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# Calculus

## Fourth EditionJon Rogawski; Colin Adams; Robert Franzosa

©2019
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## Table of Contents

**Chapter 1: Precalculus Review**

1.1 Real Numbers, Functions, and Graphs

1.2 Linear and Quadratic Functions

1.3 The Basic Classes of Functions

1.4 Trigonometric Functions

1.5 Technology: Calculators and Computers

Chapter Review Exercises

**Chapter 2: Limits**

2.1 The Limit Idea: Instantaneous Velocity and Tangent Lines

2.2 Investigating Limits

2.3 Basic Limit Laws

2.4 Limits and Continuity

2.5 Indeterminate Forms

2.6 The Squeeze Theorem and Trigonometric Limits

2.7 Limits at Infinity

2.8 The Intermediate Value Theorem

2.9 The Formal Definition of a Limit

Chapter Review Exercises

**Chapter 3: Differentiation**

3.1 Definition of the Derivative

3.2 The Derivative as a Function

3.3 Product and Quotient Rules

3.4 Rates of Change

3.5 Higher Derivatives

3.6 Trigonometric Functions

3.7 The Chain Rule

3.8 Implicit Differentiation

3.9 Related Rates

Chapter Review Exercises

**Chapter 4: Applications of the Derivative**

4.1 Linear Approximation and Applications

4.2 Extreme Values

4.3 The Mean Value Theorem and Monotonicity

4.4 The Second Derivative and Concavity

4.5 Analyzing and Sketching Graphs of Functions

4.6 Applied Optimization

4.7 Newton’s Method

Chapter Review Exercises

**Chapter 5: Integration**

5.1 Approximating and Computing Area

5.2 The Definite Integral

5.3 The Indefinite Integral

5.4 The Fundamental Theorem of Calculus, Part I

5.5 The Fundamental Theorem of Calculus, Part II

5.6 Net Change as the Integral of a Rate of Change

5.7 The Substitution Method

Chapter Review Exercises

**Chapter 6: Applications of the Integral**

6.1 Area Between Two Curves

6.2 Setting Up Integrals: Volume, Density, Average Value

6.3 Volumes of Revolution: Disks and Washers

6.4 Volumes of Revolution: Cylindrical Shells

6.5 Work and Energy

Chapter Review Exercises

**Chapter 7: Exponential and Logarithmic Functions**

7.1 The Derivative of f (x) = bx and the Number e

7.2 Inverse Functions

7.3 Logarithmic Functions and Their Derivatives

7.4 Applications of Exponential and Logarithmic Functions

7.5 L’Hopital’s Rule

7.6 Inverse Trigonometric Functions

7.7 Hyperbolic Functions

Chapter Review Exercises

**Chapter 8: Techniques of Integration**

8.1 Integration by Parts

8.2 Trigonometric Integrals

8.3 Trigonometric Substitution

8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions

8.5 The Method of Partial Fractions

8.6 Strategies for Integration

8.7 Improper Integrals

8.8 Numerical Integration

Chapter Review Exercises

**Chapter 9: Further Applications of the Integral**

9.1 Probability and Integration

9.2 Arc Length and Surface Area

9.3 Fluid Pressure and Force

9.4 Center of Mass

Chapter Review Exercises

**Chapter 10: Introduction to Differential Equations**

10.1 Solving Differential Equations

10.2 Models Involving y'=k(y-b)

10.3 Graphical and Numerical Methods

10.4 The Logistic Equation

10.5 First-Order Linear Equations

Chapter Review Exercises

**Chapter 11: Infinite Series**

11.1 Sequences

11.2 Summing an Infinite Series

11.3 Convergence of Series with Positive Terms

11.4 Absolute and Conditional Convergence

11.5 The Ratio and Root Tests and Strategies for Choosing Tests

11.6 Power Series

11.7 Taylor Polynomials

11.8 Taylor Series

Chapter Review Exercises

Chapter 12: Parametric Equations, Polar Coordinates, and Conic Sections

12.1 Parametric Equations

12.2 Arc Length and Speed

12.3 Polar Coordinates

12.4 Area and Arc Length in Polar Coordinates

12.5 Conic Sections

Chapter Review Exercises

**Chapter 13: Vector Geometry**

13.1 Vectors in the Plane

13.2 Three-Dimensional Space: Surfaces, Vectors, and Curves

13.3 Dot Product and the Angle Between Two Vectors

13.4 The Cross Product

13.5 Planes in 3-Space

13.6 A Survey of Quadric Surfaces

13.7 Cylindrical and Spherical Coordinates

Chapter Review Exercises

**Chapter 14: Calculus of Vector-Valued Functions**

14.1 Vector-Valued Functions

14.2 Calculus of Vector-Valued Functions

14.3 Arc Length and Speed

14.4 Curvature

14.5 Motion in 3-Space

14.6 Planetary Motion According to Kepler and Newton

Chapter Review Exercises

**Chapter 15: Differentiation in Several Variables**

15.1 Functions of Two or More Variables

15.2 Limits and Continuity in Several Variables

15.3 Partial Derivatives

15.4 Differentiability, Tangent Planes, and Linear Approximation

15.5 The Gradient and Directional Derivatives

15.6 Multivariable Calculus Chain Rules

15.7 Optimization in Several Variables

15.8 Lagrange Multipliers: Optimizing with a Constraint

Chapter Review Exercises

**Chapter 16: Multiple Integration**

16.1 Integration in Two Variables

16.2 Double Integrals over More General Regions

16.3 Triple Integrals

16.4 Integration in Polar, Cylindrical, and Spherical Coordinates

16.5 Applications of Multiple Integrals

16.6 Change of Variables

Chapter Review Exercises

**Chapter 17: Line and Surface Integrals**

17.1 Vector Fields

17.2 Line Integrals

17.3 Conservative Vector Fields

17.4 Parametrized Surfaces and Surface Integrals

17.5 Surface Integrals of Vector Fields

Chapter Review Exercises

**Chapter 18: Fundamental Theorems of Vector Analysis**

18.1 Green’s Theorem

18.2 Stokes’ Theorem

18.3 Divergence Theorem

Chapter Review Exercises

**Appendices **A. The Language of Mathematics

B. Properties of Real Numbers

C. Induction and the Binomial Theorem

D. Additional Proofs

ANSWERS TO ODD-NUMBERED EXERCISES

REFERENCES

INDEX

Additional content can be accessed online at www.macmillanlearning.com/calculuset4e:

**Additional Proofs:**L’Hôpital’s Rule

Error Bounds for Numerical

Integration

Comparison Test for Improper

Integrals

**Additional Content:**Second-Order Differential

Equations

Complex Numbers