Calculus With Analytic Geometry / Edition 2

by George F Simmons

Overview

Written by acclaimed author and mathematician George Simmons, this revision is designed for the calculus course offered in two and four year colleges and universities. It takes an intuitive approach to calculus and focuses on the application of methods to real-world problems. Throughout the text, calculus is treated as a problem solving science of immense capability.

Product Details

ISBN-13:	9780070576421
Publisher:	McGraw-Hill Higher Education
Publication date:	10/01/1995
Edition description:	List
Pages:	912
Product dimensions:	8.60(w) x 10.30(h) x 1.70(d)

About the Author

GEORGE F. SIMMONS has academic degree from the CAlifornia Institute of Technology, the university of chicago, and Yale University. He taught at several colleges and universities before joining the faculty of Colorado college in 1962, where he is a professor of mathematics. He is also the author of introduction to topology and Modern Analysis, Precalculus Mathematics in a Nutshell and calculus with Analytic Geometry.

Table of Contents

CHAPTER 1: Numbers, Functions, and Graphs
1-1 Introduction
1-2 The Real Line and Coordinate Plane: Pythagoras
1-3 Slopes and Equations of Straight Lines
1-4 Circles and Parabolas: Descartes and Fermat
1-5 The Concept of a Function
1-6 Graphs of Functions
1-7 Introductory Trigonometry
1-8 The Functions Sin O and Cos O

CHAPTER 2: The Derivative of a Function
2-0 What is Calculus ?
2-1 The Problems of Tangents
2-2 How to Calculate the Slope of the Tangent
2-3 The Definition of the Derivative
2-4 Velocity and Rates of Change: Newton and Leibriz
2-5 The Concept of a Limit: Two Trigonometric Limits
2-6 Continuous Functions: The Mean Value Theorem and Other Theorem

CHAPTER 3: The Computation of Derivatives
3-1 Derivatives of Polynomials
3-2 The Product and Quotient Rules
3-3 Composite Functions and the Chain Rule
3-4 Some Trigonometric Derivatives
3-5 Implicit Functions and Fractional Exponents
3-6 Derivatives of Higher Order

CHAPTER 4: Applications of Derivatives
4-1 Increasing and Decreasing Functions: Maxima and Minima
4-2 Concavity and Points of Inflection
4-3 Applied Maximum and Minimum Problems
4-4 More Maximum-Minimum Problems
4-5 Related Rates
4-6 Newtons Method for Solving Equations
4-7 Applications to Economics: Marginal Analysis

CHAPTER 5: Indefinite Integrals and Differential Equations
5-1 Introduction
5-2 Differentials and Tangent Line Approximations
5-3 Indefinite Integrals: Integration by Substitution
5-4 Differential Equations: Separation of Variables
5-5 Motion Under Gravity: Escape Velocity and Black Holes

CHAPTER 6: Definite Integrals
6-1 Introduction
6-2 The Problem of Areas
6-3 The Sigma Notation and Certain Special Sums
6-4 The Area Under a Curve: Definite Integrals
6-5 The Computation of Areas as Limits
6-6 The Fundamental Theorem of Calculus
6-7 Properties of Definite Integrals

CHAPTER 7: Applications of Integration
7-1 Introduction: The Intuitive Meaning of Integration
7-2 The Area between Two Curves
7-3 Volumes: The Disk Method
7-4 Volumes: The Method of Cylindrical Shells
7-5 Arc Length
7-6 The Area of a Surface of Revolution
7-7 Work and Energy
7-8 Hydrostatic Force
PART II

CHAPTER 8: Exponential and Logarithm Functions
8-1 Introduction
8-2 Review of Exponents and Logarithms
8-3 The Number e and the Function y = e <^>x
8-4 The Natural Logarithm Function y = ln x
8-5 Applications
Population Growth and Radioactive Decay
8-6 More Applications

CHAPTER 9: Trigonometric Functions
9-1 Review of Trigonometry
9-2 The Derivatives of the Sine and Cosine
9-3 The Integrals of the Sine and Cosine
9-4 The Derivatives of the Other Four Functions
9-5 The Inverse Trigonometric Functions
9-6 Simple Harmonic Motion
9-7 Hyperbolic Functions

CHAPTER 10 : Methods of Integration
10-1 Introduction
10-2 The Method of Substitution
10-3 Certain Trigonometric Integrals
10-4 Trigonometric Substitutions
10-5 Completing the Square
10-6 The Method of Partial Fractions
10-7 Integration by Parts
10-8 A Mixed Bag
10-9 Numerical Integration

CHAPTER 11: Further Applications of Integration
11-1 The Center of Mass of a Discrete System
11-2 Centroids
11-3 The Theorems of Pappus
11-4 Moment of Inertia

