1. Rolle’s Theorem and The Mean Value Theorem
4.2

2. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives

3. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives
Differentiable implies that the function is also continuous.

4. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives
Differentiable implies that the function is also continuous.
The Mean Value Theorem only applies over a closed interval.

5. If f (x) is a differentiable function over [a,b], then at some point between a and b:
Mean Value Theorem for Derivatives
The Mean Value Theorem says that at some point in the closed interval, the actual slope (instantaneous rate of change) equals the average slope (average rate of change).

6. Tangent parallel to chord.
Slope of tangent:
Slope of chord:

7. To find c, set derivative = to 4
Example
Suppose you have a function f(x) = x^2 in the interval [1, 3].
To find c, set derivative = to 4

8. There can be more than 1 value of c that satisfy MVTD

9. There is no value of c that satisfies this equation. Why?
f(x) is not continuous in the given interval!

12. Since f(0) = f(12), Rolle’s Theorem applies
Find the value of c that satisfies the MVTD:
Since f(0) = f(12), Rolle’s Theorem applies

13. Since f(0) = f(1), Rolle’s Theorem applies
Find the value of c that satisfies the MVTD:
Since f(0) = f(1), Rolle’s Theorem applies

14. Example
A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 miles on a toll road with a speed limit of 65 mph. Can the trucker be cited for speeding?
MVTD tells us at some point in the interval, the instantaneous velocity is equal to the average velocity! Thus she must have been speeding at some point!
Average velocity is 159/2 = 79.5