Calculus for dummies

What is calculus?

The two big ideas of calculus : differentiation and integration-plus infinite series

Why calculus works

Pre-algebra and algebra review

Funky functions and their groovy graphs

The trig tango

Limits and continuity

Evaluating limits

Differentiation orientation

Differentiation rules : yeah, man, it rules

Differentiation and the shape of curves

Your problems are solved : differentiation to the rescue!

More differentiation problems : going off on a tangent

Intro to integration and approximating area

Integration : it's backwards differentiation

Integration techniques for experts

Forget Dr. Phil : use the integral to solve problems

Taming the infinite with improper integrals

Infinite series

Ten things to remember

Ten things to forget

Ten things you can't get away with.

Introduction -- About this book -- Conventions used in this book -- How to use this book -- Foolish assumptions -- How this book is organized -- Icons used in this book -- Where to go from here -- pt. I. An overview of calculus -- 1. What is calculus? -- What calculus is not -- So what is calculus already? -- Real-world examples of calculus -- 2. The two big ideas of calculus : differentiation and integration -- Defining differentiation -- Investigating integration -- Sorting out infinite series -- 3. Why calculus works -- The limit concept : a mathematical microscope -- What happens when you zoom in -- Two caveats, or precision, preschmidgen -- pt. II. Warming up with calculus prerequisites -- 4. Pre-algebra and algebra review -- Fine-tuning your fractions -- Absolute value : absolutely easy -- Empowering your powers -- Rooting for roots -- Logarithms, this is not an event at a lumberjack competition -- Factoring schmactoring, when am I ever going to need it? -- Solving quadratic equations -- 5. Funky functions and their groovy graphs -- What is a function? -- What does a function look like? -- Common functions and their graphs -- Inverse functions -- Shifts, reflections, stretches, and shrinks -- 6. The trig tango -- Studying trig at Camp SohCahToa -- Two special right triangles -- Circling the enemy with the unit circle -- Graphing sine, cosine, and tangent -- Inverse trig functions -- Identifying with trig identities --

pt. III. Limits -- 7. Limits and continuity -- Take it to the limit, not -- Linking limits and continuity -- The 33333 limit mnemonic -- 8. Evaluating limits -- Easy limits -- The "real deal" limit problems -- Evaluating limits at infinity -- pt. IV. Differentiation -- 9. Differentiation orientation -- Differentiating : it's just finding the slope -- The derivative : it's just a rate -- 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 -- 10. Differentiation rules, yeah, man, it rules -- Basic differentiation rules -- Differentiation rules for experts, oh, yeah, I'm a calculus wonk -- Differentiating implicitly -- Getting into the rhythm with logarithmic differentiation -- Differentiating inverse functions -- Scaling the heights of higher order derivatives -- 11. Differentiation and the shape of curves -- Taking a calculus road trip -- Finding local extrema, my ma, she's like, totally extreme -- 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, GRRRRR -- 12. Your problems are solved : differentiation to the rescue! -- Getting the most (or lease) out of life : optimization problems -- Yo-yo a go-go : position, velocity, and acceleration -- Related rates, they rate, relatively -- Tangents and normals : joined at the hip -- Straight shooting with linear approximations -- Business and economics problems --

pt. V. Integration and infinite series

13. Intro to integration and approximating area

Integration : just fancy addition

Finding the area under a curve

Dealing with negative area

Approximating area

Getting fancy with summation notation

Finding exact area with the definite integral

Approximating area with the trapezoid rule and Simpson's rule

14. Integration : it's backwards differentiation

Antidifferentiation, that's differentiation in reverse

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

Finding antiderivatives : three basic techniques

Finding area with substantial problems

15. Integration techniques for experts

Integration by parts : divide and conquer

Tricky trig integrals

Your worst nightmare : trigonometric substitution

The As, Bs, and Cxs of partial fractions

16. Forget Dr. Phil : use the integral to solve problems

The mean value theorem for integrals and average value

The area between two curves, double the fun

Finding the volumes of weird solids

Analyzing arc length

Surfaces of revolution, pass the bottle 'round

L'Hôspital's rule : calculus for the sick

Improper integrals : just look at the way that integral is holding its fork!

17. Infinite series

Sequences and series : what they're all about

Convergence or divergence? That is the question

Alternating series

Keeping all the tests straight

pt. VI. The part of tens

18. Ten things to remember

19. Ten things to forget

20. Ten things you can't get away with.

Title Page; Copyright Page; Contents at a Glance; Table of Contents; Introduction; About This Book; Foolish Assumptions; Icons Used in This Book; Beyond the Book; Where to Go from Here; Part I: 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; Investigating Integration; Sorting Out Infinite Series; Chapter 3: Why Calculus Works; The Limit Concept: A Mathematical Microscope

What Happens When You Zoom InTwo Caveats, or Precision, Preschmidgen; Part II: Warming Up with Calculus Prerequisites; Chapter 4: Pre-Algebra and Algebra Review; Fine-Tuning Your Fractions; Absolute Value

Absolutely Easy; Empowering Your Powers; Rooting for Roots; Logarithms

This Is Not an Event at a Lumberjack Competition; Factoring Schmactoring

When Am I Ever Going to Need It?; Solving Quadratic Equations; Chapter 5: Funky Functions and Their Groovy Graphs; What Is a Function?; What Does a Function Look Like?; Common Functions and Their Graphs; Inverse Functions

Shifts, Reflections, Stretches, and ShrinksChapter 6: The Trig Tango; Studying Trig at Camp SohCahToa; Two Special Right Triangles; Circling the Enemy with the Unit Circle; Graphing Sine, Cosine, and Tangent; Inverse Trig Functions; Identifying with Trig Identities; Part III: Limits; Chapter 7: Limits and Continuity; Take It to the Limit

NOT; Linking Limits and Continuity; The 33333 Limit Mnemonic; Chapter 8: Evaluating Limits; Easy Limits; The "Real Deal" Limit Problems; Evaluating Limits at Infinity; Part IV: Differentiation; Chapter 9: Differentiation Orientation

Differentiating: It's Just Finding the SlopeThe Derivative: It's Just a Rate; 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; Chapter 10: Differentiation Rules

Yeah, Man, It Rules; Basic Differentiation Rules; Differentiation Rules for Experts

Oh, Yeah, I'm a Calculus Wonk; Differentiating Implicitly; Getting into the Rhythm with Logarithmic Differentiation; Differentiating Inverse Functions; Scaling the Heights of Higher Order Derivatives

Chapter 11: Differentiation and the Shape of CurvesTaking a Calculus Road Trip; Finding Local Extrema

My Ma, She's Like, Totally Extreme; 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

GRRRRR; Chapter 12: Your Problems Are Solved: Differentiation to the Rescue!; Getting the Most (or Least) Out of Life: Optimization Problems; Yo-Yo a Go-Go: Position, Velocity, and Acceleration

Related Rates

They Rate, Relatively