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Exercise 2.5: Page 1/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.6: Page 1/51 - Avaiilable at www.mathcity.org
EXERCISE 2.6 (NEW BOOK)
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Exercise 2.5: Page 2/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 2/51 - Avaiilable at www.mathcity.org
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Exercise 2.5: Page 3/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 3/51 - Avaiilable at www.mathcity.org
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Exercise 2.5: Page 4/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 4/51 - Avaiilable at www.mathcity.org
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'Exercise 2.5: Page 5/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 5/51 - Avaiilable at www.mathcity.org
EXERCISE 2.6 (NEW BOOK)
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Exercise 2.5: Page 6/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 6/51 - Avaiilable at www.mathcity.org
1 2.-... ,0kcJz_s£.@ 1
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Exercise 2.5: Page 7/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 7/51 - Avaiilable at www.mathcity.org
§£"4,k.l
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1+)
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Exercise 2.5: Page 8/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 8/51 - Avaiilable at www.mathcity.org
.1, =
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Exercise 2.5: Page 9/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 9/51 - Avaiilable at www.mathcity.org
1 1_;_,_ (u-->m ) -,1
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Exercise 2.5: Page 10/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 10/51 - Avaiilable at www.mathcity.org
v§;4;",,,,_g%._¢9.;“,_‘_,¢u¢.Q.-.n!>~§~.v ‘vs K
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Exercise 2.5: Page 11/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 11/51 - Avaiilable at www.mathcity.org
1= I--»' —‘- “(lg)-y(in) L
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Exercise 2.5: Page 12/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 12/51 - Avaiilable at www.mathcity.org
(2-E. F;~A 0. )>uU- M ‘I-
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(‘.”_,‘.“) Q53)-5 {up} K Xt
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Exercise 2.5: Page 13/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 13/51 - Avaiilable at www.mathcity.org
x
Q...“ -U. §(1>‘a= }__"_'__‘}:_ (,,),y,(¢,,
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be €_‘(°>0) ‘X 'g$(°)0)
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Exercise 2.5: Page 14/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 14/51 - Avaiilable at www.mathcity.org
Ni!-3
H.» - 1.~.. €<‘~>~>-f°'"‘
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Exercise 2.5: Page 15/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 15/51 - Avaiilable at www.mathcity.org
P
13
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1~'- '§(‘A;‘) 1 _}EA:l__
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Exercise 2.5: Page 16/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 16/51 - Avaiilable at www.mathcity.org
Q1? *&*>~>> = =3
bl?!)-9 ‘ux
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Exercise 2.5: Page 17/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 17/51 - Avaiilable at www.mathcity.org
m “§."‘L~‘ -_’:~> »;f-_>- ;%,,,,:' "3
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Exercise 2.5: Page 18/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 18/51 - Avaiilable at www.mathcity.org
----L-__. _.2_
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Exercise 2.5: Page 19/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 19/51 - Avaiilable at www.mathcity.org
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Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 39/51 - Avaiilable at www.mathcity.org
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Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 40/51 - Avaiilable at www.mathcity.org
_; L»eA>~f* 1% *_
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Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 41/51 - Avaiilable at www.mathcity.org
= ~£.____‘*"' .... ..&-J--“‘ +1*>t-3' (‘IQ “‘ %
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Exercise 2.5: Page 42/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 42/51 - Avaiilable at www.mathcity.org
