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Calculus With Analytic Geometry
We do not plan to review this book.
CHAPTER 1: Numbers, Functions, and Graphs
11 Introduction
12 The Real Line and Coordinate Plane: Pythagoras
13 Slopes and Equations of Straight Lines
14 Circles and Parabolas: Descartes and Fermat
15 The Concept of a Function
16 Graphs of Functions
17 Introductory Trigonometry
18 The Functions Sin O and Cos O
CHAPTER 2: The Derivative of a Function
20 What is Calculus ?
21 The Problems of Tangents
22 How to Calculate the Slope of the Tangent
23 The Definition of the Derivative
24 Velocity and Rates of Change: Newton and Leibriz
25 The Concept of a Limit: Two Trigonometric Limits
26 Continuous Functions: The Mean Value Theorem and Other Theorem
CHAPTER 3: The Computation of Derivatives
31 Derivatives of Polynomials
32 The Product and Quotient Rules
33 Composite Functions and the Chain Rule
34 Some Trigonometric Derivatives
35 Implicit Functions and Fractional Exponents
36 Derivatives of Higher Order
CHAPTER 4: Applications of Derivatives
41 Increasing and Decreasing Functions: Maxima and Minima
42 Concavity and Points of Inflection
43 Applied Maximum and Minimum Problems
44 More MaximumMinimum Problems
45 Related Rates
46 Newtons Method for Solving Equations
47 Applications to Economics: Marginal Analysis
CHAPTER 5: Indefinite Integrals and Differential Equations
51 Introduction
52 Differentials and Tangent Line Approximations
53 Indefinite Integrals: Integration by Substitution
54 Differential Equations: Separation of Variables
55 Motion Under Gravity: Escape Velocity and Black Holes
CHAPTER 6: Definite Integrals
61 Introduction
62 The Problem of Areas
63 The Sigma Notation and Certain Special Sums
64 The Area Under a Curve: Definite Integrals
65 The Computation of Areas as Limits
66 The Fundamental Theorem of Calculus
67 Properties of Definite Integrals
CHAPTER 7: Applications of Integration
71 Introduction: The Intuitive Meaning of Integration
72 The Area between Two Curves
73 Volumes: The Disk Method
74 Volumes: The Method of Cylindrical Shells
75 Arc Length
76 The Area of a Surface of Revolution
77 Work and Energy
78 Hydrostatic Force
PART II
CHAPTER 8: Exponential and Logarithm Functions
81 Introduction
82 Review of Exponents and Logarithms
83 The Number e and the Function y = e <^>x
84 The Natural Logarithm Function y = ln x
85 Applications
Population Growth and Radioactive Decay
86 More Applications
CHAPTER 9: Trigonometric Functions
91 Review of Trigonometry
92 The Derivatives of the Sine and Cosine
93 The Integrals of the Sine and Cosine
94 The Derivatives of the Other Four Functions
95 The Inverse Trigonometric Functions
96 Simple Harmonic Motion
97 Hyperbolic Functions
CHAPTER 10 : Methods of Integration
101 Introduction
102 The Method of Substitution
103 Certain Trigonometric Integrals
104 Trigonometric Substitutions
105 Completing the Square
106 The Method of Partial Fractions
107 Integration by Parts
108 A Mixed Bag
109 Numerical Integration
CHAPTER 11: Further Applications of Integration
111 The Center of Mass of a Discrete System
112 Centroids
113 The Theorems of Pappus
114 Moment of Inertia
CHAPTER 12: Indeterminate Forms and Improper Integrals
121 Introduction. The Mean Value Theorem Revisited
122 The Interminate Form 0/0. L'Hospital's Rule
123 Other Interminate Forms
124 Improper Integrals
125 The Normal Distribution
CHAPTER 13: Infinite Series of Constants
131 What is an Infinite Series ?
132 Convergent Sequences
133 Convergent and Divergent Series
134 General Properties of Convergent Series
135 Series on Nonnegative Terms: Comparison Tests
136 The Integral Test
137 The Ratio Test and Root Test
138 The Alternating Series Test
CHAPTER 14: Power Series
141 Introduction
142 The Interval of Convergence
143 Differentiation and Integration of Power Series
