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Calculus Details
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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
 1.2 Precalculus Review

1.2.4 Some NonEuclidean 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

2.1.7 The Squeeze Theorem

2.1.8 Continuity and Discontinuity
 2.2 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
 3.2 Using the Derivative

3.2.1 The Slope of a Tangent Line

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
 4.3 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
 5.2 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
 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 HigherOrder Derivatives and Linear Approximation

7.2.2 Using the Tangent Line Approximation Formula
 7.3 Related Rates

7.3.3 The Baseball Problem
 7.4 Optimization

7.4.1 The Connection Between Slope and Optimization

7.4.5 The WireCutting Problem
8. Curve Sketching
 8.1 Introduction

8.1.1 An Introduction to Curve Sketching
 8.2 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
 8.3 Concavity

8.3.1 Concavity and Inflection Points

8.3.2 Using the Second Derivative to Examine Concavity
 8.4 Graphing Using the Derivative

8.4.1 Graphs of Polynomial Functions

8.4.2 Cusp Points and the Derivative

8.4.3 DomainRestricted 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
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 uSubstitutions
 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
 16.5 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 CrossSections

18.2.1 Finding Volumes Using CrossSectional Slices

18.2.2 An Example of Finding CrossSectional Volumes
 18.3 Disks and Washers

18.3.1 Solids of Revolution

18.3.2 The Disk Method along the yAxis

18.3.3 A Transcendental Example of the Disk Method

18.3.4 The Washer Method across the xAxis

18.3.5 The Washer Method across the yAxis
 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
 18.7 Moments and Centers 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

19.3.4 Telescoping Series
 19.4 Convergence and Divergence

19.4.1 Properties of Convergent Series

19.4.2 The nthTerm 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
 19.6 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

19.10.2 Examples of the Ratio Test
 19.11 Polynomial Approximations of Elementary Functions

19.11.1 Polynomial Approximation of Elementary Functions

19.11.2 HigherDegree 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

19.13.2 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
 20.2 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
 20.4 Solving FirstOrder Linear Differential Equations

20.4.1 FirstOrder 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

21.1.3 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 R^{2} and R^{3}
 22.1 Vectors and the Geometry of R^{2} and R^{3}

22.1.1 Coordinate Geometry in ThreeDimensional Space

22.1.2 Introduction to Vectors

22.1.3 Vectors in R^{2} and R^{3}

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 R^{3}
 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
About the Author
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 UTAustin 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, padic 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.