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SECTION 6.5
1–8 Find the average value of the function on the given interval.
2
1. f sxd − 3x 1 8x, f21, 2g
2. f sxd − sx , f0, 4g
3. tsxd − 3 cos x, f2y2, y2g
4. tstd −
t
s3 1 t 2
5. f std − e
sin t
,
cos t,
Tstd − 50 1 14 sin
f0, y2g
18. The velocity v of blood that flows in a blood vessel
with radius R and length l at a distance r from the
central axis is
9–12
(a) Find the average value of f on the given interval.
(b) Find c such that fave − f scd.
(c) Sketch the graph of f and a rectangle whose area is the
same as the area under the graph of f .
f2, 5g
f1, 3g
; 11. f sxd − 2 sin x 2 sin 2x,
2
t
12
Find the average temperature during the period from
9 am to 9 pm.
8. hsud − sln udyu, f1, 5g
2x
; 12. f sxd − 2xe ,
(a) Use the Midpoint Rule to estimate the average
velocity of the car during the first 12 seconds.
(b) At what time was the instantaneous velocity equal
to the average velocity?
f1, 3g
7. hsxd − cos 4 x sin x, f0, g
10. f sxd − 1yx,
463
17. In a certain city the temperature (in °F ) t hours after
9 am was modeled by the function
6. f s xd − x 2ys x 3 1 3d2, f21, 1g
9. f sxd − sx 2 3d2,
Average Value of a Function
vsrd −
P
sR 2 2 r 2 d
4l
where P is the pressure difference between the ends
of the vessel and is the viscosity of the blood (see
Example 3.7.7). Find the average velocity (with respect
to r) over the interval 0 < r < R. Compare the average
velocity with the maximum velocity.
19. The linear density in a rod 8 m long is 12ysx 1 1 kgym,
where x is measured in meters from one end of the rod.
Find the average density of the rod.
f0, g
f0, 2g
3
13. If f is continuous and y1 f sxd dx − 8, show that f takes
on the value 4 at least once on the interval f1, 3g.
14. Find the numbers b such that the average value of
f sxd − 2 1 6x 2 3x 2 on the interval f0, bg is equal to 3.
20. (a) A cup of coffee has temperature 95°C and takes
30 minutes to cool to 61°C in a room with temperature 20°C. Use Newton’s Law of Cooling (Section 3.8) to show that the temperature of the coffee
after t minutes is
Tstd − 20 1 75e2kt
15. Find the average value of f on f0, 8g.
where k < 0.02.
(b) What is the average temperature of the coffee during
the first half hour?
y
1
0
2
4
6
x
16. The velocity graph of an accelerating car is shown.
√
(km/h)
60
40
21. In Example 3.8.1 we modeled the world population
in the second half of the 20th century by the equation
Pstd − 2560e 0.017185t. Use this equation to estimate the
average world population during this time period.
22. If a freely falling body starts from rest, then its displacement is given by s − 12 tt 2. Let the velocity after a time
T be v T . Show that if we compute the average of the
velocities with respect to t we get vave − 12 v T , but if we
compute the average of the velocities with respect to s
we get vave − 23 v T .
20
0
4
8
12 t (seconds)
23. Use the result of Exercise 5.5.83 to compute the
average volume of inhaled air in the lungs in one
respiratory cycle.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
464
CHAPTER 6
Applications of Integration
24. Use the diagram to show that if f is concave upward on fa, bg,
then
a1b
fave . f
2
S D
25. Prove the Mean Value Theorem for Integrals by applying
the Mean Value Theorem for derivatives (see Section 4.2)
x
to the function Fsxd − ya f std dt.
26. If fave fa, bg denotes the average value of f on the interval
fa, bg and a , c , b, show that
y
f
fave fa, bg −
0
a
a+b
2
b
APPLIED PROJECT
c2a
b2c
fave fa, cg 1
fave fc, bg
b2a
b2a
x
CALCULUS AND BASEBALL
In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are
discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by
first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is,
p − mv. Suppose an object, moving along a straight line, is acted on by a force F − Fstd
that is a continuous function of time.
(a) Show that the change in momentum over a time interval ft0 , t1 g is equal to the integral
of F from t0 to t1; that is, show that
t1
Batter’s box
An overhead view of the position of a
baseball bat, shown every fiftieth of a
second during a typical swing.
(Adapted from The Physics of Baseball)
pst1 d 2 pst 0 d − y Fstd dt
t0
This integral is called the impulse of the force over the time interval.
(b) A pitcher throws a 90-miyh fastball to a batter, who hits a line drive directly back
to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with
velocity 110 miyh. A baseball weighs 5 oz and, in US Customary units, its mass is
measured in slugs: m − wyt, where t − 32 ftys 2.
(i) Find the change in the ball’s momentum.
(ii) Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-miyh fastball by
first considering kinetic energy.
The kinetic energy K of an object of mass m and velocity v is given by K − 12 mv 2.
Suppose an object of mass m, moving in a straight line, is acted on by a force F − Fssd
that depends on its position s. According to Newton’s Second Law
Fssd − ma − m
dv
dt
where a and v denote the acceleration and velocity of the object.
(a) Show that the work done in moving the object from a position s0 to a position s1 is
equal to the change in the object’s kinetic energy; that is, show that
s1
W − y Fssd ds − 12 mv122 12 mv 02
s0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.