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Tables of Contents for Single Variable Calculus
Functions, Graphs, and Models
1
52
Functions and Mathematical Modeling
2
10
11
1
Graphs of Equations and Functions
12
12
23
1
Polynomials and Algebraic Functions
24
9
33
1
Transcendental Functions
33
12
45
1
Preview: What Is Calculus?
45
8
48
5
Tangent Lines and Slope Predictors
54
9
63
1
Project: Limits, Slopes, and Logarithms74
1
Project: Numerical Epsilon-Delta Limit Investigations87
1
The Concept of Continuity
88
13
99
2
The Derivative and Rates of Change
102
13
Basic Differentiation Rules
115
11
Derivatives of Algebraic Functions
133
8
Maxima and Minima of Functions on Closed Intervals
141
10
149
2
Applied Optimization Problems
151
13
Derivatives of Trigonometric Functions
164
11
Exponential and Logarithmic Functions
175
13
188
1
Implicit Differentiation and Related Rates
188
11
199
1
Successive Approximations and Newton's Method
199
18
211
1
Review: Formulas, Concepts, Definitions212
5
Additional Applications of the Derivative
217
84
Increments, Differentials, and Linear Approximation
218
8
Increasing and Decreasing Functions and the Mean Value Theorem
226
11
The First Derivative Test and Applications
237
10
247
1
Simple Curve Sketching
247
10
Higher Derivatives and Concavity
257
14
Curve Sketching and Asymptotes
271
12
283
1
Indeterminate Forms and L'Hopital's Rule
283
8
More Indeterminate Forms
291
10
297
4
Antiderivatives and Initial Value Problems
302
14
Elementary Area Computations
316
12
Riemann Sums and the Integral
328
11
338
1
Evaluation of Integrals
339
10
The Fundamental Theorem of Calculus
349
10
Integration by Substitution
359
9
Areas of Plane Regions
368
10
Numerical Integration
378
19
391
2
Review: Definitions, Concepts, Results393
4
Applications of the Integral
397
94
Riemann Sum Approximations
398
10
Volumes by the Method of Cross Sections
408
11
Volumes by the Method of Cylindrical Shells
419
9
427
1
Arc Length and Surface Area of Revolution
428
9
Centroids of Plane Regions and Curves
448
8
The Natural Logarithm as an Integral
456
11
466
1
Inverse Trigonometric Functions
467
10
Hyperbolic Functions
477
14
485
6
Techniques of Integration
491
56
Integral Tables and Simple Substitutions
492
4
Trigonometric Integrals
503
7
Rational Functions and Partial Fractions
510
7
Trigonometric Substitution
517
6
Integrals Involving Quadratic Polynomials
523
5
541
6
Differential Equations
547
78
Simple Equations and Models
548
12
Slope Fields and Euler's Method
560
10
569
1
Separable Equations and Applications
570
7
Linear Equations and Applications
577
12
599
1
Linear Second-Order Equations
600
9
Mechanical Vibrations
609
16
620
5
Polar Coordinates And Parametric Curves
625
58
Analytic Geometry and the Conic Sections
626
5
Area Computations in Polar Coordinates
640
5
654
1
Integral Computations with Parametric Curves
655
8
662
1
Conic Sections and Applications
663
20
681
2
Project: Nested Radicals and Continued Fractions693
1
Infinite Series and Convergence
693
11
703
1
Taylor Series and Taylor Polynomials
704
13
717
1
Project: The Number &pai;, Once and for All724
1
Comparison Tests for Positive-Term Series
724
6
Alternating Series and Absolute Convergence
730
9
Power Series Computations
752
8
760
1
Series Solutions of Differential Equations
760
13
769
4
Vectors And Matrices
773
72
Three-Dimensional Vectors
780
10
The Cross Product of Vectors
790
8
Lines and Planes in Space
798
7
Linear Systems and Matrices
805
12
816
1
Project: Automated Solution of Linear Systems831
1
Eigenvalues and Rotated Conics
832
13
842
1
Review: Definitions, Concepts, Results842
3
Curves And Surfaces in Space
845
48
Curves and Motion in Space
846
14
859
1
Curvature and Acceleration
860
13
Cylinders and Quadric Surfaces
873
10
Cylindrical and Spherical Coordinates
883
10
891
2
Partial Differentiation
893
100
Functions of Several Variables
894
10
Limits and Continuity
904
8
Multivariable Optimization Problems
922
11
Linear Approximation and Matrix Derivatives
933
11
943
1
The Multivariable Chain Rule
944
12
Directional Derivatives and the Gradient Vector
956
11
Lagrange Multipliers and Constrained Optimization
967
10
977
1
Critical Points of Multivariable Functions
977
16
988
1
Review: Definitions, Concepts, Results989
4
Project: Midpoint Sums Approximating Double Integrals1001
1
Double Integrals over More General Regions
1001
7
Area and Volume by Double Integration
1008
7
Double Integrals in Polar Coordinates
1015
7
Applications of Double Integrals
1022
11
1032
1
Project: Archimedes' Floating Paraboloid1041
1
Integration in Cylindrical and Spherical Coordinates
1042
8
Change of Variables in Multiple Integrals
1055
12
1063
4
The Fundamental Theorem and Independence of Path
1084
7
1111
1
The Divergence Theorem
1111
8
Stokes' Theorem
1119
Review: Definitions, Concepts, Results1126
A: Real Numbers and Inequalities
1
5
B: The Coordinate Plane and Straight Lines
6
7
C: Review of Trigonometry
13
6
D: Proofs of the Limit Laws
19
4
E: The Completeness of the Real Number System
23
5
F: Existence of the Integral
28
5
G: Approximations and Riemann Sums
33
3
H: L'Hopital's Rule and Cauchy's Mean Value Theorem
36
2
I: Proof of Taylor's Formula
38
1
J: Conic Sections as Sections of a Cone
39
1
K: Proof of the Linear Approximation Theorem
40
1
L: Units of Measurement and Conversion Factors
41
1
M: Formulas from Algebra, Geometry, and Trigonometry
42
2
Answers To Odd-Numbered Problems
45
40
References for Further Study
85