Thinkwell.Calculus.Course

Thinkwell.Calculus.Course




- The Basics

* 1.1 Overview
o 1.1.1 An Introduction to Thinkwell's Calculus
o 1.1.2 The Two Questions of Calculus
o 1.1.3 Average Rates of Change
o 1.1.4 How to Do Math

* 1.2 Precalculus Review
o 1.2.1 Functions
o 1.2.2 Graphing Lines
o 1.2.3 Parabolas
o 1.2.4 Some Non-Euclidean Geometry

- Limits

* 2.1 The Concept of the Limit
o 2.1.1 Finding Rate of Change over an Interval
o 2.1.2 Finding Limits Graphically
o 2.1.3 The Formal Definition of a Limit
o 2.1.4 The Limit Laws, Part I
o 2.1.5 The Limit Laws, Part II
o 2.1.6 One-Sided Limits
o 2.1.7 The Squeeze Theorem
o 2.1.8 Continuity and Discontinuity

* 2.2 Evaluating Limits
o 2.2.1 Evaluating Limits
o 2.2.2 Limits and Indeterminate Forms
o 2.2.3 Two Techniques for Evaluating Limits
o 2.2.4 An Overview of Limits

- An Introduction to Derivatives

* 3.1 Understanding the Derivative
o 3.1.1 Rates of Change, Secants, and Tangents
o 3.1.2 Finding Instantaneous Velocity
o 3.1.3 The Derivative
o 3.1.4 Differentiability

* 3.2 Using the Derivative
o 3.2.1 The Slope of a Tangent Line
o 3.2.2 Instantaneous Rate
o 3.2.3 The Equation of a Tangent Line
o 3.2.4 More on Instantaneous Rate

* 3.3 Some Special Derivatives
o 3.3.1 The Derivative of the Reciprocal Function
o 3.3.2 The Derivative of the Square Root Function

- Computational Techniques

* 4.1 The Power Rule
o 4.1.1 A Shortcut for Finding Derivatives
o 4.1.2 A Quick Proof of the Power Rule
o 4.1.3 Uses of the Power Rule

* 4.2 The Product and Quotient Rules
o 4.2.1 The Product Rule
o 4.2.2 The Quotient Rule

* 4.3 The Chain Rule
o 4.3.1 An Introduction to the Chain Rule
o 4.3.2 Using the Chain Rule
o 4.3.3 Combining Computational Techniques

- Special Functions

* 5.1 Trigonometric Functions
o 5.1.1 A Review of Trigonometry
o 5.1.2 Graphing Trigonometric Functions
o 5.1.3 The Derivatives of Trigonometric Functions
o 5.1.4 The Number Pi

* 5.2 Exponential Functions
o 5.2.1 Graphing Exponential Functions
o 5.2.2 Derivatives of Exponential Functions
o 5.2.3 The Music of Math

* 5.3 Logarithmic Functions
o 5.3.1 Evaluating Logarithmic Functions
o 5.3.2 The Derivative of the Natural Log Function
o 5.3.3 Using the Derivative Rules with Transcendental Functions

- Implicit Differentiation

* 6.1 Implicit Differentiation Basics
o 6.1.1 An Introduction to Implicit Differentiation
o 6.1.2 Finding the Derivative Implicitly

* 6.2 Applying Implicit Differentiation
o 6.2.1 Using Implicit Differentiation
o 6.2.2 Applying Implicit Differentiation

- Applications of Differentiation

* 7.1 Position and Velocity
o 7.1.1 Acceleration and the Derivative
o 7.1.2 Solving Word Problems Involving Distance and Velocity

* 7.2 Linear Approximation
o 7.2.1 Higher-Order Derivatives and Linear Approximation
o 7.2.2 Using the Tangent Line Approximation Formula
o 7.2.3 Newton's Method

* 7.3 Related Rates
o 7.3.1 The Pebble Problem
o 7.3.2 The Ladder Problem
o 7.3.3 The Baseball Problem
o 7.3.4 The Blimp Problem
o 7.3.5 Math Anxiety

* 7.4 Optimization
o 7.4.1 The Connection Between Slope and Optimization
o 7.4.2 The Fence Problem
o 7.4.3 The Box Problem
o 7.4.4 The Can Problem
o 7.4.5 The Wire-Cutting Problem

- Curve Sketching

* 8.1 Introduction
o 8.1.1 An Introduction to Curve Sketching
o 8.1.2 Three Big Theorems
o 8.1.3 Morale Moment

* 8.2 Critical Points
o 8.2.1 Critical Points
o 8.2.2 Maximum and Minimum
o 8.2.3 Regions Where a Function Increases or Decreases
o 8.2.4 The First Derivative Test
o 8.2.5 Math Magic

