Alaina's blog, 24 Oct 2010

This week we talked about several topics like Rolle’s Theorem, the Mean Value Theorem, and the First Derivative Test. The first topic we learned was Rolle’s Theorem. This tells you whether or not you have at least one max or min on the interval.
-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 fprime(c)=0.
*you must test for continuity and differentiability before using Rolle’s Theorem. You must check if the y-values match. (plug in and check).
**typically Rolle’s Theorem uses x-values [a,b] to find y-values.
***fprime(x)=0 implies max or min

Ex 1: find the 2 x-intercepts of f(x)=xsquared -3x +2 and show that fprime(x)=0 at some point between the 2 x-intercepts.
1. continuous? Yes
2. differentiable? Yes
3. xsquared -3x +2=0
(x-2)(x-1) same y-values
x=1,2 [1,2]
By Rolle’s theorem, the function is continuous, differentiable, and f(1)=f(2). Thus there must be at least one max or min (fprime(x)=0) on the interval [1,2].