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