James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 6.5 Average Value of a Function 463
1–8 Find the average value of the function on the given interval.
1. f sxd − 3x 2 1 8x, f21, 2g
2. f sxd − sx , f0, 4g
3. tsxd − 3 cos x, f2y2, y2g
t
4. tstd − , f1, 3g
2
s3 1 t
5. f std − e sin t cos t, f0, y2g
6. f s xd − x 2 ysx 3 1 3d 2 , f21, 1g
7. hsxd − cos 4 x sin x, f0, g
8. hsud − sln udyu, f1, 5g
9–12
(a) Find the average value of f on the given interval.
(b) Find c such that f ave − f scd.
(c) Sketch the graph of f and a rectangle whose area is the
same as the area under the graph of f .
9. f sxd − sx 2 3d 2 , f2, 5g
10. f sxd − 1yx, f1, 3g
; 11. f sxd − 2 sin x 2 sin 2x, f0, g
; 12. f sxd − 2xe 2x 2 , f0, 2g
13. If f is continuous and y 3 f sxd dx − 8, show that f takes
1
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.
15. Find the average value of f on f0, 8g.
y
1
0 2 4 6
16. The velocity graph of an accelerating car is shown.
√
(km/h)
60
40
20
0 4 8 12 t (seconds)
x
(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?
17. In a certain city the temperature (in °F) t hours after
9 am was modeled by the function
Tstd − 50 1 14 sin t
12
Find the average temperature during the period from
9 am to 9 pm.
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
vsrd −
P
4l sR2 2 r 2 d
where P is the pressure difference between the ends
of the ves sel 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.
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 75e 2kt
where k < 0.02.
(b) What is the average temperature of the coffee during
the first half hour?
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 − 1 2 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 v ave − 1 2 vT, but if we
compute the average of the velocities with respect to s
we get v ave − 2 3 vT.
23. Use the result of Exercise 5.5.83 to compute the
average volume of inhaled air in the lungs in one
respiratory cycle.
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