CHAPTER 12: Indeterminate Forms and Improper Integrals
12-1 Introduction. The Mean Value Theorem Revisited
12-2 The Interminate Form 0/0. L'Hospital's Rule
12-3 Other Interminate Forms
12-4 Improper Integrals
12-5 The Normal Distribution

CHAPTER 13: Infinite Series of Constants
13-1 What is an Infinite Series ?
13-2 Convergent Sequences
13-3 Convergent and Divergent Series
13-4 General Properties of Convergent Series
13-5 Series on Non-negative Terms: Comparison Tests
13-6 The Integral Test
13-7 The Ratio Test and Root Test
13-8 The Alternating Series Test

CHAPTER 14: Power Series
14-1 Introduction
14-2 The Interval of Convergence
14-3 Differentiation and Integration of Power Series
14-4 Taylor Series and Taylor's Formula
14-5 Computations Using Taylor's Formula
14-6 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
15-1 Introduction
15-2 Another Look at Circles and Parabolas
15-3 Ellipses
15-4 Hyperbolas
15-5 The Focus-Directrix-Eccentricity Definitions
15-6 (optional) Second Degree Equations

CHAPTER 16: Polar Coordinates
16-1 The Polar Coordinate System
16-2 More Graphs of Polar Equations
16-3 Polar Equations of Circles, Conics, and Spirals
16-4 Arc Length and Tangent Lines
16-5 Areas in Polar Coordinates

CHAPTER 17: Parametric Equations
17-1 Parametric Equations of Curves
17-2 The Cycloid and Other Similar Curves
17-3 Vector Algebra
17-4 Derivatives of Vector Function
17-5 Curvature and the Unit Normal Vector
17-6 Tangential and Normal Components of Acceleration
17-7 Kepler's Laws and Newton's Laws of Gravitation

CHAPTER 18: Vectors in Three-Dimensional Space
18-1 Coordinates and Vectors in Three-Dimensional Space
18-2 The Dot Product of Two Vectors
18-3 The Cross Product of Two Vectors
18-4 Lines and Planes
18-5 Cylinders and Surfaces of Revolution
18-6 Quadric Surfaces
18-7 Cylindrical and Spherical Coordinates

CHAPTER 19: Partial Derivatives
19-1 Functions of Several Variables
19-2 Partial Derivatives
19-3 The Tangent Plane to a Surface
19-4 Increments and Differentials
19-5 Directional Derivatives and the Gradient
19-6 The Chain Rule for Partial Derivatives
19-7 Maximum and Minimum Problems
19-8 Constrained Maxima and Minima
19-9 Laplace's Equation, the Heat Equation, and the Wave Equation
19-10 (optional) Implicit Functions

CHAPTER 20: Multiple Integrals
20-1 Volumes as Iterated Integrals
20-2 Double Integrals and Iterated Integrals
20-3 Physical Applications of Double Integrals
20-4 Double Integrals in Polar Coordinates
20-5 Triple Integrals
20-6 Cylindrical Coordinates
20-7 Spherical Coordinates
20-8 Areas of curved Surfaces

CHAPTER 21: Line and Surface Integrals
21-1 Green's Theorem, Gauss's Theorem, and Stokes' Theorem
21-2 Line Integrals in the Plane
21-3 Independence of Path
21-4 Green's Theorem
21-5 Surface Integrals and Gauss's Theorem
21-6 Maxwell's Equations : A Final Thought
Appendices
A: The Theory of Calculus
A-1 The Real Number System
A-2 Theorems About Limits
A-3 Some Deeper Properties of Continuous Functions
A-4 The Mean Value theorem
A-5 The Integrability of Continuous Functions
A-6 Another Proof of the Fundamental Theorem of Calculus
A-7 Continuous Curves With No Length
A-8 The Existence of e = lim h->0 (1 + h) <^>1/h
A-9 Functions That Cannot Be Integrated
A-10 The Validity of Integration by Inverse Substitution
A-11 Proof of the Partial fractions Theorem
A-12 The Extended Ratio Tests of Raabe and Gauss
A-13 Absolute vs Conditional Convergence
A-14 Dirichlet's Test
A-15 Uniform Convergence for Power Series
A-16 Division of Power Series
A-17 The Equality of Mixed Partial Derivatives
A-18 Differentiation Under the Integral Sign
A-19 A Proof of the Fundamental Lemma
A-20 A Proof of the Implicit Function Theorem
A-21 Change of Variables in Multiple Integrals
B: A Few Review Topics
B-1 The Binomial Theorem
B-2 Mathematical Induction