Contents

Part 1: The Roots of Calculus

1 What Is Calculus, Anyway?

What’s the Purpose of Calculus?

Finding the Slopes of Curves

Calculating the Area of Bizarre Shapes

Justifying Old Formulas

Calculating Complicated x-Intercepts

Visualizing Graphs

Finding the Average Value of a Function

Calculating Optimal Values

Who’s Responsible for This?

Ancient Influences

Newton vs. Leibniz

Will I Ever Learn This?

2 Polish Up Your Algebra Skills

Walk the Line: Linear Equations

Common Forms of Linear Equations

Calculating Slope

Interpreting Linear Graphs

You’ve Got the Power: Exponential Rules

Breaking Up Is Hard to Do: Factoring Polynomials

Greatest Common Factor

Special Factoring Patterns

Solving Quadratic Equations

Method One: Factoring

Method Two: Completing the Square

Method Three: The Quadratic Formula

Synthesizing the Quadratic Solution Methods

3 Equations, Relations, and Functions

What Makes a Function Tick?

Working with Graphs of Functions

Functional Symmetry

Graphs to Know by Heart

Constructing an Inverse Function

Parametric Equations

What’s a Parameter?

Converting to Rectangular Form

4 Trigonometry: Last Stop Before Calculus

Getting Repetitive: Periodic Functions

Introducing the Trigonometric Functions

Sine (Written as y = sin x)

Cosine (Written as y = cos x)

Tangent (Written as y = tan x)

Cotangent (Written as y = cot x)

Secant (Written as y = sec x)

Cosecant (Written as y = csc x)

What’s Your Sine: The Unit Circle

Incredibly Important Identities

Pythagorean Identities

Double-Angle Formulas

Solving Trigonometric Equations

Part 2: Laying the Foundation for Calculus

5 Take It to the Limit

What Is a Limit?

Can Something Be Nothing?

One-Sided Limits

When Does a Limit Exist?

When Does a Limit Not Exist?

6 Evaluating Limits Numerically

The Major Methods

Substitution Method

Factoring Method

Conjugate Method

What If Nothing Works?

Limits and Infinity

Vertical Asymptotes

Horizontal Asymptotes

Special Limit Theorems

Evaluating Limits Graphically

Technology Focus: Calculating Limits

7 Continuity

What Does Continuity Look Like?

The Mathematical Definition of Continuity

Types of Discontinuity

Jump Discontinuity

Point Discontinuity

Infinite/Essential Discontinuity

Removable vs. Nonremovable Discontinuity

The Intermediate Value Theorem

8 The Difference Quotient

When a Secant Becomes a Tangent

Honey, I Shrunk the Δx

Applying the Difference Quotient

The Alternate Difference Quotient

Part 3: The Derivative

9 Laying Down the Law for Derivatives

When Does a Derivative Exist?

Discontinuity

Sharp Point in the Graph

Vertical Tangent Line

Basic Derivative Techniques

The Power Rule

The Product Rule

The Quotient Rule

The Chain Rule

Rates of Change

Trigonometric Derivatives

Tabular and Graphical Derivatives

Technology Focus: Calculating Derivatives

10 Common Differentiation Tasks

Finding Equations of Tangent Lines

Implicit Differentiation

Differentiating an Inverse Function

Parametric Derivatives

Technology Focus: Solving Gross Equations

Using the Built-In Equation Solver

The Equation-Function Connection

11 Using Derivatives to Graph

Relative Extrema

Finding Critical Numbers

Classifying Extrema

The Wiggle Graph

The Extreme Value Theorem

Determining Concavity

Another Wiggle Graph

The Second Derivative Test

12 Derivatives and Motion

The Position Equation

Velocity

Acceleration

Vertical Projectile Motion

13 Common Derivative Applications

Newton’s Method

Evaluating Limits: L’Hôpital’s Rule

More Existence Theorems

The Mean Value Theorem

Rolle’s Theorem

Related Rates

Optimization

Part 4: The Integral

14 Approximating Area

Riemann Sums

Right and Left Sums

Midpoint Sums

The Trapezoidal Rule

Simpson’s Rule

15 Antiderivatives

The Power Rule for Integration

Integrating Trigonometric Functions

Separation

The Fundamental Theorem of Calculus

Part One: Areas and Integrals Are Related

Part Two: Derivatives and Integrals Are Opposites

u-Substitution

Tricky u-Substitution and Long Division

Technology Focus: Definite and Indefinite Integrals

16 Applications of the Fundamental Theorem

Calculating Area Between Two Curves

The Mean Value Theorem for Integration

A Geometric Interpretation

The Average Value Theorem

Finding Distance Traveled

Accumulation Functions

Arc Length

Rectangular Equations

Parametric Equations

Part 5: Differential Equations and More

17 Differential Equations

Separation of Variables

Types of Solutions

Family of Solutions

Specific Solutions

Exponential Growth and Decay

18 Visualizing Differential Equations

Linear Approximation

Slope Fields

Euler’s Method

Technology Focus: Slope Fields

19 Final Exam

Appendixes

A Solutions to “You’ve Got Problems”

B Glossary

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