Math Assignment - 01: Topic: Functions of Several Variables (Ex: 13.1 - 13.5)
Math Assignment - 01: Topic: Functions of Several Variables (Ex: 13.1 - 13.5)
Math Assignment - 01: Topic: Functions of Several Variables (Ex: 13.1 - 13.5)
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a 9 o) 2-+3
(Slo.2) 1 (es(x,4) 2r -4 +2
(b)F-Lo) = - 2- 0 +3 2-1
1
PCS4) 2.5-t+3
(c F(523) 2.5 30+3 =10-t 3
l0-3 +2 13- t
17
8 C)- 4--t
2
CF Ce.-7. 2) =e-7+2 13
d (o,-I, -) O+ 1-1
2 S(r)=4-22-2
Dama n
** 4-0
eh t
Som Ds
eqnsonn
OS S 2
28|SCy) 9-a2
in 9-6 2>0
2 9
1 O
The cdown a l l p t s that dte i oP.
x(u,v):y(u,v);z(u,v)=u;V;VA2-u^2+1
Q.43
Z
x: 0.51956
y:-0.39755
Z:-0.83853
Double-tap to reset origin
x(u,v)y(u,v);z(u,v)=u;v:(sqrt(144-16*u^2-9*v^2)/12
Q. 44
|5
x: 0.51956
y:-0.39755
Z: -0.83853
Double-tap to reset origin
A
Q. 54
-2
2
-2
A
E
Q. 57
4
--4
Ex 13. 2/
Am +3
) 1 ,1,2) - y
ddet ubsAhvHs ,
- I+2
Cs (-t,2) -I -2 3
We uve hmetn Gnd a reona
u n ton is
eorytnvous a cat ppints n s donan
Ctht is, ot all s wwe deomindor
O,
Coninuous for o *
Ca) v2)
s ) (7,4)
y dNesubdi fuMo
22 m
= &2J2
+3 +
h IS Connos
-
(M)X21) -1
r 2, 1) -3-1) (V-» D
(Nar1)
C)>(2.1)
( ) 2
m
C > C21
ubshhvh
Aria
I+
2
m
C)>Loo)
Lmitdrg - S
a
C)>o.
im O lm 1 = 11
Cos)(sO)
Gne e d /erent
pn The daant enst,
he CoSIe hine on camo egwene. SS
Let, - 2
*ro
w Co)O0) 2>0ond
Cm Co)>1
dennJor -o". So, lmit loosn n
.45JG.)
Fov h d0
.01, 001O
AsWe
Aswe ohoch =0, tke n ehan ton o e n
Hece svenidoo.
whm
2-
46 2a3
h J - ,D
) | (0. 25,o)| @00)01,0 000o1
yMs 4 1oo I000
10000 00
Henae, 15 vehed
X- 13.3
E.tE2 2
=
22. ( 2
2 2 2 41
28 al) Jn
.
Snh (2x 3y)
2 shC2«3 codl2m)2
2 cosh(2
cash 2 3
3 cosh (2a
Varama t
4n
Wwn AxO,
t 2
C
4J0
-
+
Shen 4 0
1
2
4S2
2r
2y C5y) 8 -
2+S2
03
452'2
) = 27
G5
t7 )= A4*s 2)-29 GAs.s)
)
4a22 24b
2425 -
2ne ( 5 -1y
823
3/2
,) 8
5) 27
5 Caa 4--a22
) ( 4 - ) = 0 2x-0 -2x
(4--2) 0-0 2
-dvechon = - 201) -2
=
A+
C1,1,2), skpe.
2
A ,2), dape in odreckm 2) =
ec2G(*J, 2)
I-:22-22
:G y ) =
2
V -22a-
1-22- 2 2%
2
- 2-,2) . (1-22-y2
- C--3-22) (o-2-0-o)
-Ct ) (-2)
Ct--2-22)
VI- ay-22
(1- )
-(1-
Srilov, G ( ,:) 1 - - -)
C& fCx,)3 22
,t) 3-2/
22-22 (3-22
23-2
3 -22
32-2,2)
23 -22
VB22
Cy,i): 32-22
4 2
23sa23
At.-2,)>d'
342
AH6-2.0> 2
N 3h-2
C,-2,)S J,: 2
V3+L.2
7 Ja) 4 -42lby+3
( +4 +lb +3
2 + x+0 - 4 +0 0
2% 4L-4
04 +2 -0+ lb +0
2t4-
a+2-2-0 O
2+ +8-0-)
M4A 2 & kecr
2 -40
G) 2 +8 0
-12
a 24) 2 z
48-2 =0
- 6
psw is , 2)
083|o) N 25 -- 2
25--u
2N 2s- -2
2S--a
2 - -y2
. A)
2N 2s--
V2--2
25- )
25- ) 25-
2
252
22- (0-a -2
25- - )
(2S- Y2
ro) =
N2s--
-
2c-)
25) -
2425. 2
(25-
25- +
2S--s2
2sa
2S +
(2 5- )2
2 2S
(29- 2
Sr C a-25
4,12
C29--y2z
C) (2s-2t) -
hwhnr
w-en
92 PCg,+) 2-3xy +? +*
=
2x
C2-3 -3
3 O
0 -3 +42 42-3c
(42-3-) = 0-3 - 3
on
C42-3a)) =
a.12% 2a -0.20
+0.657 2+17.70 -S1 53a +842.5
)= 0+ 1.34 +0 - 51.53+0
= 1 . 3 1 - 51.53
2 ( ) L314
(b)o a(a) - 0.24
0Amples d he deoivhve s
decYesnGnd
and o e ph Will he oncave odoiM.
