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by W. Michael Kelley
Calculus I
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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