1. Lesson 3.2 Rolle’s Theorem Mean Value Theorem 12/7/16

2. Between a and b, will f’(c) ever have to equal zero?
Given: Point A and Point B lie on a continuous differentiable function.
f(a) = f(b)
Either the function is horizontal between a & b, in which case f ’(c) = 0…
or somewhere on the interval the function increases & then decreases (or vice versa), & at that point has a f ’(c) = 0.
f(a) = f(b) a b
Between a and b, will f’(c) ever have to equal zero?
That’s a definitive YES. That YES is dependent on the conditions that f is continuous, differentiable & f(a) = f(b).

3. Dat’s just how I “Rolle”!

4. Let f(x) = sin 2x. Find all values of c in the interval
Example
Let f(x) = sin 2x. Find all values of c in the interval
such that f’(c) = 0
Does it satisfy Rolle’s Theorem?
Find any c values:
Sure And it’s continuous & differentiable.

5. “mean” slope over the interval [a, b]
Mean Value Theorem
For any secant of a function there exists a point on the function where the tangent is parallel to the secant.
“mean” slope over the interval [a, b]
slope of tangent at c

6. Example
Find a value for c within the interval [-1, 1] where the tangent line at c will be parallel to the secant line through the endpoints of the interval. f(x) = x3 – x2 – 2x

7. Day 1 Page 172 #1 – 3, 7, 9, 11, 19, 25, 27, 28
There are
kinds of people in the world: those that can do calculus and those that can not.
Day 2 #29 – 31, 35, 38, 43 – 47 odd