typed by Mr. Sean Bird, Covenant Christian High School

updated April 6, 2006

AP CALCULUS

Stuff you MUST know Cold

* means topic only on BC

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

local maximum:

    goes (+,0,) or (+,und,) or  <0

point of inflection: concavity changes

   goes from (+,0,), (,0,+),

Differentiation Rules

   Chain Rule

   Product Rule

   Quotient Rule

Approx. Methods for Integration

Trapezoidal Rule

Simpson’s Rule

Theorem of the Mean Value

i.e. AVERAGE VALUE

Basic Derivatives

“PLUS A CONSTANT”

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].

The Fundamental Theorem of Calculus

Corollary to FTC

Solids of Revolution and friends

Disk Method

Washer Method

General volume equation (not rotated)

*Arc Length  

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

.

More Derivatives

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

.

Distance, Velocity, and Acceleration

velocity =  (position)

acceleration =  (velocity)

*velocity vector =  

speed =   *

displacement =  

average velocity =

              =  

Rolle’s Theorem

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

.

l’Hôpital’s Rule

     If ,

then  

Slope of a Parametric equation

Given a x(t) and a y(t) the slope is

Values of Trigonometric

Functions for Common Angles