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

358 Chapter 10 Differentiation
y (t)
y (t 2 )
B
y
Positive
gradient
y (t 1 )
A
t 1
(t 2 – t 1 )
Figure10.2
Average rate ofchange across an interval.
u
t 2
C
y (t 2 ) – y (t 1 )
t
Zero
gradient
Negative
gradient
Figure10.3
Linescan have different gradients.
t
y (t)
B 1
Chord AB 1
Chord AB 2
A
B 2
Tangent at A
10.2.2 Rateofchangeofafunctionatapoint
t
Figure10.4
Point B ismoved nearer to Ato improve
accuracy.
Consider again Figure 10.2. Suppose we require the rate of change of the function at
pointA.WecanusethegradientofthechordABasanapproximationtothisvalue.IfB
isclosetoAthentheapproximationisbetterthanifBisnotsoclosetoA.Thereforeby
moving B nearer to A it is possible to improve the accuracy of this approximation (see
Figure 10.4).
Suppose the chord AB is extended as a straight line on both sides of AB, and B is
moved closer and closer to A until both points eventually coincide. The straight line
becomesatangenttothecurveatA.Thisisthestraightlinethatjusttouchesthecurve
atA.However,therateofchangeofthistangent,thatisitsgradient,stillcorrespondsto
the rate of change of the function, but now it is the rate of change of the function at the
point A.To summarize:
The rate of change of a function at a point A on the curve is the gradient of the
tangenttothe curve atpointA.
We have still to address the question of how the gradients of chords and tangents are
found.This requires a knowledge oflimits whichisthe topic ofthe next section.
10.3 LIMITSANDCONTINUITY
The concept of a limit is crucial to the development of differentiation. We write
t →c to denote thatt approaches, or tends to, the value ofc. Note carefully that this is
distinct from statingt = c. Ast tends tocwe consider the value to which the function
approachesand call thisvalue thelimit ofthe functionast →c.