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.

.

4.

.

To calculate this integral, use a change of variable , then . So,

6.

.

7.

.

Use a change of variable , then . So

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.

(a)

.

(b)

9.

(a)

(b)

10.

(a)   .

Use change of variable to , then . So

Also change of variable to ,

then .

.

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.

.

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?

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