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Exercise 2.5: Page 43/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 43/51 - Avaiilable at www.mathcity.org
_,,._~,
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Exercise 2.5: Page 44/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 44/51 - Avaiilable at www.mathcity.org
H7.
Sir
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Exercise 2.5: Page 45/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 45/51 - Avaiilable at www.mathcity.org
§&- Gu]M
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Exercise 2.5: Page 46/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 46/51 - Avaiilable at www.mathcity.org
2: -—k-0
§-)0
= La.
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Exercise 2.5: Page 47/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 47/51 - Avaiilable at www.mathcity.org
§ = c.§(~.~,e.) -x>$.:.t~:i> -151
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Exercise 2.5: Page 48/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 48/51 - Avaiilable at www.mathcity.org
$3,, = G135 K
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Exercise 2.5: Page 49/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 49/51 - Avaiilable at www.mathcity.org
3 .
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Exercise 2.5: Page 50/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 50/51 - Avaiilable at www.mathcity.org
@ Q-w-1; +93»; -Lu
s_._w.
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S»
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Exercise 2.5: Page 51/51 - Avaiilable at www.mathcity.org
Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.
Exercise 2.5: Page 51/51 - Avaiilable at www.mathcity.org

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Chap02 solutions-ex-2-6-calculus

  • 1. F*-~¢J;>- 4 ---1 M 5% 1 = ’§-Km‘->3 ‘ 1‘L|...~?.'v»Q--LVML o-%,.o-§'l1»- MAi¢-L)-n6,;. A’_“__M___,_____(_ -._ - __ "M Aida; ‘Q ¢k Pi-:.§k‘) ‘ "3 V x »uA'v»~>~Q~.-Q ’u..:->~.L..1.J_,, x43. ' '5~l-'~1-»‘—» '4, 1=§.U,a,u) h~..,~-'¥.'v~ Q-ink <>~f--Q Yiw. :¢..L..U,,>,3¢w "-1 uh v-1~....-~!( 0 ».».=,t..~3..1.~ >..~._q1.,o_,w,‘,_,;_ »-i-**<L>-».1».;.n ~A,;.),1'v> J=W;1 »~»'-~%~~+**-~ II W» g.. ‘z ~l‘Awz~¢9. ldu'.J~lI-0 Exercise 2.5: Page 1/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.6: Page 1/51 - Avaiilable at www.mathcity.org EXERCISE 2.6 (NEW BOOK)
  • 2. ~& =~?:~¢-"- A 2=§(~A>~>) 'v> J>¢Q_, ta M. ‘t. Q L,-._,;3: J1 wsxm)-5 (<1,,) *'~£{>*~--~;€>o>m.@ ww, cxS>u ;»~a~w.x fh,~>)_l L Q wJ~n~.w.u x-¢4_Q ,5...,L§ {A an ,x.,-(up) <>w-<,.1.....(<>.,,,) /~11 widn ick 01> -QQAA §t;‘)) =- ‘Q- (>)) '-QKQ vk ) ."ii ag L. ~°-'- {(M5) (‘|"b)'-§£6) »~:e<3¢_,k't»..~~"M’—'-¢9-i-Hv-.J< v¢.0....»,a/> (x a"))—5(¢ 2“) d-10¢-1 '.,.,,, 'Q~*~‘ §-b>">) A-0.», -)C GQLM: . (1:3)-'olmo) u ‘Ha. (<>>o)- ~__-ii-ii--q Exercise 2.5: Page 2/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 2/51 - Avaiilable at www.mathcity.org
  • 3. Paid Lu: 1‘ = §~A>~>3 ‘Io 1%,; = §-b+%» >‘o $1 =. S-(‘A-$s*>‘>§'§b">‘9) .,,,,.J p/W ~94-o'r 1"? 8* 2%. ._ w~**>~>>-%“>%>2. _L;___ it i_-’'&‘-3-'%‘>">) SK-90 8 SK--30 S“ 3____*- .&~§l'_<>-§ ' Wk M * mm 9~.-$- Qsks °-’ °* 4' Lu-)-'_-X~-‘v: So E.‘-_ = 9»-' f %u*%"’) ':‘swV”1' M 51-» $5-, $.19 b¥ Lb“ 7')'"%'>§" _-;“)_ - 8'p—Do %*:<=~¥~ .. =. l~'-Luv D Av §>Q>,»§ r- :8-::° ‘S: O 3‘: §§M_)_ §y(»,~) Qu,Y_§- &'(°HO V~ Exercise 2.5: Page 3/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 3/51 - Avaiilable at www.mathcity.org
  • 4. -gr ,;3.,,,, .,.;,,¢,_ hi %=§(x,>) n_.,, Mm AMJML-,.». ~gf:,1>->> W -3:.-A-‘-§i>= £5 = %.Hn = %*>,_ = s“ i E52. _ ‘ -'%;K )x§ " 31%; E =K§Q., =81“, I t >,~;@-i;> =l*- =%<%-> =¥»>,. -axhm) ' %i:,K%E> = if), =§(s~.. = <&..>b = ;,, °*~>‘*~1* A § -if -b... :¢..L.L1» 4; vu {AW §(¢>§=o New ;vv~ Wu <i~1.=__l%~._ a* AL ‘I~> ,.._.*_>;... “Q3 s, ,1~~-;§-.2 ~ r‘”'{."a1'.I’»‘f§.ifa ~ fy‘, - Exercise 2.5: Page 4/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 4/51 - Avaiilable at www.mathcity.org
  • 5. ivo~L-JL N. (?~hLh~.J>._.§); £LLu,_(} 3; 1- 5 (1,-,3.;g.,.) '***‘) E £ 1 $_-_!-¢1.u.m.-V1-;~Jm~. km '5-x“ Eu J. 1. SI‘:-q-- .<Jc 1. u,“-a(.,q ‘ + 1+") 1-=-~ (9-1") 9"‘*l"9(';:.) W 1-.-'-~ (‘:'5‘ 4b pa) *‘+5‘) Y-O K-Q-O A-0 .2. N <3,-1-1.-M 3:" Q‘)-J) QQ -10> 1»... €. hm‘)-o 0,45 (-)(-O 8- U :8. -l= C. 'Exercise 2.5: Page 5/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 5/51 - Avaiilable at www.mathcity.org EXERCISE 2.6 (NEW BOOK)