144 Taylor Series and Taylor's Formula
145 Computations Using Taylor's Formula
146 Applications to Differential Equations
14. 7 (optional) Operations on Power Series
14. 8 (optional) Complex Numbers and Euler's Formula
PART III
CHAPTER 15: Conic Sections
151 Introduction
152 Another Look at Circles and Parabolas
153 Ellipses
154 Hyperbolas
155 The FocusDirectrixEccentricity Definitions
156 (optional) Second Degree Equations
CHAPTER 16: Polar Coordinates
161 The Polar Coordinate System
162 More Graphs of Polar Equations
163 Polar Equations of Circles, Conics, and Spirals
164 Arc Length and Tangent Lines
165 Areas in Polar Coordinates
CHAPTER 17: Parametric Equations
171 Parametric Equations of Curves
172 The Cycloid and Other Similar Curves
173 Vector Algebra
174 Derivatives of Vector Function
175 Curvature and the Unit Normal Vector
176 Tangential and Normal Components of Acceleration
177 Kepler's Laws and Newton's Laws of Gravitation
CHAPTER 18: Vectors in ThreeDimensional Space
181 Coordinates and Vectors in ThreeDimensional Space
182 The Dot Product of Two Vectors
183 The Cross Product of Two Vectors
184 Lines and Planes
185 Cylinders and Surfaces of Revolution
186 Quadric Surfaces
187 Cylindrical and Spherical Coordinates
CHAPTER 19: Partial Derivatives
191 Functions of Several Variables
192 Partial Derivatives
193 The Tangent Plane to a Surface
194 Increments and Differentials
195 Directional Derivatives and the Gradient
196 The Chain Rule for Partial Derivatives
197 Maximum and Minimum Problems
198 Constrained Maxima and Minima
199 Laplace's Equation, the Heat Equation, and the Wave Equation
1910 (optional) Implicit Functions
CHAPTER 20: Multiple Integrals
201 Volumes as Iterated Integrals
202 Double Integrals and Iterated Integrals
203 Physical Applications of Double Integrals
204 Double Integrals in Polar Coordinates
205 Triple Integrals
206 Cylindrical Coordinates
207 Spherical Coordinates
208 Areas of curved Surfaces
CHAPTER 21: Line and Surface Integrals
211 Green's Theorem, Gauss's Theorem, and Stokes' Theorem
212 Line Integrals in the Plane
213 Independence of Path
214 Green's Theorem
215 Surface Integrals and Gauss's Theorem
216 Maxwell's Equations : A Final Thought
Appendices
A: The Theory of Calculus
A1 The Real Number System
A2 Theorems About Limits
A3 Some Deeper Properties of Continuous Functions
A4 The Mean Value theorem
A5 The Integrability of Continuous Functions
A6 Another Proof of the Fundamental Theorem of Calculus
A7 Continuous Curves With No Length
A8 The Existence of e = lim h>0 (1 + h) <^>1/h
A9 Functions That Cannot Be Integrated
A10 The Validity of Integration by Inverse Substitution
A11 Proof of the Partial fractions Theorem
A12 The Extended Ratio Tests of Raabe and Gauss
A13 Absolute vs Conditional Convergence
A14 Dirichlet's Test
A15 Uniform Convergence for Power Series
A16 Division of Power Series
A17 The Equality of Mixed Partial Derivatives
A18 Differentiation Under the Integral Sign
A19 A Proof of the Fundamental Lemma
A20 A Proof of the Implicit Function Theorem
A21 Change of Variables in Multiple Integrals
B: A Few Review Topics
B1 The Binomial Theorem
B2 Mathematical Induction

New Features 
 Revision highlights include the early introduction of trigonometry, extensive reworking of the infinite series chapters, and the addition of new exercises at varying levels of difficulty.
 New topics include firstorder nonlinear differential equations, elementary probability, and hyperbolic functions.
 Two long appendices (Variety of additional topics, Biographical notes) have been removed from the text (will be available in the text, CALCULUS GEMS).
 The text offers full coverage for the full majors on engineering calculus, but, remains shorter than most competition.


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