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