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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
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
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
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)
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