Thinkwell's Calculus with Professor Edward Burger

Calculus is advanced math for the high school student, but it's the starting point for math in the most selective colleges and universities. Thinkwell's Calculus course covers both Calculus I and Calculus II, each of which is a one-semester course in college. If you plan to take the AP® Calculus AB or AP® Calculus BC exam, you should consider our Calculus compatible with AP® courses*, which have assessments targeted to the AP® exam.

The Printed Notes (optional) are the Calculus course notes from the online subscription printed in a color, on-the-go format.

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280 engaging 5-20 minute video lessons
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Detailed, 38-week lesson plan and schedule
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What Parents Are Saying. . .
"My son used Thinkwell intermediate algebra, college algebra, trigonometry, and is now using Professor Burger's Calculus course. I can't say enough about how much he learns from this method of learning, and how much he likes it! He likes it so much that his younger sibling asked me to buy 8th grade math for her. Prof Burger makes total sense and explains very clearly. He's funny, too. I know some people, those of a serious bent, can be taken aback by his style (silly at times, or just nerdy-silly), but he grows on you until you enjoy him so much, and his humor is part of what makes you learn so well. After all, a happy, relaxed brain is going to receive information better than a stressed brain! "
– MG
"Thinkwell gave clear, concise explanations that helped me go into class the next day already knowing the concepts, or clarify any concepts that I did not fully understand when the professor taught them. The video format with illustrations didn't feel like homework. It also helped me to visualize and understand why I was doing what I was doing on my homework. It would be beneficial to any calculus student.”
– Julie G
My kids are thoroughly enjoying Thinkwell math. We are currently using the pre-algebra and calculus levels. The math is solid and well-taught, and the sprinkling of humor throughout makes the lessons enjoyable and memorable. Thank you!”
– Dawn
Course Overview

What you get

  • 12-month, online subscription to our complete Calculus course
  • 38-week, day-by-day course lesson plan
  • 275+ course lessons, each with a streaming video
  • Illustrated notes
  • Automatically graded drill-and-practice exercises with step-by-step answer feedback
  • Sample problems and solutions
  • Chapter & Practice tests, a Midterm & Final Exam
  • Animated interactivities....and more!

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  • Purchase Thinkwell's Calculus through our online store
  • Create an account username and password which will give you access to the online Calculus course section
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  • Your work is automatically tracked and updated in real-time
  • Transcripts, grade reports, and certificates of completion are available at request
About Thinkwell's Calculus Author, Edward Burger

Learn from award-winning mathematician Dr. Edward Burger

It's like having a world-class college professor right by your side teaching you Calculus.

  • "Global Hero in Education" by Microsoft Corporation
  • "America's Best Math Teacher" by Reader's Digest
  • Robert Foster Cherry Award Winner for Great Teaching
Thinkwell's Calculus Table of Contents
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1. The Basics

1.1 Overview
1.1.1 An Introduction to Thinkwell 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
5.2.3 The Music of Math
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

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

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 Related Rates
7.3.1 The Pebble Problem
7.3.2 The Ladder Problem
7.3.3 The Baseball Problem
7.3.4 The Blimp Problem
7.3.5 Math Anxiety
7.4 Optimization
7.4.1 The Connection Between Slope and Optimization
7.4.2 The Fence Problem
7.4.3 The Box Problem
7.4.4 The Can Problem
7.4.5 The Wire-Cutting Problem

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

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

11. Calculus I Review

11.1 The Close of Calculus I
11.1.1 A Glimpse Into Calculus II

12. Math Fun

12.1 Paradoxes
12.1.1 An Introduction to Paradoxes
12.1.2 Paradoxes and Air Safety
12.1.3 Newcomb's Paradox
12.1.4 Zeno's Paradox
12.2 Sequences
12.2.1 Fibonacci Numbers
12.2.2 The Golden Ratio

13. An Introduction to Calculus II

13.1 Introduction
13.1.1 Welcome to Calculus II
13.1.2 Review: Calculus I in 20 Minutes

14. L'Hôpital's Rule

14.1 Indeterminate Quotients
14.1.1 Indeterminate Forms
14.1.2 An Introduction to L'Hôpital's Rule
14.1.3 Basic Uses of L'Hôpital's Rule
14.1.4 More Exotic Examples of Indeterminate Forms
14.2 Other Indeterminate Forms
14.2.1 L'Hôpital's Rule and Indeterminate Products
14.2.2 L'Hôpital's Rule and Indeterminate Differences
14.2.3 L'Hôpital's Rule and One to the Infinite Power
14.2.4 Another Example of One to the Infinite Power

