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

Euler’s Method

If given that  and that the solution passes through (x_o, y_o),

In other words:

Polar Curve

For a polar curve r(θ), the

AREA inside a “leaf” is

where θ₁ and θ₂ are the “first” two times that r = 0.

The SLOPE of r(θ) at a given θ is

Integration by Parts

Ratio Test

The series  converges if

If the limit equal 1, you know nothing.

Trig Identities

Double Argument

Integral of Log

Use IBP and let u = ln x (Recall u=LIPET)

Taylor Series

If the function f is “smooth” at x = a, then it can be approximated by the n^th degree polynomial

Lagrange Error Bound

If  is the n^th degree Taylor polynomial of f(x) about c and  for all t between x and c, then

Pythagorean

(others are easily derivable by dividing by sin²x or cos²x)

Reciprocal

Odd-Even

sin(x) =  sin x     (odd)

cos(x) = cos x       (even)

Some more handy INTEGRALS:

Maclaurin Series

A Taylor Series about x = 0 is called Maclaurin.

Alternating Series Error Bound

If  is the N^th partial sum of a convergent alternating series, then

Geometric Series

diverges if |r|≥1;   converges to  if |r|<1