* 8.3 Concavity
o 8.3.1 Concavity and Inflection Points
o 8.3.2 Using the Second Derivative to Examine Concavity
o 8.3.3 The M?bius Band

* 8.4 Graphing Using the Derivative
o 8.4.1 Graphs of Polynomial Functions
o 8.4.2 Cusp Points and the Derivative
o 8.4.3 Domain-Restricted Functions and the Derivative
o 8.4.4 The Second Derivative Test

* 8.5 Asymptotes
o 8.5.1 Vertical Asymptotes
o 8.5.2 Horizontal Asymptotes and Infinite Limits
o 8.5.3 Graphing Functions with Asymptotes
o 8.5.4 Functions with Asymptotes and Holes
o 8.5.5 Functions with Asymptotes and Critical Points

- The Basics of Integration

* 9.1 Antiderivatives
o 9.1.1 Antidifferentiation
o 9.1.2 Antiderivatives of Powers of x
o 9.1.3 Antiderivatives of Trigonometric and Exponential Functions

* 9.2 Integration by Substitution
o 9.2.1 Undoing the Chain Rule
o 9.2.2 Integrating Polynomials by Substitution

* 9.3 Illustrating Integration by Substitution
o 9.3.1 Integrating Composite Trigonometric Functions by Substitution
o 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution
o 9.3.3 More Integrating Trigonometric Functions by Substitution
o 9.3.4 Choosing Effective Function Decompositions

* 9.4 The Fundamental Theorem of Calculus
o 9.4.1 Approximating Areas of Plane Regions
o 9.4.2 Areas, Riemann Sums, and Definite Integrals
o 9.4.3 The Fundamental Theorem of Calculus, Part I
o 9.4.4 The Fundamental Theorem of Calculus, Part II
o 9.4.5 Illustrating the Fundamental Theorem of Calculus
o 9.4.6 Evaluating Definite Integrals

- Applications of Integration

* 10.1 Motion
o 10.1.1 Antiderivatives and Motion
o 10.1.2 Gravity and Vertical Motion
o 10.1.3 Solving Vertical Motion Problems

* 10.2 Finding the Area between Two Curves
o 10.2.1 The Area between Two Curves
o 10.2.2 Limits of Integration and Area
o 10.2.3 Common Mistakes to Avoid When Finding Areas
o 10.2.4 Regions Bound by Several Curves

* 10.3 Integrating with Respect to y
o 10.3.1 Finding Areas by Integrating with Respect to y: Part One
o 10.3.2 Finding Areas by Integrating with Respect to y: Part Two
o 10.3.3 Area, Integration by Substitution, and Trigonometry

- Calculus I Review

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

- Math Fun

* 12.1 Paradoxes
o 12.1.1 An Introduction to Paradoxes
o 12.1.2 Paradoxes and Air Safety
o 12.1.3 Newcomb's Paradox
o 12.1.4 Zeno's Paradox

* 12.2 Sequences
o 12.2.1 Fibonacci Numbers
o 12.2.2 The Golden Ratio

- An Introduction to Calculus II

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

- L'Hôpital's Rule

* 14.1 Indeterminate Quotients
o 14.1.1 Indeterminate Forms
o 14.1.2 An Introduction to L'Hôpital's Rule
o 14.1.3 Basic Uses of L'Hôpital's Rule
o 14.1.4 More Exotic Examples of Indeterminate Forms

* 14.2 Other Indeterminate Forms
o 14.2.1 L'Hôpital's Rule and Indeterminate Products
o 14.2.2 L'Hôpital's Rule and Indeterminate Differences
o 14.2.3 L'Hôpital's Rule and One to the Infinite Power
o 14.2.4 Another Example of One to the Infinite Power

- Elementary Functions and Their Inverses

* 15.1 Inverse Functions
o 15.1.1 The Exponential and Natural Log Functions
o 15.1.2 Differentiating Logarithmic Functions
o 15.1.3 Logarithmic Differentiation
o 15.1.4 The Basics of Inverse Functions
o 15.1.5 Finding the Inverse of a Function

* 15.2 The Calculus of Inverse Functions
o 15.2.1 Derivatives of Inverse Functions

* 15.3 Inverse Trigonometric Functions
o 15.3.1 The Inverse Sine, Cosine, and Tangent Functions
o 15.3.2 The Inverse Secant, Cosecant, and Cotangent Functions
o 15.3.3 Evaluating Inverse Trigonometric Functions

* 15.4 The Calculus of Inverse Trigonometric Functions
o 15.4.1 Derivatives of Inverse Trigonometric Functions
o 15.4.2 More Calculus of Inverse Trigonometric Functions

* 15.5 The Hyperbolic Functions
o 15.5.1 Defining the Hyperbolic Functions
o 15.5.2 Hyperbolic Identities
o 15.5.3 Derivatives of Hyperbolic Functions

