Calculus, Single Variable Book

1	A Library of Functions	1
1.1	Functions and Change	2
1.2	Exponential Functions	9
1.3	New Functions from Old	17
1.4	Logarithmic Functions	23
1.5	Trigonometric Functions	29
1.6	Powers, Polynomials, and Rational Functions	37
1.7	Introduction to Continuity	45
2	Key Concept: The Derivative	55
2.1	How Do We Measure Speed?	56
2.2	Limits	62
2.3	The Derivative at a Point	70
2.4	The Derivative Function	78
2.5	Interpretations of the Derivative	85
2.6	The Second Derivative	89
2.7	Continuity and Differentiability	95
3	Short-Cuts to Differentiation	105
3.1	Powers and Polynomials	106
3.2	The Exponential Function	113
3.3	The Product and Quotient Rules	118
3.4	The Chain Rule	123
3.5	The Trigonometric Functions	128
3.6	Applications of the Chain Rule	133
3.7	Implicit Functions	138
3.8	Parametric Equations	141
3.9	Linear Approximation and the Derivative	150
3.10	Using Local Linearity to Find Limits	154
4	Using the Derivative	165
4.1	Using First and Second Derivatives	166
4.2	Families of Curves	176
4.3	Optimization	180
4.4	Applications to Marginality	189
4.5	Optimization and Modeling	196
4.6	Hyperbolic Functions	203
4.7	Theorems about Continuous and Differentiable Functions	207
5	Key Concept: The Definite Integral	221
5.1	How Do We Measure Distance Traveled?	222
5.2	The Definite Integral	229
5.3	Interpretations of the Definite Integral	236
5.4	Theorems about Definite Integrals	244
6	Constructing Antiderivatives	261
6.1	Antiderivatives Graphically and Numerically	262
6.2	Constructing Antiderivatives Analytically	268
6.3	Differential Equations	273
6.4	Second Fundamental Theorem of Calculus	278
6.5	The Equations of Motion	282
7	Integration	289
7.1	Integration by Substitution	290
7.2	Integration by Parts	298
7.3	Tables of Integrals	304
7.4	Algebraic Identities and Trigonometric Substitutions	309
7.5	Approximating Definite Integrals	317
7.6	Approximation Errors and Simpson's Rule	322
7.7	Improper Integrals	326
7.8	Comparison of Improper Integrals	334
8	Using the Definite Integral	345
8.1	Areas and Volumes	346
8.2	Applications to Geometry	352
8.3	Density and Center of Mass	360
8.4	Applications to Physics	368
8.5	Applications to Economics	377
8.6	Distribution Functions	383
8.7	Probability, Mean, and Median	390
9	Series	405
9.1	Geometric Series	406
9.2	Convergence of Sequences and Series	412
9.3	Tests for Convergence	417
9.4	Power Series	423
10	Approximating Functions	433
10.1	Taylor Polynomials	434
10.2	Taylor Series	441
10.3	Finding and Using Taylor Series	446
10.4	The Error in Taylor Polynomial Approximations	453
10.5	Fourier Series	457
11	Differential Equations	477
11.1	What is a Differential Equation?	478
11.2	Slope Fields	482
11.3	Euler's Method	488
11.4	Separation of Variables	492
11.5	Growth and Decay	497
11.6	Applications and Modeling	506
11.7	Models of Population Growth	514
11.8	Systems of Differential Equations	523
11.9	Analyzing the Phase Plane	532
11.10	Second-Order Differential Equations: Oscillations	537
11.11	Linear Second-Order Differential Equations	544
App. A	Roots, Accuracy, and Bounds	560
App. B	Polar Coordinates	568
App. C	Complex Numbers	570
App. D	Newton's Method	577
	Ready Reference	581
	Answers to Odd Numbered Problems	598
	Index	615