
Calculus All-in-One For Dummies (+ Chapter Quizzes Online)
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Make calculus more manageable with simplified instruction and tons of practice
Calculus All-in-One For Dummies pairs no-nonsense explanations of calculus content with practical examples and practice problems, so you can untangle the difficult concepts and improve your score in any calculus class. Plus, this book comes with access to chapter quizzes online. Dummies makes differentiation, integration, and everything in between more manageable, so you can crush calculus with confidence. Review the foundational basics, then dive into calc lessons that track your class. This book takes you through a full year of high-school calculus or a first semester of college calculus, only explained more clearly.
- Work through easy-to-understand lessons on everything in a typical calc class
- Get the score you want and need on standardized tests like AP Calculus
- Access online chapter quizzes for additional practice
- Untangle tricky problems and discover clever ways to solve them
With clear definitions, concise explanations, and plenty of helpful information on everything from limits and vectors to integration and curve-sketching, Calculus All-in-One For Dummies is the must-have resource for students who want to review for exams or just need extra help understanding the concepts from class.
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Content
- Intro
- Title Page
- Copyright Page
- Table of Contents
- Introduction
- About This Book
- Foolish Assumptions
- Icons Used in This Book
- Beyond the Book
- Where to Go from Here
- Unit 1: An Overview of Calculus
- Chapter 1 What Is Calculus?
- What Calculus Is Not
- So What Is Calculus, Already?
- Real-World Examples of Calculus
- Chapter 2 The Two Big Ideas of Calculus: Differentiation and Integration - Plus Infinite Series
- Defining Differentiation
- The derivative is a slope
- The derivative is a rate
- Investigating Integration
- Sorting Out Infinite Series
- Divergent series
- Convergent series
- Chapter 3 Why Calculus Works
- The Limit Concept: A Mathematical Microscope
- What Happens When You Zoom In
- Two Caveats
- or, Precision, Preschmidgen
- I may lose my license to practice mathematics
- What the heck does "infinity" really mean?
- Unit 2: Warming Up with Calculus Prerequisites
- Chapter 4 Pre-Algebra, Algebra, and Geometry Review
- Fine-Tuning Your Fractions
- Some quick rules
- Multiplying fractions
- Dividing fractions
- Adding fractions
- Subtracting fractions
- Canceling in fractions
- Express yourself
- The multiplication rule for canceling
- Miscellaneous Algebra
- Absolute value - absolutely easy
- Empowering your powers
- Rooting for roots
- Roots rule - make that, root rules
- Simplifying roots
- Logarithms - this is not an event at a lumberjack competition
- Factoring schmactoring - when am I ever going to need it?
- Pulling out the greatest common factor
- Looking for a pattern
- Trying some trinomial factoring
- Solving quadratic equations
- Method 1: Factoring
- Method 2: The quadratic formula
- Method 3: Completing the square
- Geometry Refresher
- Handy-dandy geometry formulas
- Formulas for two-dimensional shapes
- Formulas for three-dimensional shapes
- Coordinate geometry formulas
- Two special right triangles
- The 45° - 45° - 90° triangle
- The 30° - 60° - 90° triangle
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 4 Quiz
- Answers to Chapter 4 Quiz
- Chapter 5 Funky Functions and Their Groovy Graphs
- What Is a Function?
- The defining characteristic of a function
- Independent and dependent variables
- Function notation
- Composite functions
- What Does a Function Look Like?
