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Overview
The only way to learn calculus is to do calculus problems. Lots of them!
And that's what you get in this bookmore calculus problems than your worst nightmare—but with a BIG difference. Award-winning calculus teacher W. Michael Kelley has been through the whole book and made a ton of notes, so you get:
• 1,000 problems with comprehensive solutions
• Annotated notes throughout the text, clarifying exactly what's being asked
• Really detailed answers (no more skipped steps!)
• Extra explanations that make what's baffling perfectly clear
• Pointers to other problems that show skills you need
And all of the major players are here: limits, continuity, derivatives, integrals, tangent lines, velocity, acceleration, area, volume, infinite series—even the really tough stuff like epsilon-delta proofs and formal Riemann sums.
So dig in to your heart's content!
And that's what you get in this bookmore calculus problems than your worst nightmare—but with a BIG difference. Award-winning calculus teacher W. Michael Kelley has been through the whole book and made a ton of notes, so you get:
• 1,000 problems with comprehensive solutions
• Annotated notes throughout the text, clarifying exactly what's being asked
• Really detailed answers (no more skipped steps!)
• Extra explanations that make what's baffling perfectly clear
• Pointers to other problems that show skills you need
And all of the major players are here: limits, continuity, derivatives, integrals, tangent lines, velocity, acceleration, area, volume, infinite series—even the really tough stuff like epsilon-delta proofs and formal Riemann sums.
So dig in to your heart's content!
Product Details
ISBN-13: | 9781592575121 |
---|---|
Publisher: | DK |
Publication date: | 01/02/2007 |
Series: | Humongous Books |
Edition description: | Reissue |
Pages: | 576 |
Sales rank: | 200,461 |
Product dimensions: | 8.52(w) x 10.82(h) x 1.19(d) |
Age Range: | 18 Years |
About the Author
W. Michael Kelley is a former award-winning calculus teacher and the author of six math books, including The Complete Idiot's Guide to Algebra, Second Edition, and The Humongous Book of Calculus Problems. Kelley received an award from the Maryland Council of Teachers of Mathematics recognizing him as an Outstanding High School Mathematics Teacher and four-years-running title of Most Popular Teacher in his home school. Kelley is also the founder and editor of calculus-help.com.
Table of Contents
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
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