Chap02 Solutions Ex 2 2 Calculus

9 ;».\ Duiwmilww 7;./\ ( c\v.\=.h,\ mo. 2 )6) 1

*?_~>"~~I?»r-~s? M f<*> 1» ‘Q ~»1»~w?‘*”“'x““" ‘MM M? -f0" clwld L.‘ f'(1_\ 4 Lb

xdlamlldh ,

f(~4\ = §—(fA+\¢\)--§?(1)

\'\--90 \,\

Sen; J;tM;1@;.J. _{,a-.J1.¢__;-5 w<@

A *\ _\® — *1-'Y\-\ 7

(9 3; (13 = 1

Q K-£.(_c,?) = ¢\£‘?.»>.&m‘A

_:_‘~(§“\ = O42“

at .-: -L)-L

“ 1

@@ _j;;_(q“+\,) = 'v\(m(+\¢3 - 0L

v <

@ %;(u1-v) =éa%‘;¢‘13’;

A - . \1.é£.@. .;.;(U.v) -u¢51;+ MQ ' U__UdV

3-(<73 = "___f'i¢‘?‘-'l"T*» _“I V7»

<1 - .41@_.£(cu\ _cd*_._ _.____._-V

,. I-.-.~_~_=-="=~— V 6“Mlm

‘-45.

_ __*______ - _____ 1

Exercise 2.2: Page 1/29 - Available at www.MathCity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.


A .5¢@CAX

<61) %;(c.m t_%<M~.yi

@ = $1151

' ® 2 -c05(LLX

é-— <$¢.Cl\ :. C¢u.’b,,,,\1 dx

I

;'® %(¢.~.¢q3 = _C',5u_1_Q§.A

c

|

; _i_1

r s¢@,16> ¢§;,,¢,3 _\___%

Ax J-C-£1

@ -3; = _:L_£!!--1

i ' -2 ‘

; \+1""

I. L ‘W = ._‘:® ch (Ci IS TE‘-L

i @ é_ Q;-4‘ = __.._‘\ Ag c 1> ‘F;

@ i(_c.=.;‘ ), =J1 L1 -;\F;|’

‘ su<®6) : (L&\,\~,L

@ -3-; = 9"-¢\'\z1

_4__ (Q.-Um‘ = -C..uz¢\\~L<14

@ (g,_L\,\~,§ = - S|.L\rw\-'§n.,.\A1k

@ =- -'0-'5'.-¢»\\1-%\r\1




\L-L. ~-~ , _,_\.(D °h(S.w\\/\~A> jT___.

- SP1 @

<1 I -\ \.-on (C-AM ~A§ - --—-M_;

4 -\ \.;;( \‘aM\¢\ ~A§ .. _\_:E_

\

.\ _l___ii (Cik 1) - “llsL_ §Q<.¢\T‘~x) = -5-‘--‘A1 11 [1-1*

-\ __:_‘_______,_§:(CA1¢\¢\1) - *1-*-—'H_1L

_.._..--_.__ _ .__.__?._-._.-..__.,

$Zii\§—¢:v'1"*r;;--J-:.'» 5.» -,. M »»..;~;_..: €‘l$.E?4£LvIr - "'

’

Merging .*~*=.;*: ;§";=:¥~..w.-.-,.- ‘-,. ""r"‘>'7\'1\~»\y =.'..»,.'.,_'2-u-n~uu..¢..)-.>..-_';. . 1. .,..._ ,..,.; . ~».' r‘

+ 10' i

\'>\\L\

lv\\>!

(9 2

1

S




EXERCISE, 2.2

Dixqérqntiate w;§r.t.(P1'oblcms 1-Ilii)

Sol. Lot y =‘\/WE'

o ‘J = (a’+.x’)"’

D34{.w-A.~$;;'1\.

l1~~(&+=¢> .;¢;;;~+\.=v».

I

,

9

%




I2.,

@- -1 (a' + JP)’ "FA -1191-.dx 2-._. .l;-_.--' \

(of-v-x"\"'*' ~

. 0 ‘ “KN‘ I - V »';;;;;;&.i ~'»=*~

!3\\fa U-= J U + I M

~Jx‘+x+1 W\NV\],‘§C\

Lat y = (t'+'x+ 1)"'gig’. w..n-i-.-~\-v_|

ii- 1 {S-(1’*-n+\)y‘. §;(1+i*‘)‘ . -§H'n+|) ’. (La-\»\\.

2 -L"'—\_ *' ' I-‘§‘)'3 (x‘-uu\)"! t

v'2x+1_TT_1$“

G+I—

so

5?

_.._.-i-i.--_i--_-

T1

0+:

Let 'y=

+

7:151

-ac

51

¢I+I ~ -'N/~0.-+34,-'\j‘G—.7( .,_._

‘ '\', I», Qo\+v\~~u.\-1'.\'\l\v\$\A1'|-»\.- A r,\.0-~v:wv~ \=1 5;;-_Ia_* _J?“,,\ Lia-atJ33} + _]'¢T.'{. L |a+1-Ft‘3-61;; (QPl§+\‘\-X)—l- ‘own SA—X

' uw- L111)‘ ‘“**> - \'*-*>= ‘Lou -2_’(a.+>$La->\) La -1|“?-.~£14_

ma->\ -- A-\-1 '- Ix\j : “"'JE:7-L

*D;“,‘,,.w.,4. -X.x[0- -§_(a’- x2)" ‘(-2x) - [a-\/a"-xi]’-1

1E‘-1=

I.

¢

dx ' xx . ..—-——----a+ Ja‘-x‘

x J05-:5x

Y x‘ . x’+a'—x'v?‘_‘* °“""*"" “ff

= 'f ' 2 ="' 2 ‘X I




G2

w/T{'““_‘\- ‘A xi '02-‘? ' \3&55091- L‘-k 3 I

I ox

80\

'22

T-F

Y a l;

I.

b’~$- u-J\.t- x

'(sin_~H>= i , _

‘mi; cdlx( \/sin.1r.(c0s\/2'jams .'_ 2v;

‘\/I-;ain\/7:-(tin G)"COCK-MOI COS J1

_ IH (sin ‘/-Y)’ ‘

___‘/Y sin \/T coax-sinxcos \/ xF 2\/Tc‘\lIinx sin’\/Y

7

5.

§s:

J.;}Q-

oh ‘j-* *=/4..

