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Tables of Contents for Calculus
Chapter/Section Title
Page #
Page Count
ABOUT THE AUTHORS
ix
1
PREFACE
x
CHAPTER 1 FUNCTIONS AND GRAPHS
1
52
1.1 Functions and Mathematical Modeling
2
10
PROJECT: A Square Wading Pool
11
1
1.2 Graphs of Equations and Functions
12
12
PROJECT: A Broken Tree
23
1
1.3 A Brief Catalog of Functions, Part 1
24
9
PROJECT: A Leaning Ladder
33
1
1.4 A Brief Catalog of Functions, Part 2
33
12
PROJECT: A Spherical Asteroid
44
1
1.5 PREVIEW: What Is Calculus?
45
3
REVIEW: Definitions and Concepts
48
5
CHAPTER 2 PRELUDE TO CALCULUS
53
46
2.1 Tangent Lines and Slope Predictors
54
9
PROJECT: Numerical Slope Investigations
63
1
2.2 The Limit Concept
63
14
PROJECT: Slopes and Logarithms
76
1
2.3 More About Limits
77
9
2.4 The Concept of Continuity
86
10
PROJECT: The Broken Tree Again
95
1
REVIEW: Definitions, Concepts, Results
96
3
CHAPTER 3 THE DERIVATIVE
99
114
3.1 The Derivative and Rates of Change
100
14
PROJECT: A City's Population Growth
113
1
3.2 Basic Differentiation Rules
114
11
PROJECT: A Cold Liter of Water
125
1
3.3 The Chain Rule
125
8
3.4 Derivatives of Algebraic Functions
133
8
3.5 Maxima and Minima of Functions on Closed Intervals
141
8
PROJECT: Zooming in on Zeros of the Derivative
148
1
3.6 Applied Maximum-Minimum Problems
149
14
PROJECT A: Making a Candy Box with a Lid
162
1
PROJECT B: Power Line Design
163
1
3.7 Derivatives of Trigonometric Functions
163
11
3.8 Exponentials, Logarithms, and Inverse Functions
174
12
PROJECT: Discovering the Number e For Yourself
186
1
3.9 Implicit Differentiation and Related Rates
186
9
3.10 Successive Approximations and Newton's Method
195
13
PROJECT: How Deep Does a Floating Ball Sink?
207
1
REVIEW: Formulas, Concepts, Definitions
208
5
CHAPTER 4 ADDITIONAL APPLICATIONS OF THE DERIVATIVE
213
68
4.1 Introduction
214
1
4.2 Increments, Differentials, and Linear Approximation
214
8
4.3 Increasing and Decreasing Functions and the Mean Value Theorem
222
11
4.4 The First Derivative Test and Applications
233
10
PROJECT: Constructing a Box at Minimal Cost
243
1
4.5 Simple Curve Sketching
243
10
PROJECT: Some Exotic Graphs
252
1
4.6 Higher Derivatives and Concavity
253
13
PROJECT: Invisible Critical Points and Inflection Points
266
1
4.7 Curve Sketching and Asymptotes
266
12
PROJECT: Locating Special Points on Exotic Graphs
278
1
REVIEW: Definitions, Concepts, Results
278
3
CHAPTER 5 THE INTEGRAL
281
94
5.1 Introduction
282
1
5.2 Antiderivatives and Initial Value Problems
282
1
5.3 Elementary Area Computations
283
25
5.4 Riemann Sums and the Integral
308
12
PROJECT: Calculator/Computer Riemann Sums
316
4
5.5 Evaluation of Integrals
320
8
5.6 Average Values and the Fundamental Theorem of Calculus
328
10
5.7 Integration by Substitution
338
8
5.8 Areas of Plane Regions
346
11
PROJECT: Approximate Area Calculations
356
1
5.9 Numerical Integration
357
15
PROJECT: Approximating In2 and Pie by Numerical Integration
368
4
REVIEW: Definitions, Concepts, Results
372
3
CHAPTER 6 APPLICATIONS OF THE INTEGRAL
375
58
6.1 Setting Up Integral Formulas
376
7
6.2 Volumes by the Method of Cross Sections
383
12
PROJECT: Approximating Volumes of Solids of Revolution
394
1
6.3 Volumes by the Method of Cylinderical Shells
395
8
PROJECT: Design Your Own Ring!
