# AP Calculus AB

**Our complete AP Calculus AB package includes:**

**12-month Online Subscription**to our complete AP Calculus AB course with video lessons, day-by-day lesson plans, automatically graded exercises, and much more.**Printed Notes**(optional, AB + BC) are the AP Calculus course notes from the Online Subscription printed in a black & white, on-the-go format. These are available for purchase from the AP Calculus AB Course Site.

### AP Calculus AB details

Learn from the best! Award-winning professor Edward Burger teaches the fundamentals of Calculus AB, including calculus limits, in dynamic video tutorials. It's the best way to learn the calculus concepts needed to score a perfect 5 on the AP Calculus AB exam.

This isn't test prep that just gives exam-taking tips; it's a full course geared to AP-caliber work. Our video calculus tutorials have AP Calculus instruction that fits how students learn so they'll remember it all for the AP Calculus AB exam.

- Equivalent to 11th- or 12th-grade AP Calculus AB
- More than 180 video lessons
- 144 available contact hours (What is this?)
The number of contact hours in a course reflects the amount of time a student will typically spend completing the assignments in each course (i.e. watching videos, doing exercises, taking exams, etc...). Many people think about contact hours as the "seat time" for a course. Thinkwell provides this information so you can ensure that the amount of instruction in a Thinkwell course meets the standards and requirements for your state or region.
- 1000+ interactive AP Calculus AB problems with immediate feedback allow you to track your progress (See sample)
- Calculus AB practice chapter tests for all 12 chapters, as well as a final exam to make sure you're ready for the AP Calculus AB exam.
- Printable illustrated notes for each topic
- Real-world application examples in both lectures and exercises
- Closed captioning for all videos
- Glossary of more than 200 mathematical terms
- Engaging content to help students advance their mathematical knowledge:
- Understanding and evaluating limits and derivatives
- Computational techniques such as the power rule, product rule, quotient rule, and chain rule
- Trigonometric, exponential, and logarithmic functions
- Implicit differentiation
- Differentiation, optimization, and related rates
- Sketching curves
- Antiderivatives, integration, and the fundamental theorem of calculus
- Applications of integration such as motion, finding the area between two curves, and integrating with respect to y

### About the Author

### Table of Contents

(Expand All - Close All)#### 1. The Basics

- 1.1 Overview
- 1.1.1 An Introduction to Thinkwell's Calculus

- 1.1.2 The Two Questions of Calculus

- 1.1.3 Average Rates of Change

- 1.1.4 How to Do Math

- 1.2 Precalculus Review
- 1.2.1 Functions

- 1.2.2 Graphing Lines

- 1.2.3 Parabolas

- 1.2.4 Some Non-Euclidean Geometry

#### 2. Limits

- 2.1 The Concept of the Limit
- 2.1.1 Finding Rate of Change over an Interval

- 2.1.2 Finding Limits Graphically

- 2.1.3 The Formal Definition of a Limit

- 2.1.4 The Limit Laws, Part I

- 2.1.5 The Limit Laws, Part II

- 2.1.6 One-Sided Limits

- 2.1.7 The Squeeze Theorem

- 2.1.8 Continuity and Discontinuity

- 2.2 Evaluating Limits
- 2.2.1 Evaluating Limits

- 2.2.2 Limits and Indeterminate Forms

- 2.2.3 Two Techniques for Evaluating Limits

- 2.2.4 An Overview of Limits

#### 3. An Introduction to Derivatives

- 3.1 Understanding the Derivative
- 3.1.1 Rates of Change, Secants, and Tangents

- 3.1.2 Finding Instantaneous Velocity

- 3.1.3 The Derivative

- 3.1.4 Differentiability

- 3.2 Using the Derivative
- 3.2.1 The Slope of a Tangent Line

- 3.2.2 Instantaneous Rate

- 3.2.3 The Equation of a Tangent Line

- 3.2.4 More on Instantaneous Rate

- 3.3 Some Special Derivatives
- 3.3.1 The Derivative of the Reciprocal Function

- 3.3.2 The Derivative of the Square Root Function

#### 4. Computational Techniques

- 4.1 The Power Rule
- 4.1.1 A Shortcut for Finding Derivatives

- 4.1.2 A Quick Proof of the Power Rule

- 4.1.3 Uses of the Power Rule

- 4.2 The Product and Quotient Rules
- 4.2.1 The Product Rule

- 4.2.2 The Quotient Rule

- 4.3 The Chain Rule
- 4.3.1 An Introduction to the Chain Rule

- 4.3.2 Using the Chain Rule

- 4.3.3 Combining Computational Techniques

#### 5. Special Functions

- 5.1 Trigonometric Functions
- 5.1.1 A Review of Trigonometry

- 5.1.2 Graphing Trigonometric Functions

- 5.1.3 The Derivatives of Trigonometric Functions

- 5.1.4 The Number Pi

- 5.2 Exponential Functions
- 5.2.1 Graphing Exponential Functions

- 5.2.2 Derivatives of Exponential Functions

- 5.3 Logarithmic Functions
- 5.3.1 Evaluating Logarithmic Functions

- 5.3.2 The Derivative of the Natural Log Function

- 5.3.3 Using the Derivative Rules with Transcendental Functions

#### 6. Implicit Differentiation and the Inverse Function

- 6.1 Implicit Differentiation Basics
- 6.1.1 An Introduction to Implicit Differentiation