" 1°K1o(-I2 + 1) _ ___._L.1. 7- ‘= ’17;§(»w> = |3-(\+\2 = p"*(*+$

\ his l ll‘ -Mo ; M.

J.I"'- hM(14\)§;_“.g,.,|.J\-k"

\ \ IX2 --—-,..-----—-~ .

JI'n\v 1_§]w~(~IE+\) 55"‘,3, 4%‘. iL________.____...--- Nv~5'_y = liar! (sin I) 2.11:0:-.!]ml\‘-§\).(3\"-S!)

41 ““‘= '“""' <1-Y S01 dx_ = m’ (ainx) . J; (sinx)

2 ICC’ (Sill

I ($058.58

x). coax '

c’ (sinx).

‘1.*_ ,,_1§5'La‘a'

@

- 15' ‘T ~/T-~"" @<~-1:5»)= El 11

uh.

gz-|in\/;{;J.?W;- coax} -4\/gr§{cos\/‘:7--_"Ti_}1I




G)

7. y=arctan(——'—'-L ) b 5’1-xcosa9;“. w..\..'t-¥-

‘iivs 1sotgli \ $_ 1.__9:--\ 5

1-xcosa)\

‘+ (wLS'~\c~]" (\--'1CI7>Q).L

(\-1€A¢.)"_ \ W . **~1->‘<<v4¢:~~**““?{"*°*?- z(\-\C»»)"+(1\$'-n)" (\""\c*>\)"

(\-\\€-.q)‘-

dx (xaina 2‘$1K\-XUW+

(\-1C.s*¢.3.'i;»o. -1\$~'-Q -(-C-M)

~ sina' ______ 5""‘°‘j _:_= I ' I . 1 . '-

1 — Zr cos a + x’ cos‘ a + x‘ __s1p‘ q__,_ \_;_.‘¢-4,, _\,~;{(|_-}¢,,,;,-j.)

Aaina3 1—2xcosa+:c'

. f+x+18' y 3 lnxz-x+ 1

Sol.__y u In (§éx?0- 1?h—tln (.1-"~x+ 1). LA). . . *.

.(.‘.2+x+1)_ (x2__,_.+ 1)

dx '(x’+x + 1)dx '_ -. Qx‘.-x+1)dx -

_.- .'>.1~.+ 1 2x-1 - ' ' _

'x’+x+ 1 '-x’—x‘+ 1*‘ -V

+ 1) (P-x+ 1)-(2x.—1)(x*+x+ 1)

'( (’+ +1)("- +1) '

(x‘+‘A+\$aY' 1+- \=_1g?-{+1 -y;!Z_-1.;\_.'§‘-_--53-\ _)= --IQ’-u-2.

x2 I.9. , =1 l\~~\+nu‘-Al»$ <‘ jf ‘*7 j

O>\*1+\)(x.-1-M)

1. 'z_g!__!11(-Lt 3 = )2’ - (\

. \**+\X1"-x-M)- , -9~'\'\~ -also '--———-——————r———---M 1~3_L6'~'\

‘V01 = -in X?"

5

M. Kp.)~J\-",‘* ; VU_,,H-_,V,-‘,?.€1‘_\_v

.l_.é2. = 1‘.é_-+L~1-1*3 A-L 1 " m:2!@,,'%_!*J.!’?"i;{1,‘vi.i;~;‘}.l,.*;?f=T_}




= x+2xlnx= I3 x (1+2 nx)

=9 =y.x (2lnx+1)

. =1-'2'. x(2Inx+'1)» =x‘“". (2lnx+1)10. ln(x2 + 1)‘S01 Let y = .1"(1? * *")

- - hDierentnatnpg w.r.t. x, we ave

Q1 _.__1 (12-+3‘)dx (I +x§dx

\ _ (2.1-H)..=

'__ '2.x+\-< ‘l.('\\-H3

""¥,,;-

11. y = (arcginxSol. Taking logarithmugf boih sides

-Qhj = -Q¥\($§:»:‘A)*

1,, = »Z'.*.:,.gs~.::=.;

BL“. w.J\-t- X

%1%41 ll; \ . A ‘~1> 0I I -.

*1

Now; letu =x"" \-l'nU .-.= .9.-~,1.’*'

M M {~41-1“

}_g_u_ 1 1 1udx\q;.»;,+iInx-

1 .

='-L + In x --xi xi




a_J€,_,

1

--1.9% =-1'} ( 1 --'lnx')

_A_,_._,..~»:_:"'1-_,.,,J-A .» + 1%_ \.,v§~,; .1r * '- ' '9‘!'§

1 S=x"';5 (_1-lnx)

°'- = it-‘( \--Luz)

Putting in (1)

'3 \. ..

m; gm; 5‘~. ,,_,,.r:1

"7 »..A»= ”

iH”

vl’ IF’

1.4.2 *~ .;.______ L."‘ .ydxl. 1-w:1_X\___é+ %-~=~ 1 (v-L‘)V Q $2; . ~;.£,$___\__\_____+1“ .1“.-.),.s,.<\_,,}

$0.1. C$wg~ .5 ‘it-_“

M, y=.'r'—9, iflxl as Am_§>\ 3£"A>>a

=.-(8-9), if_|x| <3 7* ‘* “°‘A 7D1141» 04- .1\;.1

32-. 11 =~¢\~I~\>/5

. ' =..-11* 4 W43

Sim}. ,l\-1‘

/'\

F;7*‘

Val’-

'\‘i \ ‘ -\1.-__-~ +1. ~».Lv\§iM‘A-1~'(\-1;‘l‘|.- 5 :\t-Q‘ V

L.-1

_-_

N

‘Ii----_-_._..

" \]._..

13. y= \/1+ N/x+~/'4;

‘Sol. Lu-. 3 - Ix-rD:-“- tn-./\.§:~‘L

ell 1 _‘____ ____:._ _ L (~14-Ivtv6' 1I‘A-\-J3,’-\-"IX; A‘ t

:.i1+ [1-\-ji‘

-= --M

AM-

!!l.9\i= Lxxt-.R

_2. i1.+\FQ.I-1" ‘W 7-I’




I m ,

1*-Q xv:5°‘ 5" , if xjé ox‘ -

Maéz.

