The Humongous Book of Calculus Problems: For People Who Don't Speak Math

Introduction     ixLinear Equations and Inequalities: Problems containing x to the first power     1Linear Geometry: Creating, graphing, and measuring lines and segments     2Linear Inequalities and Interval Notation: Goodbye equal sign, hello parentheses and brackets     5Absolute Value Equations and Inequalities: Solve two things for the price of one     8Systems of Equations and Inequalities: Find a common solution     11Polynomials: Because you can't have exponents of I forever     15Exponential and Radical Expressions: Powers and square roots     16Operations on Polynomial Expressions: Add, subtract, multiply, and divide polynomials     18Factoring Polynomials: Reverse the multiplication process     21Solving Quadratic Equations: Equations that have a highest exponent of 2     23Rational Expressions: Fractions, fractions, and more fractions     27Adding and Subtracting Rational Expressions: Remember the least common denominator?     28Multiplying and Dividing Rational Expressions: Multiplying = easy, dividing = almost as easy     30Solving Rational Equations: Here comes cross multiplication     33Polynomial and Rational Inequalities: Critical numbers break up your number line     35Functions: Now you'll start seeing f(x) allover the place     41Combining Functions: Do the usual (+,-,x,[divide]) or plug 'em into each other     42Graphing Function Transformations: Stretches, squishes, flips, and slides     45Inverse Functions: Functions that cancel other functions out     50Asymptotes of Rational Functions: Equations of the untouchable dotted line     53Logarithmic and Exponential Functions: Functions like log, x, lu x, 4x, and e[superscript x]     57Exploring Exponential and Logarithmic Functions: Harness all those powers     58Natural Exponential and Logarithmic Functions: Bases of e, and change of base formula     62Properties of Logarithms: Expanding and sauishing log expressions     63Solving Exponential and Logarithmic Equations: Exponents and logs cancel each other out     66Conic Sections: Parabolas, circles, ellipses, and hyperbolas     69Parabolas: Graphs of quadratic equations     70Circles: Center + radius = round shapes and easy problems     76Ellipses: Fancy word for "ovals"     79Hyperbolas: Two-armed parabola-looking things     85Fundamentals of Trigonometry: Inject sine, cosine, and tangent into the mix     91Measuring Angles: Radians, degrees, and revolutions     92Angle Relationships: Coterminal, complementary, and supplementary angles     93Evaluating Trigonometric Functions: Right triangle trig and reference angles     95Inverse Trigonometric Functions: Input a number and output an angle for a change     102Trigonometric Graphs, Identities, and Equations: Trig equations and identity proofs     105Graphing Trigonometric Transformations: Stretch and Shift wavy graphs     106Applying Trigonometric Identities: Simplify expressions and prove identities     110Solving Trigonometric Equations: Solve for [theta] instead of x     115Investigating Limits: What height does the function intend to reach     123Evaluating One-Sided and General Limits Graphically: Find limits on a function graph     124Limits and Infinity: What happens when x or f(x) gets huge?     129Formal Definition of the Limit: Epsilon-delta problems are no fun at all     134Evaluating Limits: Calculate limits without a graph of the function     137Substitution Method: As easy as plugging in for x     138Factoring Method: The first thing to try if substitution doesn't work     141Conjugate Method: Break this out to deal with troublesome radicals     146Special Limit Theorems: Limit formulas you should memorize     149Continuity and the Difference Quotient: Unbreakable graphs      151Continuity: Limit exists + function defined = continuous     152Types of Discontinuity: Hole vs. breaks, removable vs. nonremovable     153The Difference Quotient: The "long way" to find the derivative     163Differentiability: When does a derivative exist?     166Basic Differentiation Methods: The four heavy hitters for finding derivatives     169Trigonometric, Logarithmic, and Exponential Derivatives: Memorize these formulas     170The Power Rule: Finally a shortcut for differentiating things like x[Prime]     172The Product and Quotient Rules: Differentiate functions that are multiplied or divided     175The Chain Rule: Differentiate functions that are plugged into functions     179Derivatives and Function Graphs: What signs of derivatives tell you about graphs     187Critical Numbers: Numbers that break up wiggle graphs     188Signs of the First Derivative: Use wiggle graphs to determine function direction     191Signs of the Second Derivative: Points of inflection and concavity     197Function and Derivative Graphs: How are the graphs of f, f[prime], and f[Prime] related?     