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

362 Chapter 10 Differentiation
10.4 RATEOFCHANGEATASPECIFICPOINT
WesawinSection10.2thattherateofchangeofafunctionatapointisthegradientof
thetangenttothecurveatthatpoint.Also,wecanthinkofatangentatAasthelimitof
an extended chord AB as B → A. We now put these two ideas together to find the rate
of change of a function atapoint.
Example10.4 Given y = f (x) = 3x 2 + 2, obtain estimates of the rate of change of y at x = 3 by
considering the intervals
(a) [3,4] (b) [3,3.1] (c) [3,3.01]
Solution (a) Consider Figure 10.8.
y(3) = 3(3) 2 +2 =29
y(4) = 3(4) 2 +2 =50
Let Abe the point (3, 29) on the curve. Let Bbe the point (4, 50).Then
average rate of change over the interval [3,4] =
=
change iny
change inx
y(4) −y(3)
4 −3
= 50−29
4 −3 = 21
ThisisthegradientofthechordABandisanestimateofthegradientofthetangent
atA.Thatis, the rate of change atAisapproximately 21.
(b) y(3.1) = 30.83 and so,
average rate of change over the interval [3,3.1] =
This isamore accurate estimateof the rate of change atA.
(c) y(3.01) = 29.1803 and so,
30.83 −29
3.1−3
29.1803 −29
average rate of change over the interval [3,3.01] =
3.01 −3
= 18.03
= 18.3
y
50
B
29
A
3 4
x
Figure10.8
Thefunction:y = 3x 2 +2.