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Stuff you MUST know Cold for the AP Calculus Exam

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## Stuff you MUST know Cold for the AP Calculus Exam

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**Curve sketching and analysis**y = f(x) must be continuous at each: • critical point: = 0 or undefined. • local minimum: goes (–,0,+) or (–,und,+) or > 0 at stationary pt • local maximum: goes (+,0,–) or (+,und,–) or < 0 at stationary pt • point of inflection: concavity changes goes from (+,0,–), (–,0,+), (+,und,–), or (–,und,+)**Basic Integrals**Plus a CONSTANT**More Derivatives**Recall “change of base”**Differentiation Rules**Chain Rule Product Rule Quotient Rule**The Fundamental Theorem of Calculus**Other part of the FTC**Intermediate Value Theorem**• If the function f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a, b) such that f(c) = y. Mean Value Theorem . . • If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that**Mean Value Theorem & Rolle’s Theorem**If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), AND f(a) = f(b), then there is at least one number x = c in (a, b) such that f '(c) = 0.**Extreme Value Theorem**If the function f(x) is continuous on [a, b], then f has both an absolute maximum and an absolute minimum on [a,b] The absolute extremes occur either at the critical points or at the endpoints.**Approximation Methods for Integration**Riemann Sum LRAM when ck is a LEFT endpoint RRAM when ck is a RIGHT endpoint MRAM when ck is a MIDPOINT Trapezoidal Rule**Theorem of the Mean Valuei.e. AVERAGE VALUE**• If the function f(x) is continuous on [a, b] and the first derivative exists on the interval (a, b), then there exists a number x = c on (a, b) such that • This value f(c) is the “average value” of the function on the interval [a, b].**Solids of Revolution and friends**• Arc Length • Disk Method • WasherMethod • General volume equation (not rotated)**Distance, Velocity, and Acceleration**velocity = (position) average velocity = (velocity) acceleration = speed = velocity vector = displacement =**Values of Trigonometric Functions for Common Angles**π/3 = 60° π/6 = 30° θ sin θ cos θ tan θ 0° 0 1 0 sine ,30° cosine ,45° 1 ,60° ,90° 1 0 ∞ π,180° 0 –1 0**Trig Identities**Double Argument Pythagorean sine cosine**Slope – Parametric & Polar**Parametric equation • Given a x(t) and a y(t) the slope is Polar • Slope of r(θ) at a given θ is What is y equal to in terms of r and θ ? x?**Polar Curve**For a polar curve r(θ), the AREA inside a “leaf” is (Because instead of infinitesimally small rectangles, use triangles) where θ1 and θ2 are the “first” two times that r = 0. We know arc length l = rθ and**l’Hôpital’s Rule**If then**Other Indeterminate forms:**Write as a ratio Use Logs**Integration by Parts**We know the product rule Antiderivative product rule (Use u = LIPET) e.g. L I P E T Logarithm Inverse Polynomial Exponential Trig Let u = ln x dv = dx du = dx v = x**Maclaurin SeriesA Taylor Series about x = 0 is called**Maclaurin. Taylor Series If the function f is “smooth” at x = a, then it can be approximated by the nth degree polynomial