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

12 Applicationsof
differentiation
Contents 12.1 Introduction 406
12.2 Maximumpointsandminimumpoints 406
12.3 Pointsofinflexion 415
12.4 TheNewton--Raphsonmethodforsolvingequations 418
12.5 Differentiationofvectors 423
Reviewexercises12 427
12.1 INTRODUCTION
In this chapter the techniques of differentiation are used to solve a variety of problems.
It is possible to use differentiation to find the maximum or minimum values of a function.
For example, it is possible to find the maximum power transferred from a voltage
source to a load resistor, as we shall show later in the chapter. Differentiation is also
usedintheNewton--Raphsonmethodofsolvingnon-linearequations.Suchanequation
needs to be solved to calculate the steady-state values of current and voltage in a series
diode--resistorcircuit.
Finallyweshowhowvectorscanbedifferentiated.Thisformsanintroductiontothe
important topic of vector calculus which is discussedinChapter 26.
12.2 MAXIMUMPOINTSANDMINIMUMPOINTS
Consider Figure 12.1. A and B are important points on the curve. At A the function
stops increasing and starts to decrease. At B it stops decreasing and starts to increase.
A is a local maximum, B is a local minimum. Note that A is not the highest point on
the curve, nor B the lowest point. However, for that part of the curve near to A, A is
the highest point. The word ‘local’ is used to stress that A is maximum in its locality.
Similarly, Bisthe lowestpoint initslocality.