- Techniques of Integration

* 16.1 Integration Using Tables
o 16.1.1 An Introduction to the Integral Table
o 16.1.2 Making u-Substitutions

* 16.2 Integrals Involving Powers of Sine and Cosine
o 16.2.1 An Introduction to Integrals with Powers of Sine and Cosine
o 16.2.2 Integrals with Powers of Sine and Cosine
o 16.2.3 Integrals with Even and Odd Powers of Sine and Cosine

* 16.3 Integrals Involving Powers of Other Trigonometric Functions
o 16.3.1 Integrals of Other Trigonometric Functions
o 16.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant
o 16.3.3 Integrals with Even Powers of Secant and Any Power of Tangent

* 16.4 An Introduction to Integration by Partial Fractions
o 16.4.1 Finding Partial Fraction Decompositions
o 16.4.2 Partial Fractions
o 16.4.3 Long Division

* 16.5 Integration by Partial Fractions with Repeated Factors
o 16.5.1 Repeated Linear Factors: Part One
o 16.5.2 Repeated Linear Factors: Part Two
o 16.5.3 Distinct and Repeated Quadratic Factors
o 16.5.4 Partial Fractions of Transcendental Functions

* 16.6 Integration by Parts
o 16.6.1 An Introduction to Integration by Parts
o 16.6.2 Applying Integration by Parts to the Natural Log Function
o 16.6.3 Inspirational Examples of Integration by Parts
o 16.6.4 Repeated Application of Integration by Parts
o 16.6.5 Algebraic Manipulation and Integration by Parts

* 16.7 An Introduction to Trigonometric Substitution
o 16.7.1 Converting Radicals into Trigonometric Expressions
o 16.7.2 Using Trigonometric Substitution to Integrate Radicals
o 16.7.3 Trigonometric Substitutions on Rational Powers

* 16.8 Trigonometric Substitution Strategy
o 16.8.1 An Overview of Trigonometric Substitution Strategy
o 16.8.2 Trigonometric Substitution Involving a Definite Integral: Part One
o 16.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two

* 16.9 Numerical Integration
o 16.9.1 Deriving the Trapezoidal Rule
o 16.9.2 An Example of the Trapezoidal Rule

- Improper Integrals

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

- Applications of Integral Calculus

* 18.1 The Average Value of a Function
o 18.1.1 Finding the Average Value of a Function

* 18.2 Finding Volumes Using Cross-Sections
o 18.2.1 Finding Volumes Using Cross-Sectional Slices
o 18.2.2 An Example of Finding Cross-Sectional Volumes

* 18.3 Disks and Washers
o 18.3.1 Solids of Revolution
o 18.3.2 The Disk Method along the y-Axis
o 18.3.3 A Transcendental Example of the Disk Method
o 18.3.4 The Washer Method across the x-Axis
o 18.3.5 The Washer Method across the y-Axis

* 18.4 Shells
o 18.4.1 Introducing the Shell Method
o 18.4.2 Why Shells Can Be Better Than Washers
o 18.4.3 The Shell Method: Integrating with Respect to y

* 18.5 Arc Lengths and Functions
o 18.5.1 An Introduction to Arc Length
o 18.5.2 Finding Arc Lengths of Curves Given by Functions

* 18.6 Work
o 18.6.1 An Introduction to Work
o 18.6.2 Calculating Work
o 18.6.3 Hooke's Law

* 18.7 Moments and Centers of Mass
o 18.7.1 Center of Mass
o 18.7.2 The Center of Mass of a Thin Plate

- Sequences and Series

* 19.1 Sequences
o 19.1.1 The Limit of a Sequence
o 19.1.2 Determining the Limit of a Sequence
o 19.1.3 The Squeeze and Absolute Value Theorems

* 19.2 Monotonic and Bounded Sequences
o 19.2.1 Monotonic and Bounded Sequences

* 19.3 Infinite Series
o 19.3.1 An Introduction to Infinite Series
o 19.3.2 The Summation of Infinite Series
o 19.3.3 Geometric Series
o 19.3.4 Telescoping Series

* 19.4 Convergence and Divergence
o 19.4.1 Properties of Convergent Series
o 19.4.2 The nth-Term Test for Divergence

* 19.5 The Integral Test
o 19.5.1 An Introduction to the Integral Test
o 19.5.2 Examples of the Integral Test
o 19.5.3 Using the Integral Test
o 19.5.4 Defining p-Series

* 19.6 The Direct Comparison Test
o 19.6.1 An Introduction to the Direct Comparison Test
o 19.6.2 Using the Direct Comparison Test