  • 6. Ga Lu 3:” “*5 7'; ($;‘Q-0 (920, $4‘- X‘) 1' , L2.-, l,,.§(o,o) xs‘ I. $:-an 3 - Um)-(~r') *5 =- En» “*1,n-mm ‘W-1--> *5 " o r. C.- :.-I -1 * '5 '5 Q_‘~_ L-.:.... _*'__...5 hm)-1!-»-7 f“'JL SJ» . - J“Mk L Ll “_,_ -L-~;' “"-"~ '5 3 L 2»... 5:2-’ L1n)—9(-.~) Xta-Y pd = AC-b0 l 3 : -‘$3-“Q =1) ,3“; , g M u»»>- (~--) , *—=» ° s. d-:- 1~l-#3‘ -'-'~‘-'* 1 L“ 136:0 -1;-‘$;2o_ A-90 A,‘ Exercise 2.5: Page 6/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 6/51 - Avaiilable at www.mathcity.org
  • 7. 1 2.-... ,0kcJz_s£.@ 1 A-00 J_*- --“M i - 9.4.. ‘ ‘ i - Jun .J.(<-so-'$¢..o) X * > - 0(€»n-Q:-~6> .9.=° 3 9,; 1..-... x- 11» + at-1‘>_% ‘l»‘>-obn.) qt‘) --H‘; -‘It -H-‘‘) $_-__=- _._g .0. L1 in J.n.g,- L;..$.$' “""* x 1 UM 1...i1.1‘a~-3;.-2‘: (‘1‘$‘.(3>3- £5 +-; -uf ..2_~, 1;... (~,:’._u~, *'.nQ‘ -1-») = _(‘oS-I“-94 1.;- 7- L 1 1,,)-;tn.)K1 5 *‘3 " cl 4'1"") 9-:-n-n.- _ z+u._u-at '-= LKi 1:}-L s~»~.u..;» a-., Akwibdc m QM w-; ¢.ai.v$¥ ' Qg 1.2.» _.._1.‘Z_ ._ (1£7) -Y (0,0) “'_,," — Exercise 2.5: Page 7/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 7/51 - Avaiilable at www.mathcity.org
  • 8. §£"4,k.l w s‘ ' 1 -, -La“ (‘>13-o(°:¢) 7"*‘)t i 5% W‘ ')>“‘~‘ “"3 %‘l>‘9) = i.-__‘:, o»;».¢du.>s ‘E- 1+) loL.m on (1,;)_.9(¢,,) M * d.i:u.<I;-A u...uv.A~, -‘,4. Q--kA_6__M* amt Lek u”>*'9(°>¢) 41,0», “.0. 1&0. j=~x u... Q = 1.2.. _}&l‘l!_._ ' hr»)-»(-,.) q@.“}‘. =_ -L:-.. __"‘_.2}E_... (‘Ia‘))-illqo) = 1~'- k_“___;___) hm)-al-,» +-~ WM va.(L.v,~@~v S¢ M L44} <1-M Md Q-X3-?>1' 1 1 '_"-""""'"“"" Q1 1.~'.- _}.i:lT _—_ ‘M‘:)—o(~n) ‘L -Q-‘1 _p_ , 14» _}‘;1‘_'._ . kw»)-» kw") "-1‘ 9% bu-L -A‘~.u rm.-Jr {um = -3;;-2;; ~=»>»-»-M vi "°‘-L-41> M Uri)-—a(<>>~) M c1.L.|.f-12nd '7~4-Lm. -/>0»-¢N’~mJ -[L-<(.I( Auomsk (AC1-AK Exercise 2.5: Page 8/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 8/51 - Avaiilable at www.mathcity.org
  • 9. .1, = "IL M ‘,,,) ____, 9,°) Q-'.--3 “L »(~v-I. ‘j 7- VMJL VKA-~ -. . Q 1-‘NVL [*A1‘>-oh») W:'-u-51$" 1;.“ XI-k-YA“) _ 1 (1,»_’(4,.) i '~}) IJ_,w_,, k-N 3 hm)-1 luv) ‘ ‘"‘k u.=.LuJ- 1» “M X- ‘5““""** "°"'“'-"¥‘“ §» M ‘§3-M 1....-Ji‘' $4» *3 ""9," l —iQ‘ 1;,‘ 0~"4' ‘Q kw-u-.-1 11-,‘ + A1 1 = 2;... .._§£.'_*.h?___ L (‘,Q_‘(a,0) §'A 9* wan M-V.» M ’§b>'>3 = 5-‘5:_—*-EL “’’*'°“,"‘* ' Q5 +<lx £2 JnL»1 M §‘*>>)--aha) €- A;;,|_¢‘I_',,,,3 Ron ha-I. Am») “J An") wui ¢»;.M:. - ax‘--»- 1. = ‘-- ( ) ln-aln ¢y}{'-Q A1 > Exercise 2.5: Page 9/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 9/51 - Avaiilable at www.mathcity.org
  • 10. 1 1_;_,_ (u-->m ) -,1 L“)-¥l'>~) Cwnu +11 0 +nw > Q c—-i.____..______I.1 0+4 -Q=L"1. 0L ¢...~..3>....1;...~.x A-u~.&uz,;£" Q 1:... K1‘-*3‘; (‘*>'$-eh») 1" *3‘ $_-3» Lu: 1 um M Q k‘?'~')‘-)L (‘M-l'»~) *':"‘3‘ k 1 L)‘ 5f w-1 J>-.» 'r..3c §£»‘>) = ‘kw 3 "'W-¢~d~n X4-j h v..!...v, M (ma)--> Lo») fu,“ oL.uI.;.~.» _....L..u. J>¢-9 “ml. 1£~.a.k cl-u»'»J<é_x;4t LQ k:“)§-—-);q) Q10“-J ‘KL 1-v'~.l. 5 =n MIIL +YV§1t) ‘§a‘!)—9¢:0) ,* ,* W? £1 . ' 1- = 1»- i‘§‘*.-~_‘..>...,. &,s_,L-,0 Q (H_Ma) 2 ‘L;~ (“'Y“")t kw»)-»»,~> H. ,,T"'-‘ ml.-<s.'~»dL“-.-.)<{-»~<H!--~7="~'~'*'~'%~v 4 4 1 § i Exercise 2.5: Page 10/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 10/51 - Avaiilable at www.mathcity.org
  • 11. v§;4;",,,,_g%._¢9.;“,_‘_,¢u¢.Q.-.n!>~§~.v ‘vs K s. m. 3;.“ A.» "rwk e.x3d<- ,.-1-__-_---Z-1-_{—-Q <32 11¢» ___.°“>‘ (3:5)-5(':s ‘I--3“ $3» Q3 1 1.1 V44, A-4, :.;...."..x “M .1 = °-'-~ _‘_*_>.‘__ um-.1-,-> 1: *3“ sq“; »~ 1..x gm,» = 131.. ,,,_,,,,_u ,0% cl!-QQ»-..& vo»9»-0, 01> (u,,)_,Z,;?) LN “A d.wx_M W“ ‘*""‘ A"-5 "vii ~U--Lt dun; -“.,- e_“;_‘.._ £‘a‘$)-9 (9)0) 01-“, K‘ Q. 3 H‘, A Q - 1Yv‘." L.~>)..,(,,,, x‘--wv‘u“ MA i-i______ ("53-l°u) --‘n:'>Q- $0 ~R*’OO (IA (x,|,)__.(.’.) “KM” °"""‘§""""*' At-1;-A M ' $§‘*>‘1) —-5 (6:0) Nude §'+§~s~. (~,~>>—-»(=,.3 “M1 1:31 ll - -L1-~ 3"-Q‘ Lu.-.;..(_.,.; 3‘~__31 (‘D5, -Qt‘;0) 2')“ Exercise 2.5: Page 11/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 11/51 - Avaiilable at www.mathcity.org
  • 12. 1= I--»' —‘- “(lg)-y(in) L S. 0'5 Q.‘))~—)(0>o A1p,~j ‘=31 W-~u ‘M. -L§w.Li <1-u. moi €-x.'~¥- QB Ll-Q €('l)'5) 0 ‘SL (‘£9 *(°”x X‘--~33 ‘—‘- O L‘. (‘£0 2(0):) $~4w A-“k § 54 ‘NJ ax “L S33 G3-- A -f(>») = §_E3:;s H’ £‘”)*("'3 4u ' Kim) =(°.v0) Rum $0») 1 O —-()3-4.‘)Nay; -'4» gq-my .. 2.-.... _l‘_‘l__. (‘>33-I(I ‘Y (o§—0(0; 0 } 1;1")‘ ~.xu.~>>-=,~.~ Q1...‘ h-. 1.2.». 5-~. it... Lg“ § (X?!) : 1‘kit I (‘h5)_,(0, Q) (‘A Pi -9(0)0) xxQ’ W}‘Q __ !L.:.-_ M1 (~wa)~o(-,. “"~‘ _ v'1 W ‘A W “duh *Y“ s. L4- §(x,~» dw. Mk ¢>(»’<- (‘>33-o('>~ .¢,,,,¢_._ ’Y0k Qi (°>o) waila§3.§»@ at .. x 3: rg I E Exercise 2.5: Page 12/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 12/51 - Avaiilable at www.mathcity.org
