Stuff you MUST know Cold for the AP Calculus Exam In preparation for Wednesday May 9, AP Physics & Calculus Covenant Christian High School 7525 West 21st Street Indianapolis, IN Phone: 317/ x104 Website: Psalm 111:2 Sean Bird Updated by Mrs. Shak May 2012

Curve sketching and analysis y = f(x) must be continuous at each: critical point: = 0 or undefined. And don’t forget endpoints for absolute min/max local minimum: goes (–,0,+) or (–,und,+) or > 0 local maximum: goes (+,0,–) or (+,und,–) or < 0 point of inflection: concavity changes goes from (+,0,–), (–,0,+) or (+,und,–), or (–,und,+) goes from incr to decr or decr to incr

Basic Derivatives

Basic Integrals

Some more handy integrals

More Derivatives Recall “change of base”

Differentiation Rules Chain Rule Product Rule Quotient Rule

The Fundamental Theorem of Calculus Corollary to FTC

Intermediate Value Theorem. 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 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.

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. Mean Value Theorem & Rolle’s Theorem

Approximation Methods for Integration Trapezoidal Rule Non-Equi-Width Trapezoids

Theorem of the Mean Value i.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].

AVERAGE RATE OF CHANGE of f(x) on [a, b] This value is the “average rate of change” of the function on the interval [a, b]. We use the difference quotient to approximate the derivative in the absence of a function

Solids of Revolution and friends Disk Method Washer Method General volume equation (not rotated) Arc Length * bc topic

Distance, Velocity, and Acceleration velocity =(position) (velocity) speed = displacement = average velocity = acceleration = *velocity vector = *bc topic

Values of Trigonometric Functions for Common Angles 0–10π,180° ∞ 01,90°,60° 4/33/54/553° 1,45° 3/44/53/537°,30° 0100° tan θcos θsin θθ π/3 = 60° π/6 = 30° sine cosine

Trig Identities Double Argument

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 θ 1 and θ 2 are the “first” two times that r = 0. and We know arc length l = r θ

l’Hôpital’s Rule If then

Integration by Parts Antiderivative product rule (Use u = LIPET) e.g. We know the product rule Let u = ln xdv = dx du = dx v = x LIPETLIPET Logarithm Inverse Polynomial Exponential Trig

Maclaurin Series A Taylor Series about x = 0 is called Maclaurin. If the function f is “smooth” at x = a, then it can be approximated by the n th degree polynomial Taylor Series