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

12.5 Differentiation of vectors 423
Withanequationofthiscomplexity,itisbettertouseacomputeroraprogrammable
calculator. Doing so gives
V 5
= 0.6895,...,V 10
= 0.5770,...,V 14
= 0.5650
which is accurate tofourdecimal places.
Itisusefultocheckthesolutionbyindependentlycalculatingthecurrentthrough
the diode using the two different expressions. So,
I = 10 −14 (e 40×0.5650 −1) = 6.53 ×10 −5 A
I = − 0.5650
2.2 ×10 + 2
4 2.2 ×10 = 6.53 4 ×10−5 A
and therefore the solution iscorrect.
EXERCISES12.4
1 Usethe Newton--Raphson technique to findthe value
ofarootofthe following equations correctto two
decimalplaces. An approximateroot,x 1 ,isgiven in
eachcase.
(a)2cosx=x 2 x 1 =0.8
(b)3x 3 −4x 2 +2x−9=0 x 1 =2
(c)e x/2 −5x=0 x 1 =6
(d)lnx= 1 x
x 1 = 1.6
(e)sinx + 2x
π =1 x 1 =0.6
2 Explain circumstances in whichthe Newton--Raphson
technique mayfail to converge to arootof f (x) = 0.
Solutions
1 (a) 1.02 (b) 1.85 (c) 7.15 (d) 1.76 (e) 0.64
12.5 DIFFERENTIATIONOFVECTORS
Consider Figure 12.16. If r represents the position vector of an object and that object
moves along a curve C, then the position vector will be dependent upon the time,t. We
writer = r(t)toshowthedependenceupontime.Supposethattheobjectisatthepoint
P with position vector r at timet and at the point Q with position vector r(t + δt) at
the later timet + δt as shown in Figure 12.17. Then PQ ⃗ represents the displacement
vectoroftheobjectduringtheintervaloftime δt.Thelengthofthedisplacementvector
represents the distance travelled while its direction gives the direction of motion. The
average velocity during the time fromt tot + δt is the displacement vector divided by
the time interval δt, thatis
PQ
average velocity = ⃗ r(t +δt)−r(t)
=
δt δt
The instantaneous velocity,v, isgiven by
r(t +δt)−r(t)
v=lim
= dr
δt→0 δt dt