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Calculus For Dummies, 2nd Edition

Book Description

Slay the calculus monster with this user-friendly guide

Calculus For Dummies, 2nd Edition makes calculus manageable—even if you're one of the many students who sweat at the thought of it. By breaking down differentiation and integration into digestible concepts, this guide helps you build a stronger foundation with a solid understanding of the big ideas at work. This user-friendly math book leads you step-by-step through each concept, operation, and solution, explaining the "how" and "why" in plain English instead of math-speak. Through relevant instruction and practical examples, you'll soon learn that real-life calculus isn't nearly the monster it's made out to be.

Calculus is a required course for many college majors, and for students without a strong math foundation, it can be a real barrier to graduation. Breaking that barrier down means recognizing calculus for what it is—simply a tool for studying the ways in which variables interact. It's the logical extension of the algebra, geometry, and trigonometry you've already taken, and Calculus For Dummies, 2nd Edition proves that if you can master those classes, you can tackle calculus and win.

  • Includes foundations in algebra, trigonometry, and pre-calculus concepts

  • Explores sequences, series, and graphing common functions

  • Instructs you how to approximate area with integration

  • Features things to remember, things to forget, and things you can't get away with

  • Stop fearing calculus, and learn to embrace the challenge. With this comprehensive study guide, you'll gain the skills and confidence that make all the difference. Calculus For Dummies, 2nd Edition provides a roadmap for success, and the backup you need to get there.

