082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

106 Chapter 2 Engineering functions
–2
0
|x| < 2
Figure2.40
The quantity |x| < 2 is equivalent to −2 <x<2.
2
–3
0 3
|x| > 3 |x| > 3
Figure2.41
Thequantity |x| 3isequivalenttox 3orx −3.
Themodulusfunctioncan beusedtodescribe regionsinthex--yplane.
Example2.24 Sketch the region defined by
(a) |x| < 2and |y| < 1
(b) |x 2 +y 2 | 9
Solution (a) The region is a rectangle as shown in Figure 2.42. The boundary is not part of the
regionasstrictinequalitieswereusedtodefineit.Theregion |x| 2, |y| 1isthe
sameasthatinFigure 2.42 with the boundaryincluded.
(b) |x 2 +y 2 | 9isequivalentto −9 x 2 +y 2 9.Note,however,thatx 2 +y 2 isnever
negative and so the region isgiven by0 x 2 +y 2 9.
Let P(x,y) be a general point as shown in Figure 2.43. Then from Pythagoras’s
theorem,the distance fromPtothe originis √ x 2 +y 2 .So,
Then,
(distance fromorigin) 2 =x 2 +y 2 9
(distance fromorigin) 3
This describes a disc, centre the origin, of radius 3 (see Figure 2.44).
y
1
y
P (x, y)
y
3
–2
–1
2
x
O
r
x
–3
–3
3 x
Figure2.42
Theregion: |x| < 2 and |y| < 1.
Figure2.43
PointP(x,y) is ageneral point.
Figure2.44
Theregion: |x 2 +y 2 | 9.
Engineeringapplication2.16
Full-waverectifier
A rectifier is an electronic circuit that is used to convert an alternating voltage to
a direct voltage, that is, one that does not change in polarity with time. Usually
the alternating voltage has the shape of a sinusoidal function. We will examine the