Preface

Guide to AP Calculus

Guide to Mathematical Practices for AP Calculus (MPACs)

AP Calculus Pacing Guide

1. Functions

1.1 Review of Functions

1.2 Representing Functions

1.3 Inverse, Exponential, and Logarithmic Functions

1.4 Trigonometric Functions and Their Inverses

Review Exercises

AP Practice Questions

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

Review Exercises

AP Practice Questions

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 Derivatives of Logarithmic and Exponential Functions

3.10 Derivatives of Inverse Trigonometric Functions

3.11 Related Rates

Review Exercises

AP Practice Questions

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

Review Exercises

AP Practice Questions

5. Integration

5.1 Antiderivatives

5.2 Approximating Areas under Curves

5.3 Definite Integrals

5.4 Fundamental Theorem of Calculus

5.5 Properties of Integrals and Average Value

5.6 Substitution Rule

5.7 Numerical Integration

Review Exercises

AP Practice Questions

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 Physical Applications

Review Exercises

AP Practice Questions

7. Integration Techniques

7.1 Basic Approaches

7.2 Integration by Parts

7.3 Partial Fractions

7.4 Improper Integrals

7.5 Trigonometric Substitutions

Review Exercises

AP Practice Questions

8. Differential Equations

8.1 Basic Ideas

8.2 Slope Fields and Euler’s Method

8.3 Separable Differential Equations

8.4 Exponential Models

Review Exercises

AP Practice Questions

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

Review Exercises

AP Practice Questions

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

Review Exercises

AP Practice Questions

11. Parametric, Polar, and Vector Curves

11.1 Parametric Equations

11.2 Calculus with Parametric Equations

11.3 Polar Coordinates

11.4 Calculus in Polar Coordinates

11.5 Vectors in the Plane

11.6 Calculus of Vector-Valued Functions

11.7 Two-Dimensional Motion

Review Exercises

AP Practice Questions

Appendix A. Algebra Review

Appendix B. Conic Sections

Appendix C. Proofs of Selected Theorems