402
1
6.4 Are Length and Surface Area of Revolution
403
9
PROJECT: Approximating Are Length and Surface Area
412
1
6.5 Separable Differential Equations
412
8
6.6 Force and Work
420
10
REVIEW: Definitions, Concepts, Results
430
3
CHAPTER 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
433
44
7.1 Exponentials, Logarithms, and Inverse Functions
434
1
7.2 The Natural Logarithm
434
10
PROJECT: Discovering the Number e by Numerical Integration
444
1
7.3 The Exponential Function
444
8
PROJECT: Discovering the Number e as a Limit
451
1
7.4 General Exponential and Logarithmic Functions
452
7
PROJECT: Going Where No One Has Gone Before
459
1
7.5 Natural Growth and Decay
459
9
PROJECT: The Rule of 72-True or False?
468
1
*7.6 Linear First-Order Equations and Applications
468
6
REVIEW: Definitions, Concepts, Results
474
3
CHAPTER 8 FURTHER CALCULUS OF TRANSCENDENTAL FUNCTIONS
477
36
8.1 Introduction
478
1
8.2 Inverse Trigonometric Functions
478
10
8.3 Indeterminate Forms and l'Hopital's Rule
488
7
PROJECT: Graphical Investigation of Indeterminate Forms
494
1
8.4 Additional Indeterminate Forms
495
5
8.5 Hyperbolic Functions and Inverse Hyperbolic Functions
500
10
PROJECT: Your First Sky-Dive
509
1
REVIEW: Definitions and Formulas
510
3
CHAPTER 9 TECHNIQUES OF INTEGRATION
513
56
9.1 Introduction
514
1
9.2 Integral Tables and Simple Substitutions
514
5
PROJECT: Comparing Different Answers
518
1
9.3 Integration by Parts
519
7
9.4 Trigonometric Integrals
526
7
9.5 Rational Functions and Partial Fractions
533
10
PROJECT: Bounded Population Growth
542
1
9.6 Trigonometric Substitution
543
5
9.7 Integrals Containing Quadratic Polynomials
548
6
9.8 Improper Integrals
554
9
PROJECT: Numerical Approximation of Improper Integrals
562
1
SUMMARY:
563
6
CHAPTER 10 POLAR COORDINATES AND PLANE CURVES
569
56
10.1 Analytic Geometry and the Conic Sections
570
5
10.2 Polar Coordinates
575
8
PROJECT: Calculator/Computer-Generated Polar Coordinates Graphs
582
1
10.3 Area Computations in Polar Coordinates
583
5
10.4 Parametric Curves
588
9
PROJECT: Calculator/Computer Graphing of Parametric Curves
596
1
10.5 Integral Computations with Parametric Curves
597
9
PROJECT: Moon Orbits and Race Tracks
604
2
10.6 The Parabola
606
4
10.7 The Ellipse
610
5
10.8 The Hyperbola
615
6
REVIEW: Concepts and Definitions
621
4
CHAPTER 11 INFINITE SERIES
625
84
11.1 Introduction
626
1
11.2 Infinite Sequences
626
10
PROJECT: Nested Radicals and Continued Fractions
635
1
11.3 Infinite Series and Convergence
636
11
PROJECT: Numerical Summation and Geometric Series
646
1
11.4 Taylor Series and Taylor Polynomials
647
14
PROJECT: Calculating Logarithms on a Deserted Island
660
1
11.5 The Integral Test
661
8
PROJECT: The Number Pie. Once and For All
668
1
11.6 Comparison Tests for Positive-Term Series
669
6
11.7 Alternating Series and Absolute Convergence
675
9
11.8 Power Series
684
12
11.9 Power Series Computations
696
9
PROJECT: Calculating Trigonometric Functions on a Deserted Island
704
1
REVIEW: Definitions, Concepts, Results
705
4
CHAPTER 12 VECTORS, CURVES, AND SURFACES IN SPACE
709
88
12.1 Vectors in the Plane
710
6
12.2 Rectangular Coordinates and Three-Dimensional Vectors
716
11
12.3 The Cross Product of Two Vectors
727
8
12.4 Lines and Planes in Space
735
9
12.5 Curves and Motion in Space
744
15
PROJECT: Does a Pitched Baseball Really Curve?
