indefinite Integrals ………………………………………………………………………………..376
More Substitution Rule ………………………………………………………………………………………………………….389
Area Problem ………………………………………………………………………………………………………………………..402
The Definition of the Definite Integral …………………………………………………………………………………….412
Computing Definite Integrals …………………………………………………………………………………………………422
Substitution Rule for Definite Integrals …………………………………………………………………………………..434
Applications of Integrals …………………………………………………………………………………………….. 445
Introduction ………………………………………………………………………………………………………………………….445
Average Function Value …………………………………………………………………………………………………………446
Area Between Curves …………………………………………………………………………………………………………….449
Volumes of Solids of Revolution / Method of Rings ………………………………………………………………….460
Volumes of Solids of Revolution / Method of Cylinders ……………………………………………………………470
Work …………………………………………………………………………………………………………………………………….478
Extras ………………………………………………………………………………………………………………………… 482
Introduction ………………………………………………………………………………………………………………………….482
Proof of Various Limit Properties …………………………………………………………………………………………..483
Proof of Various Derivative Facts/Formulas/Properties ………………………………………………………….494
Proof of Trig Limits ……………………………………………………………………………………………………………….507
Proofs of Derivative Applications Facts/Formulas …………………………………………………………………..512
Proof of Various Integral Facts/Formulas/Properties ……………………………………………………………..523
Area and Volume Formulas ……………………………………………………………………………………………………535
Types of Infinity …………………………………………………………………………………………………………………….539
Summation Notation ……………………………………………………………………………………………………………..543
Constants of Integration ………………………………………………………………………………………………………..545