    Table of Contents

      1. Introduction
        1. About This Book
        2. Foolish Assumptions
        3. Icons Used in This Book
        4. Beyond the Book
        5. Where to Go from Here
      2. Part I: An Overview of Calculus
        1. Chapter 1: What Is Calculus?
          1. What Calculus Is Not
          2. So What Is Calculus Already?
          3. Real-World Examples of Calculus
        2. Chapter 2: The Two Big Ideas of Calculus: Differentiation and Integration — plus Infinite Series
          1. Defining Differentiation
            1. The derivative is a slope
            2. The derivative is a rate
          2. Investigating Integration
          3. Sorting Out Infinite Series
            1. Divergent series
            2. Convergent series
        3. Chapter 3: Why Calculus Works
          1. The Limit Concept: A Mathematical Microscope
          2. What Happens When You Zoom In
          3. Two Caveats, or Precision, Preschmidgen
            1. I may lose my license to practice mathematics
            2. What the heck does “infinity” really mean?
      3. Part II: Warming Up with Calculus Prerequisites
        1. Chapter 4: Pre-Algebra and Algebra Review
          1. Fine-Tuning Your Fractions
            1. Some quick rules
            2. Multiplying fractions
            3. Dividing fractions
            4. Adding fractions
            5. Subtracting fractions
            6. Canceling in fractions
          2. Absolute Value — Absolutely Easy
          3. Empowering Your Powers
          4. Rooting for Roots
            1. Roots rule — make that, root rules
            2. Simplifying roots
          5. Logarithms — This Is Not an Event at a Lumberjack Competition
          6. Factoring Schmactoring — When Am I Ever Going to Need It?
            1. Pulling out the GCF
            2. Looking for a pattern
            3. Trying some trinomial factoring
          7. Solving Quadratic Equations
            1. Method 1: Factoring
            2. Method 2: The quadratic formula
            3. Method 3: Completing the square
        2. Chapter 5: Funky Functions and Their Groovy Graphs
          1. What Is a Function?
            1. The defining characteristic of a function
            2. Independent and dependent variables
            3. Function notation
            4. Composite functions
          2. What Does a Function Look Like?
          3. Common Functions and Their Graphs
            1. Lines in the plane in plain English
            2. Parabolic and absolute value functions — even steven
            3. A couple oddball functions
            4. Exponential functions
            5. Logarithmic functions
          4. Inverse Functions
          5. Shifts, Reflections, Stretches, and Shrinks
            1. Horizontal transformations
            2. Vertical transformations
        3. Chapter 6: The Trig Tango
          1. Studying Trig at Camp SohCahToa
          2. Two Special Right Triangles
            1. The 45°-45°-90° triangle
            2. The 30°-60°-90° triangle
          3. Circling the Enemy with the Unit Circle
            1. Angles in the unit circle
            2. Measuring angles with radians
            3. Honey, I shrunk the hypotenuse
            4. Putting it all together
          4. Graphing Sine, Cosine, and Tangent
          5. Inverse Trig Functions
          6. Identifying with Trig Identities
      4. Part III: Limits
        1. Chapter 7: Limits and Continuity
          1. Take It to the Limit — NOT
            1. Using three functions to illustrate the same limit
            2. Sidling up to one-sided limits
            3. The formal definition of a limit — just what you’ve been waiting for
            4. Limits and vertical asymptotes
            5. Limits and horizontal asymptotes
            6. Calculating instantaneous speed with limits
          2. Linking Limits and Continuity
            1. Continuity and limits usually go hand in hand
            2. The hole exception tells the whole story
            3. Sorting out the mathematical mumbo jumbo of continuity
          3. The 33333 Limit Mnemonic
        2. Chapter 8: Evaluating Limits
          1. Easy Limits
            1. Limits to memorize
            2. Plugging and chugging
          2. The “Real Deal” Limit Problems
            1. Figuring a limit with your calculator
            2. Solving limit problems with algebra
            3. Take a break and make yourself a limit sandwich
          3. Evaluating Limits at Infinity
            1. Limits at infinity and horizontal asymptotes
            2. Solving limits at infinity with a calculator
            3. Solving limits at infinity with algebra
      5. Part IV: Differentiation
        1. Chapter 9: Differentiation Orientation
          1. Differentiating: It’s Just Finding the Slope
            1. The slope of a line
            2. The derivative of a line
          2. The Derivative: It’s Just a Rate
            1. Calculus on the playground
            2. Speed — the most familiar rate
            3. The rate-slope connection
          3. The Derivative of a Curve
          4. The Difference Quotient
          5. Average Rate and Instantaneous Rate
          6. To Be or Not to Be? Three Cases Where the Derivative Does Not Exist
        2. Chapter 10: Differentiation Rules — Yeah, Man, It Rules
          1. Basic Differentiation Rules
            1. The constant rule
            2. The power rule
            3. The constant multiple rule
            4. The sum rule — hey, that’s some rule you got there
            5. The difference rule — it makes no difference
            6. Differentiating trig functions
            7. Differentiating exponential and logarithmic functions
          2. Differentiation Rules for Experts — Oh, Yeah, I’m a Calculus Wonk
            1. The product rule
            2. The quotient rule
            3. The chain rule
          3. Differentiating Implicitly
          4. Getting into the Rhythm with Logarithmic Differentiation
          5. Differentiating Inverse Functions
          6. Scaling the Heights of Higher Order Derivatives
        3. Chapter 11: Differentiation and the Shape of Curves
          1. Taking a Calculus Road Trip
            1. Climb every mountain, ford every stream: Positive and negative slopes
            2. I can’t think of a travel metaphor for this section: Concavity and inflection points
            3. This vale of tears: A local minimum
            4. A scenic overlook: The absolute maximum
            5. Car trouble: Teetering on the corner
            6. It’s all downhill from here
            7. Your travel diary
          2. Finding Local Extrema — My Ma, She’s Like, Totally Extreme
            1. Cranking out the critical numbers
            2. The first derivative test
            3. The second derivative test — no, no, anything but another test!
          3. Finding Absolute Extrema on a Closed Interval
          4. Finding Absolute Extrema over a Function’s Entire Domain
          5. Locating Concavity and Inflection Points
          6. Looking at Graphs of Derivatives Till They Derive You Crazy
          7. The Mean Value Theorem — GRRRRR
        4. Chapter 12: Your Problems Are Solved: Differentiation to the Rescue!
          1. Getting the Most (or Least) Out of Life: Optimization Problems
            1. The maximum volume of a box
            2. The maximum area of a corral — yeehaw!
          2. Yo-Yo a Go-Go: Position, Velocity, and Acceleration
            1. Velocity, speed, and acceleration
            2. Maximum and minimum height
            3. Velocity and displacement
            4. Speed and distance traveled