=..(,2r'-+1#;).”" ‘if M‘ o

= u=~>"* '~»»"»°]igF 0 3-L $1.0 ,'

Y»-Q. <.»~.,\‘;t\<'1, ' ‘

= lib-1)-2. = _.!,_-1-,. »J~; awe} {

T

15'_ Diu:'0n°¥§am' H =_ J;:*- ii LO

$1+x'--,1] _x'2._ ,. " -2

Klarc n ,__...;i1

Fhta .‘.~.a(I\\-q,_\_+\..

.-pup,

w ft arc cos ‘x’r_...,x

*~' - -~ ‘.1-m.>~~=<-* *-~ ?

I-Gb U:C¢‘

ya

'¢='~-C-su. w=a~q,-.

\1.1+x’+\/1-x"

'= tan- A (

*-' + 0081]. (1§'c§)3(E' '

f_ ‘(.1 +.c<>s"1Z +.~/1-1-bad. ( _ _ .

\\ u-__‘ V 2cos" 5 - '\]v2 sun’ 5

= tan H L ,i " i .—_—"

I

§\f.\/_24co‘s2 g- + ’\/ 2 sin" g

tr ' Ti-Ca.“/,_ - 3t;;..5v(,_

U"C*°h. + 3'7-$12.."/‘_7

T11: (""u’l- ' SM“);* Qw;§§MU’L

' . 1 "‘*-~w~;1‘..,-9 \-'\@MU/t

k

'*T"‘“‘uI1.5 = -tail t“‘l 't,,__u/

SO ‘j = ‘K/H__U’L[ ~- L :'1~$ -= t::'*‘~(F/5"“/1)

NM’ ‘*>-A-4:-U ,

H9’ll

..\/L M

¢




16. y = x'i“y '

S61. y- I xmwTaking logarithm ofboth ' V

cm 1-n

sldes, we haveac.“

ll’! y = -in 1 ‘J

L4»./\.-"\:-\\u

. A-Av\u.u.§1.5- _L..m ,g,JM='~“’/a1 = wag.“ L1 *5

P7. 1""-= ‘-x - 1

Sol. TakingAlogaritl'_ni\ -o§5tF sizes, we have‘N1? 1 -L“ ‘-’ I .Q

'5!“ = (1-~>).L.,.¢

v» .‘3L-m 1. 1-3

‘_'§-2-'l-P‘) :

. 3 +w;Lu.c.\5. v/4,.‘‘* "/a,-»qc.»;.9,~=¢-4“/¢, = gm 5

¢‘j;(x-u3c.»,3.L\§ ,_ §5g;_,,y,8% = ‘J$<-‘~09

" A1

‘L

ss (!m+\\ = 1

-'9

¢;.=<4*

5b.. 1A‘

@\~§~w\

3 -.-.X

\+1~\\\

54$

‘I.

\-Ix

-1'

K:4'

PK:

, ;,,._,\.§.y\_

\+Lu}.\ .. Xi(\¢\-LwA)""

\+l~.\-$\+l~m)L

La '

(\+l~.\$"

=C_

:C.




I vjInu=xln

801.1403; May‘ and u=xYTa/ifingvkigarighmikof‘ both sides <3!‘ the rst equidtion

ys

Differentiating w.r.t. x, we have

In

ah2--—'-— lLl'Z+lnudx"x'ydx y

g

.7)Q

ydx-d=y' '3-E'E+Iny

-NV 8’-Pm‘!-3 -

alt luv =ylnxNow fr_om L _v __=q..r' , takingilogarithm, we get

Differentiating w.r.t. 'x, ' we obtain

l.@_Z Pix»v dx '1 +‘]nxd§

<£__ 1 <2dx ...v[x + Inx. dx]

=.x’[l + lnx.x _dx

The given equation is‘

ll-J-U=C '

Differentiating w.r.t. x, we get

Pntting the values of 5% andgé from-(1) and (2) into

we have

*3-"J1- V ' 'or (xy"'+Jc’lnx) 3*: ~\- [y'|lny+x". ]

»»is§:--~1

2232-0dx dac-

“‘4’/ +‘s‘1~‘~‘1 "'

(*§"'+>€’-1~,_>v‘-ff; = -nli~s-1»: +3-i.‘

+

>.‘~"i<

r—--1

-{RR

+.+

Inx.'. =

RR

f.l~'.~4.¢\~>;,;m =60

X ..




. Q1 " Inor dx xy

w-.>...‘

1 Z+yx”-' ‘ "" +. y" In ' u '?~'§»:;r;,'§:i

Sol. Difforontiatirig w

" (x

.r.t. x, we have

£3) (1 +y') - (x +y) (1 _.y’) 1

(I-y)’ I " = 2r+2>:v’

°’~ 9"?)*‘*-12:?-<=~»>+§1m) um»U(-

I1.3_

Q‘\‘

3 -X-*5.) +(x-5)" _ 3

"H1-/§+xy§) = a(x+s;~;')(;__\,)"'13 * 3(“) = 2_(>\+3~;')(,_,)’~

..3+ I1“ = (""'5'>')(1-5)‘

ma-""‘~

(u-5) = x(;_-,) +3‘*-‘Jkx-a)°‘) = §g+x(x_5)?-

+ Q=J_1=LL>Tx*)'(x-y)"

'20. x+arcSiny ='xySol. x+ air?-y =vxy

Dif'Ferentiating(1) w.r.t. x, we have

".1+

.-

@ ,

O -1

$3"

O'1

Q-JG.

1-="~<

r

U\_____J

Therefore;

I-1;]

2~F:'“

3+‘?

1._

' *d= xa-‘E4-y.\

—§L%; = ,5-\-:1---——__ = -

-

/Q--\L<‘,.*<

NIi‘VaL\/

~'~<

'D-4

._x 1- .

1-)‘=y-1,

ix ;'£x_____L—1>‘/1-2dx _ 1-x‘\/1—y2 ‘

\ .. __ _,_

Q5!‘ . - E

5 1 F

x i

t




I|

»

In Problems 21-30, nd}: (;):III. f(1) = In (:41/x‘-1)

SOIl- “Q1 1 -Q-v\Kx.y.‘.*m_\)

\$~\&~-w---i-1Lf(‘l§ = .

‘ Qulgrq)‘ (W =:J'~I*'.Tu)

= (\+)(1\+[F:\) J1?-\

= -—‘-— -[_E"'§.~..:1.](‘»\;*.F\T-_n) 55"‘

22 I:f~f(x) I In Te,

801- fu>=1n

_= In e“-In (1+e‘)-H1) = x-In (1+e")bZQ.w-»A~‘Q-)4. '

‘ f,’<x)=1--—"-—-c*=I--5,-L-—1+e' \-\-Q‘__ 1+¢'~e" 1='*""*"*.i—— -it-'1+‘e" '1+¢‘

23_ ;_:.-£;.xIn.l-’ V.

