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Introduction | |

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Linear Equations and Inequalities: Problems containing x to the first power | |

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Linear Geometry: Creating, graphing, and measuring lines and segments | |

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Linear Inequalities and Interval Notation: Goodbye equal sign, hello parentheses and brackets | |

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Absolute Value Equations and Inequalities: Solve two things for the price of one | |

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Systems of Equations and Inequalities: Find a common solution | |

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Polynomials: Because you can't have exponents of I forever | |

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Exponential and Radical Expressions: Powers and square roots | |

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Operations on Polynomial Expressions: Add, subtract, multiply, and divide polynomials | |

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Factoring Polynomials: Reverse the multiplication process | |

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Solving Quadratic Equations: Equations that have a highest exponent of 2 | |

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Rational Expressions: Fractions, fractions, and more fractions | |

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Adding and Subtracting Rational Expressions: Remember the least common denominator? | |

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Multiplying and Dividing Rational Expressions: Multiplying = easy, dividing = almost as easy | |

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Solving Rational Equations: Here comes cross multiplication | |

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Polynomial and Rational Inequalities: Critical numbers break up your number line | |

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Functions: Now you'll start seeing f(x) all over the place | |

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Combining Functions: Do the usual (+,-,x,[divide]) or plug 'em into each other | |

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Graphing Function Transformations: Stretches, squishes, flips, and slides | |

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Inverse Functions: Functions that cancel other functions out | |

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Asymptotes of Rational Functions: Equations of the untouchable dotted line | |

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Logarithmic and Exponential Functions: Functions like log, x, lu x, 4x, and e[superscript x] | |

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Exploring Exponential and Logarithmic Functions: Harness all those powers | |

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Natural Exponential and Logarithmic Functions: Bases of e, and change of base formula | |

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Properties of Logarithms: Expanding and sauishing log expressions | |

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Solving Exponential and Logarithmic Equations: Exponents and logs cancel each other out | |

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Conic Sections: Parabolas, circles, ellipses, and hyperbolas | |

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Parabolas: Graphs of quadratic equations | |

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Circles: Center + radius = round shapes and easy problems | |

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Ellipses: Fancy word for "ovals" | |

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Hyperbolas: Two-armed parabola-looking things | |

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Fundamentals of Trigonometry: Inject sine, cosine, and tangent into the mix | |

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Measuring Angles: Radians, degrees, and revolutions | |

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Angle Relationships: Coterminal, complementary, and supplementary angles | |

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Evaluating Trigonometric Functions: Right triangle trig and reference angles | |

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Inverse Trigonometric Functions: Input a number and output an angle for a change | |

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Trigonometric Graphs, Identities, and Equations: Trig equations and identity proofs | |

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Graphing Trigonometric Transformations: Stretch and Shift wavy graphs | |

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Applying Trigonometric Identities: Simplify expressions and prove identities | |

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Solving Trigonometric Equations: Solve for [theta] instead of x | |

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Investigating Limits: What height does the function intend to reach | |

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Evaluating One-Sided and General Limits Graphically: Find limits on a function graph | |

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Limits and Infinity: What happens when x or f(x) gets huge? | |

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Formal Definition of the Limit: Epsilon-delta problems are no fun at all | |

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Evaluating Limits: Calculate limits without a graph of the function | |

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Substitution Method: As easy as plugging in for x | |

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Factoring Method: The first thing to try if substitution doesn't work | |

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Conjugate Method: Break this out to deal with troublesome radicals | |

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Special Limit Theorems: Limit formulas you should memorize | |

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Continuity and the Difference Quotient: Unbreakable graphs | |

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Continuity: Limit exists + function defined = continuous | |

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Types of Discontinuity: Hole vs. breaks, removable vs. nonremovable | |

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The Difference Quotient: The "long way" to find the derivative | |

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Differentiability: When does a derivative exist? | |

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Basic Differentiation Methods: The four heavy hitters for finding derivatives | |

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Trigonometric, Logarithmic, and Exponential Derivatives: Memorize these formulas | |

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The Power Rule: Finally a shortcut for differentiating things like x[Prime] | |

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The Product and Quotient Rules: Differentiate functions that are multiplied or divided | |

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The Chain Rule: Differentiate functions that are plugged into functions | |

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Derivatives and Function Graphs: What signs of derivatives tell you about graphs | |

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Critical Numbers: Numbers that break up wiggle graphs | |

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Signs of the First Derivative: Use wiggle graphs to determine function direction | |

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Signs of the Second Derivative: Points of inflection and concavity | |

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Function and Derivative Graphs: How are the graphs of f, f[prime], and f[Prime] related? | |

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Basic Applications of Differentiation: Put your derivatives skills to use | |

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Equations of Tangent Lines: Point of tangency + derivative = equation of tangent | |

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The Extreme Value Theorem: Every function has its highs and lows | |

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Newton's Method: Simple derivatives can approximate the zeroes of a function | |

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L'Hopital's Rule: Find limits that used to be impossible | |

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Advanced Applications of Differentiation: Tricky but interesting uses for derivatives | |

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The Mean Rolle's and Rolle's Theorems: Average slopes = instant slopes | |

