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

10.5 Rate of change at a general point 369
conductioncurrentwhilethatbetweenthecapacitorplatesiscalledadisplacement
current (see Figure 10.12).
Thedisplacementcurrentcanbethoughtofasavirtualcurrentthatflowsbetween
the plates due to the build-up of positive charge on one plate and negative charge
on the other plate. It doesn’t correspond to a real current flowing between the plates
as they are separated by an insulating material.
We can relate the smallchanges δx and δy tothe derivative dy
dx .
If δx isvery smallyet still finitewe can statethat
dy
dx ≈ δy
(10.1)
δx
Thisresultallowsanimportantapproximationtobemade.FromEquation(10.1)wesee
thatifasmallchange, δx,ismadetotheindependentvariable,thecorrespondingchange
inthe dependent variable isgiven by the following formula:
δy ≈ dy
dx δx
Example10.9 Ify =x 2 estimate the changeinycausedbychangingxfrom3to3.1.
Solution Ify =x 2 then dy = 2x. The approximate change inthe dependent variable isgiven by
dx
δy ≈ dy δx =2xδx
dx
Takingx = 3 and δx = 0.1 we have
δy ≈ (2)(3)(0.1) = 0.6
We conclude that at the point wherex = 3 a change inxto 3.1 causes an approximate
change of 0.6 inthe value ofy.
EXERCISES10.5
1 Calculatethe gradientofthe functionsat the specified
points.
(a) y = 2x 2 at(1,2)
(b) y=2x−x 2 at(0,0)
(c) y=1+x+x 2 at(2,7)
(d) y=2x 2 +1at(2,9)
2 Afunction,y,is such that dy is constant.
dx
What can you say about the function,y?
3 Forwhichgraphs in Figure10.13isthe derivative
always (a) positive or(b)negative?
4 Find the derivative ofy(x)whereyis
(a) x 2 (b) −x 2 +2x
5 Differentiatey = 2x 2 +9; thatis,find dy
dx .
What is the rate ofchange ofywhenx = 3, −2, 1,0?
6 Find the rate ofchange ofy = 4t −t 2 .Whatis the
value of dy whent = 2?
dt
7 Ify=x 3 −3x 2 +xthen
dy
dx =3x2 −6x+1
Estimate the change inyasxchanges from
(a) 2 to 2.05
(b) 0 to 0.025
(c) −1 to −1.05