. | I '

Sol. Taking In‘ ofboth sides, we get

In 0 <1» - In 3“M-on fl!) 2. -9»!-Ln}. '

m..A. ‘\.-1W.§(1) ‘ Luu.-%+1~\\-I,-L

fm =§(~nI'='m_£..*1'0-\\ ‘

= 1 I’i::-1Jun-\

= X '

I




H=*\9~ gm = ‘sq 1:,';;§~'.,;;I:...1£)- &“(_1t;;.-3T.1+M.§3}U4 . u.J\. . 1.

24. f(r>==1" -———-“H "1-VTV

Is

I

\S3

4

1

i

S01. /<1) . In (1+~/T)_1n (1-~13?)i

.

I \__ __L.V5(1) = tfi - (O

.. _L,.4__ + 1~‘ 1ym».y~A) =-(\—5\>

s J-—- '\'1? \-\-ii \-_F\_

= .1/[\.;£i:_\:=.Ji~_. K

1)“ <~~r»\>u-rm_, ,\___.__.__1 M»lji (\-x)7-

_]§\(\-1)

35- f(;;) = e" cos (b arc tan 1)

3ql,'f(,¢) = ¢" cos ( bt§1‘\ x)\>;4§- w-A-t-1 Q‘

;'m = 2‘. -sc.,.(\>h:2,q.b.1..= )+c»(\»*-Il*>- e. -w\ ML"

usé

= ¢ [_¢.c.~m.::1)_- %;,%=»~\\~*-51*-F]

;~}V»(\~¢) .2Q,§5,;1\ ..\>s;,..&.\»*»Ii»\-) X<@;§GT”'1‘ L % %

~ \1b+a+\/b-atan‘ i28. f(x * <1

‘ b"a ‘/.5_:z—\/_5—;a§an(-)

[ORNR

Sol.

.._[\jf;.g,,<_2§__.L. _\ \ ,_1'».'.".~Z1»-~--———-———-—1=f“) ‘ m={ J'$K+JTT;.m..-:1 ' ‘ ‘ T‘37~'F7—1"'~“!L

s

|




{ho = _~_ 31-lb-~~$~1=:‘

~r

T51

M7 1 ~~ + ' . ~

lb-Q? ‘B-HA +5: B-Q-Q - B Q mi/1}1

P‘*111'

PF

:1

"“'_1r“-""1

0’

._ -- \_ _______ .____-_ ...___ _ ._____.._____L__ _...4. __:.‘zju»-sum) J -\-A '*q~,),-K tr.»-g Jxm. -]\»-~hw1g-

‘I. 1 _______ , '= _:§i_:_.' . lb-\-Q -‘LF-A t§’~j1!~L +- i\>—\-Q i9LQ'I-L24%/Q 4%-m (,|\».\ A-{KI hm:/1)(“';'“ RE; ‘ahv

= _$i1-‘_I§_:_ n7_,d¥>+A..1ll»/» (‘>4-Ck)-Utv-A).g‘_:_T§':_

SJ 2* CL» it-_

= 7 K x W i .1 N =(\u+\\(‘,, L -(B-Q §~M 15-_

c.,>‘L1.

1-‘ 'l-1_-.C -Q1.‘ J> ;' \>c${-*+<\c.‘3§ -\>¢.-}12_ +<\%»1;1,

\

_ m (C.§1;+¢.-£}§+bLC~§-‘§—_-Sig)

\

Q a. +\=Cus~L

»27' f(-U = §ra'.5inhx84" ff" x.¢:.SJ.~\vm

.u..,\, .3;(M = x.q‘5.c.\\“ +1s.;m.8.‘-J~.¢ + I-<1-$~l~\w

4 ,, aL‘[x€.;\\1 .»xS&.L~1.l-¢ -\-Si~\w\;cos x 1 x~]f. f(x)=---fa?-;+§lntan§

Sal» Hw {(1) = ._.L. c¢*__ +J..,Q~\bM§__- . 7- SL3: 7" '“

L.)-v‘--'2.-\\ 1f(x) - -L $Q\u.L.€-;~‘*\-CJ>\\.1$r~'-\\("**_‘8+.!_ . -L-— -

1. 17> L “M3; 2. 2

1 I__\,. .

1 $,,;~‘1“>\‘ \‘ *6-~!.:

w

I‘

gvi

1

1' - sin’x - 2 cos’ x 1 co's§ x/2" 2 sin" x + 4 sin x/2 '

’ ' cos x/2 ‘

. K5

l




-sd;1—1.&'§:_. .___\»——-v"_“""_'_'% %

"‘i{"""'v=M»L 1* <~=>>»%<~»*1> "»

3"‘ . -.smsx_1_ - 'si_n’;x--~2 cos‘ x_ 1

2]+

2.(1sin is cos

sin’x+2cos’x 12 K . * . -

~ -2 sm’ x Y 2 sm x -

_sin"xi+ 2 cos” .xf_+ sin’; ”_7:‘i§>:l~5-xf-i7(__Cg1§___"§-_

5 2ain“x - ' 7.9521

j 2$s"»>=\+<;M _ M- = Q2“I 1%; ‘pi29. f(x) = arcsec (cscx +_\[; )

54“ Wu gm, ml‘ (Gu.u1+_\'T')$44. w.)\.-'\'.-'1.

é('\\ = __.‘._- ~~ ‘ ‘--;,"_~_-__*_'p”_':;;";T_ZT'_"_'I." (clli 4»-Ii) .

' (C~u<1-vl-\&')\'i (Gmm +33)! -— \ at

= _ - 1 - (-C-v.u\.€»\1+(¢,y_¢__1-+\/;)'\}(<-u.cx+\l;)2-1 Ly

4- z_F\C»u (Jun -3-J\

3 [CAI-ml -\-.5) Tau}: +1 + zfl C-H1-‘l~—lK' ly‘1-2\/;cscxcotx(ml} + J; ) \]¢-»>e"e.r + x + 2\/Zipmx-\ ~

_3()_ = 1+ ;1 Z .

Sol. Tgkinl ‘“'°“’°““=id:§» We have.

