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

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