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
