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# AP Calculus Theorems

This set goes over all those pesky theorems, rules, and properties that are useful to know when it comes to the AP test.
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Definition of Continuity
1. lim x→c f(x) exists.
2. f(c) exists.
3. lim x→c f(x) = f(c)
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
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
Definition of a Derivative
lim h→0 (f(x+h) - f(x)) / h
Product Rule
d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x)
Quotient Rule
d/dx (g(x)/ h(x)) = (h(x) g'(x) - g(x) h'(x))/ h(x)^2
Chain Rule
d/dx f(g(x)) = f'(g(x)) g'(x)
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.
The first derivative gives what?
1. critical points
2. relative extrema
3. increasing and decreasing intervals
The second derivative gives what?
1. points of inflection
2. concavity
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
Mean Value Theorem
f'(c) = (f(b) - f(a))/ (b - a)
Fundamental Theorem of Calculus
The integral on (a, b) of f(x) dx = F(b) - F(a)
Mean Value Theorem (Integrals)
The integral on (a, b) of f(x) dx = f(c) (b - a)
Average Value Theorem
1/ (b-a) times the integral on (a, b) of f(x) dx
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)
Derivative of an Inverse Function
g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x)