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Rectilinear Motion: Position, velocity, and acceleration functions | |

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Related Rates: Figure out how quickly the variables change in a function | |

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Optimization: Find the biggest or smallest values of a function | |

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Additional Differentiation Techniques: Yet more ways to differentiate | |

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Implicit Differentiation: Essential when you can't solve a function for y | |

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Logarithmic Differentiation: Use log properties to make complex derivatives easier | |

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Differentiating Inverse Trigonometric Functions: 'Cause the derivative of tan[superscript -1] x ain't sec[superscript -2] x | |

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Differentiating Inverse Functions: Without even knowing what they are! | |

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Approximating Area: Estimating the area between a curve and the x-axiz | |

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Informal Riemann Sums: Left, right, midpoint, upper, and lower sums | |

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Trapezoidal Rule: Similar to Riemann sums but much more accurate | |

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Simpson's Rule: Approximates area beneath curvy functions really well | |

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Formal Riemann Sums: You'll want to poke your "i"s out | |

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Integration: Now the derivative's not the answer, it's the question | |

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Power Rule for Integration: Add I to the exponent and divide by the new power | |

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Integrating Trigonometric and Exponential Functions: Trig integrals look nothing like trig derivatives | |

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The Fundamental Theorem of Calculus: Integration and area are closely related | |

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Substitution of Variables: Usually called u-substitution | |

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Applications of the Fundamental Theorem: Things to do with definite integrals | |

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Calculating the Area Between Two Curves: Instead of just a function and the x-axis | |

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The Mean Value Theorem for Integration: Rectangular area matches the area beneath a curve | |

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Accumulation Functions and Accumulated Change: Integrals with x limits and "real life" uses | |

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Integrating Rational Expressions: Fractions inside the integral | |

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Separation: Make one big ugly fraction into smaller, less ugly ones | |

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Long Division: Divide before you integrate | |

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Applying Inverse Trigonometric Functions: Very useful, but only in certain circumstances | |

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Completing the Square: For quadratics down below and no variables up top | |

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Partial Fractions: A fancy way to break down big fractions | |

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Advanced Integration Techniques: Even more ways to find integrals | |

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Integration by Parts: It's like the product rule, but for integrals | |

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Trigonometric Substitution: Using identities and little right triangle diagrams | |

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Improper Integrals: Integrating despite asymptotes and infinite boundaries | |

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Cross-Sectional and Rotational Volume: Please put on your 3-D glasses at this time | |

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Volume of a Solid with Known Cross-Sections: Cut the solid into pieces and measure those instead | |

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Disc Method: Circles are the easiest possible cross-sections | |

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Washer Method: Find volumes even if the "solids" aren't solid | |

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Shell Method: Something to fall back on when the washer method fails | |

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Advanced Applications of Definite Integrals: More bounded integral problems | |

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Arc Length: How far is it from point A to point B along a curvy road? | |

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Surface Area: Measure the "skin" of a rotational solid | |

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Centroids: Find the center of gravity for a two-dimensional shape | |

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Parametric and Polar Equations: Writing equations without x and y | |

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Parametric Equations: Like revolutionaries in Boston Harbor, just add + | |

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Polar Coordinates: Convert from (x,y) to (r, [theta]) and vice versa | |

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Graphing Polar Curves: Graphing with r and [theta] instead of x and y | |

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Applications of Parametric and Polar Differentiation: Teach a new dog some old differentiation tricks | |

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Applications of Parametric and Polar Integration: Feed the dog some integrals too? | |

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Differential Equations: Equations that contain a derivative | |

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Separation of Variables: Separate the y's and dy's from the x's and dx's | |

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Exponential Growth and Decay: When a population's change is proportional to its size | |

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Linear Approximations: A graph and its tangent line sometimes look a lot alike | |

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Slope Fields: They look like wind patterns on a weather map | |

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Euler's Method: Take baby steps to find the differential equation's solution | |

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Basic Sequences and Series: What's uglier than one fraction? Infinitely many | |

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Sequences and Convergence: Do lists of numbers know where they're going? | |

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Series and Basic Convergence Tests: Sigma notation and the nth term divergence test | |

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Telescoping Series and p-Series: How to handle these easy-to-spot series | |

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Geometric Series: Do they converge, and if so, what's the sum? | |

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The Integral Test: Infinite series and improper integrals are related | |

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Additional Infinite Series Convergence Tests: For use with uglier infinite series | |

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Comparison Test: Proving series are bigger than big and smaller than small | |

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Limit Comparison Test: Series that converge or diverge by association | |

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Ratio Test: Compare neighboring terms of a series | |

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Root Test: Helpful for terms inside radical signs | |

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Alternating Series Test and Absolute Convergence: What if series have negative terms? | |

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Advanced Infinite Series: Series that contain x's | |

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Power Series: Finding intervals of convergence | |

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Taylor and Maclaurin Series: Series that approximate function values | |

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Important Graphs to memorize and Graph Transformations | |

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The Unit Circle | |

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Trigonometric Identities | |

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Derivative Formulas | |

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Anti-Derivative Formulas | |

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Index | |