  • 13. (2-E. F;~A 0. )>uU- M ‘I- -§(*>‘5' = 3*‘, 5* (‘As‘>) -.1: (_°:°) 1 ) s a_ 4 (aw) =(~,- U> G1 UH‘)- Qiup» L‘ ' K > (U!)& 1:23“; qt *1 == "4°_ ' Kt») = (~20) ‘As-0. z§(~,.3 = <>~ (3%) 3*»; Now 1.-M W” = 1;-.._. ----(~m3-.k->-1 lg (‘>53-» U») *'' ¢_xLx>;_>,L=»,») o.9.~..§ W4. JL;-.1 *j=-v~ WM. (‘.”_,‘.“) Q53)-5 {up} K Xt ' Q; '5Yv‘L (>‘:)—§~>') X -I» wq‘ S‘ 25*“ §k~4 >5) = 9 (‘NS -Q (IN) $5-»u §K>">) 1» C4-llbuuus mi (°>'3 $, Li.» §-(1,5) = §5(°>n) (‘HQ -0 U2‘) :=') 0 =' a A Q-an 1 1 1 3 i Exercise 2.5: Page 13/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 13/51 - Avaiilable at www.mathcity.org
  • 14. x Q...“ -U. §(1>‘a= }__"_'__‘}:_ (,,),y,(¢,, my O __ (19)) = (0:0) e,.,,_,_;_,_,_ kn <'..L',~L.tq ~§ §¢.>(¢,.). be €_‘(°>0) ‘X 'g$(°)0) 9:1» c..:...,. g,...a.~.. 1,. '§(‘Aq‘-Q : & . (Jab) #10103 u -§. ( -1 )- ' ms) (0)0 -Ru. §[o;o -.-. O Nuw .‘ .5 _Q-W‘ gbh) = i* 3 (‘“)_,’(,’°) QM?)-!l@) "ii-3; hi k‘>">——'>(°)) M “A. f)=.M1 3 3M €L‘>,) 1.1”“ X3‘? M1 (m»)4-,-> kw»)-,1-.-> w‘ , IL-_.,. +~§’*~ Q»‘>)-sl.'»3 --w~‘ 2 JLL... (--vv?)* (‘4a‘))-5 ('9') Sh"... .. (‘$05)-g [o;)*(‘,D _ O $~'-u ~Q~i-- §-U»*a = {-9°") (*»‘>)—ak~>-3 S‘ gh-»">) 'o C-CL;-.>4-.v» £9") “~4-. Exercise 2.5: Page 14/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 14/51 - Avaiilable at www.mathcity.org
  • 15. Ni!-3 H.» - 1.~.. €<‘~>~>-f°'"‘ ‘ -50 y‘ _:_ Lu» '-° -no y‘ =9--'» 31>. -'00 h -_-_ 2.4.-. ‘H. £ S0 §x(O;b) = *- §”(.’.h _Q_-____ ‘('2 '~§ — fin1) Y~-Ibo ; .- 1.-la-A 0@- _ 1--1-n K-ac K .Q:..- Y~ K-30 K : K-30 §,,(~»Y= S‘ -gxkoyu) t ‘$’°)03 . @ Exercise 2.5: Page 15/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 15/51 - Avaiilable at www.mathcity.org
  • 16. P 13 Q1! L._* &1,)) 1: —--‘I--I‘ (‘$25) #(°)>§ 1+3 .-Q Qua) - . =2)‘Q~I-.~»4H)’d'*£° Do §_x°>v§ 5: ‘S',,°>v) €.;.»)>¥ $_3_Qx- 'vs x“: ;..;_ L*A>> -4=(~>~3 §>(*>~>3 = vi‘--3‘ (1,5) =.(n -3-bu. §§'>'§ = ° Nwu . vi“ ~> 'Q~'vu -SKXQQX 7' L“ X‘ L (1‘$)"9 kn) (‘v’)-a k. 7 .1 *3 Lu<(>~>3-e b») Q1»; Bu. LL-Q ‘j=‘~* ““~ 1~'- '§(‘A;‘) 1 _}EA:l__ QM») -a luv) QM)-akm) X“-w MR’ X‘ -_-Q.;..,. ‘~* (xm)-5 Ln») X1--~" .Q..:.- %(‘4>‘>§ = O L*>"'§-99*‘) QM); Q~m~;)--ab») <>Jl.~.-3 Rn. Q»-1w.1 7::-.) M K 9*-'* §b»~>)= """‘“ *3 M QA.a)-vb») Q,))-oli) 1 *7~ L... _3§1_. 1 L,~))-st‘-O 7'1“ Exercise 2.5: Page 16/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 16/51 - Avaiilable at www.mathcity.org
  • 17. Q1? *&*>~>> = =3 bl?!)-9 ‘ux -L2, §'(‘l~|; '-'- -if l ("am-,1-,») Q‘ -__._ ;,.,,',",g _.Q.',,,,.',,k An-IA moi Q.‘>'-J:.- -¢.,.<. § Us 'v~Ji c..t.-........>. ¢u<L~,.) Nnu L; 'S'(>b)— ‘§(°)°) §*( 3‘) M --—-C-‘————--— -90 1 .Q.:..-. 0-° —9u = Q —!u 1 O = - .vW-... 4“ *>-‘s'=°‘ ¥~»~»~>- k... 5-~——’< -_ .Q...:,- °—° K-)0 K .=l.;..-O ‘L40 7,0 Q» ~»Q-.o“- '§‘Ko;v') §, .g,[o;q FM N ,4_,_ ,,,;r,-,1 Awmd.-.~» 4, m 3%“ QPJ»1J.....», $-v_>_)- QR Exercise 2.5: Page 17/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 17/51 - Avaiilable at www.mathcity.org
  • 18. m “§."‘L~‘ -_’:~> »;f-_>- ;%,,,,:' "3 85'2.-¢_., ¥'A = §(*A>‘)) " 1. 5 - x at .. ,,. -#~, ; ~. ," M‘, u-,.£- amt‘: I. - g‘, ‘-'- Lu. L‘) 35*‘ § £. §~,= git §L‘l >5) Q‘*»‘)) : mg. ;~.n..c.., ~»-»~-’=- x -xv _—_,_._-_-11-:- X ‘1Q'~'3" ':. 7. X Q '1 2.‘). ‘$3 - '* Q,‘*)'L‘ M 1‘) iiiiiiiQ] f(~A>~>) = ‘h=I.2(»),U $3,. G‘"‘**‘"~ §b>~>) = ’..:.!(‘>),Q ‘Hi ,.».11.u., u-1-k- xv: 3'—Q’lx) Exercise 2.5: Page 18/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 18/51 - Avaiilable at www.mathcity.org
  • 19. ----L-__. _.2_ 1+2. 1‘-‘iv xi. __: é Xt-3" 1" ¥=§-"~5,31. *§ =_.__)._.__.)._(»;) 3 "*("h)")’ * -_.__.___.. ..L_ ~ 2L *xiv : ‘U’ II-"""i 0 _' X1*3; I ¥" -A =‘~*>” ___,_---_-----k-a--m Q . *2 ‘§(‘*‘>) = s'.')~(1-*0) S3’. Co»-3-u-» €(*»">) = 'tn:)¢(X-k‘) '~g;.>v.1:-1A, u-A.‘- xkb .§....§ 8 "---3)‘ M ‘M, ‘ . -=. .._.____---- . --(‘A-wgf ‘H ‘ H (‘A-~9)" Exercise 2.5: Page 19/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 19/51 - Avaiilable at www.mathcity.org
  • 20. § £7 ,_ ____.__.-C --:;(+ $1: 5:81- §Q>‘>‘ RX 1% Il L = gr-C->»>5. o =.> ftwo‘) 5‘-N - <- ~~5 ___.____. E - - (M-‘>3 -- K1--3)" <53 fa»; - ‘é‘s;..~, G.»I»A~ = m;_“__ u-A.:- xi) 1;,‘ = u: - <=~u»-...~>> Q‘ 1 -$.=..>~9 Z‘C»»~a _________--———-- Q-ii §u,~>> = 12,.g,e..,,~) 533" Gui-AM 1. ~»Q~§»€‘+~j") » ->.»:n.m, L.,.¢.#=- X4» = -‘--um x‘-Ivy" 3 $1 > 4 s E 4 ‘ ! Exercise 2.5: Page 20/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 20/51 - Avaiilable at www.mathcity.org
