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Tables of Contents for Calculus and Analytic Geometry
Chapter/Section Title
Page #
Page Count
To the Instructor
viii
To the Student
xvii
Preliminaries
Real Numbers and the Real Line
1
7
Coordinates, Lines, and Increments
8
9
Functions
17
10
Shifting Graphs
27
8
Trigonometric Functions
35
16
Questions to Guide Your Review
47
1
Practice Exercises
48
1
Additional Exercises---Theory, Examples, Applications
49
2
Limits and Continuity
Rates of Change and Limits
51
10
Rules for Finding Limits
61
5
Target Values and Formal Definitions of Limits
66
12
Extensions of the Limit Concept
78
9
Continuity
87
10
Tangent Lines
97
12
Questions to Guide Your Review
103
1
Practice Exercises
104
1
Additional Exercises---Theory, Examples, Applications
105
4
Derivatives
The Derivative of a Function
109
12
Differentiation Rules
121
10
Rates of Change
131
12
Derivatives of Trigonometric Functions
143
11
The Chain Rule
154
10
Implicit Differentiation and Rational Exponents
164
8
Related Rates of Change
172
17
Questions to Guide your Review
180
1
Practice Exercises
181
4
Additional Exercises---Theory, Examples, Applications
185
4
Applications of Derivatives
Extreme Values of Functions
189
7
The Mean Value Theorem
196
9
The First Derivative Test for Local Extreme Values
205
4
Graphing with y' and y''
209
11
Limits as x → ± ∞, Asymptotes, and Dominant Terms
220
13
Optimization
233
15
Linearization and Differentials
248
12
Newton's Method
260
15
Questions to Guide Your Review
268
1
Practice Exercises
269
3
Additional Exercises---Theory, Examples, Applications
272
3
Integration
Indefinite Integrals
275
7
Differential Equations, Initial Value Problems, and Mathematical Modeling
282
8
Integration by Substitution---Running the Chain Rule Backward
290
8
Estimating with Finite Sums
298
11
Riemann Sums and Definite Integrals
309
14
Properties, Area, and the Mean Value Theorem
323
9
The Fundamental Theorem
332
10
Substitution in Definite Integrals
342
4
Numerical Integration
346
19
Questions to Guide Your Review
356
1
Practice Exercises
357
3
Additional Exercises---Theory, Examples, Applications
360
5
Application of Integrals
Areas Between Curves
365
9
Finding Volumes by Slicing
374
5
Volumes of Solids of Revolution---Disks and Washers
379
8
Cylindrical Shells
387
6
Lengths of Plane Curves
393
7
Areas of Surfaces of Revolution
400
7
Moments and Centers of Mass
407
11
Work
418
9
Fluid Pressures and Forces
427
7
The Basic Pattern and Other Modeling Applications
434
15
Questions to Guide Your Review
444
1
Practice Exercises
444
3
Additional Exercises---Theory, Examples, Applications
447
2
Transcendental Functions
Inverse Functions and Their Derivatives
449
9
Natural Logarithms
458
9
The Exponential Function
467
7
ax and logax
474
8
Growth and Decay
482
9
L'Hopital's Rule
491
7
Relative Rates of Growth
498
6
Inverse Trigonometric Functions
504
9
Derivatives of Inverse Trigonometric Functions; Integrals
513
7
Hyperbolic Functions
520
9
First Order Differential Equations
529
12
Euler's Numerical Method; Slope Fields
541
14
Questions to Guide Your Review
547
1
Practice Exercises
548
3
Additional Exercises---Theory, Examples, Applications
551
4
Techniques of Integration
Basic Integration Formulas
555
7
Integration by Parts
562
7
Partial Fractions
569
9
Trigonometric Substitutions
578
5
Integral Tables and CAS
583
11
Improper Integrals
594
19
Questions to Guide Your Review
606
1
Practice Exercises
606
3
Additional Exercises---Theory, Examples, Applications
609
4
Infinite Series
Limits of Sequences of Numbers
613
9
Theorems for Calculating Limits of Sequences
622
8
Infinite Series
630
10
The Integral Test for Series of Nonnegative Terms
640
4
