Calculus for dummies

pt. I. An overview of calculus 1. What is calculus? 2. The two big ideas of calculus : differentiation and integration Slope Rate Plus infinite series Divergent series Convergent series 3. Why calculus works The limit concept : a mathematical microscope Precision Infinity

pt. II. Warming up with calculus prerequisites 4. Pre-algebra and algebra review Fractions Multiplying fractions Dividing fractions Adding fractions Subtracting fractions Canceling in fractions Absolute value Powers Roots Simplifying roots Logarithms Factoring GCF Trinomial factoring Solving quadratic equations Factoring The quadratic formula Completing the square 5. Funky functions and their groovy graphs Independent and dependent variables Function notation Composite functions Common functions and their graphs Lines in the plane Parabolic and absolute value functions Couple oddball functions Exponential functions Logarithmic functions Inverse functions Horizontal transformations Vertical transformations 6. The trig tango Right triangles Unit circle Measuring angles with radians Hypotenuse Graphing sine, cosine, and tangent Inverse trig functions Trig identities

pt. III. Limits 7. Limits and continuity One-sided limits Limits and vertical asymptotes Limits and horizontal asymptotes Calculating instantaneous speed with limits Linking limits and continuity 8. Evaluating limits Figuring a limit with your calculator Solving limit problems with algebra Evaluating limits at infinity Limits at infinity and horizontal asymptotes Solving limits at infinity with a calculator Solving limits at infinity with algebra

pt. IV. Differentiation 9. Differentiation orientation The slope off a line The derivative of a line The derivative : it's just a rate Calculus on the playground Speed The rate-slope connection The derivative of a curve The difference quotient Average rate and instantaneous rate 10. Differentiation rules : yeah, man, it rules Basic differentiation rules The constant rule The power rule The constant multiple rule The sum rule The difference rule Differentiating trig functions Differentiating exponential and logarithmic functions The product rule The quotient rule The chain rule Differentiating implicitly Logarithmic differentiation Differentiating inverse functions Higher order derivatives 11. Differentiation and the shape of curves Positive and negative slopes Concavity and inflection points A local minimum The absolute maximum Finding local extrema Critical numbers Finding absolute extrema on a closed interval Finding absolute extrema over a function's entire domain Locating concavity and inflection points Graphs of derivatives The mean value theorem 12. Your problems are solved : differentiation to the rescue! Optimization problems Maximum volume of a box Maximum area of a corral Position, velocity, and acceleration Velocity, speed and acceleration Maximum and minimum height Velocity and displacement Speed and distance traveled Related rates 13. More differentiation problems : going off on a tangent Tangents and normals The tangent line problem The normal line problem Linear approximations Business and economics problems Managing marginals in economics

pt. V. Integration and infinite series 14. Intro to integration and approximating area Integration : just fancy addition Finding the area under a curve Approximating area Left sums Right sums Midpoint sums Summation notation Riemann sums with sigma notation Finding exact area with the definite integral Trapezoid rule and Simpson's rule (Thomas Simpson 1710-1761) 15. Integration : it's backwards differentiation Antidifferentiation Area function Fundamental theorem of calculus Antiderivatives Finding area with substitution problems 16. Integration techniques for experts Integration by parts Trig integrals Integrals containing sines and cosines Integrals containing secants and tangents or cosecants Trigonometric substitution Partial fractions 17. Forget Dr. Phil : use the integral to solve problems The mean value theorem for integrals and average value The area between two curves Finding the volumes of weird solids Analyzing arc length Surfaces of revolution 18. Taming the infinite with improper integrals L/Hôpital's rule Improper integrals Improper integrals with vertical asymptotes Improper integrals with one or two infinite limits of integration 19. Infinite series Sequences and series Stringing sequences Summing series Convergence or divergence Alternating series

pt. VI. The part of tens

20. Ten things to remember

The product rule

The quotient rule

21. Ten things to forget

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

What is calculus?

The two big ideas of calculus: differentiation and integration

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!

Intro to integration and approximating area

Integration: it's backwards differentiation

Integration techniques for experts

Forget Dr. Phil: use the integral to solve problems

Infinite series

Ten things to remember

Ten things to forget

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