.4‘-y\§(_1)-; RM

‘I . _

us" wuhkni 1 \-£1-\-\).\ an

(\.‘» .---~

- ~__. 1» ——..1.._..§(1\\ -.: §€'._:_\..;;_. _\* (x3 U‘

'5 1_L_‘ {Q0 = 1::;!~1 4-'Lx.§-

ft!) *"“§\‘;’fA;__\ kkxjl) A 55;.-

1 (V1 + \ ‘» x. wwvv.:~":mi $1




‘*9 = §(1)“1xQ_,_(‘+'\___ ___\

_-ql 3l(3l-M)

‘Q

‘= (\*'3;)[7-"*-Qmlx-la) _. __L__. ]L “_' 10m)

Di‘aren§ia'te‘with relpgct to x" each 30!‘ U'>c"f'01]0Wi'ng

(Problems 31-42) I

81. arctan 2-xSohld’

_ _.| + L3 * *~~ ( T3‘)

lm11‘: \

. (_g.J\. -k. >1-

4 \-H1L Q;

= _ ‘ (1-_:‘_7-7--(\+u3»(--\)L ~..-‘_.._A_ .

->‘1-11 #\-b-11= \ -~ __._.____._\

..__<1-»>‘~. _@_w.>%% <=~->*(1-1‘; '-

(1/-1)‘ s s 5 \: — _ ....._..-_ __‘_ - v » ‘. _- - - __,,__,--. WW - _-—__.» _. _r -‘4-lm-m‘+\ +‘** Hi (1/J-a_\"— 5+§>\_‘ S(\-\-1"\ \-\-->2?

132. ln(arcsin e‘) + yx‘ =

\ I 3

S°1' ‘"'-~- 1~\(uI.£‘) +31." -. \h;..A.‘\'.~%\.

-_~-_~_.¢_ A. 1» =‘ d‘(‘F-.e._ + A‘ (31) O5=7~:(€‘_

‘L--—t———- -________‘ .€?‘ +§\-?.1\-¥\\.éj-~-:05-ll H?“ \\_ (£151 *

\ ex 13-~_"~-—"- _---, +. 2.1 +1» 3-_ = c9<..l(e_*> 1"'_'\_em 3 4,‘

1 A lX. ..‘L. C

. A‘ ' 1-‘Pl - —:‘—-— -- _‘ V Q-(e7'§ |\-6‘ '

@ &~ : -L __7-*\ _e__,...___‘ -~ 15' C.“Q xi, " ‘J -. I-4 ‘ = _ iv —I 'i"""""*"_"'_"_"'_':'I:-‘ ‘ - Q-~ G. F“ 1, 1 .-' 1 \

‘» > * new L0. Mg

1 IQ’




.

33_ y .-=i (arcsin 3:2)?‘.-| 1‘

Sol. Let 3 = (""1 *2)Q2“. w-A-4» ~A

%,. . 1<[$}:\‘s\‘-_\‘-’- ¢;\<==:-*>‘R

..\ K "‘= 1< (1-=.. M . -‘----_-.-.~. . L1

X \ (‘\.\\

5"~15

‘L11: \$I~3~‘ *f_')f_—‘_

_;\-uh

a4. arctan (1-')+y1“ = 1

\7|

?,£

~51‘

N1

. ,

S01. G'\\u.~»\ '\‘¢Zl(5Iv.\ -\-'3: ‘ \

'>3»\£- ""‘~'*""\ 1 Au -‘ k1&9}¢‘:!3:_\_"§ _‘_ 3.1.1 -\-"A. Tu .. r

\+\”I\\\t ‘P \A,, Q

\ ~A'\"kw‘°_A\ 4. Zi-\"*1"‘ "

' "“" ‘*% =°\ ‘*‘§""’\ + '*-*'~*'** "‘

£25; L‘ {AW = 0* “*1. -\-‘1~*‘.\* '3-I

-%t"§," "’¢!_:z_

‘~35, J’ *~g‘é:L ‘-113 L \.\"L__.§:1-;;- ,3; ‘J _’~__'}(‘ .,\ =-1.»\\bu»1.

1 ‘hi \ ‘ " ' kt'% ~ 1 1.\‘ CO5 X M \

35" 7 = 1 + cosh x \ "Ida (_*""* ‘*9 ' -.3; U5 \-‘L!\(‘A"') ,7/P§v~§t

Sol. \" “’_"*:~__f__-:~.C! G .

c;.;. .1 \--r.»,»\l‘\-M-A\~»\

9;-.A» ‘\.- K9%; . (\~c.Jm\.%;(\.f-‘A13 -. (\- <'-»&u\. (\4-W~*~)

<»~b;~\.;S’¥'"__'“ % W“q+r.,.m)g-$_..__1,,- \\§_g\.-<:.,,$§_1_\,\;.<'.'-.1.~\

(\+t.An\\‘"2 _‘:_&_1!\ll_:_§':_§‘:\__j1£_$_',J,~_§\ +G‘.\\-‘J-R‘ "2'g'\A* ._._

» <.+¢.w '** ‘ <~+¢-Mx‘, -1-z$g,§~‘\;§__.1-"lb. .-l4Q,;,,_M1;.LC-J»‘4!~L $¢..\\*h- _\,,... =..\3..\\!_uJ\_l"""' 1-. . ' - I Q q----"' ‘ ‘ '_

L=*‘-*~“‘*/1.)‘ uclm» C-\»."h. ¢~%~“/1 ‘- ‘




M1- 3 '35- y = In (¢lnh'2x) '

"0"" ('5-~\\2.1)

B_"“- U0-.A."\L- wk

.-_ \ 4.,1: **--~‘ 3'; (+A~»\\u.)

-hank2.1

1\‘ -———— . $4~¢\\11.7_"K-u\\ 11

‘ c‘u\h\ - ‘ind-3‘'11. 1,

. G1/‘\1.$ \_..._._‘._ _ ________ 2__

§'\-~\\1»‘£ cs!5\:~~|_1

1 u‘ '§T'EZ§ ‘ '""**~"'~"" W

’ ‘A 1*’ vk 7~°~§~-~~\\u\4-C;g\\1,1_

S01. ' y = ]Qg‘w'

g_>’“"\

H

$3"

ll

“.57F

'5'.8

"#3-i

4* J-m,' 1+» "X: _.._.§_._.___..

(‘+\>~LW\\7. QYH) ' 'a“;T\]'_;;O_x_1_

>-1I-Z\._/

= -= '-1¢"~u.<.\\'q\

l x + 1 - 9““\“‘* --sv. l0g1o(T) r ,.

