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Chapter 7: Q20P (page 371)

In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function fcof period 2b, the odd function fsof period 2b, and the function fpof period b, each of which equals f(x)on0 < x < b. Expand each of the three functions in an appropriate Fourier series.

20. f(x) = x2, 0 < x < 1.

Short Answer

Expert verified

The even function is the odd function is

and the proper function is

The graph of these functions is shown below:

Step by step solution

01

Given Information.

The given function is f(x) = x2, 0 < x < 1.

02

Meaning of Fourier series

A periodic function f(x) is expanded using the Fourier series formula in terms of an infinite sum of sines and cosines. Any periodic function or periodic signal is decomposed into the sum of a set of simple oscillating functions, mainly sines and cosines. The formula for a function's Fourier series is as follows:

03

Find the even function fc.

First, find the values of a0.

Now find the value of an.

As a result, the even function is

04

Find the odd function fs.

The coefficients are equal to:

Thus, the odd function is

05

Find the complete function

The Fourier series of the function itself:

Find the value of an.

Find the value of bn.

Thus, the series of the function is

06

Plotting the functions fc, fs and fp on the graph.

Now sketch the even function the odd function

, and the function

on the graph as:

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Most popular questions from this chapter

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 period2ττ . Expand the periodic function in a sine-cosine Fourier series,

role="math" localid="1659239194875" f(x)={1,-π<x<π2and0<x<π20,-π2<x<0andπ2<x<π

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.

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The displacement (from equilibrium) of a particle executing simple harmonic motion may be eithery=Asinωtory=Asin(ωt+ϕ)depending on our choice of time origin. Show that the average of the kinetic energy of a particle of mass m(over a period of the motion) is the same for the two formulas (as it must be since both describe the same physical motion). Find the average value of the kinetic energy for thesin(ωt+ϕ)case in two ways:

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