- 6.1.2 Finding the Derivative Implicitly

- 6.2 Applying Implicit Differentiation
- 6.2.1 Using Implicit Differentiation

- 6.2.2 Applying Implicit Differentiation

- 6.3 Inverse Functions
- 6.3.1 The Exponential and Natural Log Functions

- 6.3.2 Differentiating Logarithmic Functions

- 6.3.3 Logarithmic Differentiation

- 6.3.4 The Basics of Inverse Functions

- 6.3.5 Finding the Inverse of a Function

- 6.4 The Calculus of Inverse Functions
- 6.4.1 Derivatives of Inverse Functions

- 6.5 Inverse Trigonometric Functions
- 6.5.1 The Inverse Sine, Cosine, and Tangent Functions

- 6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions

- 6.5.3 Evaluating Inverse Trigonometric Functions

- 6.6 The Calculus of Inverse Trigonometric Functions
- 6.6.1 Derivatives of Inverse Trigonometric Functions

- 6.7 The Hyperbolic Functions
- 6.7.1 Defining the Hyperbolic Functions

- 6.7.2 Hyperbolic Identities

- 6.7.3 Derivatives of Hyperbolic Functions

#### 7. Applications of Differentiation

- 7.1 Position and Velocity
- 7.1.1 Acceleration and the Derivative

- 7.1.2 Solving Word Problems Involving Distance and Velocity

- 7.2 Linear Approximation
- 7.2.1 Higher-Order Derivatives and Linear Approximation

- 7.2.2 Using the Tangent Line Approximation Formula

- 7.2.3 Newton's Method

- 7.3 Optimization
- 7.3.1 The Connection Between Slope and Optimization

- 7.3.2 The Fence Problem

- 7.3.3 The Box Problem

- 7.3.4 The Can Problem

- 7.3.5 The Wire-Cutting Problem

- 7.4 Related Rates
- 7.4.1 The Pebble Problem

- 7.4.2 The Ladder Problem

- 7.4.3 The Baseball Problem

- 7.4.4 The Blimp Problem

- 7.4.5 Math Anxiety

#### 8. Curve Sketching

- 8.1 Introduction
- 8.1.1 An Introduction to Curve Sketching

- 8.1.2 Three Big Theorems

- 8.1.3 Morale Moment

- 8.2 Critical Points
- 8.2.1 Critical Points

- 8.2.2 Maximum and Minimum

- 8.2.3 Regions Where a Function Increases or Decreases

- 8.2.4 The First Derivative Test

- 8.2.5 Math Magic

- 8.3 Concavity
- 8.3.1 Concavity and Inflection Points

- 8.3.2 Using the Second Derivative to Examine Concavity

- 8.3.3 The Möbius Band

- 8.4 Graphing Using the Derivative
- 8.4.1 Graphs of Polynomial Functions

- 8.4.2 Cusp Points and the Derivative

- 8.4.3 Domain-Restricted Functions and the Derivative

- 8.4.4 The Second Derivative Test

- 8.5 Asymptotes
- 8.5.1 Vertical Asymptotes

- 8.5.2 Horizontal Asymptotes and Infinite Limits

- 8.5.3 Graphing Functions with Asymptotes

- 8.5.4 Functions with Asymptotes and Holes

- 8.5.5 Functions with Asymptotes and Critical Points

#### 9. The Basics of Integration

- 9.1 Antiderivatives
- 9.1.1 Antidifferentiation

- 9.1.2 Antiderivatives of Powers of x

- 9.1.3 Antiderivatives of Trigonometric and Exponential Functions

- 9.2 Integration by Substitution
- 9.2.1 Undoing the Chain Rule

- 9.2.2 Integrating Polynomials by Substitution

- 9.3 Illustrating Integration by Substitution
- 9.3.1 Integrating Composite Trigonometric Functions by Substitution

- 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution

- 9.3.3 More Integrating Trigonometric Functions by Substitution

- 9.3.4 Choosing Effective Function Decompositions

- 9.4 The Fundamental Theorem of Calculus
- 9.4.1 Approximating Areas of Plane Regions