758
1
12.6 Curvature and Acceleration
759
15
12.7 Cylinders and Quadric Surfaces
774
11
12.8 Cylindricals and Spherical Coordinates
785
8
PROJECT: Personal Cylindrical and Spherical Plots
792
1
REVIEW: Definitions, Concepts, Results
793
4
CHAPTER 13 PARTIAL DIFFERENTIATION
797
90
13.1 Introduction
798
1
13.2 Functions of Several Variables
798
10
PROJECT: Your Personal Portfolio of Surfaces
807
1
13.3 Limits and Continuity
808
6
13.4 Partial Derivatives
814
11
13.5 Maxima and Minima of Functions of Several Variables
825
12
PROJECT: Exotic Critical Points
836
1
13.6 Increments and Differentials
837
7
13.7 The Chain Rule
844
11
13.8 Directional Derivatives and the Gradient Vector
855
10
13.9 Lagrange Multipliers and Constrained Maximum-Minimum Problems
865
10
PROJECT: Numerical Investigation of Lagrange Multiplier Problems
874
1
13.10 The Second Derivative Test for Functions of Two Variables
875
8
PROJECT: Critical Point Investigations
882
1
REVIEW: Definitions, Concepts, Results
883
4
CHAPTER 14 MULTIPLE INTEGRALS
887
74
14.1 Double Integrals
888
7
PROJECT: Midpoint Approximation of Double Integrals
894
1
14.2 Double Integrals over More General Regions
895
6
14.3 Area and Volume by Double Integration
901
6
14.4 Double Integrals in Polar Coordinates
907
7
14.5 Applications of Double Integrals
914
12
PROJECT: Optimal Design of Downhill Race Car Wheels
924
2
14.6 Triple Integrals
926
8
PROJECT: Archimedes' Floating Paraboloid
934
1
14.7 Integration in Cylindrical and Spherical Coordinates
934
8
PROJECT: The Earth's Mantle
942
1
14.8 Surface Area
942
7
PROJECT: Computer-Generated Parametric Surfaces
948
1
14.9 Change of Variables in Multiple Integrals
949
7
REVIEW: Definitions, Concepts, Results
956
5
CHAPTER 15 VECTOR CALCULUS
961
15.1 Vector Fields
962
5
15.2 Line Integrals
967
11
15.3 The Fundamental Theorem and Independence of Path
978
8
15.4 Green's Theorem
986
9
PROJECT: Green's Theorem and Loop Areas
994
1
15.5 Surface Integrals
995
11
PROJECT: Surface Integrals and Rocket Nose Cones
1005
1
15.6 The Divergence Theorem
1006
7
15.7 Stokes' Theorem
1013
7
REVIEW: Definitions, Concepts, Results
1020
APPENDICES
A-1
49
A: Real Number and Inequalities
A-1
5
B: The Coordinate Plane and Straight Lines
A-6
8
C: Review of Trigonometry
A-14
6
D: Proofs of the Limit Laws
A-20
6
E: The Completeness of the Real Number System
A-26
5
F: Proof of the Chain Rule
A-31
1
G: Existence of the Integral
A-32
6
H: Approximations and Riemann Sums
A-38
3
I: L' Hopital's Rule and Cauchy's Mean Value Theorem
A-41
3
J: Proof of Taylor's Formula
A-44
1
K: Conic Sections as Sections of a Cone
A-45
1
L: Units of Measurement and Conversion Factors
A-46
1
M: Formulas from Algebra, Geometry, and Trigonometry
A-47
2
N: The Greek Alphabet
A-49
1
ANSWERS TO ODD-NUMBERED PROBLEMS
A-50
42
REFERENCES FOR FURTHER STUDY
A-92
INDEX
1-1