  • 21. §* ¥1. .¥_& -, _2'l'_..- , »¢~;- i gv‘“'§;""='-'(X 4-)) X"-+‘)" I =‘-‘> at-*3" Si" G-Alum- §£>~a) =. ~Q-K <§(1,->3 = ’°“"K *6 _._._._.__ (S1‘~T>‘-*3 i _.____._.."-‘) -_-1»-_-1-ii... Q...“ §b>») = .Q.,&_§§{~~2‘ -1 S$Q‘a-3" '' * X T‘-x§ -3-U‘*-3‘+‘I> 5;“. .,.;.k- x<‘) ‘- _K-7-.-A---'l‘A-> _(_.1....__.":.x *- .. ----—-——---' F-_l ——-Q-——~ 1 1, L ’ Kg‘:-‘) -1) "v) *1‘) K:-S )---‘_-_ <";iI.T~**>(mm F3 Q 1-/S xx‘--‘a‘" Q >§ 7’Sx‘--;‘-) (SE1-*3 wr'~,~> Q m¢ms1<;~ > -- _ __.L._--- -=. - _7;__.. ':. 1--—{>l X X1.~31. $56.‘. 31. Exercise 2.5: Page 21/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 21/51 - Avaiilable at www.mathcity.org
  • 22. _L__..*' £9 : _-l--—-—--'&1Fkf‘:>,_7~‘)} _____________$.z§T‘,‘:} X -'4-X(F1? **) ( yf"r; :2 -—-——-—-—Q V -' 3 ’ I 4;:@:;-*»u:~:;~> <m=~»~sLI»I=-l;~> V 5 - 2 " .F>*~=~ = ‘>.___.._ Kyzzbi +1 - .i¥*‘)t ‘** V X 51!-...,)'¥ &y;1_*‘)L __x KS~k._~). __-*9 =2&Sign .. 1.x‘) '2 "--""""':._~_=- ISX ‘> ~3‘ Zhk____,, iaif-*5“ ~:—¢— _._. (Au >">2= -_= -~ - ‘* ’ > gr ‘I ' L I L §b>))2§ ____ -.: §x’*».~)-»%.§ ,§X1‘--3"--:1-L -3/_'_ gx _i‘_-k_£__‘5w.__-;L1§ _ Q8_k_K‘£“—-‘)1-'%_ 3 _ ____:l________ ~L '- -;_ 1. 1 3,‘2_§x--3--1) Exercise 2.5: Page 22/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 22/51 - Avaiilable at www.mathcity.org
  • 23. Nan ._'_ §~,=' 1c»¢~,uw"~ .___.__..-1--_-—---- _ » ~- & 81 " -,_ x -,_ 311. Lu.--‘)-wt) _.-| ‘£7 T. _-_:_T_~ 3h. (X--5 +1‘) 11) . .3; Qx‘--5*-v ‘ .13 _ 3/L + gz = _¢{@e..5*~m>-~_>E§»¢-;‘».;;'~. £1“ ..}/L 11) . ‘L1 ___E__._._ y_ ‘L 13,1bx.--§ +1 ) =_-__,1 N J;~.»_¢,,,_; .uh.~. ¢11l-~‘L:Mu k?.»,.&.1-....2» 1'5-1;)-. Q13 E‘, 5.-3» M 2. mg. 1%.. = in B11 ‘K-“> @_ G)____.-_--—-- 6) u~>-A.':- ‘Ix '1-‘9 Q_. 1-‘D ____-=5,-‘bk!’ Nu-.» B1: ‘A-‘Q E3./‘ma -.'. -- E ___$_; ¢_ i(—?Zi>77” 2. 1) I s ; 1 i Exercise 2.5: Page 23/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 23/51 - Avaiilable at www.mathcity.org
  • 24. Bl1 "K-3 is M -- = —¢ -£- <‘ 53 . Mm» ii. = .::§>;§~) bx»; R = -3;-*4-1i"> 5 V x-> -: -—-9..- *- _b,‘__._% T. - &‘ Q-1}‘ X-+2 ’ .._._ SJ» Ldr 2 = 1*-“i’-- V ---—-~® *-3 mg. ® Lo-A . 'k- X bid QK-Z)__-_ ?‘§-kb)-‘T 7* K1-5)” = >.(-~>-7K-‘a (1-*0‘ >1 = _ i‘>_ M U-~>)" ._.. ‘L s bk») *&1= jl>____ M1 £1-*1)“ NM <>'~q§.® >»1;.1s., Q»-¢.:.~, ii -= -1.‘; ),1.- Exercise 2.5: Page 24/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 24/51 - Avaiilable at www.mathcity.org
  • 25. R g_~,). - (§>)- (-0 3.9 ' ' bx-w)" i 1 . 1 ‘- (1 ‘>) 2--5- 1% ,,,) u-~>>‘" E >3» (1-5) 1- _ _,____}_..--K 2 % “ '3} £1-~91’ Na.» 1 2 33:. >»~>¢ 3 xx '37. K kx-5)‘) -= 2.‘, §‘*'T°3‘i" ‘**'}&‘JLK k bx-‘:D‘ '_ 1' :6-116+ ‘J"—'-§'--”'{‘> n (X-‘=)“ = -LF ‘J“"L LS L x-~o)“ ___.LK,L__,)1.. _________________._. K1-*>>“ -1.L-‘>3 ""°3 _ bl-~>)“ Y‘% ':. Em)“; QM bf’ _'__;.-_-—-v- Exercise 2.5: Page 25/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 25/51 - Avaiilable at www.mathcity.org
  • 26. Q15 95;. Lek 1= 5-Z M @- m._“.@ <.¢~)--- * 2?..= 2» $- 2}. = ‘$1. - E 3* s 3 s- X. 4 y=_ =~,K~.-.. e.-‘av. +e..a»->x N’ v; V )0 New bi = *6 75‘ 1 L1 ‘D ‘I-L Q; ,[5:;:‘.: +‘9-)~'$- ' ‘ -i ‘L 5-‘L >Z_§__-= E:&')- at: +>L~»-0-* ‘X w-»..t- 5 ‘O . *-‘lml >23 ‘O u s ~, x 1-L.a~- Q. . {Lax £5; = -- New #1 ,__ any s #2 "“-X __.,.Q.1.,.u.ax + 1--°-~-&-‘>1'~ 1 ' Us- -- ‘>1?-9-~x + ‘Pk --9--‘*3 s 2 an. ‘>- Exercise 2.5: Page 26/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 26/51 - Avaiilable at www.mathcity.org
  • 27. " Qli 'b~'~‘~I-3* -vb:-‘H 3*.1* La 1 = 1"-.Lh>2‘.* 4-*1-.:.‘.~; oh -L ?z,..U-':),Q+*i...W=2‘ca _ _-....u.;~.*> . ’t¢...U-$23 ‘E K 2- 1+‘? -@ -X‘) 1 m_“_® ‘,,,‘_g,;,__q,q,,, w-,.¥.-1. R u-»~,>“>-k~5~~>>-<—*>L ‘M Q-xv)‘ -2:‘) +1» +5‘ Q -xv)'“ _§_23_ ._ __-xi. M (v-*3)‘ 37'___2'- -:. (M-‘:"Y '7‘ ('5)bx’!- U-xv)’ 31% -L5 (u-9") me u-m>’ u:-.“,.® §.,.1.-.1», ¢,,.,,.+¢- a EL *($-1»)-->~~hL:"~3- *9 Q-10)" -sg/; -wl“ +‘AZ§_ U-uh) " >2. = .1112; 5‘) (-xv)‘ 1 K*i). ._.-L?-_-----a‘ (“*3 (-aw) Exercise 2.5: Page 27/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 27/51 - Avaiilable at www.mathcity.org
  • 28. Ndw V we _ I9... 8 “ii " x",_ _§_ K __1'__._ 31 (-aw)‘ L-xvi Ix -U *193- 1 §"*")Pg L -xv)“ zxh-u») +1» Kw-x‘-) L -us)‘ 31-1% 4.9: -5-I.~Q() _ Q-xv)‘ 78% fz.Qa--~>) any (-ma)‘ n-i ,1: un~$" -.< *5‘ V L-M3‘ I 5 1 Qv» “»‘-n~.~,..), '11- 31 , V44.-(.45 Kui 51 = }-‘S’ ‘$059) ‘ 2’ QMKML--L) -———-G “'43-Q u-A-‘L ‘AL {L = ‘§”__;=_u~q.,+c»g»~<.>.Z'- 1 1». g§_ = ‘§’&~;c.»,Q.~¢_ _,s;_».».<)} 2* .§"_§__=é__ EL )5,‘ = ;q_ ,§’'~ac.»&s+¢-$$;..{.~o] 7") W3) B5 bk Exercise 2.5: Page 28/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 28/51 - Avaiilable at www.mathcity.org
