Table of Contents
Introduction 1
What You'll Find 1
Beyond the Book 2
Where to Go for Additional Help 2
Part 1 The Questions 5
Chapter 1 Algebra Review 7
The Problems You'll Work On 7
What to Watch Out For 7
Simplifying Fractions 8
Simplifying Radicals 8
Writing Exponents Using Radical Notation 9
The Horizontal Line Test 9
Find Inverses Algebraically 10
The Domain and Range of a Function and Its Inverse 10
Linear Equations 10
Quadratic Equations 11
Solving Polynomial Equations by Factoring 11
Absolute Value Equations 12
Solving Rational Equations 12
Polynomial and Rational Inequalities 12
Absolute Value Inequalities 13
Graphing Common Functions 13
Domain and Range from a Graph 14
End Behavior of Polynomials 15
Adding Polynomials 15
Subtracting Polynomials 15
Multiplying Polynomials 16
Long Division of Polynomials 16
Chapter 2 Trigonometry Review 17
The Problems You'll Work On 17
What to Watch Out For 17
Basic Trigonometry 18
Converting Degree Measure to Radian Measure 18
Converting Radian Measure to Degree Measure 19
Finding Angles in the Coordinate Plane 19
Finding Common Trigonometric Values 21
Simplifying Trigonometric Expressions 21
Solving Trigonometric Equations 22
Amplitude, Period, Phase Shift, and Midline 23
Equations of Periodic Functions 24
Inverse Trigonometric Function Basics 26
Solving Trigonometric Equations Using Inverses 27
Chapter 3 Limits and Rates of Change 29
The Problems You'll Work On 29
What to Watch Out For 29
Finding Limits from Graphs 30
Evaluating Limits 31
Applying the Squeeze Theorem 32
Evaluating Trigonometric Limits 33
Infinite Limits 33
Limits from Graphs 36
Limits at Infinity 37
Horizontal Asymptotes 38
Classifying Discontinuities 38
Continuity and Discontinuities 39
Making a Function Continuous 40
The Intermediate Value Theorem 41
Chapter 4 Derivative Basics 43
The Problems You'll Work On 43
What to Watch Out For 43
Determining Differentiability from a Graph 44
Finding the Derivative by Using the Definition 45
Finding the Value of the Derivative Using a Graph 46
Using the Power Rule to Find Derivatives 47
Finding All Points on a Graph Where Tangent Lines Have a Given Value 48
Chapter 5 The Product, Quotient, and Chain Rules 49
The Problems You'll Work On 49
What to Watch Out For 49
Using the Product Rule to Find Derivatives 50
Using the Quotient Rule to Find Derivatives 51
Using the Chain Rule to Find Derivatives 53
More Challenging Chain Rule Problems 54
Chapter 6 Exponential and Logarithmic Functions and Tangent Lines 55
The Problems You'll Work On 55
What to Watch Out For 55
Derivatives Involving Logarithmic Functions 55
Logarithmic Differentiation to Find the Derivative 56
Finding Derivatives of Functions Involving Exponential Functions 57
Finding Equations of Tangent Lines 58
Finding Equations of Normal Lines 58
Chapter 7 Implicit Differentiation 59
The Problems You'll Work On 59
What to Watch Out For 59
Using Implicit Differentiation to Find a Derivative 60
Using Implicit Differentiation to Find a Second Derivative 60
Finding Equations of Tangent Lines Using Implicit Differentiation 61
Chapter 8 Applications of Derivatives 63
The Problems You'll Work On 63
What to Watch Out For 63
Finding and Evaluating Differentials 64
Finding Linearizations 64
Using Linearizations to Estimate Values 64
Understanding Related Rates 65
Finding Maxima and Minima from Graphs 66
Using the Closed Interval Method 67
Finding intervals of Increase and Decrease 68
Using the First Derivative Test to Find Local Maxima and Minima 68
Determining Concavity 69
Identifying Inflection Points 69
Using the Second Derivative Test to Find Local Maxima and Minima 69
Applying Rolle's Theorem 70
Using the Mean Value Theorem 70
Applying the Mean Value Theorem to Solve Problems 70
Relating Velocity and Position 71
Finding Velocity and Speed 71
Solving Optimization Problems 72
Doing Approximations Using Newton's Method 73
Approximating Roots Using Newton's Method 74
Chapter 9 Areas and Riemann Sums 75
The Problems You'll Work On 75
What to Watch Out For 75
Calculating Riemann Sums Using Left Endpoints 76
Calculating Riemann Sums Using Right Endpoints 76
Calculating Riemann Sums Using Midpoints 77
Using Limits and Riemann Sums to Find Expressions for Definite Integrals 77
Finding a Definite Integral from the Limit and Riemann Sum Form 78
Using Limits and Riemann Sums to Evaluate Definite Integrals 78
Chapter 10 The Fundamental Theorem of Calculus and the Net Change Theorem 79
The Problems You'll Work On 79
What to Watch Out For 80
Using the Fundamental Theorem of Calculus to Find Derivatives 80
Working with Basic Examples of Definite Integrals 81
Understanding Basic Indefinite Integrals 82
Understanding the Net Change Theorem 84
Finding the Displacement of a Particle Given the Velocity 85
Finding the Distance Traveled by a Particle Given the Velocity 85
Finding the Displacement of a Particle Given Acceleration 86
Finding the Distance Traveled by a Particle Given Acceleration 86
Chapter 11 Applications of Integration 87
The Problems You'll Work On 87
What to Watch Out For 87
Areas between Curves 88
Finding Volumes Using Disks and Washers 89
Finding Volume Using Cross-Sectional Slices 91
Finding Volumes Using Cylindrical Shells 92
Work Problems 94
Average Value of a Function 98
Chapter 12 Inverse Trigonometric Functions, Hyperbolic Functions, and L'Hôpital's Rule 101
The Problems You'll Work On 101
What to Watch Out For 102
Finding Derivatives Involving Inverse Trigonometric Functions 102
Finding Antiderivatives by Using Inverse Trigonometric Functions 103
Evaluating Hyperbolic Functions Using Their Definitions 104
Finding Derivatives of Hyperbolic Functions 104
Finding Antiderivatives of Hyperbolic Functions 105
Evaluating Indeterminate Forms Using L'Hôpital's Rule 105
Chapter 13 U-Substitution and Integration by Parts 109
The Problems You'll Work On 109
What to Watch Out For 109
Using u-Substitutions 110
Using Integration by Parts 111
Chapter 14 Trigonometric Integrals, Trigonometric Substitution, and Partial Fractions 115
The Problems You'll Work On 115
What to Watch Out For 116
Trigonometric Integrals 116
Trigonometric Substitutions 118
Finding Partial Fraction Decompositions (without Coefficients) 119
Finding Partial Fraction Decompositions (including Coefficients) 120
Integrals Involving Partial Fractions 120
Rationalizing Substitutions 121
Chapter 15 Improper Integrals and More Approximating Techniques 123
The Problems You'll Work On 123
What to Watch Out For 123
Convergent and Divergent Improper Integrals 124
The Comparison Test for Integrals 125
The Trapezoid Rule 126
Simpson's Rule 126
Part 2 The Answers 127
Chapter 16 Answers and Explanations 129
Index 581
