General Calculus I

Chapter

Topics

Subtopics

The Basics

• Overview
• Precalculus Review

• Welcome to Calculus
• The Two Questions of Calculus
• Average Rates of Change
• How to Do Math
• Functions
• Rational functions
• Complex number
• Zeros of polynomial
• Graphing Lines
• Parabolas
• Some Non-Euclidean Geometry

Limits

• The Concept of the Limit
• Evaluating Limits

• Finding Rate of Change over an Interval
• Finding Limits Graphically
• The Formal Definition of a Limit
• The Limit Laws, Part I
• The Limit Laws, Part II
• One-Sided Limits
• The Squeeze Theorem
• Continuity and Discontinuity
• Evaluating Limits
• Limits and indeterminate Forms
• Two Techniques for Evaluating Limits
• An Overview of Limits

An Introduction to Derivatives

• Understanding the Derivative
• Using the Derivative
• Some Special Derivatives

• Rates of Change, Secants, and Tangents
• Finding Instantaneous Velocity
• The Derivative
• Differentiability
• The Slope of a Tangent Line
• Instantaneous Rate
• The Equation of a Tangent Line
• More on Instantaneous Rate
• The Derivative of the Reciprocal Function
• The Derivative of the Square Root

Computational Techniques

• The Power Rule
• The Product and Quotient Rules
• The Chain Rule

• A Shortcut for Finding Derivatives
• A Quick Proof of the Power Rule
• Uses of the Power Rule
• The Product Rule
• The Quotient Rule
• An Introduction to the Chain Rule
• Using the Chain Rule
• Combining Computational Techniques

Special Functions

• Trigonometric Functions
• Exponential Functions
• Logarithmic Functions

• A Review of Trigonometry
• Graphing Trigonometric Functions
• The Derivatives of Trigonometric Functions
• The Number Pi
• Graphing Exponential Functions
• Derivatives of Exponential Functions
• The Music of Math
• Evaluating Logarithmic Functions
• The Derivative of the Natural Log Function
• Using the Derivative Rules with Transcendental Functions

Implicit Differentiation

• Implicit Differentiation Basics
• Applying Implicit Differentiation

• An Introduction to Implicit Differentiation
• Finding the Derivative Implicitly
• Using Implicit Differentiation
• Applying Implicit Differentiation

Applications of Differentiation

• Position and Velocity
• Linear Approximation
• Optimization
• Related Rates

• Acceleration and the Derivative
• Solving Word Problems Involving Distance and Velocity
• Higher-Order Derivatives and Linear Approximation
• Using the Tangent Line Approximation Formula
• Newton’s Method
• The Connection Between Slopes and Optimization
• The Fence Method
• The Box Problem
• The Can Problem
• The Wire-Cutting Problem
• The Pebble Problem
• The Ladder Problem
• The Baseball Problem
• The Blimp Problem
• Math Anxiety

Curve Sketching

• Introduction
• Critical Points
• Concavity
• Graphing Using the Derivative
• Asymptotes

• An Introduction to Curve Sketching
• Three Big Theorems
• Morale Moment
• Critical Points
• Maximum and Minimum
• Regions Where a Function Increases or Decreases
• The First Derivative Test
• Magic Math
• Concavity and Inflection Points
• Using the Second Derivative to Examine Concavity
• The Möbius Band
• Graphs of Polynomial Functions
• Cusp Points and the Derivative
• Romain-Restricted Functions and the Derivative
• The Second Derivative Test
• Vertical Asymptotes
• Horizontal Asymptotes and Infinite Limits
• Graphing Functions with Asymptotes
• Functions with Asymptotes and Holes
• Functions with Asymptotes and Critical Points

The Basics of Integration

• Antiderivatives
• Integration by Substitution
• Illustrating Integration by Substitution
• The FUndamental Theorem of Calculus

• Antidifferentiation
• Antiderivatives of Powers of x
• Antiderivatives of Trigonometric and Exponential Functions
• Undoing the Chain Rule
• Integrating Polynomials by Substitution
• Integrating Composite Trigonometric Functions by Substitution
• Integrating Composite Exponential and Rational Functions by Substitution
• More Integrating Trigonometric Functions by Substitution
• Choosing Effective Function Decompositions

Applications of Integration

• Motion
• Finding the Area between Two Curves
• Integrating with Respect to y

• Antiderivatives and Motion
• Gravity and Vertical Motion
• Solving Vertical Motion Problems
• The Area between Two Curves
• Limits of Integration and Area
• Common Mistakes to Avoid When Finding Areas
• Regions Bound by Several Curves
• Finding Areas by Integrating with Respect to y: Part One
• Finding Areas by Integrating with Respect to y: Part Two
• Area, Integration by Substitution, and Trigonometry

L’Hôpital’s Rule

• Indeterminate Quotients

• Indeterminate Forms
• An Introduction to L’Hôpital’s Rule
• Basic Uses of L’Hôpital’s Rule
• More Exotic Examples of Indeterminate Forms

Elementary Functions and Their Inverses

• Inverse Functions
• The Calculus of Inverse Functions
• Inverse Trigonometric Functions
• The Calculus of Inverse Trigonometric Functions

• The Exponential and Natural Log Functions
• Differentiating Logarithmic Functions
• Logarithmic Differentiation
• The Basics of Inverse Functions
• Finding the Inverse of a Function
• Derivatives of Inverse Functions
• The Inverse Sine, Cosine, and Tangent Functions
• The Inverse Secant, Cosecant, and Cotangent Functions
• Evaluating Inverse Trigonometric Functions
• Derivatives of Inverse Trigonometric Functions
• More Calculus of Inverse Trigonometric Functions

Techniques of Integration

• An Introduction to Integration by Partial Fractions
• Integration by Parts

• Finding Partial Fraction Decompositions
• Partial Fractions
• Long Division
• An Introduction to Integration by Parts
• Applying Integration by Parts to the Natural Log Function
• Inspirational Examples of Integration by Parts
• Repeated Application of Integration by Parts
• Algebraic Manipulation and Integration by Parts

Improper Integrals

• Improper Integrals

• The First Type of Improper Integral
• The Second Type of Improper Integral
• Infinite Limits of Integration, Convergence, and Divergence

Differential Equations

• Separable Differential Equations
• Solving First-Order Linear Differential Equations

• An Introduction to Differential Equations
• Solving Separable DIfferential Equations
• Finding a Particular Solution
• Direction Fields
• Euler’s Method
• First-Order Linear Differential Equations

Review and Final Exam

Review and Final Exam

• Review and Final Exam