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

2.3 Basic concepts of functions 59
square
x x
the input
2
f
f(x)
9
4
–2 0 3 x
Figure2.5
Thefunction: ‘square the input’.
Manyvariablesofinteresttoengineers,forexamplevoltage,resistanceandcurrent,can
be related by means of functions. We try to choose an appropriate letter for a particular
variable; so, forexample,t is usedfor time andPforpower.
Engineeringapplication2.1
Functiontomodelthepowerdissipationinaresistor
Recall from Engineering application 1.1 that the power, P, dissipated by a resistor
depends on the current,I, flowing through the resistance,R. The relationship is
given by
P=I 2 R
The power dissipated in the resistor depends on the square of the current passing
throughit.InthiscaseI istheindependent variableandPisthedependent variable,
assumingRremainsconstant.Thefunctionisgivenbytherule:‘squaretheinputand
multiply by the constantR’, and the input to the function isI. The output from the
function isP. This isillustratedinFigure 2.6,for the casesR = 4 andR = 2.
I
square the
input and
multiply
by R
Figure2.6
The function:P =I 2 R.
P
P
16
8
4
2
R = 4
R = 2
–1 2 I
This model for a resistor only approximates the behaviour of the device. In practice,changesinthetemperatureoftheresistorleadtoslightchangesintheresistance
value. If the current through the resistor is excessively high then the resistor overheatsandispermanentlydamaged.Itnolongerhasthecorrectresistancevalue.The
amount of power that a resistor can handle depends on the materials that have been
usedinitsconstruction.Agoodcircuitdesignerwouldcalculatetheamountofpower
tobedissipatedandthenallowasuitablesafetymargintoensurethattheresistorcannot
beoverloaded.