Calculus of One Variable

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