Review Session Continuity, Differentiability and Major Theorems

Topic 1Continuity

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

Common areas were functions are not continuous

Holes

Vertical Asymptotes

Jumps in graphs

Topic 2Differentiability

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.

Topic 3Mean Value Theorem

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:

Topic 4Intermediate Value Theorem

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

Topic 5Extreme Value Theorem

Extreme Value Theorem

If f is continuous on a closed interval [a, b] then f has both a minimum and a maximum on the interval

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Review SessionFree Response

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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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