The Mean Value Theorem and
Rolle’s Theorem
5-6, pg. 202-207
Lucas Guarino
Introduction
I
chose this topic because I was not up to par in my
knowledge of the MVT and Rolle’s Theorem
 A real world application of the section would be
traveling through the New Jersey Turnpike
Topics of Discussion
 Learn
MVT and Rolle’s Theorem and learn how to
find the point in an interval at which the
instantaneous rate of change equals the average
rate of change.
Relate to Earlier Lesson
 Big
Picture: Use the MVT to state why the velocity
is true and use Rolle’s Theorem to prove MVT.
 In 5-7 and 5-8 the MVT leads to an algerbraic
method for finding exact definite integrals.
Typical Problem
F(x)=x1/3 on (0,8)
F’(x)=1/3x-2/3 F’(0) would be 0-2/3= 1/02/3 =1/0
which is infinite
F is differentiable at (0,8) and continuous at 0 and 8
Msec=81/3-01/3/8-0=1/4
F’(c)=1/4
1/3c-2/3 = ¼ then c-2/3 = ¾
C = +- (3/4) -3/2 = +-1.5396 only + val. In (0,8) so
c=1.5396
Step by Step Procedure
F(x) = xsinx
 F(x) = 0 at x = 0 and at x = pi because sin x is 0 at those
points, establish differentiability by, f’(x) = sinx + xcosx
which exists for all x
 It is differentiable on (0,pi). The function is continuous at
this point. F’(c)= 0 if and only if sinc + ccosc = 0
c = 2.02875
 F’(0) is also 0 but c cannot = 0 because 0 is not in the
open interval (0,pi)

Summarize Investigation or Detail the
Procedure
 Mean
 If
Value Theorem
f is differentiable for all values of x in the open
interval (a,b) and
 If f is continuous at x=a and at x=b,
Then there is at least one number x=c in (a,b) such that
f’(c) = f(b)-f(a)/b-a
Rolle’s Theorem
 If
f is differentiable for all values of x in the open
interval (a,b) and
 If f is continuous at x = a and at x = b, and
 If f (a) = f(b) = 0
 Then there is at least one number x = c in (a,b)
such that f’(c) = 0
Real Life Examples
 When
you enter the New Jersey Turnpike you
receive a card that indicates your entrance point
and the time at which you entered. When you exit,
therefore, it can be determined how far you went
and how long it took, and thus what your average
speed was.
Solution
velocity = f(b) – f(a)/b – a
 Displacement a = 0m, displacement b = 1057m,
time a = 0s, time b = 147s
 1057m – 0m/147s – 0s = 7.19m/s
 Average velocity = 7.19m/s
 Average
Conclusion
The Mean Value Theorem and Rolle’s Theorem
 When you enter the New Jersey Turnpike you receive a
card that indicates your entrance point and the time at
which you entered. When you exit, therefore, it can be
determined how far you went and how long it took, and
thus what your average speed was.
 The audience learned what the MVT and Rolle’s
Theorems are and how to relate them to the real world.
