Warm-Up:
1. What are the three things that need to be true to apply Rolle’s Theorem?
2. What is the equation of the tangent line at x=1 to the curve y  x 4  4 x  5 ?
ln x
3. Find the derivative of y 
(x  3) 2


Objective: Students will be able
to understand and use Rolle’s
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 number c in (a,b) such that
f’(c)=0.
How does it work?
1.
Check first to be sure your function is continuous and differentiable
on the interval given.
2.
Check the y-values for the endpoints of the interval (plug both interval
endpoints into the original f(x))
IF the y-values in step two are equal – continue to Step 3
3.
Take the derivative of the original function.
4.
Set the derivative equal to zero
5.
Solve for x. This is the value(s) of “c” the x-value location of the
horizontal tangent line(s)
Example 1: The table below gives values of the position of a car from its origin over
a time period 0  t  5 . x(t) is a continuous and differentiable function in the given
time interval.
t (hours)
0 1
2
3
4
5
x(t)
(position in miles)
0 70 102 150 140 70
Is there a time interval upon which the velocity of the car is equal to zero? Explain
why or why not, justify your answer.
Example 2: Determine whether or not Rolle’s Theorem applies to this situation. If it
does not, explain why. If it does, find the value(s) of c that must exist:
the interval [-1, 1]

2
f x   x 3 on
Example 3: Determine whether or not Rolle’s Theorem applies to this situation. If it
does not, explain why. If it does, find the value(s) of c that must exist:
x 2  x  2

f x  
on the interval [-1, 1]
x 2

Example 4: Explain why Rolle’s Theorem does not applies to this situation:
 3 
f x  sin x on the interval 0, . Show that there is a value of c such that
 4 
f c  0 . Does this contradict Rolle’s Theorem? Why or why not



Example 5: Find the two x-intercepts of f (x)  x 2  3x  2 and show that f’(x)=0 at
some point between the two x-intercepts.

Warm-Up:
1. Let f (x)  tan x . Show that f(0)=f(π) but there is no number c in the interval
(0,π) such that f ‘(c)=0. Why does this contradict Rolle’s Theorem?
1
2. Verify that f (x)  x  x satisfies Rolle’s Theorem on the interval [0,9].
3
 Find all numbers c that satisfy the conclusion of Rolle’s Theorem.

Objective: Students will be able to understand
and use the Mean Value Theorem.
Mean Value Theorem: If f is continuous on the closed interval [a,b] and
differentiable on the open interval (a,b), then there exists a number c in (a,b) such
f (b)  f (a)
that f '(c) 
ba
What this says:

What this means:
Example 1: Let f be the function given by f (x)  x 3  7x  6 . Find the number c that
satisfies the conclusion of the Mean Value Theorem on [1,3].

4
. What is the value of c for
x
which the instantaneous rate of change of f at x=c is the same as the average rate of
change over [1,4]?
Example 2: Let f be the function defined by f (x)  5 

x2
on the interval [−1, 1]. Find the exact value of
x+2
every c that satisfies the conclusion of the Mean Value Theorem.
Example 3: Consider f (x) =
Example 4: Two stationary patrol cars equipped with radar are 5 miles apart on a
highway. As a truck passes the first patrol car, its speed is clocked at 55 miles per
hour. Four minutes later, when the truck passes the second patrol car, its speed is
clocked at 50 miles per hour. Prove that the truck must have exceeded the speed
limit of 55 miles per hour at some time during the four minutes.
Example 5: A driver passed through the Weston Tolls on the Mass Pike at 4pm. At
5:45pm, he passed through the Lee Tolls, 124 miles from Weston. If the speed limit
on the Mass Pike is 60mph, was the driver cited for speeding? Why?
   
Example 6: Given the function f ( )  sin  on the interval  , . Does the Mean
 2 2 
Value Theorem apply for the given interval? If so, calculate the value(s) of c. If not,
state the reason why.


Calculus
Homework – Mean Value Theorem
Name
Does the Mean Value Theorem apply for the given interval? If so, calculate the
value(s) of c. If not, state the reason why.
1) f x   x 2  3x 1
2) f  x  
1 3
x  x  4
3
3) f  x  
x
x 3
4) f    sin 
  ,  
 
5
3
 1,2
0,2
5) f x  3x 2  5x  1
6) f  x   x
3,1
1,1
2,5