1. Table of Contents
21. Section 4.3 Mean Value Theorem

2. Mean Value Theorem
Essential Question – What 2 concepts that we have learned about does the mean value theorem connect?

3. Mean Value Theorem
Connects concepts of average rate of change over an interval (secant line) with the instantaneous rate of change at a point within that interval (tangent line)
Somewhere between
points a and b on a differentiable curve,
there is at least one
tangent line parallel
to secant AB.

4. Mean Value Theorem
If f(x) is continuous at every point on [a,b] and differentiable on (a,b), then there is at least one point c in (a,b) where
Rolle’s Theorem is the special case of the MVT where f(a) = f(b)
Slope of tangent line
Slope of secant line

5. Example Show that f(x) = x2 satisfies the Mean Value Theorem on [0,2].
Continuous on [0,2] and differentiable on [0,2]

6. Example
Why does the mean value theorem not work on [-1,1] for the following functions?
Not differentiable at x=0
Not continuous at x=1

7. Example
Find a tangent that is parallel to secant AB

8. Uses of Mean Value Theorem
Speed
Mean Value Theorem says that the instantaneous rate of change (velocity) at some point must equal the average rate of change over the whole interval
Example
A car accelerating from zero takes 8 sec to go 352 feet. The average velocity is 352/8 = 44 ft/sec or 30 mph. At some point during the acceleration, the car’s speed must be exactly 30 mph.

9. Uses of Mean Value Theorem
Increasing and Decreasing
When the derivative is positive, function is increasing
When the derivative is negative, function is decreasing
If function only increases or decreases on an interval, then it is called monotonic
Example
Where is y = x2 increasing and decreasing?

10. Example Where is f(x)=x3-4x increasing and decreasing? Increasing

11. Testing Critical points
A function’s first derivative tells how the graph looks
At points where f has a minimum value, f’<0 to the left and f’>0 to the right
Similarly, where f has a maximum value, f’>0 to the left and f’<0 to the right

12. 1st derivative test for local extrema
At a critical point c
If f’ changes sign from positive to negative, then f has a local max
If f’ changes sign from negative to positive, then f has a local min
If f’ doesn’t change sign, then no local extrema
At left endpoint if f’<0 on interval to the right, f has a local max
At left endpoint if f’>0 on interval to the right, f has a local min
At right endpoint if f’<0 on interval to the left, f has a local min
At right endpoint if f’>0 on interval to the right, f has a local max

13. Example -2 2
Plug in values in each interval to f’
Local max Local min

14. Example -3 1 Plug in values in each interval to f’ 3 -5
Min max min max

15. Assignment
Pg. 236: #1-9 odd, odd, 41, 45, 57