## Summary

A revision of McGraw-Hill's leading calculus text for the 3-semester sequence taken primarily by math, engineering, and science majors. The revision is substantial and has been influenced by students, instructors in physics, engineering, and mathematics, and participants in the national debate on the future of calculus. Revision focused on these key areas: Upgrading graphics and design, expanding range of problem sets, increasing motivation, strengthening multi-variable chapters, and building a stronger support package.

## Table of Contents

# Calculus and Analytic Geometry

### 1. An Overview of Calculus.

#### 1.1 The Derivative

#### 1.2 The Integral

#### 1.3 Survey of the Text

### 2. Functions, Limits, and Continuity.

#### 2.1 Functions

#### 2.2 Composite Functions

#### 2.3 The Limit of a Function

#### 2.4 Computations of Limits

#### 2.5 Some Tools for Graphing

#### 2.6 A Review of Trigonometry

#### 2.7 The Limit of (sin Ø)/Ø as Ø Approaches 0

#### 2.8 Continuous Functions

#### 2.9 Precise Definitions of "lim(x->infinity)f(x)=infinity" and "lim(x->infinity)f(x)=L"

#### 2.10 Precise Definition of "lim(x->a)f(x)=L"

#### 2.S Summary

### 3. The Derivative.

#### 3.1 Four Problems with One Theme

#### 3.2 The Derivative

#### 3.3 The Derivative and Continuity

#### 3.4 The Derivative of the Sum, Difference, Product, and Quotient

#### 3.5 The Derivatives of the Trigonometric Functions

#### 3.6 The Derivative of a Composite Function

#### 3.S Summary

### 4. Applications of the Derivative.

#### 4.1 Three Theorems about the Derivative

#### 4.2 The First Derivative and Graphing

#### 4.3 Motion and the Second Derivative

#### 4.4 Related Rates

#### 4.5 The Second Derivative and Graphing

#### 4.6 Newton's Method for Solving an Equation

#### 4.7 Applied Maximum and Minimum Problems

#### 4.9 The Differential and Linearization

#### 4.10 The Second Derivative and Growth of a Function

#### 4.S Summary

### 5. The Definite Integral.

#### 5.1 Estimates in Four Problems

#### 5.2 Summation Notation and Approximating Sums

#### 5.3 The Definite Integral

#### 5.4 Estimating the Definite Integral

#### 5.5 Properties of the Antiderivative and the Definite Integral

#### 5.6 Background for the Fundamental Theorems of Calculus

#### 5.7 The Fundamental Theorems of Calculus

#### 5.S Summary

### 6. Topics in Differential Calculus.

#### 6.1 Logarithms

#### 6.2 The Number e

#### 6.3 The Derivative of a Logarithmic Function

#### 6.4 One-to-One Functions and Their Inverses

#### 6.5 The Derivative of b^x

#### 6.6 The Derivatives of the Inverse Trigonometric Functions

#### 6.7 The Differential Equation of Natural Growth and Decay

#### 6.8 l'Hopital's Rule

#### 6.9 The Hyperbolic Functions and Their Inverses

#### 6.S Summary

### 7. Computing Antiderivatives.

#### 7.1 Shortcuts, Integral Tables, and Machines

#### 7.2 The Substitution Method

#### 7.3 Integration by Parts

#### 7.4 How to Integrate Certain Rational Functions

#### 7.5 Integration of Rational Functions by Partial Fractions

#### 7.6 Special Techniques

#### 7.7 What to Do in the Face of an Integral

#### 7.S Summary

### 8. Applications of the Definite Integral.

#### 8.1 Computing Area by Parallel Cross Sections

#### 8.2 Some Pointers on Drawing

#### 8.3 Setting Up a Definite Integral

#### 8.4 Computing Volumes

#### 8.5 The Shell Method

#### 8.6 The Centroid of a Plane Region

#### 8.7 Work

#### 8.8 Improper Integrals

#### 8.S Summary

### 9. Plane Curves and Polar Coordinates.