1“ (‘A4-\__._:'=§l

J“/\ \0

‘-1

- (1-».>.>]xi

c»x --\ -1

_..\"" ____

38. arccoa ‘J 1 -xi _ (.4,r"'3-]‘1,_\,_, .3; .

S01. Let y I _¢$§$1__,,W’

.é:2A _ -\A - —---- i=‘ ‘-—--—\_(“Y_‘_:_,M, Aihn >0)

= ——-—-- - -‘-»-t-11> _ =1

-s.{W;=T 1-F31 "




A.-

. 2 v ¢ ,A 4» ‘ -15‘-4 *""

1; ' - " " -;~ >_"'~ 0.717?§i“i{?“!'_'.‘4%%Z"g‘i.."§i‘.~ Q51

VJV\f‘N.if'i~§Fz :;;}[email protected]&i},",&Il=%’§‘

\ .' 7-I

'r?5'W‘= *_._‘__._

3‘!-

\><s F-I1

cum 92¢ ($1;/v\\'\‘)

$3" I-u< j= <.¢z‘(s¢“\A>\)bL1/1],. Q)./~..E-‘L

4»ch

- \ ‘ A §%iM\A_x §*'*$~.~;\\“A1*\l§4.~/\\v\‘A-\

-_- __L____..__._< \ CJ»\n\

s'i"'\\'“( l&g»§»\/\\I\L"l-"\

CJ<\~~_A____

X?“-\K0. c>\Ac$L~$<uL¢c,9<,QM»$

M-

$»9\. Luk %‘_/;.$C£‘JQMM_§

»'~\s- W-~*~"‘* -all , __\ ___;_:.-. . i-(c.;x,‘$lM§4* §\-§c.£‘nM»\" A‘

\ _\ . ,\ 4

’ 1\-\-Ll-ML 7‘

._ "\ ;"""x(_\+9l.1§ J \ -" (C-i"~Q-*\ __’/____‘_/,.

C.a\(‘(u-1‘)

s.&.- Lu $3 = ¢~!~\/?‘§\**“)

A;

g,q.J\.¥:"k

__ \ __ .11‘-(_\+‘k‘"\

Lil-S-91-

AA»~X?\‘;’_tI3f:\"3K1* 1* _... :I--;. ' :: -—--'—'_”"' ‘L.____,__W _), V“ F‘: ~

\

L-2.} ‘j%?§.‘u.A.£.f ___¢;£_5;\Q6g__|__'

[R L +...\§~\ A1 , f§I¥5§\§"v.

a

K




‘I-L

Differentiate, (logarithmically) with respect to x(Problems 43-47)

I-—-¢——-—_.-1-. I 2

I

43- Y = “*1

sols-La 5\(1"+ 3 "33 I

3 ll (1-0*]bu.‘ "'N\ IV\. an Mu‘

J/“*5 = 1“ L *(\'*4\_)__}V'i(1-1)‘ '

\\(1‘+|_§‘ 1

- (vs-1)‘,' 'l5'l’-l-at +.l~.(\\'-+\\_.X~.-.(_\-.\f‘.s

‘*3 " J5 l_-‘l’“‘\ +-l-(1‘-H.\ ~11“ (1-0']\1\%- \'°'~'\-‘E-K

- -\----2.1 1 l' '-’—*"\ 1- L.-LL--_7;__l ,3“ 1" 3l ‘* wk‘-a-\ ‘A-\

wl'

¥°

W

wl’

3 d‘

\(\‘+\3(1-$ ll ll ’ ‘_.__\‘*..l‘_T'*l'}_"‘,“}~",'7_"‘§,§.l'rL‘¥-l

' J=-l.;{(\‘+\§(1-\\+1~A-x(_1_\)-'z1(‘5H)] \ [1 2 1 1- 3, _3

3 $~1‘+\\l7n_,).l;.‘é$?_-k = M1

- ‘*(\\"-\-$(\\-\) ,l

An '1 - ‘I ‘I'% _,_.. _A, ' X-E£__~"‘~.‘ =

13-(x‘+\)1' 1”’-M33‘--=\~\'\.,‘ 7‘*0 v ~14- "'“””""'“' “"“" '+ >( U 3(_X-\}' Y (X-H H1-\l'3 1.

= L’-§:;:1_'l __ __ x.3- 31'.‘-1,-_\|_\,1 §7\§‘ |_\I - .__...__~_.._.____.l,._. _.-----V-M -- - I5.-\.\3* '(“"7 '(‘L*‘) 3 3:15. (1--\\.q3-‘(x‘+1lh44, y_ \r§(1-ix)"

_(2"';_1)" (3-4x)'"S°]' Tam"! l.08arithm of‘ both sides we get

"“'\‘j : .1/v, [I*T('-11;“. _(**"$""""'“"v|1-an ". (3~ln\) ‘l

= 1,, "; 2,,’ '1);<5 (\-11) l... 1“ (V1 gx) _(.5_\m>

= (jx +l“("“?l ' ll“ (’-~3\;j"+ L (a- w ll/El l= -R11 + l~.(\-1\'\A)’3' -1\»$2_3“)"~_ 1M(_5_‘w)"I3 ‘V

‘M3 = -%~j~~X -\- (\-Ml ) - (z_},$ (3_q‘1)




bLgg~ U-A-{V-K

: .\_. \ 7-3 6‘ ,~-;_- +-3--T‘-I-4-1)- .1‘.-l_-kn -1- l--(-*0

— 1 H Z pl. '3 3-‘I1

-3 _ *.\1 ‘.$\-11) \<(z-1,~;) 3 (3_“ i)

2 -3~—1+ " A + Z“ '7 M

** "<1-»1> =<~-1» * '-;rT"1- 1)

= 1“-3‘3,*§} 4_ _-‘4('5--‘<\\+\'-(\-u\\M‘ ) ’>(\_§_‘) ‘)7

, “""\*"* + -n +\£1 +\L-31%

‘“‘ U4‘) 's(\-1.» )0, -R1)

-23- : ‘H-35K H_\L,\(

“‘*‘\(1-'51) ‘5(\-11) ('3 -K1)

- '& ‘A i +3‘ _ + t\—-\L\J1 Riki‘-$1) '5(\__1$\3_\n‘)1

: _'Ei__(_~""'x) I’ * ----‘K + \$\_\$____

(14; 3”" ('5-M1)" “* (7--3*) 3(\-.u\)(3-R X)

