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Calculus AB compatible with AP*

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Table of Contents

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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

About the Author

Author BUR

Edward Burger
Williams College

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest listed him in the "100 Best of America". After completing his tenure as Gaudino Scholar at Williams, he was named Lissack Professor for Social Responsibility and Personal Ethics.

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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