#### 9.1 Polar Coordinates

#### 9.2 Area in Polar Coordinates

#### 9.3 Parametric Equations

#### 9.4 Arc Length and Speed on a Curve

#### 9.5 The Area of a Surface of Revolution

#### 9.6 Curvature

#### 9.7 The Reflection Properties of the Conic Sections

#### 9.S Summary

### 10. Series.

#### 10.1 An Informal Introduction to Series

#### 10.2 Sequences

#### 10.3 Series

#### 10.4 The Integral Test

#### 10.5 Comparison Tests

#### 10.6 Ratio Tests

#### 10.7 Tests for Series with Both Positive and Negative Terms

#### 10.S Summary

### 11. Power Series and Complex Numbers.

#### 11.1 Taylor Series

#### 11.2 The Error in Taylor Series

#### 11.3 Why the Error in Taylor Series Is Controlled by a Derivative

#### 11.4 Power Series and Radius of Convergence

#### 11.5 Manipulating Power Series

#### 11.6 Complex Numbers

#### 11.7 The Relation between the Exponential and the Trigonometric Functions

#### 11.S Summary

### 12. Vectors.

#### 12.1 The Algebra of Vectors

#### 12.2 Projections

#### 12.3 The Dot Product of Two Vectors

#### 12.4 Lines and Planes

#### 12.5 Determinants

#### 12.6 The Cross Product of Two Vectors

#### 12.7 More on Lines and Planes

#### 12.S Summary

### 13. The Derivative of a Vector Function.

#### 13.1 The Derivative of a Vector Function

#### 13.2 Properties of the Derivative of a Vector Function

#### 13.3 The Acceleration Vector

#### 13.4 The Components of Acceleration

#### 13.5 Newton's Law Implies Kepler's Laws

#### 13.S Summary

### 14. Partial Derivatives.

#### 14.1 Graphs

#### 14.2 Quadratic Surfaces

#### 14.3 Functions and Their Level Curves

#### 14.4 Limits and Continuity

#### 14.5 Partial Derivatives

#### 14.6 The Chain Rule

#### 14.7 Directional Derivatives and the Gradient

#### 14.8 Normals and the Tangent Plane

#### 14.9 Critical Points and Extrema

#### 14.10 Lagrange Multipliers

#### 14.11 The Chain Rule Revisited

#### 14.S Summary

### 15. Definite Integrals over Plane and Solid Regions.

#### 15.1 The Definite Integral of a Function over a Region in the Plane

#### 15.2 Computing |R f(P) dA Using Rectangular Coordinates

#### 15.3 Moments and Centers of Mass

#### 15.4 Computing |R f(P) dA Using Polar Coordinates

#### 15.5 The Definite Integral of a Function over a Region in Space

#### 15.6 Computing |R f(P) dV Using Cylindrical Coordinates

#### 15.7 Computing |R f(P) dV Using Spherical Coordinates

#### 15.S Summary

### 16. Green's Theorem.

#### 16.1 Vector and Scalar Fields

#### 16.2 Line Integrals

#### 16.3 Four Applications of Line Integrals

#### 16.4 Green's Theorem

#### 16.5 Applications of Green's Theorem

#### 16.6 Conservative Vector Fields

#### 16.S Summary

### 17. The Divergence Theorem and Stokes' Theorem.

#### 17.1 Surface Integrals

#### 17.2 The Divergence Theorem

#### 17.3 Stokes' Theorem

#### 17.4 Applications of Stokes' Theorem

#### 17.S Summary

## Appendices:

#### A. Real Numbers.

#### B. Graphs and Lines.

#### C. Topics in Algebra.

#### D. Exponents.

#### E. Mathematical Induction.

#### F. The Converse of a Statement.

#### G. Conic Sections.

#### H. Logarithms and Exponentials Defined through Calculus.

## I. The Taylor Series for f(x,y).

#### J. Theory of Limits.

#### K. The Interchange of Limits.

#### L. The Jacobian.

#### M. Linear Differential Equations with Constant Coefficients.

### Answers to Selected Odd-Numbered Problems and to Guide Quizzes

### List of Symbols

### Index