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1 Foundation for Calculus: Functions and Limits 1
1.1 Functions and Change 2
1.2 Exponential Functions 13
1.3 New Functions from Old 23
1.4 Logarithmic Functions 32
1.5 Trigonometric Functions 39
1.6 Powers, Polynomials, and Rational Functions 49
1.7 Introduction to Limits and Continuity 58
1.8 Extending the Idea of a Limit 67
1.9 Further Limit Calculations Using Algebra 75
1.10 Optional Preview of the formal Definition of a Limit Online
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2 Key Concept: the Derivative 83
2.1 How Do We Measure Speed? 84
2.2 The Derivative at a Point 91
2.3 The Derivative Function 99
2.4 Interpretations of the Derivative 108
2.5 The Second Derivative 115
2.6 Differentiability 123
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3 Short-Cuts to Differentiation of9
3.1 Powers and Polynomials 130
3.2 The Exponential Function 140
3.3 The Product and Quotient Rules 144
3.4 The Chain Rule 151
3.5 The Trigonometric Functions 158
3.6 The Chain Rule and inverse Functions 164
3.7 Implicit Functions 171
3.8 Hyperbolic Functions 174
3.9 Linear Approximation and the Derivative 178
3.10 theorems About Differentiable Functions 186
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4 Using the Derivative 191
4.1 Using First and Second Derivatives 192
4.2 Optimization 203
4.3 Optimization and Modeling 212
4.4 Families of Functions and Modeling 224
4.5 Applications to Marginality 233
4.6 Rates and Related Rates 243
4.7 L’hopital’s Rule, Growth, and Dominance 252
4.8 Parametric Equations 259
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5 Key Concept: the Definite Integral 271
5.1 How Do We Measure Distance Traveled? 272
5.2 The Definite integral 283
5.3 The Fundamental theorem and interpretations 292
5.4 theorems About Definite integrals 302
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6 Constructing Antiderivatives 315
6.1 Antiderivatives Graphically and Numerically 316
6.2 Constructing Antiderivatives Analytically 322
6.3 Differential Equations and Motion 329
6.4 Second Fundamental theorem of Calculus 335
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7 integration 341
7.1 integration By Substitution 342
7.2 integration By Parts 353
7.3 Tables of Integrals 360
7.4 Algebraic Identities and Trigonometric Substitutions 366
7.5 Numerical Methods for Definite integrals 376
7.6 Improper Integrals 385
7.7 Comparison of Improper Integrals 394
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8 Using the Definite Integral 401
8.1 Areas and Volumes 402
8.2 Applications to Geometry 410
8.3 Area and Arc Length in Polar Coordinates 420
8.4 Density and Center of Mass 429
8.5 Applications to Physics 439
8.6 Applications to Economics 450
8.7 Distribution Functions 457
8.8 Probability, Mean, and Median 464
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9 Sequences and Series 473
9.1 Sequences 474
9.2 Geometric Series 480
9.3 Convergence of Series 488
9.4 Tests for Convergence 494
9.5 Power Series and interval of Convergence 504
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10 Approximating Functions Using Series 513
10.1 Taylor Polynomials 514
10.2 Taylor Series 523
10.3 Finding and Using Taylor Series 530
10.4 The Error in Taylor Polynomial Approximations 539
10.5 Fourier Series 546
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11 Differential Equations 561
11.1 What is a Differential Equation? 562
11.2 Slope Fields 567
11.3 Euler’s Method 575
11.4 Separation of Variables 580
11.5 Growth and Decay 586
11.6 Applications and Modeling 597
11.7 The Logistic Model 606
11.8 Systems of Differential Equations 616
11.9 Analyzing the Phase Plane 626
11.10 Second-Order Differential Equations: Oscillations 632
11.11 Linear Second-Order Differential Equations 640
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Appendices
A Roots, Accuracy, and Bounds
B Complex Numbers
C Newton’s Method
D Vectors in the Plane
Ready Reference 651
Answers to Odd-Numbered Problems 663
Index 707
