4.2 Rolle's Theorem and Mean Value Theorem
Rolle's Theorem: Let f be continuous on the closed interval [a,b] and
differentiable on the open interval (a,b).
If f(a) = f(b) , then there is at least one c in (a,b)
such that f '(c) = 0.
Example: Find the two x-intercepts of f(x) = x2 - 3x + 2 and
show that f '(x)= 0 at some point between the 2 intercepts.
Goal: To look over several cases where
the Mean Value Theorem applies and
attempt to formulate the text of the
theorem on our own and have a better
idea of the applications of MVT
Mean Value Theorem
1. For f(x) = -.01x3 +1.2x2 - 3.6x + 5,
there is a value x = c between 3 and 7 at which
the tangent to the graph is parallel to the
secant through (3, f(3)) and (7, f(7)).
The secant line is drawn in ______________
The tangent line is drawn in _____________
For the graph, c ________
Is f differentiable on (3,7)?
____________
Is f continuous on [3,7]? ____________
2. The function f (x) from problem 1 has two
values of x = c between x = 1 and x=7 at which
f '(c) equals the slope of the corresponding
secant line. In other words,
The tangent line is PARALLEL to the secant line
c ______ and c _______
Is f differentiable on (1,7)? _______
Is f continuous on [1,7]? ________
3. For g(x) = 6 - 2(x - 4)2/3
The secant line through (1, g(1)) and
(5, g(5)) is drawn in green.
Is g differentiable on (1,5)? _________
Is g continuous on [1,5]? _________
Is there a c value in [1,5] such that g'(c) equals
the slope of this green secant line? Why or
what not?
From the graph, c _________
Is g differentiable on (1,4)? _______
Is g continuous on [1,4]? ________
How is it possible that a c exists?
The secant line through (5, h(5)) and (7,h(7))
is drawn in green.
Is h differentiable on (5,7)? __________
Is h continuous on [5,7]? __________
Why is there no value x=c in (5,7) for which
h'(c) equals the slope of the secant line?
6. The graph below is the function h from #5.
The secant line through (5,h(5)) and (9, h(9))
is drawn in green.
Is h differentiable on (5,9)? ________
Is h continuous on [5,9]? __________
There is a point x = c in (5,9) where h '(c)
equals the slope of the secant line. The
tangent line is drawn in red.
Estimate the value of c _________
The Mean Value Theorem
If f(x) is continuous at every point on the closed interval [a,b]
and differentiable at every point on the open interval (a,b),
then there is a least one point c in (a,b) at which
f '(c) = f(b) - f(a)
b-a
What
about
#6?
Sufficient means if you meet the condition of the hypothesis, you are guaranteed the conclusion.
Not necessary means that if you do not meet the conditions of the hypothesis, you may still reach the conclusion, but you are not guaranteed to reach the conclusion.
If Mean Value Theorem applies, find the values of c such that f '(c) =
slope of the secant line in the given interval.
45
Using your calculator to find the exact value for c in #1....
1. Graph the function
2. What is the slope of the secant line on the interval [3,7]?
3. What is f'(x) ?
4. How can the calculator help?
Two stationary patrol cars equipped with radar are 5 miles apart
on the highway. As a truck passes, its speed is clocked at 55 miles
per hour. Four minutes later, its speed is clocked at 50 miles per
hour. Prove that the truck must have exceeded the posted speed
limit (of 55 mph) at some time during the 4 minutes.
Find both c values for problem #2
f(x) = -0.1x 3 + 1.2x 2 - 3.6 x + 5 between [1, 7]