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Preface

To the Instructor

1 Preliminaries

1.1 Sets of Real Numbers

1.2 Absolute Value and Inequalities

1.3 The Cartesian Plane

1.4 Lines

1.5 Equations of a Straight Line

1.6 Functions

1.7 Operations with Functions

1.8 Shifting the Graphs of Functions (Optional)

Review Exercises for Chapter One

2 Limits And Derivatives

2.1 Introduction to the Derivative

2.2 The Calculation of Limits

2.3 The Limit Theorems

2.4 Infinite Limits and Limits at Infinity

2.5 Tangent Lines and Derivatives

2.6 The Derivative as a Rate of Change

2.7 Continuity

2.8 The Theory of Limits (Optional)

Review Exercises for Chapter Two

3 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 the Trigonometric Functions

3.6 Implicit Differentiation

3.7 Higher-Order Derivatives

3.8 Approximation and Differentials

Review Exercises for Chapter Three

4 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 (Optional)

4.8 Newton's Method for Solving Equations

Review Exercises for Chapter Four

5 The Integral

5.1 Introduction

5.2 Antiderivatives

5.3 The Σ Notation

5.4 Approximations to Area

5.5 The Definite Integral

5.6 The Fundamental Theorem 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

6 Exponentials And Logarithms

6.1 Inverse Functions

6.2 The Exponential and Logarithmic Functions I

6.3 The Derivatives and Integrals of logax and ax

6.4 The Exponential and Logarithmic Functions II

6.5 Differentiation and Integration of More General Exponential and Logarithmic Functions

6.6 Differential Equations of Exponential Growth and Decay

6.7 Applications in Economics (Optional)

6.8 A Model for Epidemics (Optional)

Review Exercises for Chapter Six

7 More On Trigonometric Functions And The Hyperbolic Functions

7.1 Integration of Trigonometric Functions

7.2 The Inverse Trigonometric Functions

7.3 Periodic Motion (Optional)

7.4 The Hyperbolic Functions

7.5 The Inverse Hyperbolic Functions (Optional)

Review Exercises for Chapter Seven

8 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

9 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

10 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

11 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

12 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

13 Taylor Polynomials And Approximation

13.1 Taylor's Theorem and Taylor Polynomials

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

13.3 Approximation Using Taylor Polynomials

Review Exercises for Chapter Thirteen

14 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

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

Tables

A.1 Exponential Functions

A.2 Natural Logarithms

A.3 Hyperbolic Functions

A.4 Integrals

Answers to Odd-Numbered Problems and Review Exercises

Index

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2nd Edition - November 7, 1985

Author: Stanley I. Grossman

Language: EnglisheBook ISBN:

9 7 8 - 1 - 4 8 3 2 - 6 2 4 6 - 8

Calculus of One Variable, Second Edition presents the essential topics in the study of the techniques and theorems of calculus. The book provides a comprehensive introduction to… Read more

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Calculus of One Variable, Second Edition presents the essential topics in the study of the techniques and theorems of calculus. The book provides a comprehensive introduction to calculus. It contains examples, exercises, the history and development of calculus, and various applications. Some of the topics discussed in the text include the concept of limits, one-variable theory, the derivatives of all six trigonometric functions, exponential and logarithmic functions, and infinite series. This textbook is intended for use by college students.

Preface

To the Instructor

1 Preliminaries

1.1 Sets of Real Numbers

1.2 Absolute Value and Inequalities

1.3 The Cartesian Plane

1.4 Lines

1.5 Equations of a Straight Line

1.6 Functions

1.7 Operations with Functions

1.8 Shifting the Graphs of Functions (Optional)

Review Exercises for Chapter One

2 Limits And Derivatives

2.1 Introduction to the Derivative

2.2 The Calculation of Limits

2.3 The Limit Theorems

2.4 Infinite Limits and Limits at Infinity

2.5 Tangent Lines and Derivatives

2.6 The Derivative as a Rate of Change

2.7 Continuity

2.8 The Theory of Limits (Optional)

Review Exercises for Chapter Two

3 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 the Trigonometric Functions

3.6 Implicit Differentiation

3.7 Higher-Order Derivatives

3.8 Approximation and Differentials

Review Exercises for Chapter Three

4 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 (Optional)

4.8 Newton's Method for Solving Equations

Review Exercises for Chapter Four

5 The Integral

5.1 Introduction

5.2 Antiderivatives

5.3 The Σ Notation

5.4 Approximations to Area

5.5 The Definite Integral

5.6 The Fundamental Theorem 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

6 Exponentials And Logarithms

6.1 Inverse Functions

6.2 The Exponential and Logarithmic Functions I

6.3 The Derivatives and Integrals of logax and ax

6.4 The Exponential and Logarithmic Functions II

6.5 Differentiation and Integration of More General Exponential and Logarithmic Functions

6.6 Differential Equations of Exponential Growth and Decay

6.7 Applications in Economics (Optional)

6.8 A Model for Epidemics (Optional)

Review Exercises for Chapter Six

7 More On Trigonometric Functions And The Hyperbolic Functions

7.1 Integration of Trigonometric Functions

7.2 The Inverse Trigonometric Functions

7.3 Periodic Motion (Optional)

7.4 The Hyperbolic Functions

7.5 The Inverse Hyperbolic Functions (Optional)

Review Exercises for Chapter Seven

8 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

9 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

10 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

11 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

12 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

13 Taylor Polynomials And Approximation

13.1 Taylor's Theorem and Taylor Polynomials

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

13.3 Approximation Using Taylor Polynomials

Review Exercises for Chapter Thirteen

14 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

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

Tables

A.1 Exponential Functions

A.2 Natural Logarithms

A.3 Hyperbolic Functions

A.4 Integrals

Answers to Odd-Numbered Problems and Review Exercises

Index

- No. of pages: 906
- Language: English
- Edition: 2
- Published: November 7, 1985
- Imprint: Academic Press
- eBook ISBN: 9781483262468

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