  • 29. ‘§ ="~2_'C-'>(1.-»c§+‘»_‘:4'-»U»~n+n ->*$‘~U""“ '—"" ® 1 ':. ~€,{_c¢6-Iv ¢.§ ''&'5¢A+<. - $;*u"‘“"‘3 ' 2*‘ Y‘ ti ‘ haw N,“ .5.-.g;.@ >~.1.-..u., u-..*.» 5 ‘ EL ,_ X ‘Q?-CnU»+ L3 5) ->z%_321 an», "‘ D$~)5 53.- §(=,-D = Lké ""2 ‘AL ,_ C»U»x+ Q} aw ‘"1 _ 12”’. _s:..~.~¢.» +1?-»£~¢-Q--s -- ¢ -Q-L‘-‘*‘3 X‘) 1 Q &c.,u.~¢ ,,~»,c.,L~.¢ -»$:.U»-~u} --® FM» ® < © __?1. = ZLuh bu - ¢_,E_;_ §q*,~,> = 1.~(e~<? Gin)-on w&.@ ;.,.1;-.11.-, w-»'~"°'* ‘X _.e.7* K8.-r¢:) 2*} ,______,___ 1» N04) Exercise 2.5: Page 29/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 29/51 - Avaiilable at www.mathcity.org
  • 30. >’-§- ;__ Q __m ,,, K ‘H X = '3? Y : i___._"' . e- ' (£3-0:’ )1' 1' ‘M.-‘Q ii. = _ _¢.__ ________ @ 1 >7)‘ (e.“+é’)" | Nwu Q to-A-'2' 3 ii - ._l___ - €? D‘, L‘+2§ b§» _ Q’ -£7‘ _ Qt.‘-#2) New -Li}; = 3w1'Qé;'»>) _ a - DA (an-9:’) v __ '* "-L Q - --——-'-" I: tQt‘--2) 11¢‘: b‘§ -_' - K e _ M, wm‘ 'Q"""‘ ® k , 3% is . **7"> ”‘°z‘* Exercise 2.5: Page 30/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 30/51 - Avaiilable at www.mathcity.org
  • 31. Q__1' §u,~» = .w.i~=_._:3)I) g? G£ww A‘ Qb5)"‘&“£!:st6-~> §(*,_,) = 1..(¢.»§-)-1..u~~>> ----' -.;,.@ >.~x,=.u., ->-1~-*' ’~ 5 ,,zx--L---'5 % - 1%)" N’ 1 2 -ii“ - L“ 3;: 1‘~»‘ ‘“ _°_‘i_.-_a_ 2%. bbh _ Wk“) _ ._3.-l‘~....._.L. ' Q8‘. "¢~>" A = v_x.:_----"-‘P 29-; ,___‘1.b.2.._ ,_ -—-——--—— ®nu b‘+>‘) NM» A-'-0,.® >~cb..u., w..a-t- 5 Ei = __.z~; __L_.X )9 fFf' *3 §§ : z§ 3'; 1‘-0‘ " _BiS__,>§.>.§_)aha»z .3.-2.. _.1_ 7' -7}: 1‘--3" " 13 _ 11 be-a*>‘ Exercise 2.5: Page 31/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 31/51 - Avaiilable at www.mathcity.org
  • 32. in b"£ : Q1‘; © >5 (*‘+a‘>‘ Q»-»®+ >1; 2?; M» <32 _ D55‘; (_‘L>‘)) -; )2 $_~l»- Gu-'u~ fc»,~>> = 1”-»5‘ _----CD §.'.“_.® L»-A-i-‘K 9%‘: : 351.‘ .33.. = :;;;-‘$9 35% z‘; bah ‘ 5 75:‘ +‘}1~‘o3 1 ‘J-K;--1-*) +33‘. x- La. "M; 1 ~:‘§‘>!.-.1-- ~+ ;'(--x!..~,) -___ ?.§_..M- ‘> x- 7;-9--‘|L -- "IN; .>“1.- =;_ EL ‘omx, Mk M) ) ' = 7~,~*-' § -3 . ‘ X- 5- — 33-l—-a-9.-1.‘aX +7‘-5-"L‘”3'*‘7 "" x Exercise 2.5: Page 32/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 32/51 - Avaiilable at www.mathcity.org
  • 33. ‘S _ 1- — QC‘ -vi»!-‘9rL‘ -"i-*5 'l”“3 ‘*5 ._._ 1"k+;l-x -N1?‘ k~*'Q-‘>+ 8'8’ .:_ ‘;:k5.!-~x+§+;-‘§+Xl~;) ———--6) 7*)? &-~®+© $5 ,‘§ hr. )5 Z‘: 3* qii-n--—---i__ Q2 $(‘*>‘1§ 1 sin Guy“ , '$'§“>‘9§ = §*:K‘I;Q ---""“ Q ‘>4-N-Q >w1.w.9.-, ‘-°-’*-"-'* N,bc El >1 B“;- _____....-=—-- >~>>u. 4 §-Q‘|'=)‘ 5 _ __L.__. . L. ' ‘9 i9-x‘- 5 , _.?__. 5'1’ _ .)_. ‘9 "' .---_—-II'1-1 T--.--_ Q1 -3/1 Exercise 2.5: Page 33/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 33/51 - Avaiilable at www.mathcity.org
  • 34. ‘M an (>‘-9)" um AR!‘-Q w--‘~-‘K’--*3 1L = -l-—-- ‘L : ‘ _________.® ' ~{ 1 s I 1 1 i 7‘) i-Ufa)‘ 5‘ 1 I ‘L ' |_ x73» 3‘ ‘a --‘A 1_;yi_i__;._;: 3? = _ _‘S_.___ av 5551,‘? _3lL= -L 951- M015 “K M) = g_£__-_2.>.._ } hg 5S,_‘L I = -*;~%rA) (-1.1) = - *~;S”"“' " "B‘-1L" L‘21"iL)--XL X 5 U5X91 >'"-x 1 _ .L. £ ‘$" 3 Kn‘-wC~)"‘ I ii”2 - _-L‘-111 __'___ ®~ 31$; $15 BR 1 Exercise 2.5: Page 34/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 34/51 - Avaiilable at www.mathcity.org
  • 35. Q.§3' -$L'l>‘)) ':- -------—-M3 “L g_,§_,. i -~1C‘~~)"' G.~.3¢-A.»/ - ' §u,~>u = { 7*"? — ~@ w.~® ‘v~T-*”*> M-‘<-* ____Z-._._ %~i: = “~-% Pi N. z‘% bah .11 F - -*~ ‘I ; ] = 3 L §~~i“~3‘*‘> . , ____1l:_._._- = 51 _S""&"'3 “, ,J-;-:¢$¢‘z!-Ax K L—'¢-1‘--3") =5 F +~£{--3‘-><t X L Q--i."+3")u"' 3+3}= i'_ 3!. (+1f-§'§ " ..____§5_-;,> ;_ _§‘9--33) X 2"’& Q--uh-§"')t I_L . u*_£__‘_3,-;‘_- (H351>_(’*)'5)_ _%§--§Q'...)'-) .71.‘) " '* ” gJ.¢+3=)" ‘ (+1‘-5") t §,‘*3‘J’Il"3(~ 5‘? 5“ L - Y (-nC"-3")‘ M ya; /3% -@{*‘;*?8*3;~‘3 "'i?“ 7 ( —1 Q j i V L+w1."'+~;“)&' Exercise 2.5: Page 35/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 35/51 - Avaiilable at www.mathcity.org
  • 36. -31-; ‘%"‘3*>‘*3*>’“, - ?-@ vs ‘ 9 “’ ‘Ia ‘I 353* (--7Z‘+~j“)3 “N 44!--Q wd.<..QJl.-,_ u>-.~£- *3 25.-. 2 >» BE. = ‘IX _-__l‘ -..-~>-11$.~"" x. AK ------35 _§+1§~f'§ (-1"--*3‘-) ‘£‘""'3 (->< x‘-w 3‘-) *- “::‘:3i;€‘ K =__ —‘:L3 D” U--x-‘+j‘) h‘ Na.» ‘ - _i..12.... - ;;0,,> B17»; 3 _ A 1--‘L ' "" =1 hi U--~»'“+~j") t -. ~. ~,- *1 >1 J1 (M-x"--j’-) L- (--31") - §+‘2)- %~ "'*L“*f) ) - ‘Lx 7 ,__ _ M Z1 (-x-Q‘-n-*J"-) ‘L 1.3,1' 3 ‘L LV7“_ (+x+3 ) - (_+'$vE' — 3X(-~1§U-§1+)) mg * H (+w“C-vj‘ )5 L+*.§"-»~3‘)(+'$1‘i)_ 3xK1+x3 )‘ Y W 1.33/1. (—-1!‘-P5 i ! “> 1 Exercise 2.5: Page 36/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 36/51 - Avaiilable at www.mathcity.org