Q5» y = (tanx°°"' + (cot. x)“'"" G) '

Sol. Let ui= (tan x)“""‘ _

mg 2“ PW 1.». #1»lml.) =In u =‘<;0t'x In (tan"x)

DiITero;I_1tiating, w.r.t. x, we have

1_.é1 . c.»\=\._;_..s¢t~H-L»('\a....3.-c.ué1\U 3' *3.-1

iii; = U {C4\\.(-.**_ué‘ __ L-(‘\'n~.\\§.O\|,;"\\=(*--3‘Kc,i‘,.~;.h - 1-(\'-~)-C~u¢§\‘S

' 1(+..8‘[' :6; - 1..(v,.,;,¢,.,},

*5; ‘<*~§3‘[¢..,¢* -1_(x.m-uh}

A \} ':

12“, = L.(c.¢n **‘*“_= \:~.*.1..<c¢1)

‘*1-u-*--k-v.a 'b...~g...L,..-G4-é‘1 +1'*((:*1\,S¢l

elm (‘J1ii‘: = V‘_—‘\'l-M-'\'4.~i-G-1:! -\-1~(C~u\_.$lr}*\

= v f__1.1.1.c-.48» + -°~(_c.¢,\.s¢b\\

. (Q,:~[_ ~1..Lc.m. QZJ]

,/J/A z (_(,§,3 £- $1511 -+ L (U<\\- <‘»u\u‘§




New ‘Q7,

3=U+\J1*‘!Q1“ - 1£1. .,'

d1 *,

Pmtq vJ-u>A» QYTJ ’ (*““‘*3 ‘[Q“2“'£‘“(*"~*>"c‘-“é“i + ((J=‘:;M‘{"‘iI¢}\ Q-~9~~(r‘.h).%u'I1L

46_ y=x‘.-e‘ sinx-.lnx '. “MS91. Taking logarithm of‘ both sides, we get

in = 1» (13.8-s;..,.Q_.)

g J'““?*!*'*¢‘+1-$L.1+-l-(-1-'il)s3\-Ln! Jr‘!-~Q +1mS.'».'a-Q-L.(_l~,\A)

-3»; 1 ‘A-Lu +1 -1» .1-'i;..~. .;.1.,$.lM\,)$4,‘. hi--A-k-K

' x"\i"'L*"l+1 + -L--~-C-M 4. .L_. .\_.<‘-.-.‘ 1,,‘ 1

. 5[l+l-\+ i +c,&,\ 4, _i___

L‘ i . “*1= 1.¢.&..1.1~.,)[1 Q+ Mi +C'Jn\+ £71-ii} A_____..9M\

Q.-

i“l<‘2~

In Pmbleins 48 - so, n

(1 + 2)’47- y=""“""'-*1:-——';r

‘.(x+1_)(x +3)

Sol. Taking In of both sides we have;

. z 3_ 154 , (‘A4r,\3V\’\-\-\_)(\1'_* vi-‘_*lL\_ \%1L J!‘

48' ‘G + '\{_- _ ‘J-5 *3)Sol. Differentiating, both" sides w.r.t. x , we get. '

L - 3 'iv)‘ &“[ j ' 'q"‘(“’“l"'Q““(“*$~»°-~(\‘-n\1-is - 11-_u+_1)-l_41+\)-z.L.(_»3‘+ 3)M. \o.\-v\.§. '1‘-~- . 1..L_ _ _.1_» _3..L_“-_-21

.5 4‘ an 1+1 £4; x 1Q; - ~,{ ._ .L. __ ix f zumu‘-;) -(1+g)(\m-¢$».;;,,,)+1 ‘Nu ‘us * '1 ( ‘, ‘ ‘ L \\+$bu-1.) 1+1.) J- '1 Pl‘ “"“”*‘:\ *<'3:('* -21:‘;-1_§;1f_§*(1‘+s1+1_>‘AM; _\\-gm){ll tax _“ __§‘$:ug§2) 1

y (~A+\)(\q 3)} (‘I4-|)(~‘+1)(Il

-\__ _\____¢_k; _ 't1.j~1\*:._F;A1\ ‘O

\_ _\_..§__ _5*?» 3* '°‘AL. - _.\....N £1 YA




e\:_ =-—~-ch

Q1 .

O!‘ dx=-49. x‘+y"-.-.3axy‘= 0"

.15’

"u$"l‘I\.' - .. w~n¢\‘J-.'

I

--mu-»-r

in‘"-v‘MW' .*(

‘ -wi \f;.-iv§_I

3;

.5‘,~§

A=_,

_;,,v.p

,..‘.-,<».

,u ' ‘ ' -N,‘

w...-. >-_ ,1vyuu»-am... ‘ .~

>

Sol. Differehtiating w.r.t,. x, we get V.

d - J _

3x'+8y’ -‘:5-80 [ac 5% +.y.1]=6d

0; x'+y'g'x -a[x %+y.1]=0I

O'1

&+s~'H»*», G:'|' t‘I

(1 -1.y’-a x

50. y-cos (x+y) = Q

Sol. It canberewritten as

y = cos (x+y)

ghldu -A1-4°74‘ -mfg -.= '6'd‘) .3 Q3, *1

1

Differentiatnn jboth sides w.r.t. 'x, we gqt,

' d l-‘z'=—sin(.r+y) 1+‘—'Z-x ‘d [ 1

at - -*~<-~+»» - $**(*+=)-A”/Au

dx

#3; *$5-(\\+\>\-‘$3; 1 - Q;-\(\\-\-Q.

at \*‘5~'-(\-aw) . - $~:-h-\-‘=3.

Q sin (x + X) .

or (1x_ “ l+sin(X +7)SI. arctan (x+_\') =4 arcsin (4-” +x)

“4‘- I-ulna-"15"" 121-'2(1u-s) - $~.:'.‘( Z-\-1)

\_\-L_{*_»'(‘*".k¢) - --»»--_i._@T~,<=1’~a=:@~\>* “*3 \\\-(em)

._‘_._ + ..l__.. Ala. ,\+hu»>)‘ \+(“\,§~ A;

\__ .

