Higher Engineering Mathematics, Sixth Edition

Revision Test 19

This Revision Test covers the material contained in Chapters 66 to 71.The marks for each question are shown in
brackets at the end of each question.

1. Obtain a Fourier series for the periodic function
   f(x)defined as follows:

f(x)=

{

− 1 , when−π≤x≤ 0

1 , when 0≤x≤π

The function is periodic outside of this range with

period 2π. (13)

2. Obtain a Fourier series to representf(t)=tin the
   range−πto+π. (13)


3. Expand the function f(θ )=θ in the range
   0 ≤θ≤πinto (a) a half range cosine series, and
   (b) a half range sine series. (18)


4. (a) Sketch the waveform defined by:

f(x)=

⎧

⎨

⎩

0 , when− 4 ≤x≤− 2

3 , when− 2 ≤x≤ 2

0 , when 2≤x≤ 4

and is periodicoutsideof thisrange of period8.

(b) State whether the waveform in (a) is odd, even

or neither odd nor even.

(c) Deduce the Fourier series for the function

defined in (a). (15)

5. Displacement y on a point on a pulley when
   turned through an angle ofθdegrees is given by:

θ y

30 3.99

60 4.01

90 3.60

120 2.84

150 1.84

180 0.88

210 0.27

240 0.13

270 0.45

300 1.25

330 2.37

360 3.41

Sketch the waveform and construct a Fourier series

for the first three harmonics (23)

6. A rectangular waveform is shown in Fig. RT19.1.
   (a) State whether the waveform is an odd or even
   function.
   (b) Obtain the Fourier series for the waveform in
   complex form.
   (c) Show that the complex Fourier series in (b) is
   equivalent to:

f(x)=

20

π

(

sinx+

1

3

sin3x+

1

5

sin5x

+

1

7

sin7x+···

)

(18)

5

f(x)

(^0) x
25
22  2   2  3 
Figure RT19.1