Keeping Your Averages Straight:
Mean Value Theorem/Average Value Theorem
Scott Pass
Travis High School
Austin ISD, Austin TX
[email protected]
Mean Value Theorem & the Average Value of f ( x)
MVT
If f ( x) is continuous on [a, b] and differentiable on (a, b) then
f (b)  f (a )
there must exist a number c in (a, b) such that f (c) 
ba
f(x)
a
c
b

f(x)

The Average Value of a function

If f ( x) is integrable on [a, b], then the average value of f ( x) on
the interval is

b
1
f ( x) dx
b  a a






First Example
The graph of the velocity v (t ) , in ft/sec, of a car traveling on a straight road, for 0  t  50, is shown above. A
table of values for v (t ) , at 5 second intervals of time t, is shown to the right of the graph.
 Find the average acceleration of the car, in ft/sec 2 , over the interval 0  t  50.
Language of Instantaneous Rates of Change compared to Average Rates of Change
f (b)  f (a )
ba
 The average rate of change of f ( x) on [a, b] is
 The rate of change of f ( x) at x  c is f (c )
b
 The average value of f ( x) on [a, b] is
1
f ( x) dx
b  a a
b
 The average velocity v (t ) on [a, b] is
1
v(t ) dt
b  a a
b
 The average acceleration a(t ) on [a, b] is
1
a(t ) dt
b  a a
b
1
 The average velocity anything on [a, b] is
 anything  dt
b  a a
Examples in Context
If f (t ) represents the rate of earnings of an individual born in 1965 measured in dollars/year and t is measured
in years since 1965.
13
a) Interpret
 f (t ) dt  0 .
0
40
b) Interpret
 f (t ) dt  1000000 .
0
40
c) Interpret
1
f (t ) dt
40  0 0
W (t ) is the rate at which a baby gains weight after birth where W is measured in pounds per month and t is
measured in months since the baby is born.
1
d) Interpret  W (t ) dt  .127
0
12
e) Interpret  W (t ) dt  9.7
0
12
1
W (t ) dt  9.7
f) Interpret
12 0
If a(t ) represents the acceleration of a drag racer on a straight track measured in m/sec 2 and t is measured in
seconds. a(t ) is given in the graph.
2
g) Interpret
 a(t ) dt in terms of the drag
0
racer? Use correct units.

a(t)




h) When is the velocity of the dragster the
greatest?



10
1
a(t ) dt
i) Interpret
10  0 0











 
Let the velocity of a particle moving on the x-axis be given by v (t ) , measured in meter/second and t is
measured in seconds, where v (t ) is given in the graph to the right.
3
(a) Describe the meaning of  v(t ) dt in terms of the particles
0
movement, use appropriate units.


v(t)


3
1
(b) Describe the meaning of  v(t ) dt in terms of the particles
30
movement, use appropriate units.












7
(c) Describe the meaning of  v(t ) dt in terms of the particles

0
movement, use appropriate units.
7
(d) Describe the meaning of
1
v(t ) dt in terms of the particles movement, use appropriate units.
7 0
10
(e) Evaluate
 v(t ) dt , does this mean that the particle has not moved? Explain
0
10
(f) Describe the meaning of
1
v(t ) dt in terms of the particles movement, use appropriate units.
10 0
(g) Find the total distance traveled by the particle from t = 0 to t =10.
10
(h) Determine
 v(t ) dt
0


 
2004 AB1
1. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per
minute. The traffic flow at a particular intersection is modeled by the function F defined by
t
F (t )  82  4sin   for 0  t  30 ,
2
where F (t ) is measured in cars per minute and t is measured in minutes.
(a) To the nearest whole number, how many cars pass through the intersection over the 30-minute
period?
(b) Is the traffic flow increasing or decreasing at t  7 ? Give the reason for your answer.
(c) What is the average value of the traffic flow over the time interval 10  t  15 ? Indicate units of
measure.
(d) What is the average rate of change of the traffic flow over the time interval 10  t  15 ? Indicate
units of measure.
2003 AB 3
The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twicedifferentiable and strictly increasing function R of time t. The graph of R and a table of selected values of R(t),
for the time interval 0  t  90 minutes, are shown above.
(a)
Use data from the table to find an approximation for R(45) . Show the computations that lead to your
answer. Indicate units of measure.
(b)
The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of R(45) ?
Explain your reasoning.
(c)
Approximate the value of

90
0
R (t ) dt using a left Riemann sum with the five subintervals indicated by the
data in the table. Is this numerical approximation less than the value of

90
0
R (t ) dt ? Explain your
reasoning.
(d)
For 0  b  90 minutes, explain the meaning of
Explain the meaning of
in both answers.

b
0
R (t ) dt in terms of fuel consumption for the plane.
1 b
R (t ) dt in terms of fuel consumption for the plane. Indicate units of measure
b0
Practice applying the Mean Value Theorem.
For the following determine if the Mean Value Theorem applies, if it does explain verbally what is known, if it
does not state why?
1. g ( x)  4 x3  x 2  4 on the interval [-1, 1]
2. f ( x) 
1
on the interval [0, 3]
x 1
3. h( x ) 
1
on the interval [0, 3]
x 1

4. f ( x) on the interval [0, 5]

f(x)











5. g ( x ) on the interval [-2, 2]


g(x)














6. The rate at which fuel flows into a tank R (t ) , in gallons per hour, is given by a differentiable function over
time. The table shows rates measured every two hours for a given 12-hour period. Is there some time t,
0  t  12 , such that R(t )  0 ? Justify your answer.
T
0
2
4
6
8
10
12
R(t)
76
84
88
92
94
93
88
Worksheet 3 Practice applying the Mean Value Theorem.-Key
For the following determine if the Mean Value Theorem applies, if it does explain verbally what is known, if it
does not state why?
1. g ( x)  4 x3  x 2  4 on the interval [-1, 1]
Since g(x) is continuous and differentiable on [-1,1] we know that there is at least one x-value where the
instantaneous rate of change of g(x) is equal to the average rate of change of g(x) on [-1,1].
1
on the interval [0, 3]
x 1
Since f(x) is not differentiable at x = 1 the Mean Value Theorem does not apply
2. f ( x) 
1
on the interval [0, 3]
x 1
Since h(x) is continuous and differentiable on [0,3] we know that there is at least one x-value where the
instantaneous rate of change of h(x) is equal to the average rate of change of h(x) on [0,3].
3. h( x ) 

4. f ( x) on the interval [0, 5]

f(x)


Since f(x) is not differentiable at x = 3 the Mean Value Theorem does not apply.







5. g ( x ) on the interval [-2, 2]


g(x)

Since g(x) is continuous and differentiable on [-2, 2] we know that there is at
least one x-value where the instantaneous rate of change of g(x) is equal to
the average rate of change of g(x) on [-2, 2].













6. The rate at which fuel flows into a tank R (t ) , in gallons per hour, is given by a differentiable function over
time. The table shows rates measured every two hours for a given 12-hour period. Is there some time t,
0  t  12 , such that R(t )  0 ? Justify your answer.
Since R(t)) is differentiable, which implies continuity, and the average rate of change on
the interval [4, 12] is 0, then there must be at least one time t where the instantaneous
rate of change of R(t) is zero.
T
0
2
4
6
8
10
12
R(t)
76
84
88
92
94
93
88

