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

by Kelley, W. Michael

• ISBN13: 9781592575121
• ISBN10: 1592575129
• Format: Paperback
• Copyright: 2007-01-02
• Publisher: PENGUIN

Summary

Now students have nothing to fear …Math textbooks can be as baffling as the subject they’re teaching. Not anymore. The best-selling author of The Complete Idiot’s Guide to Calculushas taken what appears to be a typical calculus workbook, chock full of solved calculus problems, and made legible notes in the margins, adding missing steps and simplifying solutions. Finally, everything is made perfectly clear. Students will be prepared to solve those obscure problems that were never discussed in class but always seem to find their way onto exams. --Includes 1,000 problems with comprehensive solutions --Annotated notes throughout the text clarify what’s being asked in each problem and fill in missing steps --Kelley is a former award-winning calculus teacher

Author Biography

W. Michael Kelley is a former award-winning calculus teacher and author of The Complete Idiot-'s Guide to Calculus, The Complete Idiot-'s Guide to Precalculus, and The Complete Idiot-'s Guide to Algebra. He is also the founder and editor of calculus-help.com, which helps thousands of students conquer their math anxiety every month.

Table of Contents

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