2413 Calculus
Chapter 4(6)
Mean Value Theorem
Rolle’s Theorem
Mean Value Theorem
If you have a continuous function (solid) on an interval
with a derivative (no sharp points) then someplace on
the interval the tangent line is parallel to the line
through the ends.
1) Show f(x) is continuous on [a, b]
2) Show f(x) has a derivative on [a, b]
3) Set f (c) 
f (b)  f (a)
(slope between endpoints)
ba
If f(x) is a continuous function at every point of the
closed interval [a,b] and it has a derivative at every
point in (a,b), then there is some point c in (a, b) were:
f (c) 
f (b)  f (a)
ba
Continuous on [a,b] ?
Derivative on (a,b) ?
c has slope of ab
a
c
b
Physical interpretations of the MVT
f (c )
f (b )  f ( a )
ba
is the change at an instant (when x = c)
is the change between the endpoints, or
the average change over the interval
If you start at 0 ft and take 6 seconds to go 300 feet your
average velocity (average speed) is 50 ft/sec. From the
MVT there must be some time where your velocity is
exactly 50 ft/sec.
Physical Interpretations II
Assume you start in Houston at noon and arrive in
Dallas (240 miles north) at 3:30. The MVT says you
68.571 mph at some time during the
were going _____________
trip.
240  0
f (c) 
 68 .571
3. 5  0
Example 3: Determine if the MVT applies and find c
f ( x)  x  7 x  4 on [1,3]
2
Continuous because:
Polynomials are continuous
Find c:
Differentiable because:
Polynomials always have derivatives
f (b)  f (a)
f (c) 
ba
26  4
2c  7 
3 1
2c  7  11
c2
for (1,4) and (3,26)
Example 4: Determine if the MVT applies and find c
1
f ( x) 
on [1,3]
x2
Continuous because:
The vert. Asymp. Is outside
the interval [-1,3]
Differentiable because:
Derivative exists inside (-1,3)
f (b)  f (a)
Show : f (c) 
for (1,  1) and (3, 51 )
ba
1
1
5 1

2
c2  5
(c  2)
3 1
1
1

c  2  5
2
(c  2)
5
Rolle’s Theorem
If you have a continuous (solid) function on an interval
with a derivative (no sharp points) and the endpoints
are the same height, then someplace on the interval the
tangent line has a slope of zero
1) Show f(x) is continuous on [a, b]
2) Show f(x) has a derivative on [a, b]
3) Verify the f(a) = f(b)
4) Set
f (c)  0
(horizontal tangent, slope is zero)
Example 5:
Determine if Rolle’s Theorem applies and find c
x2  1
f ( x) 
x
on [1, 1]
Not Continuous because:
Undefined at x = 0
Rolle’s Theorem Does Not Apply
Example 6:
Determine if Rolle’s Theorem applies and find c
f ( x)  x  2ln x
Continuous because:
Ln is defined for x > 0
f (1)  1 and f (3)  3  2ln 3
on [1,3]
Differentiable because:
Has a derivative if x ≠ 0
Endpoints are NOT
the same
Rolle’s Theorem Does Not Apply
Example 7:
Determine if Rolle’s Theorem applies and find c
f ( x)  x  x
3
on [0, 1]
Differentiable because:
Has a derivative if x ≠ 0
Continuous because:
Cube-roots are continuous
f (0)  0 and f (1)  0
Show: f (c)  0
f '( x)  1 
1
1
3 3 c2
1
3 3 x2
Endpoints are the same
on [0, 1]
=0
c 2  13
1
c
27
3