Warm-Up
If f(x) = x√2x - 3, then f '(x) =
A) 3x - 3
√2x - 3
B)
x
√2x - 3
C)
1
√2x - 3
D) -x + 3
√2x - 3
E)
5x - 6
2√2x - 3
Problem of the Day
If f(x) = x√2x - 3, then f '(x) =
A) 3x - 3
√2x - 3
B)
x
√2x - 3
C)
1
√2x - 3
D) -x + 3
√2x - 3
E)
5x - 6
2√2x - 3
3-2: Rolle’s Theorem &
The Mean Value Theorem
Objectives:
•Introduce two important
Calculus Theorems
•Work problems which
illustrate these theorems
©2002 Roy L. Gover (www.mrgover.com)
Review
Extreme Value Theorem:
If f is continuous on a closed
interval, then f has a both a
minimum and a maximum on
the interval.
Rolle’s Theorem
Let f be continuous on [a,b]
& differentiable on (a,b). If
f(a)=f(b), then there exists a
number c in (a,b) such that
f’(c)=0.
Rolle’s Theorem
f '(c)  0
f (c )
f ( x)
f (a)  f (b)
a c
b
Important Idea
•Rolle’s Theorem is an
extension of the Extreme
Value Theorem
•Rolle’s Theorem provides a
connection between
continuity and
differentiability
Important Idea
•Extreme Value Theorem
guarantees a maximum and
a minimum
•Rolle’s Theorem provides a
way to find where the
maximum and minimum
occur.
Example
If Rolle’s Theorem applies to
2
f ( x)  x  3x  2 in the
interval [1,2], find all values
c in the interval such that
f’(c)=0.
IsDoes
fIscontinuous
f(1)=f(2)
on
?
[1,2]?
f differentiable on (1,2)?
Try This
If Rolle’s Theorem applies to
f ( x)  x  3  2 in the interval
[1,4], find all values c in the
interval such that f’(c)=0.
Doesn’t apply: f (1)  f (4)
Not differentiable at x=3
Try This
2
4
Let f(x)=2x -x . Confirm
that Rolle’s Theorem
applies and find all
values c in the interval
(-2,2) such that f’(c)=0
x  0, 1
Mean Value Theorem
If f is continuous on
[a,b] & differentiable on
(a,b), then there exists a
number c in (a,b) such
that:
f (b)  f (a)
f '(c) 
ba
Mean Value Theorem
If f is continuous on [a,b] &
differentiable on (a,b), there
is a number c in the interval
(a,b) where the slope of the
tangent line, f '(c ) , equals
the slope of the secant line
through f ( a ) & f (b ) .
Mean Value Theorem
Instantaneous Tangent line
rate of change
Average
rate of
change
a
Secant line
c
b
Warm-Up
4
Given f ( x)  5  , find
x
all values c in (1,4) such
f
(4)

f
(1)
that: f '(c) 
4 1
…or, where does the
instantaneous value equal
the average value?
Try This
x 1
Given f ( x) 
, find
x
1 

all values c in  , 2  such
2 
that:
f (b)  f (a)
f '(c) 
ba
Solution
1
f '( x)   2 ;
x
1 3
f (2)  f  
3
2  2
 1
1
3
2
2
2
1
 2  1  x  1
x
Why is x=-1 not a solution?
Try This
Two police cars with radar
are 5 miles apart. A truck
passes the first patrol car at
55 mph. Four minutes later
the truck passes the second
patrol car at 50 mph. Why
did the truck driver get a
ticket for exceeding the
speed limit of 55 mph?
Lesson Close
In your own words and
without looking at your
notes, what is:
Rolle’s
Theorem
The
Extreme
Value
The Mean Value Theorem
Theorem
Assignment
176/1-3, 5, 11,13,15,19,
29,33,34,35,39,41,43,45