Calculus

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
  • 5.2.3 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