  • 37. ‘ " }"§- i+~£.‘+‘)L+'§/3:.-'*3b‘:*3:_‘:-'3L'?‘7A':~ "" ( J 1 Sm, g».»e».~,=3”*- 31$. i+~ft+3" +*'5i5'“ ’~"‘**"> k -v~3’~ )3" F~A-a ® 4, w>‘~% _ _.__.-WY “.3”; ' a~_~,2>u S~»+..-J: ~»~+“- ‘(*“*“*'3 PMjsokbivus L“?b~u,;, u,- £95 . Plygi = o(’~!-1-~ 3':-an) Q32 fiim) = $‘~'~*-i"“"‘5 %(x,'>3 = $5-*-c"‘;"‘“3 """_'_® , u).).£'* *0‘; 1 C)>'»..$$~»~‘6 '31 W B“: =_ -S~"»x~$*~‘J """'_‘® "L ‘»LNu~> M»-® ‘="*'**"** “*"‘)M ‘_bi_ : 9,,-_,.~1~C’-*>r‘*) 1") % 3,1 1 ;,;.,@,-.~> @ EL '' §2L"z. 1- O ~ 7,.» *5 .-.__.. Exercise 2.5: Page 37/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 37/51 - Avaiilable at www.mathcity.org
  • 38. Q-it fvlf) = -Q}cr‘) 539 Gvlva» W,» ‘ 2% .___ CD 5"“ EL.>~ B‘? - _'¢l-~>.c».~, T5- §j- ~:. QC»;3,}. New ‘zi >3 ?i : -_Q‘c0$‘j hi- AM--;®-® Pl~!'_L'-Ome z-,~ ' -Q Lo-A-‘- -36$» ---—-— Q) g,.,n_u., <,.,.A.+.~, .;iQ}§ {um = 34- 6.1»... Y §(*v> =- km’ {(1,3) . L(=..3~§"‘ ‘$(‘l>5‘ =' ;.“.@) ‘>¢.1=,.JJ.1 w-2..-bx lb}: , .L__..L_..__. 1* Q1-.-. R "'. x‘-v) * é ~»--—-@ Exercise 2.5: Page 38/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 38/51 - Avaiilable at www.mathcity.org
  • 39. 31.9: ‘—L~&'~.~§').~*L7"’ ‘ *2»:- £"->‘3 R L ‘L ‘I- =‘:_j*_>:_’:1“_-% K‘z~,5'L>‘ ‘LQ, 3‘.-it 7® I M‘ b¢~>‘> 1. 1 N... J--w -»Iw- -»-*'*"’ é __§__ . _L--‘_. 13 2;; ”' 1 Q34-3) 2. = -9-. N 1‘-v)" ' NO-9 w »¢~:*> 1"--)'“ - ‘UL "-''i')L *2._ ‘)1. "E? _" *‘~a‘)"? ML‘-»g®‘ E3---3-‘-§':‘=° 21‘ “J _'_______,____.. 5 Q3 §-um) = ‘MILK _-:::3»3 six-' ¢,_;,,_,_,,_,, 11» _) ,6) gum = ‘Y-ilk 16-3 { j w.® k-~1>»”*» “""1"* Exercise 2.5: Page 39/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 39/51 - Avaiilable at www.mathcity.org
  • 40. Z1‘) =&% N- ___.__ : _ - £._’x) ilk‘) +--—-$5‘ ' UL‘-i).- 10-‘O :~. _Z_‘___.;_:-_ . 7-5 k‘_)._“ + §I'_§_.. ha‘-3‘) N/5)‘ 1., ‘-:"—'-" ' w-~:>‘~w>* W/>*>‘ 1*: (-‘- 5‘) 1“+3‘-‘l"i§'-¥ ‘&")' -7-‘) Q1‘-W3‘) ~i‘~3“+1~|L‘3" _; -7-‘J 1'"+ 2‘-3 (‘xI.*>)" 3_§'_'.:._"'.'__‘.".‘2._-_ Us N» b"§ 1- -gr‘! ‘~)") vs’ = Q-15). _T_‘_-- - L1 (1‘-v3‘) Rx 5 1; _- @ 310- (If-)‘) "°"“‘ ‘MVQ w-»-V ‘9 Pi-. =5 ' 1.$.f2§ > 2'- if-3*). §- *5 (f7-5*) ' + ._.>__*~** w-»~>* t ,,d_'.1~.-‘H... .,,, ....,,. gs @155», 1“? 5,._ WW. _ ,5 ;'5% , ...., . E L §~’1(-I“ " LL“.t:':.. § 1 3 » Exercise 2.5: Page 40/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 40/51 - Avaiilable at www.mathcity.org
  • 41. _; L»eA>~f* 1% *_ (x‘-';‘§z-- '11 K1,“--5‘) £49- 1"kj+‘tXzj‘ 1% K1‘-3" X“--3“ + {Lmy § _. 3~__*£::il_ aw)‘ _>_._§_=_2z~_..... >0 ‘)"‘L-‘r gi. :11 _. (X +3‘>>1» '-‘ '2. ‘> W/J‘) 153- = = —“‘-3 “ * ”"® M‘ h~‘+>‘)‘ RAJ;-Q ® < ?L'.'5_».3’3-_; 51¢ >5 :.O (§_'g1 ,4; .§u,~>>= ~l*<@=;~>m -~§’<~=W""») '“- "="““’ “J )“r A ,. , an ‘L. ‘Ma % **‘J" 532‘ G.~'u~lw M,» = »&ww=~> - >‘*~=1“'~>‘ B-4;‘, v,,,§,_,u,, u-A-’c-“> L‘ 16% ;___;__;___,__X_,*,._i_.‘.>_.-:§’..-L A-:.W>).1> --5/1" x 3" 1 K E 1 3"--1.3"} ; ‘°‘ ‘ - l '2 | § § E Exercise 2.5: Page 41/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 41/51 - Avaiilable at www.mathcity.org
  • 42. = ~£.____‘*"' .... ..&-J--“‘ +1*>t-3' (‘IQ “‘ % ‘*>, )"*‘ = ‘>9 _,.*v>‘ = _z*;‘Yp.‘:.}(‘I>) 3}»; 1‘-Ar)" -_ __}:_'.*_L - 151374 Q‘/>3 X‘-Iv)" 1 ‘* K1“-‘>"3 _ 24125.‘. HI») §1‘~>") EL 2 X - 13142 (‘[5, *7 S“§ _ z Pi Ex)‘; >5) = sjiix--qw&w»] _~._ - -1: ‘*3: 5 SK : - ..__.7T_——-— -+31>- Y 7.5“ = - ‘_ ‘$-xx‘ =f_>';?_1c“_'l'3:- 5"- x“ )1. 8 xi. DL x*§‘L -—s——z--z-_-—-—z-—-—i- Gig. 9* §(1>~>)= 133$): *‘*“‘ “""‘ k¥x‘ %~:)L = ‘*0’ §1"§‘>) Exercise 2.5: Page 42/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 42/51 - Avaiilable at www.mathcity.org
  • 43. ‘ii: ’§ >;.(;,. m.».{- - ‘A 40 ‘3 5‘ ‘Q New §> ‘§~, Nw» G.~l.u-A Q‘! >5) : x-_-——l*51- 1--‘; Q;-»~';).7.*A-i-(*:@'=|-)")~‘} (+~>)‘“ ____ >_1?»n-w -1‘— u“ bx +'a)L {+11%-3‘_ ______________ K-W)‘- QA,,~,)_ 1‘; - MC‘-x-5‘) - -1 b*»>" L 11‘: -w 15‘ -- X7“-3- .._.._-Q--._..---¢ ,-nu- --_ _- Lx--‘a)" - L1--~a)" wk’-+ um - 3‘ 5‘-r P-‘>~ 1‘ gx-S‘) 1' _ K Z ‘x--5)" b~b)" »¢+=-24» -3‘ — 3‘-=->0» +=~‘~ L+~a)" ?_~(’“...'z.§’*- (‘A ->91" 1 (1-~>)h+~a (‘A--*2)" D Exercise 2.5: Page 43/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 43/51 - Avaiilable at www.mathcity.org
