Calculus, Early Transcendentals Book

TABLE OF CONTENTS

About the Authors

Preface

1 Functions, Graphs, and Models

1.1 Functions and Mathematical Modeling

Investigation: Designing a Wading Pool

1.2 Graphs of Equations and Functions

1.3 Polynomials and Algebraic Functions

1.4 Transcendental Functions

1.5 Preview: What Is Calculus?

REVIEW — Understanding: Concepts and Definitions

Objectives: Methods and Techniques

2 Prelude to Calculus

2.1 Tangent Lines and Slope Predictors

Investigation: Numerical Slope Investigations

2.2 The Limit Concept

Investigation: Limits, Slopes, and Logarithms

2.3 More About Limits

Investigation: Numerical Epsilon-Delta Limits

2.4 The Concept of Continuity

REVIEW – Understanding: Concepts and Definitions

Objectives: Methods and Techniques

3 The Derivative

3.1 The Derivative and Rates of Change

3.2 Basic Differentiation Rules

3.3 The Chain Rule

3.4 Derivatives of Algebraic Functions

3.5 Maxima and Minima of Functions on Closed Intervals

Investigation: When Is Your Coffee Cup Stablest?

3.6 Applied Optimization Problems

3.7 Derivatives of Trigonometric Functions

3.8 Exponential and Logarithmic Functions

Investigation: Discovering the Number e for Yourself

3.9 Implicit Differentiation and Related Rates

Investigation: Constructing the Folium of Descartes

3.10 Successive Approximations and Newton's Method

Investigation: How Deep Does a Floating Ball Sink?

REVIEW — Understanding: Concepts, Definitions, and Formulas

Objectives: Methods and Techniques

4 Additional Applications of the Derivative

4.1 Introduction

4.2 Increments, Differentials, and Linear Approximation

4.3 Increasing and Decreasing Functions and the Mean Value Theorem

4.4 The First Derivative Test and Applications

Investigation: Constructing a Candy Box With Lid

4.5 Simple Curve Sketching

4.6 Higher Derivatives and Concavity

4.7 Curve Sketching and Asymptotes

Investigation: Locating Special Points on Exotic Graphs

4.8 Indeterminate Forms and L'Hôpital's Rule

4.9 More Indeterminate Forms

REVIEW – Understanding: Concepts, Definitions, and Results

Objectives: Methods and Techniques

5 The Integral

5.1 Introduction

5.2 Antiderivatives and Initial Value Problems

5.3 Elementary Area Computations

5.4 Riemann Sums and the Integral

Investigation: Calculator/Computer Riemann Sums

5.5 Evaluation of Integrals

5.6 The Fundamental Theorem of Calculus

5.7 Integration by Substitution

5.8 Areas of Plane Regions

5.9 Numerical Integration

Investigation: Trapezoidal and Simpson Approximations

REVIEW — Understanding: Concepts, Definitions, and Results

Objectives: Methods and Techniques

6 Applications of the Integral

6.1 Riemann Sum Approximations

6.2 Volumes by the Method of Cross Sections

6.3 Volumes by the Method of Cylindrical Shells

Investigation: Design Your Own Ring!