Fm&), 2
1.314
, 0 ,indes t t h deryhre s ineres
2
an he p h l l concare P
129sp J()= +32
3 2
(oe 2( )
2
Spr2-(s)
2
2 )
bta0-
ha(h-
3
Sooh)-J(n)
ah(a32
242
h>o Ca+2)
ho (+)
3
22
.os)= O
hyo h
h
,0)-lb)
h h>
open k R ca Lo,0
Hrmevt ee medns ave Couiuou c non drign
2 n
a 2) +
(G02s co
C2) +
a&)
22 +J *
+2yat jey) ol
- ) . (**) -
Crs) 2 (2-3,)
2
2-3
-2)
-
2-33
D -3) n ) - (a) 2 - 3)
-3
-3a) )3)
-
-3+ 3 (*s)
-3,2
L-3,)- 3 ) ()-_C) s(a-4)
-3) O -
)
3)2
2
tta rdanaid,
Ttat anaAA, olwa
ds d 3,)+3 Cea) As- d2
-3 -3y)
J( 6--
I
Ca S2.1 16 - 4- } -
S(D)
=10.437g- )
- 0. 5 25
w C 1 6 y * ) 0- 0-2 =- 2
=
d e -2x dx d
=
-
0
-, 2. 1-2 : 0.
d
. 1.0S-J =0.05
-0.5
2
CL21): e 2.
f2. ), ).0S ) 1.05e
=
(e) e
,d ge*da +
ed -
d 2 -*, s
0,||
0.OS
Od2: e:(o.+¬(o.os)
= 0.1Se?
1o84
-3o) I-30s) S.oc
2
1-(3.0.05) 3 ?
2
(-0.5
Trevrse 2-f() - - 2
ra0.0S, 81- OS
2 (1-3)
2)
2%
y2
= -)-2y3
2 (2-
3
dt 2 da 3 dy
=-20-Co.o 2( -o
(o )
N-0.012 04
23 en oum i h CL
wih S i (W
hay12 i ( H)
Voke, V» zLx VMx H
SL WH
WL
V LH
. dV= W L + LHd w +WdH
As
dV- 5x12 xo.02 x12 XO.02|+|BA O.02
1.2 1,92 0.
392o
2.92
xlo 0.81672
8xSx 2
Ta 32 ( ert a 224
b 20J
Tom cse law, 2 20beos
hrcosre 2 2 -
2g cos9
2 2 cos9 -yeos9
2 23-2js N -2g cos 9
2-2 os
2N 27- 2ss4 2 -2aJeos9
2 n
2-2s
a 2eos
Wen -3 3 1#443 9 2
8.27 f4
Ex-13.5
t-1
d - 2t
O r)+)2t -+)
-+t * 2(t-1t)+ (t-1t-1)
t-1+t 2t2+ 1+ 2t 4+t-2
t-3
CI 22 + 2
1 t t + +- t
3
2t-3t 1
2
dt [2 34+ 6t-3
S- t
hn (2a1%=2cos(2)A
S (s+4) 1
D
2 3) S cay (2
-) 1
9 ) 8as(2 )
Cos (2xBy
Saos2 (s+t) r3 (6-:)
Ses (2s+ 2t+ -3t
5 cs (s-t
Scos(-3)
NoWz
t
++) 1
-t= -1
C - 2s (2r33) -
Bcas(2
(2n3)
N
cos(2n)
-
cos
2 G£)+36-+)
-cas 2 2 t +3s-9t
At seo,g t»
c9s
ot
C
2 2 We cosge
g2
2 S - 2t
+
S -
ESt coS
] 2
C-zen ya)r= -s
2)
(as2) =*.-(cosge) +Cosy2
asnJr) - a
6-24) 1
2s coy-*Sng*
23ss-2t) n{t°6-26)
2sos42-213) -ssin (G2
Cos JF
S Cas Cs2-2)
'os (a-2)
-s ost-2)]+cost2') 6)
n (st-23) ()+ 2ssst213)
2sas (23) - t n s4- 2)
7 4
>dn(%+32) + t - 4 a 0
Le,
Fl) Jn* )*a *d-4
2 2e +1
2
- n ) +t -
=
a2 1
+ta2
.29 22=- 1
2
(2y 1) =2*
.raa) (+3+t-) :
2y+2
Thesrem 13.8 m+ drdA
.(va 2c
()
+2
22
21+22
84 2