15. Elementary Functions and Their Inverses

15.1 Inverse Functions
15.1.1 The Exponential and Natural Log Functions
15.1.2 Differentiating Logarithmic Functions
15.1.3 Logarithmic Differentiation
15.1.4 The Basics of Inverse Functions
15.1.5 Finding the Inverse of a Function
15.2 The Calculus of Inverse Functions
15.2.1 Derivatives of Inverse Functions
15.3 Inverse Trigonometric Functions
15.3.1 The Inverse Sine, Cosine, and Tangent Functions
15.3.2 The Inverse Secant, Cosecant, and Cotangent Functions
15.3.3 Evaluating Inverse Trigonometric Functions
15.4 The Calculus of Inverse Trigonometric Functions
15.4.1 Derivatives of Inverse Trigonometric Functions
15.4.2 More Calculus of Inverse Trigonometric Functions
15.5 The Hyperbolic Functions
15.5.1 Defining the Hyperbolic Functions
15.5.2 Hyperbolic Identities
15.5.3 Derivatives of Hyperbolic Functions

16. Techniques of Integration

16.1 Integration Using Tables
16.1.1 An Introduction to the Integral Table
16.1.2 Making u-Substitutions
16.2 Integrals Involving Powers of Sine and Cosine
16.2.1 An Introduction to Integrals with Powers of Sine and Cosine
16.2.2 Integrals with Powers of Sine and Cosine
16.2.3 Integrals with Even and Odd Powers of Sine and Cosine
16.3 Integrals Involving Powers of Other Trigonometric Functions
16.3.1 Integrals of Other Trigonometric Functions
16.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant
16.3.3 Integrals with Even Powers of Secant and Any Power of Tangent
16.4 An Introduction to Integration by Partial Fractions
16.4.1 Finding Partial Fraction Decompositions
16.4.2 Partial Fractions
16.4.3 Long Division
16.5 Integration by Partial Fractions with Repeated Factors
16.5.1 Repeated Linear Factors: Part One
16.5.2 Repeated Linear Factors: Part Two
16.5.3 Distinct and Repeated Quadratic Factors
16.5.4 Partial Fractions of Transcendental Functions
16.6 Integration by Parts
16.6.1 An Introduction to Integration by Parts
16.6.2 Applying Integration by Parts to the Natural Log Function
16.6.3 Inspirational Examples of Integration by Parts
16.6.4 Repeated Application of Integration by Parts
16.6.5 Algebraic Manipulation and Integration by Parts
16.7 An Introduction to Trigonometric Substitution
16.7.1 Converting Radicals into Trigonometric Expressions
16.7.2 Using Trigonometric Substitution to Integrate Radicals
16.7.3 Trigonometric Substitutions on Rational Powers
16.8 Trigonometric Substitution Strategy
16.8.1 An Overview of Trigonometric Substitution Strategy
16.8.2 Trigonometric Substitution Involving a Definite Integral: Part One
16.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two
16.9 Numerical Integration
16.9.1 Deriving the Trapezoidal Rule
16.9.2 An Example of the Trapezoidal Rule

17. Improper Integrals

17.1 Improper Integrals
17.1.1 The First Type of Improper Integral
17.1.2 The Second Type of Improper Integral
17.1.3 Infinite Limits of Integration, Convergence, and Divergence

18. Applications of Integral Calculus

18.1 The Average Value of a Function
18.1.1 Finding the Average Value of a Function
18.2 Finding Volumes Using Cross-Sections
18.2.1 Finding Volumes Using Cross-Sectional Slices
18.2.2 An Example of Finding Cross-Sectional Volumes
18.3 Disks and Washers
18.3.1 Solids of Revolution
18.3.2 The Disk Method along the y-Axis
18.3.3 A Transcendental Example of the Disk Method
18.3.4 The Washer Method across the x-Axis
18.3.5 The Washer Method across the y-Axis
18.4 Shells
18.4.1 Introducing the Shell Method
18.4.2 Why Shells Can Be Better Than Washers
18.4.3 The Shell Method: Integrating with Respect to y
18.5 Arc Lengths and Functions
18.5.1 An Introduction to Arc Length
18.5.2 Finding Arc Lengths of Curves Given by Functions
18.6 Work
18.6.1 An Introduction to Work
18.6.2 Calculating Work
18.6.3 Hooke's Law
18.7 Moments and Centers of Mass
18.7.1 Center of Mass
18.7.2 The Center of Mass of a Thin Plate