- 9.4.2 Areas, Riemann Sums, and Definite Integrals

- 9.4.3 The Fundamental Theorem of Calculus, Part I

- 9.4.4 The Fundamental Theorem of Calculus, Part II

- 9.4.5 Illustrating the Fundamental Theorem of Calculus

- 9.4.6 Evaluating Definite Integrals

- 9.5 Trigonometric Substitution Strategy
- 9.5.1 An Overview of Trigonometric Substitution Strategy

- 9.5.2 Trigonometric Substitution Involving a Definite Integral: Part One

- 9.5.3 Trigonometric Substitution Involving a Definite Integral: Part Two

- 9.6 Numerical Integration
- 9.6.1 Deriving the Trapezoidal Rule

- 9.6.2 An Example of the Trapezoidal Rule

#### 10. Applications of Integration

- 10.1 Motion
- 10.1.1 Antiderivatives and Motion

- 10.1.2 Gravity and Vertical Motion

- 10.1.3 Solving Vertical Motion Problems

- 10.2 Finding the Area between Two Curves
- 10.2.1 The Area between Two Curves

- 10.2.2 Limits of Integration and Area

- 10.2.3 Common Mistakes to Avoid When Finding Areas

- 10.2.4 Regions Bound by Several Curves

- 10.3 Integrating with Respect to y
- 10.3.1 Finding Areas by Integrating with Respect to y: Part One

- 10.3.2 Finding Areas by Integrating with Respect to y: Part Two

- 10.3.3 Area, Integration by Substitution, and Trigonometry

- 10.4 The Average Value of a Function
- 10.4.1 Finding the Average Value of a Function

- 10.5 Finding Volumes Using Cross-Sections
- 10.5.1 Finding Volumes Using Cross-Sectional Slices

- 10.5.2 An Example of Finding Cross-Sectional Volumes

- 10.6 Disks and Washers
- 10.6.1 Solids of Revolution

- 10.6.2 The Disk Method along the y-Axis

- 10.6.3 A Transcendental Example of the Disk Method

- 10.6.4 The Washer Method across the x-Axis

- 10.6.5 The Washer Method across the y-Axis

- 10.7 Shells
- 10.7.1 Introducing the Shell Method

- 10.7.2 Why Shells Can Be Better Than Washers

- 10.7.3 The Shell Method: Integrating with Respect to y

- 10.8 Work
- 10.8.1 An Introduction to Work

- 10.8.2 Calculating Work

- 10.8.3 Hooke's Law

- 10.9 Moments and Centers of Mass
- 10.9.1 Center of Mass

- 10.9.2 The Center of Mass of a Thin Plate

- 10.10 Arc Lengths and Functions
- 10.10.1 An Introduction to Arc Length

- 10.10.2 Finding Arc Lengths of Curves Given by Functions

#### 11. Differential Equations

- 11.1 Separable Differential Equations
- 11.1.1 An Introduction to Differential Equations

- 11.1.2 Solving Separable Differential Equations

- 11.1.3 Finding a Particular Solution

- 11.1.4 Direction Fields

- 11.1.5 Euler's Method for Solving Differential Equations Numerically

- 11.2 Growth and Decay Problems
- 11.2.1 Exponential Growth

- 11.2.2 Logistic Growth

- 11.2.3 Radioactive Decay

#### 12. L'Hôpital's Rule and Improper Integrals

- 12.1 Indeterminate Quotients
- 12.1.1 Indeterminate Forms

- 12.1.2 An Introduction to L'Hôpital's Rule

- 12.1.3 Basic Uses of L'Hôpital's Rule

- 12.1.4 More Exotic Examples of Indeterminate Forms

- 12.2 Other Indeterminate Forms
- 12.2.1 L'Hôpital's Rule and Indeterminate Products

- 12.2.2 L'Hôpital's Rule and Indeterminate Differences

- 12.2.3 L'Hôpital's Rule and One to the Infinite Power

- 12.2.4 Another Example of One to the Infinite Power

- 12.3 Improper Integrals
- 12.3.1 The First Type of Improper Integral

- 12.3.2 The Second Type of Improper Integral

- 12.3.3 Infinite Limits of Integration, Convergence, and Divergence

#### 13. Math Fun

- 13.1 Paradoxes
- 13.1.1 An Introduction to Paradoxes

- 13.1.2 Paradoxes and Air Safety

- 13.1.3 Newcomb's Paradox

- 13.1.4 Zeno's Paradox

- 13.2 Sequences
- 13.2.1 Fibonacci Numbers

- 13.2.2 The Golden Ratio

- 13.3 The Close of Calculus AB
- 13.3.1 A Glimpse Into Calculus II