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

10.2 Graphical approach to differentiation 357
y (t)
1 2 3 4 5 6 7 8 9 10 11 12 t
Figure10.1
Thefunctiony(t)hasdifferentratesof
change over different regionsoft.
10.2 GRAPHICALAPPROACHTODIFFERENTIATION
Differentiation is concerned with the rate at which a function is changing, rather than
the actual change itself. We can explore the rate of change of a function by examining
Figure 10.1. There are several regions to this curve corresponding to different intervals
oft. In the interval [0, 5] the function does not change at all. The rate of change of y
is zero. Fromt = 5 tot = 7 the function increases slightly. Thus the rate of change of
y ast increases is small. Sinceyis increasing, the rate of change ofyis positive. From
t = 7 tot = 8 there is a rapid rise in the value of the function. The rate of change ofy
islargeandpositive.Fromt = 8tot = 9thevalueofydecreasesveryrapidly.Therate
ofchangeofyislargeandnegative.Finallyfromt = 9tot = 12thefunctiondecreases
slightly. Thus the rate ofchangeofyissmalland negative.
The aim of differential calculus is to specify the rate of change of a function precisely.Itisnotsufficienttosay‘therateofchangeofafunctionislarge’.Werequirean
exactvalueorexpressionfortherateofchange.Beforebeingabletodothisweneedto
introduce two conceptsconcerningthe rate ofchangeofafunction.
10.2.1 Averagerateofchangeofafunctionacrossaninterval
ConsiderFigure10.2.Whent =t 1
,thefunctionhasavaluey(t 1
).ThisisdenotedbyA
on the curve. Whent =t 2
, the function has a value ofy(t 2
). This point is denoted by B
on the curve. The function changes by an amounty(t 2
) −y(t 1
) over the interval [t 1
,t 2
].
Thusthe average rate ofchangeofthe function over the intervalis
changeiny
change int = y(t 2 ) −y(t 1 )
t 2
−t 1
The straight line joining A and B is known as a chord. Graphically,y(t 2
) −y(t 1
) is the
verticaldistanceandt 2
−t 1
thehorizontaldistancebetweenAandB,sothatthegradient
of the chord AB isgiven by
BC
AC = y(t 2 ) −y(t 1 )
t 2
−t 1
Thegradientorslopeofalineisameasureofitssteepnessandlinesmayhavepositive,
negative orzero gradients as shown inFigure 10.3.
Thus the gradient of the chord AB corresponds to the average rate of change of the
function between Aand B. To summarize:
TheaveragerateofchangeofafunctionbetweentwopointsAandBisthegradient
ofthe chord AB.