‘ \-(_i’-+1.)‘ — ' WSL

in3'3'

+

__‘____..§ _ e" _____ 41 1 \

\*'u'*"’>t A‘ _!"(e*-13‘ 6* F-(€‘.-1>{ -. V\+(‘.\»'\$1K

"1 \ 1: \ _m..H‘*<*~»>’E311.» ="_'__"""‘ " “"""..".;.-_-I-'~ - ’ Ci

A* _ \'\'§‘\\'?'§t \-(e."-\-I3‘ g - ¢s1\l- \+l~A»u\\

>4@r_<;;‘_;, ~:

E




5'2. x

-'7—[J\-(L’-+1)‘ -¢\°[!+Q-0-3)") [l+(lL+~s)‘- \_- 1 “"""“““"-"-——————-»~-- :_ ,_\ 1 “*““ M"-[ "(“"’5’ )J'T(¢“’+»\>‘ I*_—E’+ ] .

%1"('“”“*>‘-¢”(\*'(:*+»3")} 2 N b**‘°\1 - J \-(Z-WWX .

:1

:1-1)w L] 1-‘

Itqr

F"-1

of £3 I +<x+vi- xll-(<'*i+~>_’ ifI" \/I"-<¢'+x>’-¢-'<1+<x+.v>*>,

= a(t-sinl),y = a (1—cost)S01. Differentiating w.r.t. 1, wk: get

Nun

b

‘£5 = a (1-cost)

DJ

H5$153“

= a(-sin 1) = >a$:...'L

=.<iY 2!dt "dx

asinl\__ -._i.___._..__...-.

ijsm 5 cos 5

oz

bi“-

Zsin ""

55. x=acoa‘t,y=bsin"t_

a(1-cost) '

. t t

QO

U)

v\h)'\

‘C="__"'~‘-'COt';.sin

N"

Sk@L 71 I OKi:¥- §- 3 I'\D§£1t,

P-

8.2-gU

“‘\A)£,--Ié2—~#-Au

I

2

~ \»-m-t.- K“*3Q; ¢ a . '3Ca>£. (-g;_,¢)At 1-3aCJ>t.$;,,J¢

=\>s§’¢.u-A-4:-‘ll

‘O (3%i.£-cu‘)I '3kS$:\¥(;,¥

A-E 'd4's\.$.ZtC,

~ ~ a-aaaiea-Me.

_L_.‘~'-=._-£2 . -k_’q\..+,CL C;t_ CL

\+(1+~>)‘




2.1

30! aa5‘? x ’ 1+‘! 0 y "' 1 + £2 _;'q_'s_;’-r“['!f"*fb

‘A; $1,; (- '

>¢ ’ ~ 'Sol. \>'~&&- ~»-~\-’<' ‘

§L . @.{¢~+~‘>-\-1-tw}** (\+x>1* .

= &»[Lb§kF](\-Hr‘-)1

Q1 _, $¢(\-V‘)dk‘. ,(\-Q-£1)‘

New él = $a{_(“\e§'1_'E'_.:E?_gf_'Q]C“ (\+§:,")1'

, 3q[1u19-v¢‘3(\+k?)" -

= 3Q Qt.)

(\+i,‘-3"‘

A’: (mi: at _(H_t_\)"

'“"‘ ‘éjf; = 'Ié"3I( ' 0+0->‘ '_s<~<\.u>

‘ ( 1.

55- y = (séc 1:3 + arcsec ac)‘

3 ..\ 1“

Sol. 5 = ($1:-x +s-L10“ n;.\g.u....k-1

~ £5)» = z(g¢,q-_’.,.g,1‘1§,§..($¢¢~;+¢¢r_x)

' ‘(M1 +<~1n.[sm.+.....;¢+ J.»-1, (st \ -\ 1 ( , *,§1L"-|.- '2. cx_~<L¢e.a){35¢_\_§;,__‘ _\_ \ \______ 5,5,

56. y = exP(arccsc 13”"\

4'-W-(‘I-A)Q

W.$-A-&-1h = ‘E“"*’_;1_._._ =1-6“ .L. Lu. 1‘

¢é¥0n) \’i“) \ oJ&{> \= =€.-_--

'1 !;*Zt‘E177 J v

Sol. ‘5 =




A‘, Cvl;0:(\|~;) \

:7 ’ "1

=31..1

J2.A1

“P.*1‘

l 1 J \-‘iTML40-: f

57. _y_ 1 arcsin (ln_x) - In (arctan x)‘SJ: 5 ‘ -‘‘i\-_(-QM) — 3»~(i.?.‘.1)W. U-J-~\.£.‘_

S

\-(L.~¢)§ i - Rx"\ \

“F-(.3'-~\)‘ (\+v§§§1J.l.:a

58. y arc sin x - x arctan y = 1

S°1- ‘j$C.‘\x -1+cC.‘\\§ = \ ‘

\u~“5

' .\,._\._,\.t. -K

----*"+%'Z:1-Q2. 1._\.__._1\.1.,i\-RC’ J1 C M3‘ da _ 3 - = O

<52. (<2.-:31 _. _L~.> + 1.. 4&1.) ,0A1 H-91 J'§';;\ "

3*) (‘*‘;“"":}:l. = --:9-—--+'1"‘\;$4 ) . M,(S-s~3>") 53?

,3 .._..

-_ 1 .-. Y. " vii‘.

AL-((\*‘1'>S1E~_‘_};;__”L> . -u+§t;=\:>::»A‘ __....M

a ‘ ‘

= h‘::$~‘>) ' ‘qt;" IT? ““““"i"" ”—" "“*'

5_9. arcsin (In xy) = x-1 3/"‘K"*“")';’~’:‘ ‘ ‘*3' 1 ‘4.?;'

- i“(gMJT:U.~»\‘1§1 61' *3) = ‘*7-"3'é2'

d~A

—-L---_-.~_..=_, -.l-..-- \&2""\")~\ : \ '63.,1!-U»~\ss)" 7*‘) K 4' “tr, 1:‘ él

1% +3 * "*3JT;(_9»u~a)" S_\-\-1‘:-AJ

d4:_ ..---. ..___*5’;-113 ]\-(L‘»3‘-4: = *5,&\-(Lax-;)1 -3

‘Q.

‘L é; +3 = 13 _v'3T'_"(T'9.':;~:)-" + zx 3‘ ]\._(11,.,,¢,§:‘.§:’.




x%( \-1§1tt:T;~3,) = 5(1y.:7_.@;,.,;¢_\) L_ 5x\(1;A

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Chap02 Solutions Ex 2 2 Calculus