Table of Contents
Part 1 The Roots of Calculus 1
1 What Is Calculus, Anyway? 3
What's the Purpose of Calculus? 4
Finding the Slopes of Curves 4
Calculating the Area of Bizarre Shapes 4
Justifying Old Formulas 5
Calculating Complicated x-Intercepts 5
Visualizing Graphs 5
Finding the Average Value of a Function 6
Calculating Optimal Values 6
Who's Responsible for This? 7
Ancient Influences 7
Newton vs. Leibniz 9
Will I Ever Learn This? 11
2 Polish Up Your Algebra Skills 13
Walk the Line: Linear Equations 14
Common Forms of Linear Equations 14
Calculating Slope 16
Interpreting Linear Graphs 18
You've Got the Power: Exponential Rules 21
Breaking Up Is Hard to Do: Factoring Polynomials 22
Greatest Common Factor 23
Special Factoring Patterns 23
Solving Quadratic Equations 24
Method 1 Factoring 25
Method 2 Completing the Square 25
Method 3 The Quadratic Formula 26
Synthesizing the Quadratic Solution Methods 27
3 Equations, Relations, and Functions 31
What Makes a Function Tick? 31
Working with Graphs of Functions 36
Functional Symmetry 39
Graphs to Know by Heart 43
Constructing an Inverse Function 45
Parametric Equations 47
What's a Parameter? 47
Converting to Rectangular Form 48
4 Trigonometry: Last Stop Before Calculus 51
Getting Repetitive: Periodic Functions 51
Introducing the Trigonometric Functions 53
Sine (Written as y = sin x) 54
Cosine (Written as y = cos x) 54
Tangent (Written as y = tan x) 55
Cotangent (Written as y = cot x) 56
Secant (Written as y = sec x) 57
Cosecant (Written as y = csc x) 57
What's Your Sine: The Unit Circle 59
Incredibly important Identities 61
Pythagorean identities 62
Double-Angle Formulas 63
Solving Trigonometric Equations 64
Part 2 Laying the Foundation for Calculus 67
5 Take It to the Limit 69
What Is a Limit 70
Can Something Be Nothing? 71
One-Sided Limits 74
When Does a Limit Exist? 78
When Does a Limit Not Exist? 79
6 Evaluating Limits Numerically 85
The Major Methods 86
Substitution Method 86
Factoring Method 87
Conjugate Method 88
What If Sorting Works? 90
Limits and Infinity 90
Vertical Asymptotes 90
Horizontal Asymptotes 92
Special Limit Theorems 96
Evaluating Limits Graphically 97
Technology Focus: Calculating Limits 99
7 Continuity 103
What Does Continuity Look Like? 104
The Mathematical Definition of Continuity 104
Types of Discontinuity 109
Jump Discontinuity 109
Point Discontinuity 113
Infinite Essential Discontinuity 117
Removable vs. Nonremovable Discontinuity 117
The Intermediate Value Theorem 118
8 The Difference Quotient 121
When a Secant Becomes a Tangent 122
Honey, I Shrunk the δx 123
Applying the Difference Quotient 127
The Alternate Difference Quotient 129
Part 3 The Derivative 131
9 Laying Down the Law for Derivatives 133
When Docs a Derivative Exist? 134
Discontinuity 134
Sharp Point in the Graph 134
Vertical Tangent Line 135
Basie Derivative Techniques 136
The Power Rule 136
The Product Rule 138
The Quotient Rule 139
The Chain Rule 140
Rates of Change 141
Trigonometric Derivatives 144
Tabular and Graphical Derivatives 145
Technology Focus: Calculating Derivatives 150
10 Common Differentiation Tasks 155
Finding Equations of Tangent Lines 156
Implicit Differentiation 159
Differentiating an Inverse Function 161
Parametric Derivatives 164
Technology Focus: Solving Gross Equations 166
Using the Built-In: Equation Solver 166
The Equation-Function Connection 170
11 Using Derivatives to Graph 173
Relative Extrema 174
Finding Critical Numbers 175
Classifying Extrema 176
The Wiggle Graph 178
The Extreme Value Theorem 180
Determining Concavity 182
Another Wiggle Graph 183
The Second Derivative Test 184
12 Derivatives and Motion 187
The Position Equation 188
Velocity 190
Acceleration 191
Vertical Projectile Motion 193
13 Common Derivative Applications 195
Newton's Method 196
Evaluating Limits: L'Hôpital's Rule 199
More Existence Theorems 200
The Mean Value Theorem 201
Rolle's Theorem 203
Related Rates 204
Optimization 208
Part 4 The Integral 215
14 Approximating Area 217
Riemann Sums 218
Right and Left Sums 219
Midpoint Sums 221
The Trapezoidal Rule 222
Simpson's Rule 225
15 Antiderivatives 227
The Power Rule for integration 228
Integrating Trigonometric Functions 230
Separation 232
The Fundamental Theorem of Calculus 233
Part 1 Areas and Integrals Are Related 233
Part 2 Derivative: and Integrals Are Opposites 235
u-Substiution 236
Tricky u-Substitution and Long Division 237
Technology Focus: Definite and Indefinite Integrals 239
16 Applications of the Fundamental Theorem 245
Calculating Area Between Two Curves 246
The Mean Value Theorem for Integration 249
A Geometric Interpretation 249
The Average Value Theorem 251
Finding Distance Traveled 253
Accumulation Functions 255
Arc Length 256
Rectangular Equations 256
Parametric Equations 257
Part 5 Differential Equations and More 259
17 Differential Equations 261
Separation of Variables 262
Types of Solutions 263
Family of Solutions 264
Specific Solutions 266
Exponential Growth and Decay 267
18 Visualizing Differential Equations 275
Linear Approximation 276
Slope Fields 277
Euler's Method 281
Technology Focus: Slope Fields 285
19 Final Exam 289
Appendixes
A Solutions to "You've Got Problems" 301
B Glossary 317
Index 323