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

10 Differentiation
Contents 10.1 Introduction 356
10.2 Graphicalapproachtodifferentiation 357
10.3 Limitsandcontinuity 358
10.4 Rateofchangeataspecificpoint 362
10.5 Rateofchangeatageneralpoint 364
10.6 Existenceofderivatives 370
10.7 Commonderivatives 372
10.8 Differentiationasalinearoperator 375
Reviewexercises10 385
10.1 INTRODUCTION
Differentiation is a mathematical technique for analysing the way in which functions
change. In particular, it determines how rapidly a function is changing at any specific
point.Asthefunctioninquestionmayrepresentthemagneticfieldofamotor,thevoltage
acrossacapacitor,thetemperatureofachemicalmix,etc.,itisoftenimportanttoknow
howquicklythesequantitieschange.Forexample,ifthevoltageonanelectricalsupply
network is falling rapidly because of a short circuit, then it is necessary to detect this
in order to switch out that part of the network where the fault has occurred. However,
the system should not take action for normal voltage fluctuations and so it is important
to distinguish different types and rates of change. Another example would be detecting
a sudden rise in the pressure of a fermentation vessel and taking appropriate action to
stabilize the pressure.
Differentiationwillbeintroducedinthischapter.Weshallderiveaformulawhichcan
be used to find the rate of change of a function. To avoid always having to resort to the
formulaengineersoftenuseatableofderivatives;suchatableisgiveninSection10.7.
The chapter closes with a discussion of an important property of differentiation -- that
of linearity.