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

Review exercises 12 427
REVIEWEXERCISES12
1 Determine the position ofall maximum points,
minimumpointsand pointsofinflexion of
(a)y=2t 3 −21t 2 +60t+9
(b)y =t(t 2 −1)
2 InSection 9.8 weshowed that the impedance ofan
LCRcircuitcan be written as
( )
Z=R+j ωL − 1
ωC
(a) Find |Z|.
(b) Foragiven circuit,R,L andC are constants,and
ω can be varied. Find d|Z|
dω .
(c) Forwhat value of ω will |Z| have amaximum or
minimum value? Does thisvalue give a
maximum orminimum value of |Z|?
3 Usetwo iterationsofthe Newton--Raphson technique
to findan improved estimateofthe rootof
t 3 =e t
givent = 1.8 is an approximate root.
4 Given
a=(t 2 +1)i−j+tk
b=2tj−k
find
da
(a)
dt
(c)
d
dt (a·b)
(b)
db
dt
(d) d
dt (a×b)
5 Use two iterationsofthe Newton--Raphson method
to find animproved estimateofthe rootof
sint=1− t ,0 t π,givent=0.7isan
2
approximate root.
6 Determine the position ofallmaximumpoints,
minimum pointsandpointsofinflexion of
(a) y = e −x2
(b) y =t 3 e −t
(c) y=x 3 −3x 2 +3x−1
(d) y=e x +e −x
(e) y=|t|−t 2
Solutions
( 7
1 (a) (2,61) maximum, (5,34) minimum,
2 , 95 )
2
point ofinflexion
( ) ( )
1
(b) √ ,− 2
3 3 √ minimum, −√ 1 2
,
3 3 3 √ 3
maximum, (0,0) point ofinflexion
√
2 (a) R 2 +ω 2 L 2 − 2L
C + 1
ω 2 C 2
(b)
ωL 2 −1/ω 3 C 2
√
R 2 + ω 2 L 2 −2L/C +1/(ω 2 C 2 )
(c) ω = 1 √
LC
producesaminimumvalue ofZ
3 1.859,1.857
4 (a) 2ti+k
(b) 2j
(c) −3
(d) −4ti+2tj+ (6t 2 +2)k
5 0.705, 0.705
6 (a) (0,1)is amaximum
√
2
Points ofinflexion whenx = ±
2
(b) (0,0)point ofinflexion, (3,1.34) maximum.
Further pointsofinflexion whent = 4.73,1.27
(c) (1,0)point ofinflexion
(d) (0,2)minimum
(e) (0.5,0.25) maximum, (−0.5,0.25) maximum,
minimum at (0,0)(y ′ doesnot existhere)