## Table of Contents

**1. Functions **

1.1 Review of functions

1.2 Representing functions

1.3 Trigonometric functions

1.4 Trigonometric functions

**2. Limits**

2.1 The idea of limits

2.2 Definitions of limits

2.3 Techniques for computing limits

2.4 Infinite limits

2.5 Limits at infinity

2.6 Continuity

2.7 Precise definitions of limits

**3. Derivatives**

3.1 Introducing the derivative

3.2 Working with derivatives

3.3 Rules of differentiation

3.4 The product and quotient rules

3.5 Derivatives of trigonometric functions

3.6 Derivatives as rates of change

3.7 The Chain Rule

3.8 Implicit differentiation

3.9 Related rates

**4. Applications of the Derivative**

4.1 Maxima and minima

4.2 What derivatives tell us

4.3 Graphing functions

4.4 Optimization problems

4.5 Linear approximation and differentials

4.6 Mean Value Theorem

4.7 L’Hôpital’s Rule

4.8 Newton’s Method

4.9 Antiderivatives

**5. Integration**

5.1 Approximating areas under curves

5.2 Definite integrals

5.3 Fundamental Theorem of Calculus

5.4 Working with integrals

5.5 Substitution rule

**6. Applications of Integration**

6.1 Velocity and net change

6.2 Regions between curves

6.3 Volume by slicing

6.4 Volume by shells

6.5 Length of curves

6.6 Surface area

6.7 Physical applications

**7. Logarithmic and Exponential Functions**

7.1 Inverse functions

7.2 The natural logarithmic and exponential functions

7.3 Logarithmic and exponential functions with other bases

7.4 Exponential models

7.5 Inverse trigonometric functions

7.6 L’ Hôpital’s Rule and growth rates of functions

7.7 Hyperbolic functions

**8. Integration Techniques**

8.1 Basic approaches

8.2 Integration by parts

8.3 Trigonometric integrals

8.4 Trigonometric substitutions

8.5 Partial fractions

8.6 Other integration strategies

8.7 Numerical integration

8.8 Improper integrals

8.9 Introduction to differential equations

**9. Sequences and Infinite Series**

9.1 An overview

9.2 Sequences

9.3 Infinite series

9.4 The Divergence and Integral Tests

9.5 The Ratio, Root, and Comparison Tests

9.6 Alternating series

**10. Power Series**

10.1 Approximating functions with polynomials

10.2 Properties of Power series

10.3 Taylor series

10.4 Working with Taylor series

**11. Parametric and Polar Curves **

11.1 Parametric equations

11.2 Polar coordinates

11.3 Calculus in polar coordinates

11.4 Conic sections

**12. Vectors and Vector-Valued Functions**

12.1 Vectors in the plane

12.2 Vectors in three dimensions

12.3 Dot products

12.4 Cross products

12.5 Lines and curves in space

12.6 Calculus of vector-valued functions

12.7 Motion in space

12.8 Length of curves

12.9 Curvature and normal vectors

**13. Functions of Several Variables**

13.1 Planes and surfaces

13.2 Graphs and level curves

13.3 Limits and continuity

13.4 Partial derivatives

13.5 The Chain Rule

13.6 Directional derivatives and the gradient

13.7 Tangent planes and linear approximation

13.8 Maximum/minimum problems

13.9 Lagrange multipliers

**14. Multiple Integration**

14.1 Double integrals over rectangular regions

14.2 Double integrals over general regions

14.3 Double integrals in polar coordinates

14.4 Triple integrals

14.5 Triple integrals in cylindrical and spherical coordinates

14.6 Integrals for mass calculations

14.7 Change of variables in multiple integrals

**15. Vector Calculus**

15.1 Vector fields

15.2 Line integrals

15.3 Conservative vector fields

15.4 Green’s theorem

15.5 Divergence and curl

15.6 Surface integrals

15.6 Stokes’ theorem

15.8 Divergence theorem

Appendix A. Algebra Review

Appendix B. Proofs of Selected Theorems

**D1 Differential Equations (online)**

D1.1 Basic Ideas

D1.2 Direction Fields and Euler’s Method

D1.3 Separable Differential Equations

D1.4 Special First-Order Differential Equations

D1.5 Modeling with Differential Equations

**D2 Second-Order Differential Equations (online)**

D2.1 Basic Ideas

D2.2 Linear Homogeneous Equations

D2.3 Linear Nonhomogeneous Equations

D2.4 Applications

D2.5 Complex Forcing Functions