Calculus All-in-One For Dummies

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