Frction of several variables
Lt .
J= fl)
d dy 0
a fa,y)
3)
J'
2
* Find the and nd por kal dert botine fol-xay
3ay
-]tufo)s-(ha)o
&y zax
(sy2 Sar)
6y
(u3-*),4
)
1-4),g+
te
2izrhrr)
(2hr) -
-
Jy
Jy
Y0-y)t). t Mly )o) +y/2-j'
log (ytny? p' Buye)
Ju
Jyz.
Ju
Jyz-y?
3p'3y
)
:-3(34y+z) 3y) 3(349+2j
log (ton+ fony + ton ?)
Sin z 5in 2y
syin 22
Jy
tone fanytton 2
(sin 22) (2 Sin ost) ( B
fan tony sfon
2 tonx
tonl tony ton.
tary tany +be
1Sin 2x) du - (2siny (os) (3er y)
tony +tony 4 fon?
2 tany
tonz fany+ ton?
tony toy+ lan?
(2 snz cos 2) (sec 2)
fany tany tonz
+fslz) zlan
tan) lany hon 2
2
bp
Func tlon a foncton
Ju
2
wen Y'-a (y-6)
Shou that
(t Ju Ju
y-6)
= 2 (u - a)
Siniloy (y:
y-6
/vu'-ur
2(o-ay? -
4
he
2
6
np
+
71744p7
np be cp t he
. Jq dy
(y+e) et+ (*4 2) (-e) +
yx)e)
-t
(ete)
2 Sinht 2co sht.i.
ti t
43 7 (yZ, , X-y), shoç that à,u
lot uif lrs,t).
() + J4
Ju
+
+
4 +
) +
Sp
Sp
(7s4):h
() (H)
n te
4
((
(J)
de Ju
dy
lagt, y: , Find du
J4.
y t
Jy +
22y°+vyet
2 lgyt et ,3(lg t)etiet
lag t et.
e' lhyt (+ slgt)
2oloal23
.4)
y=rsino
Show that
du
du (sin
Cos 'o t
+
V
(ha)
ne
he
(re)
he O-(
ne
re
-6
+
2
rly
y-) 3l-a) +
2 Jye
3!
y Jx
2
(oty Casy log
Siny log tanx'+ y
siny
dy
Casy
dx
dy
0 logsihy asx-x 9 y
smy l-I
og (osJ
b6 oh
log
(siny
26/o9/23
lay lev's apansion
[a-a)? + 2Q-a) y-6) ly-6)
y
dy 2
3/x-a)ry-607
(y-6) .)
dy
at le) upto hiid degee erms
- Expand ß cody
CoSy
J :e CoS9
osy
ecosy
esiny)
2
=esiny
1
: [e' (siny) dydy (Oo) o
dyds
Jyda?
- :|-
+ +
Cr),6
2
he
Eyand tan ( in porS ad a) ond (y-)
upto Somd
Sraond
foy): tan'
/
Ju
2?/09/2....
minia. o the fnctlenS
Maxima.i and
inde pevdent vaioBle
fay) bove velotive
s Sald to have
A fonrion
a avttcular sel o vche3o o n d
haimu m o
-lo+k,
ho Smql -ve valves of
ve 'or -ye
also
Colked ids extrcme vale(marimum and mihiìmym)
# Sa lle point neither max
Necrs Sovy can di tion ker may
ddicient (adton or exttemevale od )
let
t J
) it s>0, eo (a,6) -) marlmuy
Sa dle peivl
i)t- s'-o father inneslgalbr requ
kle toto fnd
Wordmg Rule fnd ma,min
min o fla, y)
Step 1: Find a Save the eguaton:f.
ld la, 6), (a, 6)..
which ahe
slotlvavy. gid: doy)
’ Find the cstreme vales y3 3x- 7 y 20
let a,y) y 2y t20
3-3
3y'- 12
3y 3:6 2 12: 0
Stoliona iy points , ,,(-,d, y
S
-6
4y'r-
Yy
vakes etremy the find
39 2,
35
24 -)f-y
-2yt26 +y' Jo 20
1y'3-
+12y
mazimyn -l--)
pint
minimym (,2)-
pdint maximym ’
>0. 2) (-) rE-s
point Saddle ,)>
point Sadle f,-z)’
Co -z) 6) t-s
minimym fl,)’
9XX
(er) (er-
(2r2r-)
(o'e).
o4lrol23
Molti pliort
Lagrarg s Molhad of Uhdetemind! Mbi
nar/mym
f(a9, 2)
9 by ) Coñditon /ooshiaint
in the fom od belahon
)-’ is unhngwn.
F,y,2) - foy ) +g (y 2) -o
Do not inte chang nd
y
* Arectangulor bar oper ot he top is to have alume ot
32 (ubic f!. Fnd theimensins jn orler thot the
total suaco qea iS, minimum.
given,
32
Jy - 32 =0-0
f(u 9, z ) +22 +2y2
F(oye)
+(R2.t
)K2 (icX 0: 9-
(at+h)-
(krz)-
+7)2 (62
