Stuff you MUST know Cold for the AP

Stuff you MUST know Cold for the AP Calculus Exam in the morning of Wednesday, May 7, 2008. Sean Bird AP Physics & Calculus Covenant Christian High School 7525 West 21 st Street Indianapolis, IN 46214 Phone: 317/390. 0202 x 104 Email: seanbird@covenantchristian. org Website: http: //cs 3. covenantchristian. org/bird Updated April 24, 2009 Psalm 111: 2

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

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 n . . 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 n 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 If the function f(x) is continuous on [a, b], Theorem AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that 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.

Approximation Methods for Trapezoidal Rule Integration Simpson’s Rule Simpson only works for Even sub intervals (odd data points) 1/3 (1 + 4 + 2 + 4 + 1 )

Theorem of the Mean Value i. e. AVERAGE VALUE n 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 n This value f(c) is the “average value” of the function on the interval [a, b].

Solids of Revolution and friends n Disk Method n Arc L engt h n Washer Method n General volume equation (not rotated) * bc topic

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

Values of Trigonometric Functions for Common Angles π/3 = 60° θ sin θ cos θ tan θ 0° 0 1 0 , 30° 37° sine cosine 3/5 4/5 , 45° 53° π/6 = 30° 3/4 1 4/5 3/5 4/3 , 90° 1 0 ∞ π, 180° 0 – 1 0 , 60°

Trig Identities Double Argument

Trig Identities Double Argument Pythagorean sine cosine

Slope – Parametric & Polar Parametric equation n Given a x(t) and a y(t) the slope is Polar n 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. We know arc length l = r θ and

l’Hôpital’s Rule If then

Integration by Parts We know the product rule L Logarithm I Inverse P Polynomial E Exponential T Trig Antiderivative product rule (Use u = LIPET) Let u = ln x e. g. du = dx dv = dx v=x

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