Calculus

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