More on the Mean Value Theorem.

1. The Mean
Value Theorem
or
quot;There's got to
be something
in the middle!
It's what you do inbetween that
counts by flickr user Rob Gallop

2. Derivative SLOPE of
of ƒ at 'c' SECANT Line
a c c b

3. The Mean Value Theorem (MVT)
If the function ƒ(x) is differentiable on the closed interval
[a, b]. Then there exists a point c in (a, b) such that
i.e. there exists a point, c, somewhere in the interval [a, b]
such that the slope of the tangent line through (a, ƒ(a)) and
(b, ƒ(b)) is equal to the derivative at that point.
Let's see the proof at Visual Calculus

4. Rolle’s Theorem
Suppose ƒ(x) is a function that satisfies all of the following.
1. ƒ(x) is continuous on the closed interval [a,b].
2. ƒ(x) is differentiable on the open interval (a,b).
3. ƒ(a) = ƒ(b)
Then there is a number c such that a < c < b and ƒ'(c) = 0.
Or, in other words ƒ(x) has a critical point in (a,b).

5. Find the number in the given interval which satisfies the conclusion of the Mean
Value Theorem.
x=1

6. Find the number in the given interval which satisfies the conclusion of the Mean
Value Theorem.
x = 3.154700539 and x = 0.845299461