1.
Average Value Theorem: 1/ (b-a) times the integral on (a, b) of f(x) dx
2.
Chain Rule: d/dx f(g(x)) = f'(g(x)) g'(x)
3.
Definition of a Derivative: lim h→0 (f(x+h) - f(x)) / h
4.
Definition of Continuity: 1. lim x→c f(x) exists.
2. f(c) exists.
3. lim x→c f(x) = f(c)
5.
Derivative of an Inverse Function: g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x)
6.
Extrema Value Theorem: If f is continuous on the closed interval [a, b], then f has both a maximum and a minimum on the interval.
7.
Fundamental Theorem of Calculus: The integral on (a, b) of f(x) dx = F(b) - F(a)
8.
Intermediate Value Theorem: If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b) then there is at least one number c in [a, b] such that f(c) = k
9.
Mean Value Theorem: f'(c) = (f(b) - f(a))/ (b - a)
10.
Mean Value Theorem (Integrals): The integral on (a, b) of f(x) dx = f(c) (b - a)
11.
Product Rule: d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x)
12.
Quotient Rule: d/dx (g(x)/ h(x)) = (h(x) g'(x) - g(x) h'(x))/ h(x)^2
13.
Rolle's Theorem: Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c)= 0
14.
Second Fundamental Theorem of Calculus: If f is continuous on an open interval containing a, then for every x in the interval the derivative of the the integral of f(x) dx on said interval is equal to f(x)
15.
The first derivative gives what?: 1. critical points
2. relative extrema
3. increasing and decreasing intervals
16.
The second derivative gives what?: 1. points of inflection
2. concavity
17.
When does the limit not exist?: 1. f(x) approaches a different number from the right as it does from the left as x→c
2. f(x) increases or decreases without bound as x→c
3. f(x) oscillates between two fixed values as x→c
