Calculus 1

1. Introduction

·          Precalculus Review

1.       Functions

2.       Graphing Lines

3.       Parabolas

4.       Some Non-Euclidean Geometry

 

 2. Limits

·          The Concept of the Limit

1.       Finding Rate of Change over an Interval

2.       Finding Limits Graphically

3.       The Formal Definition of a Limit

4.       The Limit Laws

5.       One-Sided Limits

6.       Continuity and Discontinuity

·          Evaluating Limits

1.       Evaluating Limits

2.       Limits and Indeterminate Forms

3.       Two Techniques for Evaluating Limits

4.       An Overview of Limits

 

3. Introduction to Derivatives

·          Understanding the Derivative

1.       Rates of Change, Secants, and Tangents

2.       Finding Instantaneous Velocity

3.       The Derivative

4.       Differentiability

·          Using the Derivative

1.       The Slope of a Tangent Line

2.       Instantaneous Rate

3.       The Equation of a Tangent Line

4.       More on Instantaneous Rate

·          Some Special Derivatives

1.       The Derivative of the Reciprocal Function

2.       The Derivative of the Square Root Function

 

 4. Computational Techniques

·          The Power Rule

1.       A Shortcut for Finding Derivatives

2.       A Quick Proof of the Power Rule

3.       Uses of the Power Rule

·          The Product and Quotient Rules

1.       The Product Rule

2.       The Quotient Rule

·          The Chain Rule

1.       An Introduction to the Chain Rule

2.       Using the Chain Rule

3.       Combining Computational Techniques

 

5. Special Functions

·          Trigonometric Functions

1.       A Review of Trigonometry

2.       Graphing Trigonometric Functions

3.       The Derivatives of Trigonometric Functions

4.       The Number Pi

·          Exponential Functions

1.       Graphing Exponential Functions

2.       Derivatives of Exponential Functions

3.       The Music of Math

·          Logarithmic Functions

1.       Evaluating Logarithmic Functions

2.       The Derivative of the Natural Log Function

3.       Using the Derivative Rules with Transcendental Functions

 

 6. Implicit Differentiation

·          Implicit Differentiation Basics

1.       An Introduction to Implicit Differentiation

2.       Finding the Derivative Implicitly

·          Applying Implicit Differentiation

1.       Using Implicit Differentiation

2.       Applying Implicit Differentiation

 

7. Practical Application of the Derivative

·          Position and Velocity

1.       Acceleration and the Derivative

2.       Solving Word Problems Involving Distance and Velocity

·          Linear Approximation

1.       Higher-Order Derivatives and Linear Approximation

2.       Using the Tangent Line Approximation Formula

3.       Newton's Method

·          Optimization

1.       The Connection Between Slope and Optimization

·          Related Rates

 

8. Curve Tracing

·          Introduction

1.       An Introduction to Curve Tracing

2.       Three Big Theorems

3.       Morale Moment

·          Critical Points

1.       Critical Points

2.       Regions Where a Function Increases or Decreases

3.       The First Derivative Test

·          Concavity

1.       Concavity and Inflection Points

2.       Using the Second Derivative to Examine Concavity

3.       The Möbius Band

·          Graphing Using the Derivative

1.       Graphs of Polynomial Functions

2.       Cusp Points and the Derivative

3.       Domain-Restricted Functions and the Derivative

4.       The Second Derivative Test

·          Asymptotes

1.       Vertical Asymptotes

2.       Horizontal Asymptotes and Infinite Limits

3.       Graphing Functions with Asymptotes

4.       Functions with Asymptotes and Holes

5.       Functions with Asymptotes and Critical Points