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Chapter 7: Q12P (page 350)

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

cos27πx2on(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 cos27πx2on0,87.

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∫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.

=∫08/7cos27πx2dx

04

The period of the function

The period of the function that is integrated is:

2πT=7π2T=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

In Problems 17to 20,find the Fourier sine transform of the function in the indicated problem, and write f(x)as a Fourier integral [use equation (12.14)]. Verify that the sine integral for f(x)is the same as the exponential integral found previously.

Problem 10.

For each of the periodic functions in Problems5.1to 5.11, use Dirichlet's theorem to find the value to which the Fourier series converges at x=0,±π/2,±π,±2π.

In each of the following problems you are given a function on the interval -π<x<π .Sketch several periods of the corresponding periodic function of period 2ττ . Expand the periodic function in a sine-cosine Fourier series,

f(x)=1+x,-π<x<π.

For each of the periodic functions in Problems 5.1to 5.11.use Dirichlet's theorem to find the value to which the Fourier series converges atx=0,±π/2,±π,±2π.

We have said that Fourier series can represent discontinuous functions although power series cannot. It might occur to you to wonder why we could not substitute the power series for sinnxand cosnx(which converge for all x) into a Fourier series and collect terms to obtain a power series for a discontinuous function. As an example of what happens if we try this, consider the series in Problem 9.5. Show that the coefficients of x, if collected, form a divergent series; similarly, the coefficients of x3form a divergent series, and so on.
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