- Common Functions and Their Graphs
- Lines in the plane in plain English
- Hitting the slopes
- Graphing lines
- Slope-intercept and point-slope forms
- Parabolic and absolute value functions - even-steven
- A couple of oddball functions
- Exponential functions
- Logarithmic functions
- Inverse Functions
- Shifts, Reflections, Stretches, and Shrinks
- Horizontal transformations
- Vertical transformations
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 5 Quiz
- Answers to Chapter 5 Quiz
- Chapter 6 The Trig Tango
- Starting off with SohCahToa
- Two Important Trig Triangles
- Circling the Enemy with the Unit Circle
- Angles in the unit circle
- Measuring angles with radians
- Honey, I shrunk the hypotenuse
- Putting it all together
- Graphing Sine, Cosine, and Tangent
- Inverse Trig Functions
- Identifying with Trig Identities
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 6 Quiz
- Answers to Chapter 6 Quiz
- Unit 3: Limits
- Chapter 7 Limits and Continuity
- Take It to the Limit - NOT
- Using three functions to illustrate the same limit
- Sidling up to one-sided limits
- The formal definition of a limit - just what you've been waiting for
- Limits and vertical asymptotes
- Limits and horizontal asymptotes
- Calculating instantaneous speed with limits
- Linking Limits and Continuity
- Continuity and limits usually go hand in hand
- The hole exception tells the whole story
- Sorting out the mathematical mumbo jumbo of continuity
- The 33333 Limit Mnemonic
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 7 Quiz
- Answers to Chapter 7 Quiz
- Chapter 8 Evaluating Limits
- Easy Does It - Easy Limits
- Limits to memorize
- Plugging and chugging
- The "Real Deal" Limit Problems
- Figuring a limit with your calculator
- Method one
- Method two
- Solving limit problems with algebra
- Take a break and make yourself a limit sandwich
- Evaluating Limits at ±Infinity
- Limits of rational functions at ±infinity
- Solving limits at infinity with a calculator
- Solving limits at ±infinity with algebra
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 8 Quiz
- Answers to Chapter 8 Quiz
- Unit 4: Differentiation
- Chapter 9 Differentiation Orientation
- Differentiating: It's Just Finding the Slope
- The slope of a line
- The derivative of a line
- The Derivative: It's Just a Rate
- Calculus on the playground
- Speed - the most familiar rate
- The rate-slope connection
- The Derivative of a Curve
- The Difference Quotient
- Average Rate and Instantaneous Rate
- To Be or Not to Be? Three Cases Where the Derivative Does Not Exist
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 9 Quiz
- Answers to Chapter 9 Quiz
- Chapter 10 Differentiation Rules - Yeah, Man, It Rules
- Basic Differentiation Rules
- The constant rule
- The power rule
- The constant multiple rule
- The sum rule - hey, that's some rule you got there
- The difference rule - it makes no difference
- Differentiating trig functions
- Differentiating exponential and logarithmic functions
- Exponential functions
- Logarithmic functions
- Differentiation Rules for Experts - Oh, Yeah, I'm a Calculus Wonk
- The product rule
- The quotient rule
- Linking up with the chain rule
- Differentiating Implicitly
- Differentiating Inverse Functions
- Scaling the Heights of Higher-Order Derivatives
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 10 Quiz
- Answers to Chapter 10 Quiz
- Chapter 11 Differentiation and the Shape of Curves
- Taking a Calculus Road Trip
- Climb every mountain, ford every stream: Positive and negative slopes
- I can't think of a travel metaphor for this section: Concavity and inflection points
- This vale of tears: A local minimum
- A scenic overlook: The absolute maximum
- Car trouble: Teetering on the corner
- It's all downhill from here
- Your travel diary
- Finding Local Extrema - My Ma, She's Like, Totally Extreme
- Cranking out the critical numbers
- The first derivative test
- The second derivative test - no, no, anything but another test!
- Finding Absolute Extrema on a Closed Interval
- Finding Absolute Extrema over a Function's Entire Domain
- Locating Concavity and Inflection Points
- Looking at Graphs of Derivatives Till They Derive You Crazy
- The Mean Value Theorem - Go Ahead, Make My Day
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 11 Quiz
- Answers to Chapter 11 Quiz
- Chapter 12 Your Problems Are Solved: Differentiation to the Rescue!
- Getting the Most (or Least) Out of Life: Optimization Problems
- The maximum volume of a box
- The maximum area of a corral - yeehaw!
- Yo-Yo a Go-Go: Position, Velocity, and Acceleration
- Velocity, speed, and acceleration
- Maximum and minimum height
- Velocity and displacement
- Total displacement
- Average velocity
- Maximum and minimum velocity
- Speed and distance traveled
- Total distance traveled
- Average speed
- Maximum and minimum speed
- Burning some rubber with acceleration
- Positive and negative acceleration
- Speeding up and slowing down
- Tying it all together
- Related Rates - They Rate, Relatively
- Blowing up a balloon
- Filling up a trough
- Fasten your seat belt: You're approaching a calculus crossroads
- Try this at your own risk
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 12 Quiz
- Answers to Chapter 12 Quiz
- Chapter 13 More Differentiation Problems: Going Off on a Tangent
- Tangents and Normals: Joined at the Hip
- The tangent line problem
- The normal line problem
- Straight Shooting with Linear Approximations
- Business and Economics Problems
- Managing marginals in economics
- Marginal cost
- Marginal revenue
- Marginal profit
- Maximum profit
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 13 Quiz
- Answers to Chapter 13 Quiz
- Unit 5: Integration and Infinite Series
- Chapter 14 Intro to Integration and Approximating Area
- Integration: Just Fancy Addition
- Finding the Area Under a Curve
- Approximating Area
- Approximating area with left sums
- Approximating area with right sums
- Approximating area with midpoint sums
- Getting Fancy with Summation Notation
- Summing up the basics
- Writing Riemann sums with sigma notation
- Finding Exact Area with the Definite Integral
- Approximating Area with the Trapezoid Rule and Simpson's Rule
- The trapezoid rule
- Simpson's rule - that's Thomas (1710-1761), not Homer (1987-)
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 14 Quiz
- Answers to Chapter 14 Quiz
- Chapter 15 Integration: It's Backwards Differentiation
- Antidifferentiation
- Vocabulary, Voshmabulary: What Difference Does It Make?