            5. Burning some rubber with acceleration
            6. Tying it all together
          3. Related Rates — They Rate, Relatively
            1. Blowing up a balloon
            2. Filling up a trough
            3. Fasten your seat belt: You’re approaching a calculus crossroads
        5. Chapter 13: More Differentiation Problems: Going Off on a Tangent
          1. Tangents and Normals: Joined at the Hip
            1. The tangent line problem
            2. The normal line problem
          2. Straight Shooting with Linear Approximations
          3. Business and Economics Problems
            1. Managing marginals in economics
      6. Part V: Integration and Infinite Series
        1. Chapter 14: Intro to Integration and Approximating Area
          1. Integration: Just Fancy Addition
          2. Finding the Area Under a Curve
          3. Approximating Area
            1. Approximating area with left sums
            2. Approximating area with right sums
            3. Approximating area with midpoint sums
          4. Getting Fancy with Summation Notation
            1. Summing up the basics
            2. Writing Riemann sums with sigma notation
          5. Finding Exact Area with the Definite Integral
          6. Approximating Area with the Trapezoid Rule and Simpson’s Rule
            1. The trapezoid rule
            2. Simpson’s rule — that’s Thomas (1710–1761), not Homer (1987–)
        2. Chapter 15: Integration: It’s Backwards Differentiation
          1. Antidifferentiation
          2. Vocabulary, Voshmabulary: What Difference Does It Make?
          3. The Annoying Area Function
          4. The Power and the Glory of the Fundamental Theorem of Calculus
          5. The Fundamental Theorem of Calculus: Take Two
            1. Why the theorem works: Area functions explanation
            2. Why the theorem works: The integration-differentiation connection
            3. Why the theorem works: A connection to — egad! — statistics
          6. Finding Antiderivatives: Three Basic Techniques
            1. Reverse rules for antiderivatives
            2. Guessing and checking
            3. The substitution method
          7. Finding Area with Substitution Problems
        3. Chapter 16: Integration Techniques for Experts
          1. Integration by Parts: Divide and Conquer
            1. Picking your u
            2. Integration by parts: Second time, same as the first
            3. Going around in circles
          2. Tricky Trig Integrals
            1. Integrals containing sines and cosines
            2. Integrals containing secants and tangents or cosecants and cotangents
          3. Your Worst Nightmare: Trigonometric Substitution
            1. Case 1: Tangents
            2. Case 2: Sines
            3. Case 3: Secants
          4. The As, Bs, and Cxs of Partial Fractions
            1. Case 1: The denominator contains only linear factors
            2. Case 2: The denominator contains irreducible quadratic factors
            3. Case 3: The denominator contains repeated linear or quadratic factors
            4. Bonus: Equating coefficients of like terms
        4. Chapter 17: Forget Dr. Phil: Use the Integral to Solve Problems
          1. The Mean Value Theorem for Integrals and Average Value
          2. The Area between Two Curves — Double the Fun
          3. Finding the Volumes of Weird Solids
            1. The meat-slicer method
            2. The disk method
          4. The Washer Method
            1. The matryoshka-doll method
          5. Analyzing Arc Length
          6. Surfaces of Revolution — Pass the Bottle ’Round
        5. Chapter 18: Taming the Infinite with Improper Integrals
          1. L’Hôpital’s Rule: Calculus for the Sick
            1. Getting unacceptable forms into shape
          2. Improper Integrals: Just Look at the Way That Integral Is Holding Its Fork!
            1. Improper integrals with vertical asymptotes
            2. Improper integrals with one or two infinite limits of integration
            3. Blowing Gabriel’s horn
        6. Chapter 19: Infinite Series
          1. Sequences and Series: What They’re All About
            1. Stringing sequences
            2. Summing series
          2. Convergence or Divergence?That Is the Question
            1. A no-brainer divergence test: The nth term test
            2. Three basic series and their convergence/divergence tests
            3. Three comparison tests for convergence/divergence
            4. The two “R” tests: Ratios and roots
          3. Alternating Series
            1. Finding absolute versus conditional convergence
            2. The alternating series test
          4. Keeping All the Tests Straight
      7. Part VI: The Part of Tens
        1. Chapter 20: Ten Things to Remember
          1. Your Sunglasses
          2. a<sup xmlns="" xmlns:epub="">2</sup> - b - b<sup xmlns="" xmlns:epub="">2</sup>=(a-b)(a+b)=(a-b)(a+b)
          3. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq1902a_fmt.png" alt="9781118791295-eq1902.tif"></img>, but , but <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq1903a_fmt.png" alt="9781118791295-eq1903.tif"></img> Is Undefined Is Undefined
          4. Anything<sup xmlns="" xmlns:epub="">0</sup>=1=1
          5. SohCahToa
          6. Trig Values for 30, 45, and 60 Degrees
          7. sin<sup xmlns="" xmlns:epub="">2</sup>&#952;+cosθ+cos<sup xmlns="" xmlns:epub="">2</sup>&#952;=1θ=1
          8. The Product Rule
          9. The Quotient Rule
          10. Where You Put Your Keys
        2. Chapter 21: Ten Things to Forget
          1. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2001a_fmt.png" alt="9781118791295-eq2001.tif"></img> &#8212; Wrong! — Wrong!
          2. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2005a_fmt.png" alt="9781118791295-eq2005.tif"></img> &#8212; Wrong! — Wrong!
          3. Slope = <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2008a_fmt.png" alt="9781118791295-eq2008.tif"></img> &#8212; Wrong! — Wrong!
          4. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2010a_fmt.png" alt="9781118791295-eq2010.tif"></img> &#8212; Wrong! — Wrong!
          5. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2012a_fmt.png" alt="9781118791295-eq2012.tif"></img> &#8212; Wrong! — Wrong!
          6. If k Is a Constant, <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2016a_fmt.png" alt="9781118791295-eq2016.tif"></img> &#8212; Wrong! — Wrong!
          7. The Quotient Rule Is <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2020a_fmt.png" alt="9781118791295-eq2020.tif"></img> &#8212; Wrong! — Wrong!
          8. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2021a_fmt.png" alt="9781118791295-eq2021.tif"></img> &#8212; Wrong! — Wrong!
          9. <img xmlns="" xmlns:epub="" src="../images/9781118791295-eq2022a_fmt.png" alt="9781118791295-eq2022.tif"></img> &#8212; Wrong! — Wrong!
          10. Green’s Theorem
        3. Chapter 22: Ten Things You Can’t Get Away With
          1. Give Two Answers on Exam Questions
          2. Write Illegibly on Exams
          3. Don’t Show Your Work on Exams
          4. Don’t Do All of the Exam Problems
          5. Blame Your Study Partner for Low Grade
          6. Tell Your Teacher You Need an “A” in Calculus to Impress Your Significant Other
          7. Claim Early-Morning Exams Are Unfair Because You’re Not a “Morning Person”
          8. Protest the Whole Idea of Grades
          9. Pull the Fire Alarm During an Exam
          10. Use This Book as an Excuse
      8. About the Author
      9. Cheat Sheet
      10. More Dummies Products