6.4 Arc Length and Surface Area of Revolution

6.5 Force and Work

6.6 Centroids of Plane Regions and Curves

6.7 The Natural Logarithm as an Integral

Investigation: Natural Functional Equations

6.8 Inverse Trigonometric Functions

6.9 Hyperbolic Functions

REVIEW – Understanding: Concepts, Definitions, and Formulas

Objectives: Methods and Techniques

7 Techniques of Integration

7.1 Introduction

7.2 Integral Tables and Simple Substitutions

7.3 Integration by Parts

7.4 Trigonometric Integrals

7.5 Rational Functions and Partial Fractions

7.6 Trigonometric Substitutions

7.7 Integrals Involving Quadratic Polynomials

7.8 Improper Integrals

SUMMARY — Integration Strategies

REVIEW – Understanding: Concepts and Techniques

Objectives: Methods and Techniques

8 Differential Equations

8.1 Simple Equations and Models

8.2 Slope Fields and Euler's Method

Investigation: Computer-Assisted Slope Fields and Euler's Method

8.3 Separable Equations and Applications

8.4 Linear Equations and Applications

8.5 Population Models

Investigation: Predator-Prey Equations and Your Own Game Preserve

8.6 Linear Second-Order Equations

8.7 Mechanical Vibrations

REVIEW — Understanding: Concepts, Definitions, and Methods

Objectives: Methods and Techniques

9 Polar Coordinates and Parametric Curves

9.1 Analytic Geometry and the Conic Sections

9.2 Polar Coordinates

9.3 Area Computations in Polar Coordinates

9.4 Parametric Curves

Investigation: Trochoids Galore

9.5 Integral Computations with Parametric Curves

Investigation: Moon Orbits and Race Tracks

9.6 Conic Sections and Applications

REVIEW – Understanding: Concepts, Definitions, and Formulas

Objectives: Methods and Techniques

10 Infinite Series

10.1 Introduction

10.2 Infinite Sequences

Investigation: Nested Radicals and Continued Fractions

10.3 Infinite Series and Convergence

Investigation: Numerical Summation and Geometric Series

10.4 Taylor Series and Taylor Polynomials

Investigation: Calculating Logarithms on a Deserted Island

10.5 The Integral Test

Investigation: The Number p, Once and for All

10.6 Comparison Tests for Positive-Term Series

10.7 Alternating Series and Absolute Convergence

10.8 Power Series

10.9 Power Series Computations

Investigation: Calculating Trigonometric Functions on a Deserted Island

10.10 Series Solutions of differential Equations

REVIEW — Understanding: Concepts, and Results

Objectives: Methods and Techniques

11 Vectors, Curves, and Surfaces in Space

11.1 Vectors in the Plane

11.2 Three-Dimensional Vectors

11.3 The Cross Product of Two Vectors

11.4 Lines and Planes in Space

11.5 Curves and Motion in Space

Investigation: Does a Pitched Baseball Really Curve?

11.6 Curvature and Acceleration

11.7 Cylinders and Quadric Surfaces

11.8 Cylindrical and Spherical Coordinates

REVIEW – Understanding: Concepts, Definitions, and Results

Objectives: Methods and Techniques

12 Partial Differentiation

12.1 Introduction

12.2 Functions of Several Variables

12.3 Limits and Continuity

12.4 Partial Derivatives

12.5 Multivariable Optimization Problems

12.6 Increments and Linear Approximation

12.7 The Multivariable Chain Rule

12.8 Directional Derivatives and the Gradient Vector

12.9 Lagrange Multipliers and Constrained Optimization

Investigation: Numerical Solution of Lagrange Multiplier Systems

12.10 Critical Points of Functions of Two Variables

Investigation: Critical Point Investigations

REVIEW — Understanding: Concepts, Definitions, and Results

Objectives: Methods and Techniques

13 Multiple Integrals

13.1 Double Integrals

Investigation: Midpoint Sums Approximating Double Integrals

13.2 Double Integrals over More General Regions

13.3 Area and Volume by Double Integration

13.4 Double Integrals in Polar Coordinates

13.5 Applications of Double Integrals

Investigation: Optimal Design of Race Car Wheels

13.6 Triple Integrals

Investigation: Archimedes' Floating Paraboloid

13.7 Integration in Cylindrical and Spherical Coordinates

13.8 Surface Area

13.9 Change of Variables in Multiple Integrals

REVIEW – Understanding: Concepts, Definitions, and Results

Objectives: Methods and Techniques

14 Vector Calculus

14.1 Vector Fields

14.2 Line Integrals

14.3 The Fundamental Theorem and Independence of Path

14.4 Green's Theorem

14.5 Surface Integrals

Investigation: Surface Integrals and Rocket Nose Cones

14.6 The Divergence Theorem

14.7 Stokes' Theorem

REVIEW — Understanding: Concepts, Definitions, and Results

Objectives: Methods and Techniques

Appendices

A: Real Numbers and Inequalities

B: The Coordinate Plane and Straight Lines

C: Review of Trigonometry

D: Proofs of the Limit Laws

E: The Completeness of the Real Number System

F: Existence of the Integral

G: Approximations and Riemann Sums

H: L'Hôpital's Rule and Cauchy's Mean Value Theorem

I: Proof of Taylor's Formula

J: Conic Sections as Sections of a Cone

K: Proof of the Linear Approximation Theorem

L: Units of Measurement and Conversion Factors

M: Formulas from Algebra, Geometry, and Trigonometry

N: The Greek Alphabet

Answers to Odd-Numbered Problems

References for Further Study

Index