Chapter 02: Calculus With Analytics Geometry

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.xif1‘j:-§(1)h-I.Lu:fw- 1I:- ,o.cu'v M301.11‘);,'43fhidx ‘Jr"1"A5 7- I \xCwAA'Lhlw J3 481:-LiiP(“l ‘))11.§-5,.‘ 3G(‘l §J.,f}- 93)‘M-Ll.QmajbutNA '3:,S.(1) a §mT‘!PK 3 :\-V‘ 23 Q )1IoN“5(1)' t \\ ‘ 5A3}Pm»1ss. Mn"3 5.2.WK‘‘\’7:s.;@ % ; .T’ ')A§:.k§AIS.Q3S.S3#43. .,‘0-5%i le.\o1'»'»l.§ou0.p &1)U ,.)\:‘P-U,Th., ‘\)J0-IJ»5'\' 1 .- w.wuA AP'v m w» .\ \Pw.w1d-f.' Puu.h1o.eAJ»aL ,.P 9 ? x\on/Mlu, .ua».J.ox»\ h . Y. . ; \,,j3} ;\ ,h,-Uka.M \ .t.;.Jab. J;Exercise 2.3: Page 1 of 17- §- IV ’**l-Vb- " ’*-“\Avaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.-. e\w: n:uu w.w, --» » ---»».),',-V.‘mm--v. —.-\.1 m .---., -\.Pv'!& »' -'—‘ "4\I\rV.-EXERCISE 2.3Find ‘Ay, dy, A)’ "dy if./'l1.0)y x“-1,Y (u)301. -0-5x 1" .%Ax 0-3' 2. y Here1).A‘)I3’,[(1 n)3-1] -[Q-J: (ii-Q-51),-4 -1‘ \(‘ b‘)3 "330“)2‘J8' ‘“5 ’("‘"'§)A\5""‘J *( .s)3 ‘26-1 15-I'.:1'.:-O - 75buy:317TA'\l 11-3x-2-O"-‘K011 1;.- - g 3(ii) Now(\- .s)1-(U3 L3(\) (-0-5) 55 §(a n)—S(-1) ». . Kw»-.i7.::‘-7-)5)‘O-3Q.‘S&‘.- *s-F5-1A31§(1 M.3-§-(M‘I’PM‘,5G), ,/ . ,.,.-1*1571-“"I'** 1 31.1.1\';(-its)-1 "Si"-ZA3.' . . »-3» Q- O-'l\3§Ag--—-""": O-IlseMUM?z ; A , gExercise 2.3: Page 2 of 17-@-'z.\s§»0-119 """'9w-5Avaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.32.2.Use di“f'erentials to -approxiamte(i)V2624'Sol. We considerwith xrmy 25 . .-- . 1; /IandAx 1.2From (1), we have-“-*dx2 E1d ySubstituting x 25,A) -1 .L—-.(\.7.)-‘'*(1.; Ax 1.2(2,in (2), we get25?g ., O-\'L.3.-(\-1):1‘!-5\-2.—-":‘A-‘O-VI.-\ § 0-\‘L"‘ ‘ ‘A3chA5 f(1 z 1)-§( \,)-5-A3 (ii) 0.\7- 33 -XI;‘ll.-7. -5V80.9301- LetH-u1‘A Q\onVJ’andA3/'(r)\/E-—O.1 zlx Ax: NowA51: A 3 §(1 m)-§l 1)LA‘HA 5 §1 A12;?I-—-——(L-o.\)zri-‘.-0-005 !JQ\. .\—o-ooS§§ ‘Qh-Q-0-‘2*‘S30-WW 30-'\ -6-\Exercise 2.3: Page 3 of 17 --F'- ET-q-Q : -0-4 0§§ ' ,-0-0\'§S§ qAvaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.3is§i I'5;‘W1»1'5‘at‘Q;-;\. -' » .1 .H.-5-,.» . ,.t(iii) .-r.v.l.\'\4.‘§-;»-.*1Lu 5 §(-n .\- \1-S(V2.3) ‘- A;-'.- A 3:{'3‘b"'z'\A3‘*3 - %;‘I -0-O1‘-L‘I;15; 9WM9'3""a“*1 123Sol.l-u;.\\.M.J3\-:‘ 1.3- A‘.1.§"‘h§(1-vmn-*§( n\'‘I('a m\)‘- x3 ‘I3‘I-o-oz“(I152-Z)’-(\1 )- -1(Q)‘I ’(myI],'1Q -- .n.u, (n3)\‘: ---—-1'MM),("7- 3 - 1 ct‘(na)" S ‘.V3“"’-'1-1 -o-o'1(.( § g3(5)‘(iv)“maQ»cos 61'Ixtu cos;Nan-a.L2 - ».L .!‘ Ho3A1Q2 \:- 1 ‘Kxu' Lus»3‘.‘(1-1 1 (‘\ A\)- '(\\):, )QM -'-L(1-\13'3‘\ -%A(r1;\.-1%“‘- QM“- -§-o \Sn.G,Q\. t S -;‘K\B 0q(Z,,g\' -O-\§\L-O-\S\1 o o-‘\3‘1 lExercise 2.3: Page 4 of 171:A‘) 14'-'o.' ((5 \. -C09‘.'§ -LNM A3 ., ‘ .a h\.;, -Jji bf)-1%,.---1Avaiilable at www.mathcity.orgO