Comparison Tests for Series of Nonnegative Terms
644
5
The Ratio and Root Tests for Series of Nonnegative Terms
649
6
Alternating Series, Absolute and Conditional Convergence
655
8
Power Series
663
9
Taylor and Maclaurin Series
672
6
Convergence of Taylor Series; Error Estimates
678
10
Applications of Power Series
688
21
Questions to Guide Your Review
699
1
Practice Exercises
700
3
Additional Exercises---Theory, Examples, Applications
703
6
Conic Sections, Parametrized Curves, and Polar Coordinates
Conic Sections and Quadratic Equations
709
14
Classifying Conic Sections by Eccentricity
723
5
Quadratic Equations and Rotations
728
6
Parametrizations of Plane Curves
734
10
Calculus with Parametrized Curves
744
7
Polar Coordinates
751
5
Graphing in Polar Coordinates
756
8
Polar Equations for Conic Sections
764
6
Integration in Polar Coordinates
770
17
Questions to Guide Your Review
777
1
Practice Exercises
778
5
Additional Exercises---Theory, Examples, Applications
783
4
Vectors and Analytic Geometry in Space
Vectors in the Plane
787
8
Cartesian (Rectangular) Coordinates and Vectors in Space
795
11
Dot Products
806
9
Cross Products
815
7
Lines and Planes in Space
822
7
Cylinders and Quadric Surfaces
829
12
Cylindrical and Spherical Coordinates
841
14
Questions to Guide Your Review
847
1
Practice Exercises
848
3
Additional Exercises---Theory, Examples, Applications
851
4
Vector-Valued Functions and Motion in Space
Vector-Valued Functions and Space Curves
855
13
Modeling Projectile Motion
868
8
Arc Length and the Unit Tangent Vector T
876
5
Curvature, Torsion, and the TNB Frame
881
12
Planetary Motion and Satellites
893
16
Questions to Guide Your Review
902
1
Practice Exercises
902
3
Additional Exercises---Theory, Examples, Applications
905
4
Multivariable Functions and Partial Derivatives
Functions of Several Variables
909
8
Limits and Continuity
917
7
Partial Derivatives
924
9
Differentiability, Linearization, and Differentials
933
11
The Chain Rule
944
8
Partial Derivatives with Constrained Variables
952
5
Directional Derivatives, Gradient Vectors, and Tangent Planes
957
13
Extreme Values and Saddle Points
970
10
Lagrange Multipliers
980
9
Taylor's Formula
989
12
Questions to Guide Your Review
993
1
Practice Exercises
994
4
Additional Exercises---Theory, Examples, Applications
998
3
Multiple Integrals
Double Integrals
1001
11
Areas, Moments, and Centers of Mass
1012
8
Double Integrals in Polar Form
1020
6
Triple Integrals in Rectangular Coordinates
1026
8
Masses and Moments in Three Dimensions
1034
5
Triple Integrals in Cylindrical and Spherical Coordinates
1039
9
Substitutions in Multiple Integrals
1048
13
Questions to Guide Your Review
1055
1
Practice Exercises
1056
2
Additional Exercises---Theory, Examples, Applications
1058
3
Integration in Vector Fields
Line Integrals
1061
6
Vector Fields, Work, Circulation, and Flux
1067
9
Path Independence, Potential Functions, and Conservative Fields
1076
8
Green's Theorem in the Plane
1084
12
Surface Area and Surface Integrals
1096
10
Parametrized Surfaces
1106
8
Stokes's Theorem
1114
9
The Divergence Theorem and a Unified Theory
1123
Questions to Guide Your Review
1134
1
Practice Exercises
1134
3
Additional Exercises---Theory, Examples, Applications
1137
Appendices
A.1 Mathematical Induction
A-1
A.2 Proofs of Limit Theorems in Section 1.2
A-4
A.3 Complex Numbers
A-7
A.4 Simpson's One-Third Rule
A-17
A.5 Cauchy's Mean Value Theorem and the Stronger Form of l'Hopital's Rule
A-18
A.6 Limits That Arise Frequently
A-20
A.7 The Distributive Law for Vector Cross Products
A-21
A.8 Determinants and Cramer's Rule
A-22
A.9 Euler's Theorem and the Increment Theorem
A-29
Answers
A-35
Index
I-1
A Brief Table of Integrals
T-1