19. Sequences and Series

19.1 Sequences
19.1.1 The Limit of a Sequence
19.1.2 Determining the Limit of a Sequence
19.1.3 The Squeeze and Absolute Value Theorems
19.2 Monotonic and Bounded Sequences
19.2.1 Monotonic and Bounded Sequences
19.3 Infinite Series
19.3.1 An Introduction to Infinite Series
19.3.2 The Summation of Infinite Series
19.3.3 Geometric Series
19.3.4 Telescoping Series
19.4 Convergence and Divergence
19.4.1 Properties of Convergent Series
19.4.2 The nth-Term Test for Divergence
19.5 The Integral Test
19.5.1 An Introduction to the Integral Test
19.5.2 Examples of the Integral Test
19.5.3 Using the Integral Test
19.5.4 Defining p-Series
19.6 The Direct Comparison Test
19.6.1 An Introduction to the Direct Comparison Test
19.6.2 Using the Direct Comparison Test
19.7 The Limit Comparison Test
19.7.1 An Introduction to the Limit Comparison Test
19.7.2 Using the Limit Comparison Test
19.7.3 Inverting the Series in the Limit Comparison Test
19.8 The Alternating Series
19.8.1 Alternating Series
19.8.2 The Alternating Series Test
19.8.3 Estimating the Sum of an Alternating Series
19.9 Absolute and Conditional Convergences
19.9.1 Absolute and Conditional Convergence
19.10 The Ratio and Root Tests
19.10.1 The Ratio Test
19.10.2 Examples of the Ratio Test
19.10.3 The Root Test
19.11 Polynomial Approximations of Elementary Functions
19.11.1 Polynomial Approximation of Elementary Functions
19.11.2 Higher-Degree Approximations
19.12 Taylor and Maclaurin Polynomials
19.12.1 Taylor Polynomials
19.12.2 Maclaurin Polynomials
19.12.3 The Remainder of a Taylor Polynomial
19.12.4 Approximating the Value of a Function
19.13 Taylor and Maclaurin Series
19.13.1 Taylor Series
19.13.2 Examples of the Taylor and Maclaurin Series
19.13.3 New Taylor Series
19.13.4 The Convergence of Taylor Series
19.14 Power Series
19.14.1 The Definition of Power Series
19.14.2 The Interval and Radius of Convergence
19.14.3 Finding the Interval and Radius of Convergence: Part One
19.14.4 Finding the Interval and Radius of Convergence: Part Two
19.14.5 Finding the Interval and Radius of Convergence: Part Three
19.15 Power Series Representations of Functions
19.15.1 Differentiation and Integration of Power Series
19.15.2 Finding Power Series Representations by Differentiation
19.15.3 Finding Power Series Representations by Integration
19.15.4 Integrating Functions Using Power Series

20. Differential Equations

20.1 Separable Differential Equations
20.1.1 An Introduction to Differential Equations
20.1.2 Solving Separable Differential Equations
20.1.3 Finding a Particular Solution
20.1.4 Direction Fields
20.2 Solving a Homogeneous Differential Equation
20.2.1 Separating Homogeneous Differential Equations
20.2.2 Change of Variables
20.3 Growth and Decay Problems
20.3.1 Exponential Growth
20.3.2 Radioactive Decay
20.4 Solving First-Order Linear Differential Equations
20.4.1 First-Order Linear Differential Equations
20.4.2 Using Integrating Factors

21. Parametric Equations and Polar Coordinates

21.1 Understanding Parametric Equations
21.1.1 An Introduction to Parametric Equations
21.1.2 The Cycloid
21.1.3 Eliminating Parameters
21.2 Calculus and Parametric Equations
21.2.1 Derivatives of Parametric Equations
21.2.2 Graphing the Elliptic Curve
21.2.3 The Arc Length of a Parameterized Curve
21.2.4 Finding Arc Lengths of Curves Given by Parametric Equations
21.3 Understanding Polar Coordinates
21.3.1 The Polar Coordinate System
21.3.2 Converting between Polar and Cartesian Forms
21.3.3 Spirals and Circles
21.3.4 Graphing Some Special Polar Functions
21.4 Polar Functions and Slope
21.4.1 Calculus and the Rose Curve
21.4.2 Finding the Slopes of Tangent Lines in Polar Form
21.5 Polar Functions and Area
21.5.1 Heading toward the Area of a Polar Region
21.5.2 Finding the Area of a Polar Region: Part One
21.5.3 Finding the Area of a Polar Region: Part Two
21.5.4 The Area of a Region Bounded by Two Polar Curves: Part One
21.5.5 The Area of a Region Bounded by Two Polar Curves: Part Two

22. Vector Calculus and the Geometry of R2 and R3

22.1 Vectors and the Geometry of R2 and R3
22.1.1 Coordinate Geometry in Three-Dimensional Space
22.1.2 Introduction to Vectors
22.1.3 Vectors in R2 and R3
22.1.4 An Introduction to the Dot Product
22.1.5 Orthogonal Projections
22.1.6 An Introduction to the Cross Product
22.1.7 Geometry of the Cross Product
22.1.8 Equations of Lines and Planes in R3
22.2 Vector Functions
22.2.1 Introduction to Vector Functions
22.2.2 Derivatives of Vector Functions
22.2.3 Vector Functions: Smooth Curves
22.2.4 Vector Functions: Velocity and Acceleration
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A typical sequence of secondary math courses completed by a college-bound student is: Grade 6 Math > Grade 7 Math > Grade 8 Math > Algebra 1 > Geometry > Algebra 2 > Precalculus. For students looking to include Calculus as part of their high school curriculum and are able to complete Grade 7 Math in 6th grade, the sequence can be: Grade 7 Math > Prealgebra > Algebra 1 > Geometry > Algebra 2 > Precalculus > Calculus.

Is Calculus a college course?

Yes, but it is also taught in high school. Thinkwell Calculus is a two-semester course of Calculus 1 and Calculus 2.

Is Thinkwell Calculus appropriate for my accelerated student?

Yes, there is no “honors” course for Calculus. If you are interested preparing for the Calculus AP exams®, you’ll want to investigate Thinkwell’s Calculus AB compatible with AP® and Thinkwell’s Calculus BC compatible with AP®courses.

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