Stuff you MUST know Cold for the AP Calculus Exam in the morning of Wednesday, May 7, AP Physics & Calculus Covenant Christian High School 7525 West 21st Street Indianapolis, IN Phone: 317/ x104 Website: Psalm 111:2 Sean Bird Updated April 24, 2009

Curve sketching and analysis y = f(x) must be continuous at each: critical point: = 0 or undefined. And don’t forget endpoints 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,+), (+,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. 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

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].

Area “Under” and Between Curves Sketch the Area Find Points of Intersection Integrate “top” curve – “bottom” curve (or right curve – left curve) Use Geometry as appropriate

Solids of Revolution and friends Disk Method Washer Method General volume equation (not rotated- cross sections)

Distance, Velocity, and Acceleration velocity =(position) (velocity) speed = displacement = average velocity = acceleration =

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

l’Hôpital’s Rule (BC but nice to know) If then