Mean Value Theorem

1. The Mean Value Theorem
1

2. Road Trip!
2

3. no
yes
5.7
yes
yes
For 1≤x≤4, tangent slopes are positive
but steeper than secant.
For 4≤x≤5, tangent slopes are negative.
5.7
2.3 5.7
yes
yes 3.1
yes
yes
2.3 5.7
3.1
3

4. yes
yes
no
For continuity at an endpoint, only need to
look at 1­sided limit (in this case left­hand
Slope of secant is negative. limit as x approaches 5.
Slope of tangents in (5, 7) are positive
and increasing
3
yes
no
1, 2, 4, 7
5.8 3 and 5
6
If a function satisfies the hypotheses, then
that is enough (sufficient) to make the
function satisfy the conclusion.
But for a function that satisfies the
5.8 conclusion, it is not necessary for the
hypotheses to be satisfied.
Example: Problem 5: For [2, 8] the function
is not continuous, but the conclusion is
satisfied.
4

5. f(x) is continuous on the closed interval [a, b] and
f(x) is differentiable on the open interval (a, b)
there exists at least one x = c value such that
derivative at x = c = slope of the secant from a to b.
in the domain of f(x)
A constant function has a zero derivative
for each x­value in its domain.
If two functions have the same derivative for
each x­value in (a, b), then the functions differ
by a constant.
Function f is a vertical translation of function g.
5

6. Informal proof of Mean Value Theorem
Rotate f(x) in interval [a, b]
and
Appy Rolle's Theorem
f ' (c) = 0
a c b
6

7. 1) f(x) is continuous on [0, 8].
2) f(x) is differentiable on (0, 8)
(Note: f(x) is not differentiable at x = 0)
7