* 19.7 The Limit Comparison Test
o 19.7.1 An Introduction to the Limit Comparison Test
o 19.7.2 Using the Limit Comparison Test
o 19.7.3 Inverting the Series in the Limit Comparison Test

* 19.8 The Alternating Series
o 19.8.1 Alternating Series
o 19.8.2 The Alternating Series Test
o 19.8.3 Estimating the Sum of an Alternating Series

* 19.9 Absolute and Conditional Convergences
o 19.9.1 Absolute and Conditional Convergence

* 19.10 The Ratio and Root Tests
o 19.10.1 The Ratio Test
o 19.10.2 Examples of the Ratio Test
o 19.10.3 The Root Test

* 19.11 Polynomial Approximations of Elementary Functions
o 19.11.1 Polynomial Approximation of Elementary Functions
o 19.11.2 Higher-Degree Approximations

* 19.12 Taylor and Maclaurin Polynomials
o 19.12.1 Taylor Polynomials
o 19.12.2 Maclaurin Polynomials
o 19.12.3 The Remainder of a Taylor Polynomial
o 19.12.4 Approximating the Value of a Function

* 19.13 Taylor and Maclaurin Series
o 19.13.1 Taylor Series
o 19.13.2 Examples of the Taylor and Maclaurin Series
o 19.13.3 New Taylor Series
o 19.13.4 The Convergence of Taylor Series

* 19.14 Power Series
o 19.14.1 The Definition of Power Series
o 19.14.2 The Interval and Radius of Convergence
o 19.14.3 Finding the Interval and Radius of Convergence: Part One
o 19.14.4 Finding the Interval and Radius of Convergence: Part Two
o 19.14.5 Finding the Interval and Radius of Convergence: Part Three

* 19.15 Power Series Representations of Functions
o 19.15.1 Differentiation and Integration of Power Series
o 19.15.2 Finding Power Series Representations by Differentiation
o 19.15.3 Finding Power Series Representations by Integration
o 19.15.4 Integrating Functions Using Power Series

- Differential Equations

* 20.1 Separable Differential Equations
o 20.1.1 An Introduction to Differential Equations
o 20.1.2 Solving Separable Differential Equations
o 20.1.3 Finding a Particular Solution
o 20.1.4 Direction Fields

* 20.2 Solving a Homogeneous Differential Equation
o 20.2.1 Separating Homogeneous Differential Equations
o 20.2.2 Change of Variables

* 20.3 Growth and Decay Problems
o 20.3.1 Exponential Growth
o 20.3.2 Radioactive Decay

* 20.4 Solving First-Order Linear Differential Equations
o 20.4.1 First-Order Linear Differential Equations
o 20.4.2 Using Integrating Factors

- Parametric Equations and Polar Coordinates

* 21.1 Understanding Parametric Equations
o 21.1.1 An Introduction to Parametric Equations
o 21.1.2 The Cycloid
o 21.1.3 Eliminating Parameters

* 21.2 Calculus and Parametric Equations
o 21.2.1 Derivatives of Parametric Equations
o 21.2.2 Graphing the Elliptic Curve
o 21.2.3 The Arc Length of a Parameterized Curve
o 21.2.4 Finding Arc Lengths of Curves Given by Parametric Equations

* 21.3 Understanding Polar Coordinates
o 21.3.1 The Polar Coordinate System
o 21.3.2 Converting between Polar and Cartesian Forms
o 21.3.3 Spirals and Circles
o 21.3.4 Graphing Some Special Polar Functions

* 21.4 Polar Functions and Slope
o 21.4.1 Calculus and the Rose Curve
o 21.4.2 Finding the Slopes of Tangent Lines in Polar Form

* 21.5 Polar Functions and Area
o 21.5.1 Heading toward the Area of a Polar Region
o 21.5.2 Finding the Area of a Polar Region: Part One
o 21.5.3 Finding the Area of a Polar Region: Part Two
o 21.5.4 The Area of a Region Bounded by Two Polar Curves: Part One
o 21.5.5 The Area of a Region Bounded by Two Polar Curves: Part Two

- Vector Calculus and the Geometry of R2 and R3

* 22.1 Vectors and the Geometry of R2 and R3
o 22.1.1 Coordinate Geometry in Three Dimensional Space
o 22.1.2 Introduction to Vectors
o 22.1.3 Vectors in R2 and R3
o 22.1.4 An Introduction to the Dot Product
o 22.1.5 Orthogonal Projections
o 22.1.6 An Introduction to the Cross Product
o 22.1.7 Geometry of the Cross Product
o 22.1.8 Equations of Lines and Planes in R3

* 22.2 Vector Functions
o 22.2.1 Introduction to Vector Functions
o 22.2.2 Derivatives of Vector Functions
o 22.2.3 Vector Functions: Smooth Curves
o 22.2.4 Vector Functions: Velocity and Acceleration