# Calculus

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Calculus, Second Edition discusses the techniques and theorems of calculus. This edition introduces the sine and cosine functions, distributes ?-? material over several chapters, and includes a detailed account of analytic geometry and vector analysis. This book also discusses the equation of a straight line, trigonometric limit, derivative of a power function, mean value theorem, and fundamental theorems of calculus. The exponential and logarithmic functions, inverse trigonometric functions, linear and quadratic denominators, and centroid of a plane region are likewise elaborated. Other topics include the sequences of real numbers, dot product, arc length as a parameter, quadric surfaces, higher-order partial derivatives, and Green's theorem in the plane. This publication is a good source for students learning calculus.

## Table of Contents

Preface

To the Instructor

One Preliminaries

1.1 Sets of Real Numbers

1.2 Absolute Value and Inequalities

1.3 The Cartesian Plane

1.4 Lines

1.5 The Equation of a Straight Line

1.6 Circles

1.7 Functions

1.8 Operations with Functions

1.9 Shifting the Graphs of Functions

1.10 Second-Degree Equations

Review Exercises for Chapter One

Two Limits And Derivatives

2.1 Introduction to Limits

2.2 The Calculation of Limits

2.3 The Limit Theorems

2.4 Infinite Limits and Limits at Infinity

2.5 One-Sided Limits

2.6 A Trigonometric Limit

2.7 Tangent Lines and Derivatives

2.8 Tangent Lines and Derivatives (Continued)

2.9 The Derivative as a Rate of Change

2.10 Continuity

2.11 The Theory of Limits (Optional)

Review Exercises for Chapter Two

Three More About Derivatives

3.1 Some Differentiation Formulas

3.2 The Product and Quotient Rules

3.3 The Derivative of Composite Functions—The Chain Rule

3.4 The Derivative of a Power Function

3.5 The Derivatives of Sines and Cosines

3.6 Implicit Differentiation

3.7 Higher-Order Derivatives

3.8 Approximation and Differentials

Review Exercises for Chapter Three

Four Applications Of The Derivative

4.1 Related Rates of Change

4.2 The Mean Value Theorem

4.3 Elementary Curve Sketching I: Increasing and Decreasing Functions and the First Derivative Test

4.4 Elementary Curve Sketching II: Concavity and the Second Derivative Test

4.5 The Theory of Maxima and Minima

4.6 Maxima and Minima: Applications

4.7 Some Applications in Economics

4.8 Newton's Method for Solving Equations

Review Exercises for Chapter Four

Five The Integral

5.1 The Area Problem

5.2 The Σ Notation

5.3 Approximations to Area

5.4 The Definite Integral

5.5 The Antiderivative

5.6 The Fundamental Theorems of Calculus

5.7 Integration by Substitution

5.8 The Area Between Two Curves

5.9 Work, Power, and Energy (Optional)

5.10 Additional Integration Theory (Optional)

Review Exercises for Chapter Five

Six Exponentials And Logarithms

6.1 The Exponential and Logarithmic Functions

6.2 The Derivatives and Integrals of logax and ax

6.3 The Exponential and Logarithmic Functions II

6.4 Differentiation and Integration of More General Exponential and Logarithmic Functions

6.5 Differential Equations of Exponential Growth and Decay

6.6 Applications in Economics (Optional)

6.7 Inverse Functions

Review Exercises for Chapter Six

Seven The Trigonometric And Hyperbolic Functions

7.1 Differentiation of Trigonometric Functions

7.2 Integration of Trigonometric Functions

7.3 The Inverse Trigonometric Functions

7.4 Periodic Motion (Optional)

7.5 The Hyperbolic Functions

7.6 The Inverse Hyperbolic Functions (Optional)

Review Exercises for Chapter Seven

Eight Techniques Of Integration

8.1 Review of the Basic Formulas of Integration

8.2 Integration by Parts

8.3 Integrals of Certain Trigonometric Functions

8.4 The Idea Behind Integration by Substitution

8.5 Integrals Involving √a2 — x2, √a2 + x2, and √x2 — a2; Trigonometric Substitutions

8.6 The Integration of Rational Functions I: Linear and Quadratic Denominators

8.7 The Integration of Rational Functions II: The Method of Partial Fractions

8.8 Other Substitutions

8.9 Using the Integral Tables

8.10 Numerical Integration

Review Exercises for Chapter Eight

Nine Further Applications Of The Definite Integral

9.1 Volumes

9.2 Arc Length

9.3 Surface Area

9.4 Center of Mass and the First Moment

9.5 The Centroid of a Plane Region

9.6 Moments of Inertia and Kinetic Energy (Optional)

9.7 Fluid Pressure (Optional)

Review Exercises for Chapter Nine

Ten Topics In Analytic Geometry

10.1 The Ellipse and Translation of Axes

10.2 The Parabola

10.3 The Hyperbola

10.4 Second-Degree Equations and Rotation of Axes

Review Exercises for Chapter Ten

Eleven Polar Coordinates

11.1 The Polar Coordinate System

11.2 Graphing in Polar Coordinates

