3. Definitions
A function f is said to be continuous at x = c provided the following conditions are
satisfied:
1. f(c) is defined
2. The limit as x approaches c for f(x) exist
3. The limit form #2 and the the value from #1 are the same
6. Differentiability
A function is said to be differentiable at all places where the derivative exist.
Functions are generally differentiable as long as the function doesn’t have a cusp
or point, a vertical tangent line, or a jump in the graph.
8. Mean Value Theorem, MVT
If f is continuous on the closed interval [a, b] and differentiable on the open interval
(a, b), then there exist a number c in (a, b) such that:
10. Intermediate Value Theorem, IVT
If f is continuous on the closed interval [a, b] and w 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
The IVT is often used to show that there must be an x-intercept on an interval
40. Thank you for attending
I hope that this presentation has been helpful.
Don’t forget about the upcoming sessions:
BC only, this Thursday 3/30 at 7 PM on Polar
AB/BC combined next Tuesday 4/4 at 7 PM on Applications of Derivatives
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