6.5 Average Value of a Function by SGLee - HSKim, YJLim
1-7. Find the average value of the function on the given interval.
1. 
, 
http://matrix.skku.ac.kr/cal-lab/cal-6-5-1.html
.
Calculus-Sec-6-5-Solution
6.5 Average Value of a Function by SGLee - HSKim, YJLim
1-7. Find the average value of the function on the given interval.
1. ,
http://matrix.skku.ac.kr/cal-lab/cal-6-5-1.html
.
4. 
, 
.
To calculate this integral, use a change of variable 
, then 
. So,
6. 
, 
.
8-10. (a) Find the average value of 
on the given interval.
(b) Find 
such that 
.
(c) Sketch the graph of 
and a rectangle whose area is the same as the area under the graph of 
.
8. 
, 
http://matrix.skku.ac.kr/cal-lab/cal-6-5-8.html
(a)
.
(b)
9. 
, 
(a)
(b)
10. 
, 
(a) 
.
Use change of variable to 
, then 
. So
Also change of variable to 
,
then 
.
.
(b)http://matrix.skku.ac.kr/cal-lab/cal-6-5-10.html
11. If 
is continuous and 
, show that 
takes on the value 
at least once on the interval 
.
Since 
is continuous and 
, there exists a constant 
in 
such that
.
12. Find the numbers 
such that the average value of 
on the interval 
is equal to 
.
.
That is 
.
13. In a certain city the temperature (in 
) 
hours after 
A.M. is modeled by the function 
.
Find the average temperature during the period from 
A.M. to 
P.M.
http://matrix.skku.ac.kr/cal-lab/cal-6-5-13.html
.
14. If a cup of coffee has temperature 
in a room where the temperature is 
then, according to Newton's Law of Cooling,
the temperature of the coffee after 
minutes is 
.
What is the average temperature of the coffee during the first half hour?
http://matrix.skku.ac.kr/cal-lab/cal-6-5-14.html
15. The linear density in a rod 
long is 
, where 
is measured in meters from one end of the rod.
Find the average density of the rod.
Let 
. Then
. Change of variable to 
, then 
. So
16. If 
denotes the average value of 
on the interval 
and 
,
show that 
, 
, 
, 
.
, 
.
, then 
. So
