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

414 Chapter 12 Applications of differentiation
ClearlyPdependsonthevalueofR L
.Differentiatingw.r.t.R L
andusingthequotient
rule,weobtain
dP
=V 21(R S +R L )2 −R L
2(R S
+R L
)
dR L
(R S
+R L
) 4
=V 2 (R S +R L )−2R L
(R S
+R L
) 3
=V 2 R S
−R L
(R S
+R L
) 3
Equating dP
dR L
tozero toobtain the turning pointgives
thatis,
V 2 R S
−R L
(R S
+R L
) 3 = 0
R L
=R S
So a turning point occurs when the load resistance equals the source resistance. We
need tocheck ifthisisamaximum turning point,so
dP
dR L
=V 2 R S
−R L
(R S
+R L
) 3
d 2 P
dR 2 L
=V 2 −1(R S +R L )3 − (R S
−R L
)3(R S
+R L
) 2
(R S
+R L
) 6
=V 2 −(R S +R L )−3(R S −R L )
(R S
+R L
) 4
=V 2 2R L −4R S
(R S
+R L
) 4
= 2V 2 (R L −2R S )
(R S
+R L
) 4
WhenR L
= R S
, this expression is negative and so the turning point is a maximum.
Therefore, maximum power transfer occurs when the load resistance equals the
source resistance.
EXERCISES12.2
1 Locatethe position ofany turningpointsofthe
following functionsanddeterminewhether they are
maxima orminima.
(a) y=x 2 −x+6 (b) y=2x 2 +3x+1
(c) y=x−1 (d) y=1+x−2x 2
(e) y=x 3 −12x (f) y=7+3x
2 Locateandidentify allturningpointsof
(a) y= x3
3 −3x2 +8x+1
(b) y =te −t
(c) y=x 4 −2x 2