ap review session continuity, differentiability and major theorems

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

Example

Answer

Example

Answer

Example

Answer

Example

Example

Example

Answer

Example

Answer

Example

Answer

Example

Answer

Example

Answer

Example

Example

Review SessionFree Response

Example

Example Scoring Guide

Example

Example Scoring Guide

Example

Example Scoring Guide

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

https://docs.google.com/a/ncpublicschools.gov/forms/d/e/1FAIpQLSdPqmIGa0ctT6IaPViMmuKk4GhLiahri7S75eQqLWr16_x0JA/viewform?c=0&w=1