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Calculus I

Book Description

Let's face it: most students don't take calculus because they find it intellectually stimulating. It's not ... at least for those who come up on the wrong side of the bell curve! There they are, minding their own business, working toward some non-science related degree, when ... BLAM! They get next semester's course schedule in the mail, and first on the list is the mother of all loathed college courses ... CALCULUS!

Not to fear--Idiot's Guides: Calculus I is a curriculum-based companion book created with this audience in mind. This new edition continues the tradition of taking the sting out of calculus by adding more explanatory graphs and illustrations and doubling the number of practice problems! By the time readers are finished, they will have a solid understanding (maybe even a newfound appreciation) for this useful form of math. And with any luck, they may even be able to make sense of their textbooks and teachers.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Introduction
  5. Part 1: The Roots of Calculus
    1. 1 What Is Calculus, Anyway?
      1. What’s the Purpose of Calculus?
        1. Finding the Slopes of Curves
        2. Calculating the Area of Bizarre Shapes
        3. Justifying Old Formulas
        4. Calculating Complicated x-Intercepts
        5. Visualizing Graphs
        6. Finding the Average Value of a Function
        7. Calculating Optimal Values
      2. Who’s Responsible for This?
        1. Ancient Influences
        2. Newton vs. Leibniz
      3. Will I Ever Learn This?
    2. 2 Polish Up Your Algebra Skills
      1. Walk the Line: Linear Equations
        1. Common Forms of Linear Equations
        2. Calculating Slope
        3. Interpreting Linear Graphs
      2. You’ve Got the Power: Exponential Rules
      3. Breaking Up Is Hard to Do: Factoring Polynomials
        1. Greatest Common Factor
        2. Special Factoring Patterns
      4. Solving Quadratic Equations
        1. Method One: Factoring
        2. Method Two: Completing the Square
        3. Method Three: The Quadratic Formula
        4. Synthesizing the Quadratic Solution Methods
    3. 3 Equations, Relations, and Functions
      1. What Makes a Function Tick?
      2. Working with Graphs of Functions
      3. Functional Symmetry
      4. Graphs to Know by Heart
      5. Constructing an Inverse Function
      6. Parametric Equations
        1. What’s a Parameter?
        2. Converting to Rectangular Form
    4. 4 Trigonometry: Last Stop Before Calculus
      1. Getting Repetitive: Periodic Functions
      2. Introducing the Trigonometric Functions
        1. Sine (Written as y = sin x)
        2. Cosine (Written as y = cos x)
        3. Tangent (Written as y = tan x)
        4. Cotangent (Written as y = cot x)
        5. Secant (Written as y = sec x)
        6. Cosecant (Written as y = csc x)
      3. What’s Your Sine: The Unit Circle
      4. Incredibly Important Identities
        1. Pythagorean Identities
        2. Double-Angle Formulas
      5. Solving Trigonometric Equations
  6. Part 2: Laying the Foundation for Calculus
    1. 5 Take It to the Limit
      1. What Is a Limit?
      2. Can Something Be Nothing?
      3. One-Sided Limits
      4. When Does a Limit Exist?
      5. When Does a Limit Not Exist?
    2. 6 Evaluating Limits Numerically
      1. The Major Methods
        1. Substitution Method
        2. Factoring Method
        3. Conjugate Method
        4. What If Nothing Works?
      2. Limits and Infinity
        1. Vertical Asymptotes
        2. Horizontal Asymptotes
      3. Special Limit Theorems
        1. Evaluating Limits Graphically
      4. Technology Focus: Calculating Limits
    3. 7 Continuity
      1. What Does Continuity Look Like?
      2. The Mathematical Definition of Continuity
      3. Types of Discontinuity
        1. Jump Discontinuity
        2. Point Discontinuity
        3. Infinite/Essential Discontinuity
      4. Removable vs. Nonremovable Discontinuity
      5. The Intermediate Value Theorem
    4. 8 The Difference Quotient
      1. When a Secant Becomes a Tangent
      2. Honey, I Shrunk the Δx
      3. Applying the Difference Quotient
      4. The Alternate Difference Quotient
    5. Part 3: The Derivative
      1. 9 Laying Down the Law for Derivatives
        1. When Does a Derivative Exist?
          1. Discontinuity
          2. Sharp Point in the Graph
          3. Vertical Tangent Line
        2. Basic Derivative Techniques
          1. The Power Rule
          2. The Product Rule
          3. The Quotient Rule
          4. The Chain Rule
        3. Rates of Change
        4. Trigonometric Derivatives
        5. Tabular and Graphical Derivatives
        6. Technology Focus: Calculating Derivatives
      2. 10 Common Differentiation Tasks
        1. Finding Equations of Tangent Lines
        2. Implicit Differentiation
        3. Differentiating an Inverse Function
        4. Parametric Derivatives
        5. Technology Focus: Solving Gross Equations
          1. Using the Built-In Equation Solver
          2. The Equation-Function Connection
      3. 11 Using Derivatives to Graph
        1. Relative Extrema
          1. Finding Critical Numbers
          2. Classifying Extrema
        2. The Wiggle Graph
        3. The Extreme Value Theorem
        4. Determining Concavity
          1. Another Wiggle Graph
          2. The Second Derivative Test
      4. 12 Derivatives and Motion
        1. The Position Equation
        2. Velocity
        3. Acceleration
        4. Vertical Projectile Motion
      5. 13 Common Derivative Applications
        1. Newton’s Method
        2. Evaluating Limits: L’Hôpital’s Rule
        3. More Existence Theorems
          1. The Mean Value Theorem
          2. Rolle’s Theorem
        4. Related Rates
        5. Optimization
    6. Part 4: The Integral
      1. 14 Approximating Area
        1. Riemann Sums
          1. Right and Left Sums
          2. Midpoint Sums
        2. The Trapezoidal Rule
        3. Simpson’s Rule
      2. 15 Antiderivatives
        1. The Power Rule for Integration
        2. Integrating Trigonometric Functions
        3. Separation
        4. The Fundamental Theorem of Calculus
          1. Part One: Areas and Integrals Are Related
          2. Part Two: Derivatives and Integrals Are Opposites
        5. u-Substitution
        6. Tricky u-Substitution and Long Division
        7. Technology Focus: Definite and Indefinite Integrals
      3. 16 Applications of the Fundamental Theorem
        1. Calculating Area Between Two Curves
        2. The Mean Value Theorem for Integration
          1. A Geometric Interpretation
          2. The Average Value Theorem
        3. Finding Distance Traveled
        4. Accumulation Functions
        5. Arc Length
          1. Rectangular Equations
          2. Parametric Equations
    7. Part 5: Differential Equations and More
      1. 17 Differential Equations
        1. Separation of Variables
        2. Types of Solutions
          1. Family of Solutions
          2. Specific Solutions
        3. Exponential Growth and Decay
      2. 18 Visualizing Differential Equations
        1. Linear Approximation
        2. Slope Fields
        3. Euler’s Method
        4. Technology Focus: Slope Fields
      3. 19 Final Exam
  7. Appendixes
    1. A Solutions to “You’ve Got Problems”
    2. B Glossary
  8. About the Author