202Basic Applications of Differentiation: Put your derivatives skills to use     205Equations of Tangent Lines: Point of tangency + derivative = equation of tangent      206The Extreme Value Theorem: Every function has its highs and lows     211Newton's Method: Simple derivatives can approximate the zeroes of a function     214L'Hopital's Rule: Find limits that used to be impossible     218Advanced Applications of Differentiation: Tricky but interesting uses for derivatives     223The Mean Rolle's and Rolle's Theorems: Average slopes = instant slopes     224Rectilinear Motion: Position, velocity, and acceleration functions     229Related Rates: Figure out how quickly the variables change in a function     233Optimization: Find the biggest or smallest values of a function     240Additional Differentiation Techniques: Yet more ways to differentiate     247Implicit Differentiation: Essential when you can't solve a function for y     248Logarithmic Differentiation: Use log properties to make complex derivatives easier     255Differentiating Inverse Trigonometric Functions: 'Cause the derivative of tan[superscript -1] x ain't sec[superscript -2] x     260Differentiating Inverse Functions: Without even knowing what they are!     262Approximating Area: Estimating the area between a curve and the x-axiz     269Informal Riemann Sums: Left, right, midpoint, upper, and lower sums     270Trapezoidal Rule: Similar to Riemann sums but much more accurate     281Simpson's Rule: Approximates area beneath curvy functions really well     289Formal Riemann Sums: You'll want to poke your "i"s out     291Integration: Now the derivative's not the answer, it's the question     297Power Rule for Integration: Add I to the exponent and divide by the new power     298Integrating Trigonometric and Exponential Functions: Trig integrals look nothing like trig derivatives     301The Fundamental Theorem of Calculus: Integration and area are closely related     303Substitution of Variables: Usually called u-substitution     313Applications of the Fundamental Theorem: Things to do with definite integrals     319Calculating the Area Between Two Curves: Instead of just a function and the x-axis     320The Mean Value Theorem for Integration: Rectangular area matches the area beneath a curve     326Accumulation Functions and Accumulated Change: Integrals with x limits and "real life" uses     334Integrating Rational Expressions: Fractions inside the integral     343Separation: Make one big ugly fraction into smaller, less ugly ones     344Long Division: Divide before you integrate     347Applying Inverse Trigonometric Functions: Very useful, but only in certain circumstances     350Completing the Square: For quadratics down below and no variables up top     353Partial Fractions: A fancy way to break down big fractions     357Advanced Integration Techniques: Even more ways to find integrals     363Integration by Parts: It's like the product rule, but for integrals     364Trigonometric Substitution: Using identities and little right triangle diagrams     368Improper Integrals: Integrating despite asymptotes and infinite boundaries     383Cross-Sectional and Rotational Volume: Please put on your 3-D glasses at this time     389Volume of a Solid with Known Cross-Sections: Cut the solid into pieces and measure those instead     390Disc Method: Circles are the easiest possible cross-sections     397Washer Method: Find volumes even if the "solids" aren't solid     406Shell Method: Something to fall back on when the washer method fails     417Advanced Applications of Definite Integrals: More bounded integral problems     423Arc Length: How far is it from point A to point B along a curvy road?     424Surface Area: Measure the "skin" of a rotational solid     427Centroids: Find the center of gravity for a two-dimensional shape     432Parametric and Polar Equations: Writing equations without x and y     443Parametric Equations: Like revolutionaries in Boston Harbor, just add +     444Polar Coordinates: Convert from (x,y) to (r, [theta]) and vice versa     448Graphing Polar Curves: Graphing with r and [theta] instead of x and y     451Applications of Parametric and Polar Differentiation: Teach a new dog some old differentiation tricks     456Applications of Parametric and Polar Integration: Feed the dog some integrals too?     462Differential Equations: Equations that contain a derivative     467Separation of Variables: Separate the y's and dy's from the x's and dx's     468Exponential Growth and Decay: When a population's change is proportional to its size     473Linear Approximations: A graph and its tangent line sometimes look a lot alike     480Slope Fields: They look like wind patterns on a weather map     482Euler's Method: Take baby steps to find the differential equation's solution     488Basic Sequences and Series: What's uglier than one fraction? Infinitely many     495Sequences and Convergence: Do lists of numbers know where they're going?     496Series and Basic Convergence Tests: Sigma notation and the nth term divergence test     498Telescoping Series and p-Series: How to handle these easy-to-spot series     502Geometric Series: Do they converge, and if so, what's the sum?      505The Integral Test: Infinite series and improper integrals are related     507Additional Infinite Series Convergence Tests: For use with uglier infinite series     511Comparison Test: Proving series are bigger than big and smaller than small     512Limit Comparison Test: Series that converge or diverge by association     514Ratio Test: Compare neighboring terms of a series     517Root Test: Helpful for terms inside radical signs     520Alternating Series Test and Absolute Convergence: What if series have negative terms?     524Advanced Infinite Series: Series that contain x's     529Power Series: Finding intervals of convergence     530Taylor and Maclaurin Series: Series that approximate function values     538Important Graphs to memorize and Graph Transformations     545The Unit Circle     551Trigonometric Identities     553Derivative Formulas     555Anti-Derivative Formulas     557Index     559