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Chapter 7: Q12P (page 350)

To find the average value of the function on the given interval.

cos27蟺虫2on(0,87).

Short Answer

Expert verified

The average value of the given function over two periods is 12.

Step by step solution

01

Given

The given function on the interval is cos27蟺x2on0,87.

02

The concept of the average value of a function over a particular interva

The average value of a function over a particular interval can be found with an expression involving an integral.

Let's say that interval is[a,b|.

Then the average value off(x)over said interval is1b-a鈭�abf(x)dx.

03

Averagefunction on the given interval

From the given information, the average value is zero since all the functions are periodic with periodsT=87.

The average function on the given interval is shown below.

=鈭�08/7cos27蟺x2dx

04

The period of the function

The period of the function that is integrated is:

2蟺T=7蟺2T=4/7

Thus, the average value of the given function over two periods is 12.

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Most popular questions from this chapter

Sketch several periods of the corresponding periodic function of period2蟺. Expand the periodic function in a sine-cosine Fourier series.

f(x)={1,-蟺<x<0,...0,0<x<蟺,....

Following a method similar to that used in obtaining equations(12.11) to (12.14), show that if f(x)is even, theng伪is even too. Show that in this case f(x)andg(伪)can be written as Fourier cosine transforms and obtain (12.15).

Use Parseval鈥檚 theorem and the results of the indicated problems to find the sum of the series in Problems 5 to 9. The series 鈭�n=odd1n4 ,using problem 9.10.

Using problem 17,show that

鈭�0鈭�1鈭�cos蟺伪伪sin伪d伪=蟺2鈭�0鈭�1鈭�cos蟺伪伪sin伪d伪=蟺4

In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the x axis are the same.

x-cos26xon(0,蟺6)
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