typed by Mr. Sean Bird, updated 
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 undefinedlocal 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 . 
BC TOPICS and important TRIG identities and values
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 θ_{1} and θ_{2} 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

Pythagorean
(others are easily derivable by dividing by sin^{2}x or cos^{2}x)
Reciprocal
OddEven 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 
This is available at http://covenantchristian.org/bird/Smart/Calc1/StuffMUSTknowColdNew.htm