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.Mk)?Y v .§,%:’ a?t'S01-Hy-1x)I4 .3d.t‘with:- 4x‘x 3 and ‘Ax 0.02‘ 4Sim"! 413 2.16.ii?"3“ (0.02)‘5'71 §(1 l A)--u)4:%u M‘§ d (3 - z)".( 3)"1.\LJ.-.(3-o1)“(3-01 -an9 "L“‘jM4W0‘. heI A‘) .'@., ,\, 3.“ 6:-0-07.3’) ‘ ' » § » * -,- -0133 A3‘‘L' 14“/1)-(-3 81)\ . - -1'1cm/4\(-5;)‘%3. 6A» -'5»-(u z 1)-'lm.»\.'8 0WeNvw '%*T§T§l1)M M; . 3 AI* A12-IL-\-8! ' 2-\(, (E)M) “mm. 1-0-@133 'Ta.z1'1'A \1.q'. 3 H W1I-*1»-1.1.fwqlq“-.L5'-L--:' -54,13‘;H“ 11 - ns18.,Exercise 2.3: Page 5 of 171’-('s§.-1')-'12.3,‘ - - -31 -o- n ' ' o- 733 0- 5-Avaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.3)’The side 0 a cube is measured with a possiblmerror of 3: 2 %.Find tho percerxtage error in the surfaceerea of one face of the3.-cube.Sol. Letx be edge ofthe cubemMArn A of a face isA 8' I-‘L1-'- 1.12t,&1M- IruJJS'»4w-\,§1But-"'" Z § 4 *.\ 2. d .xx)L* :t0.02Therefore,2(i 0.02), ,' -3% ' 3‘ :t 0.04The percentage; error in the s urfaoo gre§ ia 'l — ' \ I ‘-*- ' "“‘ ' t.‘1A box with osquare base hos it; height,twige.jtst ,w4idth. if the4.inches (in.) with‘ a pos‘s ible error of 0.3width of the box isin, find the po sible‘er ror in th'e'v'olume‘ of the box.Sol. Letx in be the width of the box. Then its 3/olume V ist 'V L-w-MavBut- 1-1.2“:-xi‘ 6x2.dx'(R i0-'57-7'-Therefore change in volume'dV- ‘ -s(s".s "bx(10.13 ,.* (6)(1z-zs)( -'5)'1'\3o-4:-SExercise 2.3: Page 6 of 17Culnlt-Avaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.365.The radius x of a circle increases from ac": 10 cm to x *' -10.1‘cm. Find the corresponding change in the area of the circle.Also nd the percentage change in the area.-ol. LetAVof the circleiof‘ radius x. Thenbe areaA 9K12H 211 d-IINow, x-' 10 cm "and Ax: dx 0.1t'mChange in the area ér the circ'le"is\CIA 21! (10) (0.1), *i. -35 21: cu:ative c ange in t e area‘\-/.v.»9" ' 3"‘, Q72A1r(10)190 0.02'Percentage change 1361100 2%.The diameter of a tree was 8 inches.’ §After one year thecircumference of the tree increased by 2 inches. How much did(i) the diameter of the tree increase?(ii) the cross-section area of the tree change?Sol. Ifx is the radius of the tree, then its circumference.I'"»C 21:1:Therefore, dC 21c”dx''Changeih,cii'cuinfe,rence’is dC ' 2'and so the ,c hange Ax in radius is given‘ ;by'Zi.2 2rca'x1or-. air u//, .//"Xab.1!I.\Thus the diameter increased by 2E inches.\\Area A of the cross-section at Otu. bx A ,- When21cxdx(L4 x 4, dx. i27 ’Exercise 2.3: Page 7 of 17and change in area is1'dA iR124.1‘;- 8919-Lad/ AAAvaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.3'1Sand! popng; ;o1jn ' a,Ach ut'o' fdrms‘e conica! pile "whogeq eltftude.ie enlyvn yja-equ§1 l ?to the radius. If the radiué of the p'lle" ir10 cm, 'Ifind the approximate change in ' ra iig;,a. ;wh n.t.yeluzneincreases by 2 cm“.Sol. The volume V of‘ ,thie *conica1V pile of radius r and height r,ia7.*I, .1--.1V 51:13V a\v .&51:.'sJ§fAA"‘“mu \\'-- mfdut5V'5A\l1s» .: .\.\,.d .1 -\.a.A;.)\Ho-uf,,, oL\1\l4lu\ A':‘Q2 2-Cam,A)», ,1,, ,31 K(‘.\LM2-laid)» Qch- §"'*'}:(c"""S‘ C-"‘**-3‘A domeiiu"-in th'e'*'Ihape»of a ‘hemisphere wnth radius 60 ft.‘The"dome is to bepainted with a layer of 0.01 inch th'ickne' 9s.'Usedifferentials to estimate the amount of ‘.the, paint, required.Sol. If V is Volume of hemisphereztvith radiuvs rr, thenW . * . .§i . . . . . *3? .go (L5 L. .‘,{- V 2l' '.8.‘V.' 9 Av 13;-‘K.3J\?'oDlAvL»-RMM-1 ‘\ '\Lu\-,11 11uZ' \J\t-/@r. g;.;. Av* ah stAV ‘ 0'0 } -‘--ge:1!-lav." .3‘ 5 \S‘it 1"A(6 \‘.T!1-V,3o‘o\nx,\1n.Pqxt»,-.; vo-Lu.» 3 \ 3.Ree21 3aft‘.1The side of-n building is in the shape of u square aurmountedby an cqpilhteral triangle. If the length of the base is 15:n withan er:-or'of‘ 1%, hd the percentage en-of in the area of the"‘ “'side.the side isApfSol. L e\', : Q;1'p b!; e. t l\ e;l‘ength of the base. Then area9.\'-@"@"l '.'A* Mo v) \p-\1 e»4. . !,‘f vZ4. \,l1 xz, , §{ .\\"1-A - : L. . .-'1‘.1L i . El‘\:‘.E -*--17.{ 5.131“ “AA (1-1 u §-.zu)d1Exercise 2.3: Page 8 of 17'-VAvaiilable at www.mathcity.org