11.3 Points of Intersection of Graphs of Polar Equations

11.4 Derivatives and Tangent Lines

11.5 Areas in Polar Coordinates

Review Exercises for Chapter Eleven

Twelve Indeterminate Forms And Improper Integrals

12.1 The Indeterminate Form 0/0 and L'Hôpital's Rule

12.2 Proof of L'Hôpital's Rule (Optional)

12.3 Other Indeterminate Forms

12.4 Improper Integrals

Review Exercises for Chapter Twelve

Thirteen Taylor Polynomials And Approximation

13.1 Taylor's Theorem and Taylor Polynomials

13.2 A Proof of Taylor's Theorem and Estimates on the Remainder Term (Optional)

13.3 Approximation Using Taylor Polynomials

Review Exercises for Chapter Thirteen

Fourteen Sequences And Series

14.1 Sequences of Real Numbers

14.2 Bounded and Monotonic Sequences

14.3 Geometric Series

14.4 Infinite Series

14.5 Series with Nonnegative Terms I: Two Comparison Tests and the Integral Test

14.6 Series with Nonnegative Terms II: The Ratio and Root Tests

14.7 Absolute and Conditional Convergence: Alternating Series

14.8 Power Series

14.9 Differentiation and Integration of Power Series

14.10 Taylor and Maclaurin Series

Review Exercises for Chapter Fourteen

Fifteen Vectors In The Plane

15.1 Vectors and Vector Operations

15.2 The Dot Product

15.3 Some Applications of Vectors (Optional)

Review Exercises for Chapter Fifteen

Sixteen Vector Functions, Vector Differentiation, And Parametric Equations

16.1 Vector Functions and Parametric Equations

16.2 The Equation of the Tangent Line to a Parametric Curve

16.3 The Differentiation and Integration of a Vector Function

16.4 Some Differentiation Formulas

16.5 Arc Length Revisited

16.6 Arc Length as a Parameter

16.7 Velocity, Acceleration, Force, and Momentum (Optional)

16.8 Curvature and the Acceleration Vector (Optional)

Review Exercises for Chapter Sixteen

Seventeen Vectors In Space

17.1 The Rectangular Coordinate System in Space

17.2 Vectors in R3

17.3 Lines in R3

17.4 The Cross Product of Two Vectors

17.5 Planes

17.6 Quadric Surfaces

17.7 Vector Functions and Parametric Equations in R3

17.8 Cylindrical and Spherical Coordinates

Review Exercises for Chapter Seventeen

Eighteen Differentiation Of Functions Of Two And Three Variables

18.1 Functions of Two and Three Variables

18.2 Limits and Continuity

18.3 Partial Derivatives

18.4 Higher-Order Partial Derivatives

18.5 Differentiability and the Gradient

18.6 The Chain Rule

18.7 Tangent Planes, Normal Lines, and Gradients

18.8 Directional Derivatives and the Gradient

18.9 Conservative Vector Fields and the Gradient (Optional)

18.10 The Total Differential and Approximation

18.11 Exact Differentials or How to Obtain a Function from Its Gradient

18.12 Maxima and Minima for a Function of Two Variables

18.13 Constrained Maxima and Minima—Lagrange Multipliers

Review Exercises for Chapter Eighteen

Nineteen Multiple Integrations

19.1 Volume Under a Surface and the Double Integral

19.2 The Calculation of Double Integrals

19.3 Density, Mass, and Center of Mass (Optional)

19.4 Double Integrals in Polar Coordinates

19.5 Surface Area

19.6 The Triple Integral

19.7 The Triple Integral in Cylindrical and Spherical Coordinates

Review Exercises for Chapter Nineteen

Twenty Introduction To Vector Analysis

20.1 Vector Fields

20.2 Work, Line Integrals in the Plane, and Independence of Path

20.3 Green's Theorem in the Plane

20.4 Line Integrals in Space

20.5 Surface Integrals

20.6 Divergence and Curl of a Vector Field in R3

20.7 Stokes' Theorem

20.8 The Divergence Theorem

20.9 Changing Variables in Multiple Integrals and the Jacobian

Review Exercises for Chapter Twenty

Twenty-One Ordinary Differential Equations

21.1 Introduction

21.2 First-Order Equations—Separation of Variables

21.3 First-Order Linear Equations

21.4 Second-Order Linear, Homogeneous Equations with Constant Coefficients

21.5 Second-Order Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients

21.6 Vibratory Motion (Optional)

Review Exercises for Chapter Twenty-One

Appendix 1 Review Of Trigonometry

1.1 Angles and Radian Measure

1.2 The Trigonometric Functions and Basic Identities

1.3 Other Trigonometric Functions

1.4 Triangles

Appendix 2 Mathematical Induction

Appendix 3 Determinants

Appendix 4 The Binomial Theorem

Appendix 5 The Proofs Of Some Theorems On Limits, Continuity, And Differentiation

Appendix 6 Complex Numbers

Tables A.1 Exponential Functions

A.2 Natural Logarithms

A.3 Common Logarithms

A.4 Trigonometric Functions

A.5 Hyperbolic Functions

A.6 Integrals

Answers to Odd-Numbered Problems and Review Exercises

Index

## Product details

- No. of pages: 1174
- Language: English
- Copyright: © Academic Press 1981
- Published: January 1, 1981
- Imprint: Academic Press
- eBook ISBN: 9781483262437