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Miscellaneous Exercises 125

Chapter Review

See the CD for review questions.

Miscellaneous Exercises

1.Find the Fourier sine series of the trapezoidal function given for

0 <x<πby

f(x)=

{x/α, 0 <x<α,

1 ,α<x<π−α,

(π−x)/α, π−α<x<π.

2.Show that the series found in Exercise 1 converges uniformly.

3.Whenαapproaches 0, the function of Exercise 1 approaches a square

wave. Do the sine coefficients found in Exercise 1 approach those of a

square wave?

4.Find the Fourier cosine series of the function

F(x)=

∫x

0

f(t)dt,

wherefdenotes the function in Exercise 1. Sketch.

5.Find the Fourier sine series of the function given in the interval 0<x<a

by the formula (αis a parameter between 0 and 1)

f(x)=







hx

αa

, 0 <x<αa,

h(a−x)

( 1 −α)a,αa<x<a.

6.Sketch the function of Exercise 5. To what does its Fourier sine series

converge atx=0? atx=αa?atx=a?

7.Suppose thatf(x)=1, 0<x<a. Sketch and find the Fourier series of

the following extensions off(x):

a. even extension;

b.odd extension;

c. periodic extension (perioda);

d.even periodic extension;