3.2
Calculus 1
Rolle’s Theroem
And the
Mean Value Theorem
for Derivatives
Mrs. Kessler
Rolle’s Theorem for Derivatives (1691)
Let f (x) be continuous function over [a,b],
and differentiable on (a, b).
If f(a) = f(b),
then there is at least one number c in (a, b)
such that f ′(c) = 0
Differentiable implies that the function is also continuous,
but notice continuous is closed, differentiable interval is open
Rolle’s Theorem for Derivatives
Example: Determine whether Rolle’s Theorem can be
applied to f(x) = (x - 3)(x + 1)2 on [-1,3]. Find all
values of c such that f ′(c )= 0.
f(-1)= f(3) = 0 AND f is continuous on [-1,3] and diff on
(1,3) therefore Rolle’s Theorem applies.
f ′(x )= (x-3)(2)(x+1)+ (x+1)2
FOIL and Factor
f ′(x )= (x+1)(3x-5) , set = 0
c = -1 ( not interior on the interval) or 5/3
c = 5/3
Mean Value Theorem for Derivatives
If f (x) is continuous on [a, b] and a differentiable
function over (a,b), then a point c between a and
b:
f b  f  a 
ba
 f c
This goes one step further than Rolle’s Theorem
Mean Value Theorem for Derivatives
If f (x) is a differentiable function over (a, b), then
at some point between a and b:
f b  f  a 
ba
 f c
The Mean Value Theorem says that at some point
in the interior of the closed interval, the slope
equals the average slope.

Tangent parallel
to chord.
y
Slope of tangent:
f  c
B
Slope of chord:
f b   f  a 
ba
A
0
y  f  x
a
c
x
b

Alternate form of
the Mean Value Theorem for Derivatives
f (b)  f (a)  (b  a) f '(c)
Determine if the mean value theorem applies, and if
so find the value of c.
x 1
1 
f ( x) 
on  , 2 
x
2 
f is continuous on [ 1/2, 2 ], and differentiable on (1/2, 2).
1
f (2)  f  
2
1
2
2

3
3
2
3
2
 1
This should equal f ’(x)
at the point c. Now
find f ’(x).
x(1)  ( x  1)(1)
1
f '( x) 
 2
2
x
x
Determine if the mean value theorem applies, and if so
find the value of c.
x 1
1 
f ( x) 
on  , 2 
x
2 
1
f (2)  f  
2
1
2
2

3
3
2
3
2
1
 2  1
x
x2  1
x 1
c 1
 1
x(1)  ( x  1)(1)
1
f '( x) 
 2
2
x
x
Application of
the Mean Value Theorem for Derivatives
You are driving on I 595 at 55 mph when you pass a
police car with radar. Five minutes later, 6 miles down the
road, you pass another police car with radar and you are
still going 55 mph. She pulls you over and gives you a
ticket for speeding citing the mean value theorem as
proof.
WHY ?
Application of the Mean Value Theorem for Derivatives
You are driving on I 595 at 55 mph when you pass a police car with radar. Five minutes later,
6 miles down the road you pass another police car with radar and you are still going 55mph.
He pulls you over and gives you a ticket for speeding citing the mean value theorem as proof.
Let t = 0 be the time you pass PC1. Let s = distance
traveled. Five minutes later is 5/60 hour = 1/12 hr. and
6 mi later, you pass PC2. There is some point in time
c where your average velocity is defined by
f b   f  a 
ba
s(1/12)  s(0)
6mi
Average Vel. =


(1/12  0)
1/12hr
72 mph