OF

THE

TABLE OF CONTENTS

UNIT 1 REVIEW, LIMITS, CONTINUITY ............................................................................. 1

1-1 Slopes, Lines, Calculator Review .................................................................................... 1 1-2 Functions, Inverses, Graphing Adjustments .................................................................... 4 1-3 Intercepts, Symmetry, Even/Odd, Intersections ............................................................. 10 1-4 Limits, Continuity 14 ................................................................................................................................. Unit 1 Summary .......................................................................................................................... 18 1-5 Review ........................................................................................................................... 19 UNIT 2 LIMITS, CONTINUITY, DIFFERENTIATION ...................................................... 21

2-1 More Limits, More Continuity, Intermediate Value Theorem ....................................... 21 2-2 Infinite Limits, Limits at Infinity ................................................................................... 24 2-3 Curve Sketching ............................................................................................................. 28 2-4 Limit Definition of the Derivative, Alternate Form of the Limit Definition.................. 31 2-5 Derivative Rules (Short Cuts), Tangent Lines, Nondifferentiability, Rate of Change... 34 Unit 2 Summary .......................................................................................................................... 39 2-6 Review ........................................................................................................................... 40 UNIT 3 DIFFERENTIATION .................................................................................................. 43

3-1 Position, Velocity, Acceleration .................................................................................... 43 3-2 Product and Quotient Rules, Calculator Differentiation ................................................ 47 3-3 The Chain Rule and the General Power Rule ................................................................ 50 3-4 Implicit Differentiation .................................................................................................. 52 3-5 Related Rates .................................................................................................................. 54 Unit 3 Summary .......................................................................................................................... 58 3-6 Review ........................................................................................................................... 59 UNIT 4 APPLICATIONS OF DIFFERENTIATION ............................................................ 61

4-1 Absolute Extrema and the Mean Value Theorem .......................................................... 61 4-2 Increasing/Decreasing Functions, First Derivative Test for Relative Extrema .............. 65 4-3 Concavity and Points of Inflection, the Second Derivative Test for Relative Extrema . 68 4-4 Curve Sketching with Extrema and Points of Inflection ................................................ 72 4-5 Graphing Derivatives and Antiderivatives from Graphs ................................................ 75 Unit 4 Summary .......................................................................................................................... 78 4-6 Review ........................................................................................................................... 79 UNIT 5 APPLICATIONS OF DIFFERENTIATION, INTEGRATION.............................. 81

5-1 Max/Min Applications (Optimization)........................................................................... 81 5-2 More Max/Min Applications.......................................................................................... 84 5-3 Approximating with the Tangent Line ........................................................................... 86 5-4 Antidifferentiation, Indefinite Integrals ......................................................................... 90 5-5 The General Power Rule for Integrals and U-Substitution ............................................ 95 5-6 The Fundamental Theorem of Calculus, Definite Integrals, Calculator Integration ...... 99 Unit 5 Summary ........................................................................................................................ 104 5-7 Review ......................................................................................................................... 105

UNIT 6 INTEGRATION .......................................................................................................... 108

6-1 The Second Fundamental Theorem of Calculus, Interpretation of “Rate” Graphs ...... 108 6-2 Riemann Sums ............................................................................................................. 112 6-3 The Trapezoidal Rule ................................................................................................... 116 6-4 Area Between Curves, Average Value of a Function .................................................. 120 6-5 Volumes of Solids with Known Cross Sections ........................................................... 124 6-6 Volumes of Solids of Revolution: Discs and Washers ............................................... 127 Unit 6 Summary ........................................................................................................................ 131 6-7 Review ......................................................................................................................... 132 UNIT 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS ....................................... 135

7-1 Exponential Functions .................................................................................................. 135 7-2 Differentiating and Integrating Exponential Functions ................................................ 138 7-3 Derivatives of Inverse Functions ................................................................................. 141 7-4 Logarithmic Functions ................................................................................................. 145 7-5 Differentiating Logarithmic Functions ........................................................................ 150 7-6 Integration Involving the Natural Log Function .......................................................... 154 Unit 7 Summary ........................................................................................................................ 157 7-7 Review ......................................................................................................................... 158 UNIT 8 DIFFERENTIAL EQUATIONS, TRIGONOMETRY .......................................... 160

8-1 Differential Equations .................................................................................................. 160 8-2 Exponential Growth and Decay ................................................................................... 163 8-3 Slope Fields .................................................................................................................. 167 8-4 L’Hopital’s Rule .......................................................................................................... 172 8-5 Review of Basic Trigonometry .................................................................................... 175 Unit 8 Summary ........................................................................................................................ 180 8-6 Review ......................................................................................................................... 181 UNIT 9 TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS ........ 184

9-1 Trigonometry with a Calculator, Graphs of Trigonometric Functions ........................ 184 9-2 Differentiating Trigonometric Functions ..................................................................... 188 9-3 Integrating Trigonometric Functions ........................................................................... 192 9-4 Inverse Trigonometric Functions, Differentiating Inverse Trigonometric Functions .. 195 9-5 Integration Involving Inverse Trigonometric Functions .............................................. 200 9-6 Review of Basic Integration Formulas ......................................................................... 203 9-7 Final Integration Techniques, Review ......................................................................... 206 Unit 9 Summary ........................................................................................................................ 209

List of all of the Differentiation Formulas ................................................................................... 211 List of all of the Integration Formulas .......................................................................................... 212 APPENDIX A ........................................................................................................................... 213

A-1,2 Shell Method Volume .................................................................................................. 213 A-3,4 Integration by Parts ...................................................................................................... 217 Answers to Selected Exercises ..................................................................................................... 219 Index ............................................................................................................................................. 247