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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.
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
d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x)
d/dx (g(x)/ h(x)) = (h(x) g'(x) - g(x) h'(x))/ h(x)^2
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
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)