- The Annoying Area Function
- The Power and the Glory of the Fundamental Theorem of Calculus
- The Fundamental Theorem of Calculus: Take Two
- Why the theorem works: Area functions explained
- Why the theorem works: The integration- differentiation connection
- Why the theorem works: A connection to - egad! - statistics
- Finding Antiderivatives: Three Basic Techniques
- Reverse rules for antiderivatives
- No-brainer reverse rules
- The slightly more difficult reverse power rule
- Guessing and checking
- The substitution method
- Finding Area with Substitution Problems
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 15 Quiz
- Answers to Chapter 15 Quiz
- Chapter 16 Integration Techniques for Experts
- Integration by Parts: Divide and Conquer
- Picking your u
- Integration by parts: Second time, same as the first
- Tricky Trig Integrals
- Integrals containing sines and cosines
- Case 1: The power of sine is odd and positive
- Case 2: The power of cosine is odd and positive
- Case 3: The powers of both sine and cosine are even and nonnegative
- Integrals containing secants and tangents (or cosecants and cotangents)
- Case 1: The power of tangent (or cotangent) is odd
- Case 2: The power of secant (or cosecant) is even
- Your Worst Nightmare: Trigonometric Substitution
- Case 1: Tangents
- Case 2: Sines
- Case 3: Secants
- The A's, B's, and Cx's of Partial Fractions
- Case 1: The denominator contains only linear factors
- Case 2: The denominator contains irreducible quadratic factors
- Bonus: Equating coefficients of like terms
- Case 3: The denominator contains one or more factors raised to a power greater than 1
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 16 Quiz
- Answers to Chapter 16 Quiz
- Chapter 17 Who Needs Freud? Using the Integral to Solve Your Problems
- The Mean Value Theorem for Integrals and Average Value
- The Area between Two Curves - Double the Fun
- Volumes of Weird Solids: No, You're Never Going to Need This
- The meat-slicer method
- The disk method
- The washer method
- Analyzing Arc Length
- Surfaces of Revolution - Pass the Bottle 'Round
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 17 Quiz
- Answers to Chapter 17 Quiz
- Chapter 18 Taming the Infinite with Improper Integrals
- L'Hôpital's Rule: Calculus for the Sick
- Getting unacceptable forms into shape
- Looking at three more unacceptable forms
- Improper Integrals: Just Look at the Way That Integral Is Holding Its Fork!
- Improper integrals with vertical asymptotes
- A vertical asymptote at one of the limits of integration
- A vertical asymptote between the limits of integration
- Improper integrals with one or two infinite limits of integration
- Integrals with one infinite limit of integration
- Integrals with two infinite limits of integration
- Blowing Gabriel's horn
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 18 Quiz
- Answers to Chapter 18 Quiz
- Chapter 19 Infinite Series: Welcome to the Outer Limits
- Sequences and Series: What They're All About
- Stringing sequences
- Convergence and divergence of sequences
- Sequences and functions go hand in hand
- Determining limits with L'Hôpital's rule
- Summing series
- Partial sums
- The convergence or divergence of a series - the main event
- Convergence or Divergence? That Is the Question
- A no-brainer divergence test: The nth term test
- Three basic series and their convergence/ divergence tests
- Geometric series
- p-series
- Telescoping series
- Three comparison tests for convergence/ divergence
- The direct comparison test
- The limit comparison test
- The integral comparison test
- The two "R" tests: Ratios and roots
- Alternating Series
- Finding absolute versus conditional convergence
- The alternating series test
- Keeping All the Tests Straight
- Practice Questions Answers and Explanations
- Whaddya Know? Chapter 19 Quiz
- Answers to Chapter 19 Quiz
- Index
- EULA
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