**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 Inverse Functions

1.6 Exponential and Logarithmic Functions

1.7 Technology Calculators and Computers

**Chapter 2: Limits**2.1 Limits, Rates of Change, and Tangent Lines

2.2 Limits: A Numerical and Graphical Approach2.3 Basic Limit Laws

2.4 Limits and Continuity

2.5 Evaluating Limits Algebraically

2.6 Trigonometric Limits

2.7 Limits at Infinity

2.8 Intermediate Value Theorem

2.9 The Formal Definition of a Limit

**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 Derivatives of Inverse Functions

3.9 Derivatives of General Exponential and Logarithmic Functions

3.10 Implicit Differentiation

3.11 Related Rates

**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 Shape of a Graph

4.5 L’Hopital’s Rule

4.6 Graph Sketching and Asymptotes

4.7 Applied Optimization

4.8 Newton’s Method

4.9 Antiderivatives

**Chapter 5: The Integral**

5.1 Approximating and Computing Area

5.2 The Definite Integral

5.3 The Fundamental Theorem of Calculus, Part I

5.4 The Fundamental Theorem of Calculus, Part II

5.5 Net Change as the Integral of a Rate

5.6 Substitution Method

5.7 Further Transcendental Functions

5.8 Exponential Growth and Decay

**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

6.4 The Method of Cylindrical Shells

6.5 Work and Energy

**Chapter 7: Techniques of Integration**

7.1 Integration by Parts

7.2 Trigonometric Integrals

7.3 Trigonometric Substitution

7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions

7.5 The Method of Partial Fractions

7.6 Improper Integrals

7.7 Probability and Integration

7.8 Numerical Integration

**Chapter 8: Further Applications of the Integral and Taylor Polynomials**

8.1 Arc Length and Surface Area

8.2 Fluid Pressure and Force

8.3 Center of Mass

8.4 Taylor Polynomials

**Chapter 9: Introduction to Differential Equations**

9.1 Solving Differential Equations

9.2 Models Involving *y’ = k (y-b)*

9.3 Graphical and Numerical Methods

9.4 The Logistic Equation

9.5 First-Order Linear Equations

**Chapter 10: Infinite Series**

10.1 Sequences

10.2 Summing an Infinite Series

10.3 Convergence of Series with Positive Terms

10.4 Absolute and Conditional Convergence

10.5 The Ratio and Root Tests

10.6 Power Series

10.7 Taylor Series

**Chapter 11: Parametric Equations, Polar Coordinates, and Conic Sections**

11.1 Parametric Equations

11.2 Arc Length and Speed

11.3 Polar Coordinates

11.4 Area and Arc Length in Polar Coordinates

11.5 Conic Sections

**Chapter 12: Vector Geometry**

12.1 Vectors in the Plane

12.2 Vectors in Three Dimensions

12.3 Dot Product and the Angle Between Two Vectors

12.4 The Cross Product

12.5 Planes in Three-Space

12.6 A Survey of Quadric Surfaces

12.7 Cylindrical and Spherical Coordinates

**Chapter 13: Calculus of Vector-Valued Functions**

13.1 Vector-Valued Functions

13.2 Calculus of Vector-Valued Functions

13.3 Arc Length and Speed

13.4 Curvature

13.5 Motion in Three-Space

13.6 Planetary Motion According to Kepler and Newton

**Chapter 14: Differentiation in Several Variables**

14.1 Functions of Two or More Variables

14.2 Limits and Continuity in Several Variables

14.3 Partial Derivatives

14.4 Differentiability and Tangent Planes

14.5 The Gradient and Directional Derivatives

14.6 The Chain Rule

14.7 Optimization in Several Variables

14.8 Lagrange Multipliers: Optimizing with a Constraint

**Chapter 15: Multiple Integration**

15.1 Integration in Variables

15.2 Double Integrals over More General Regions

15.3 Triple Integrals

15.4 Integration in Polar, Cylindrical, and Spherical Coordinates

15.5 Applications of Multiplying Integrals

15.6 Change of Variables

**Chapter 16: Line and Surface Integrals**

16.1 Vector Fields

16.2 Line Integrals

16.3 Conservative Vector Fields

16.4 Parametrized Surfaces and Surface Integrals

16.5 Surface Integrals of Vector Fields

**Chapter 17: Fundamental Theorems of Vector Analysis**

17.1 Green’s Theorem

17.2 Stokes’ Theorem

17.3 Divergence Theorem

**Appendices**

A. The Language of Mathematics

B. Properties of Real Numbers

C. Mathematical Induction and the Binomial Theorem

D. Additional Proofs of Theorems

E. Taylor Polynomials

**Answers to Odd-Numbered Exercises**

References

Photo Credits

Index