  • 44. _,,._~, £25 lkx v) '2 ___.__'.-_-—-— ;,’;’ },§.,%’_i". ;I_ s‘) (‘*3’) _€-y-Q~¢‘=~ U1 - §‘2)1 '= —---—-—‘(Q‘-’)L (“")>L n New 64¢“ "‘§1"§3 ; i3Q"~L'>‘J~‘JL 1 ?*ziw3“*1< (X _,,_,)'*- L1 ~*:)L L~M.~,)‘._{‘.. 1109-3‘) - ?‘>"- xx‘; -X‘) U*9)" géf.-L7(‘3+§_;é-17(‘>+)ji_X€_1‘2+X‘ Q+‘9)L ._; x"-PA‘: + 5"_i,_1,,__-i-—-i—? (X--‘9)L U->>’*'2 _,_______._.--.-L--~ U~>) S~:L‘Ksxd§>> 1 git.-LX~~>>'“ “Mu K%"'§‘>$L 1" ‘0'§“'S") Q-50‘ gym “A vb, >»§= 5*-»~(1‘>} /50¢-'~ £§t_‘ --315.33 :'. ° § L F Exercise 2.5: Page 44/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 44/51 - Avaiilable at www.mathcity.org
  • 45. H7. Sir §u,»> = %».~u»» ------65 >=nco>-t¢~v», -~-’~-‘<~ * 8* 2 COQQXW3-3 st —.: ‘jcL*")) §-wk = ‘j--8‘-~(1~">3-‘$ §“ = ..~3"9.~,..&~m d__4_ ® ;?a_.1_,_,_U,.’ u:.».-k- “5 S’ : cJ>(‘l‘9~* g 1'. c4nU")) 5” = 1.-$L.-L1‘a-‘* § = -~19:-4 WW3 A W NW1 39),“ {Q1 - ‘Jx §~,~> Q-§_~}q,;_..Q1~, - ~j"Q—wZ‘$:..L‘»~~@) ._ _;~,;-,5[<',,_-,.u~, + £2;/%;~L*“») 1. o = 0 ' Q1») =Lv>=~) <32 M %<‘,~,>= ha-:»1*>-~ »**-=:w»> '4 W» +<=»~> *2 %- *u:k 3'-_mL<>>'~) =i= §'>£0>~3 Exercise 2.5: Page 45/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 45/51 - Avaiilable at www.mathcity.org
  • 46. §&- Gu]M New ¥1 %(‘A>‘)) : x.LJQ:'‘(,h _‘)?.*._:(x’ ) ‘LO Z = X-___._.___ . =_. -H2: K1-.3 *1?‘-("la)-In -3? _ I 5 -7Q-"~ :1." “T-T xi) +2u'c.2.(~.~;,3 _ 5_ ‘:1 '~'- —- ~12-— +7-x'L'.'l(b; ) ‘:3‘aw’; 1 - -—--— -.~. 'n'T4:I.(by.‘, __ __3___ X x1431} 1+3 X. $49» % U‘ = "~=¢<~'1_l§*»m - §§ ‘New * 9» = 11,_._‘.___._..-3;; +’'¢:',1:¢;.-Q} 3'? (X-'5) ‘-'- (°>0 3"--x" *9 fag‘ + ii + 5.. XI bl L =-__._.l‘.._..._+.l‘l__. ___-Mai-,;3,_ 1 ‘L1. ‘9- 5+1xi- __ 3 1 - —_’-“-——- + —--Y -- 15%»? L ‘£4-3‘ X‘-v3" “’ = ~-1-' k“¥*a") ~ 1-War) 2*; @9453 v» “"=*‘"=*% New ; ,,,_ §,c~,»> - M»,-> 1» = B-In v ‘° *~£ bu») ¢(.,.§ > } , ._ E Exercise 2.5: Page 46/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 46/51 - Avaiilable at www.mathcity.org
  • 47. 2: -—k-0 §-)0 = La. M-an K") ¥xb(.,.)= - NM» <1; .9-r,_.. §'2LK’°)*'§”§'5"l bx = " ” * ( $1-D» K =_ .04.. ’~-¢> ¥~-->0 y_ »$L‘..- _- ¥.--to &) §vuL"°> Z $1: -F‘-5£0>°) 5‘: 'S,.*L°)°) ____________._.---1--v Q__‘§}_ ms bur §-_L~,»,%.3 -.—. vP+'>3¥. +%¢-(1‘a1) P-hum h¢k gx-,1 2' gray §£>‘w,?:) = X;-u-‘5§%. -- $;..(v,,1) nu 1 §* = 3x+‘)iCJ>(151) £5 = ii‘)--%¢-L~»~>+_).u +0-sk~A»2>_s] .§“ = ?-C-abb2:)-x;£'<5.-.,.~,~;a_) V ,-Igwoi -;, '1. ‘*5 '" _ Exercise 2.5: Page 47/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 47/51 - Avaiilable at www.mathcity.org
  • 48. § = c.§(~.~,e.) -x>$.:.t~:i> -151 § = ¢»u»-as-hark-£»*>->‘v*">“>" "191 l N no '§;_ = 35 -'>‘) C-wvai) -ax = ‘:[X--$¢-§*‘>?-)-‘a-1 -Q,(x51). it {H = 5?-Mwsl) ->~,‘z S.a.(1~,z) Una) -1x~,1&:..(x 5-1) gun = 3.-$.a.(1‘>-l3.xi+Q(1b?.). -IIKELQ-Ls1-xi +$2~(~;1).1a] = -15%$:..(‘1U+ ¢-.(1» 13 - <1-(um - P->>1%=~k-a1 ‘Y = Q,b~,?. -3'1*;%$.;..(~A~;1.) ~ 7L'“5"1"<;.,@~,1) ____by PM @ ~x © §'*‘ai = ¥i‘A‘) __..___w sg §Lx,'>,=L>u) = _.1i">_ ,,s~M '-ok ?.-1-ho _g, 1.. ii-§X.*~) (1*u)'; Si» G-Q-AM V '1‘? . '§h",’%’u ‘ 2.--u 54;. Lu-A-''.' X ‘$1 =‘-521?: $ < ® J } 1 Exercise 2.5: Page 48/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 48/51 - Avaiilable at www.mathcity.org
  • 49. $3,, = G135 K _ _.!._.. g1» _ 1+“, gxul = K51») 1 _ -~ %“‘>1 - (1--no)" “IMO &‘,1L_, = 13?. X, k-) K-17 K1-v ‘~31 _ _?=____ (Q-v w); Qw iu1J4~..Js N1-5 v -‘$2; Q~_‘L'1 ')"‘--".Q~) --obi‘ -.= o S_~§»- ~ LA-§(1>~>) = 3"-ni‘~9+°~“ -—-—-—» Y,‘ ~; zx-,+k¢x” gs " 7'°*x N.».§_~_._,____*:._. ch $5 Exercise 2.5: Page 49/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 49/51 - Avaiilable at www.mathcity.org
  • 50. 3 . <1» bx‘;--‘-e.~A. ' n1 A‘ '1-‘>--x‘- -Ii>-@--_i___ QQ3 1. 3 "” "3 --X. = ° Q3!» hi -§(X>">§= 37:“-31--13 -__.____@ D3-{Q--® m-A.‘:- *‘) 5;‘ -; Gx--’>x'" S» 1 ‘D’ *“""é:>..=.__2cL. <11 gt, (n-»'sx" _.7_,~) ex-+315: '7-‘D E "3‘~-*9 +5‘--cox —-o_‘ = w 9,33- Vi §U>b) = 1'“--3‘3-vjw“ -*-QX-+>) 1“ __-A< W?-® >-»~Iu».Q1-,- m.;,_.‘E:- xx») £1 = 11+‘)--0. ¥., = ‘A~-1~;--> “~ é:»_. = _ .11. A1 g., .. ’-‘**‘>f*!“.. _~u- ‘ 1+1‘) +> ____m_._____-._ 4 ! s i Exercise 2.5: Page 50/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 50/51 - Avaiilable at www.mathcity.org
  • 51. @ Q-w-1; +93»; -Lu s_._w. ‘A S» Na» Lil §@A>“>) = ‘E-'1§ --$~'~'*‘) *-"Q uw-® >-**'~’~°-» “"’*""”““’ 11-+5 = '2.‘A‘j --<‘»‘o <;2=__i=L_ 61 9» L rx+b Q12... : -" An 1->~>~ 0»; E-nan ‘Q C-""L TYl»~Ju> T» “*3-3""°$ “Q“°‘L _?_,_,-_-Q-@—@'_"' 1 1- Exercise 2.5: Page 51/51 - Avaiilable at www.mathcity.org Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore. Exercise 2.5: Page 51/51 - Avaiilable at www.mathcity.org