Calculus I by Kelley, W. Michael -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Introduction 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

About the Author

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