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.AA 11» -§ } ) \*C"[(2(\5\““w»@m AA"’w %M» §L,M,- -I-:1(» \]""no-o,p. .,1.1,Now“W ---sun.K'Ait: -U"\ ,.“j‘\§1ZwZ‘[I5-ts3 —i-—15 '§ . ."‘'* L J," H:(,7' ‘ '1li§)‘ L--A (" 55-l‘ ) \1*‘»\‘--7-,Q-x‘I1a §u*1.x !1u"‘*,s'7‘H.‘»-MI?L2WM* m'‘- Ml-atL"‘gt' d7“’ ’ :27'A boy makes a paper cupin the shape ,0!‘ a right circularconewith height four times its radius. If thqrndiui in changed Y'rom2 cm to 1.5 cm but the height remains four times theradiixs,nd the approxiamte decrease‘ inytho-c apn'city of-t lwcup'.Sol. -Ifr i the radius-of theibue and h in height of the. C\1p,'.\8.it8;voYume V isgiwen by‘10.1.1".-'V —a#h — anV 7 dvAb dc -v fy-u-. 9 Q”"“6‘*.§.x.3;.‘ \/x AvI\ x(z)‘(- -g) r.AV‘:‘Hi. \ .:l.Z‘-4'K(-1\-—%K-Ht Ci-“ 1( -“NCuk.The - ve sign, shoves that there is decregpe in the capaci ty of theHut.‘.h.‘in JV HKIe'J/\.;, :K A (M\ s z -V3 §(‘ ‘ q:(A1’L§cup .1 ‘1.To estimate the’he'ight of Minar-i-Pakistan, the shado 'a ,‘3 "m pole placed 24 m from the Miner in measured. !§‘.the‘1eg'r'gthof the shadow is 1 m with a percezitgge error of 1 %, red theheight of the Minar. Also nd the"percentage error’ in theheight so found.VExercise 2.3: Page 9 of 17-7Avaiilable at www.mathcity.org'

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.--"9-YM mLye oM» a,.\.»,.If4 Ac 1“. anPg‘3.1x,,m. is’ beightwof Hthe“Minar',"the’n'f§dm“A The gure5. 2m" 1Therefore, x 25 x 3 75.I Ieig}xto-H110 minar 75 m.1:", 3; a; “ném1'1 n'g1h of‘V,XACthe shadow ofthe pole, then" 24‘:1 'n-13— — 31,.ororavor'3y 72",,‘*2yheight ewav§ xyi 0.01. When x 75, relative error in theV43-.in1} u-\ . ;. -(3-'I '§(' -M)1.[ 1Z)(iO-oi) .-* 1s,ze ::.2. ‘I ‘\ -é — ‘ 0-oozPercentage error \.AA- -1 . »5k) 12.024 xdy ydx(3-:r')dy 'ydx“(3-x)%Z dx .---@3d yNowSo 7IA :.\m to-on‘! (-Y.\6'o- s -q q(,-‘-/Oi} spilled from a tanker spreads in a circle whose radiusincreases at the rate of'2 ft/sec. How fast is the area increasingwhen {he r'adii § ofthe circle is 40 1‘?‘Sol. Let r be the radius of the circle at any instant t. Than area A ofthe circle isA 1cr2bl“. \.u-.A. L- §-A.,,,at‘dk it44?.)5.‘K-‘L11- !)';. 37%.;1u . 'B-\.fk.-.: - .‘ 5;2: ;*;"‘. :;§:7‘.Exercise 2.3: Page 10 of 17-Avaiilable at www.mathcity.org:5‘Vi

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.'10we have to ndan-(T when(1), we have-dr4: 2 andr 4 0. Substituting intb,'2% 21z 40x2 . . 1601:.Thus area of'the\‘cii-pie changqs z 1t;th e,,rqtg of 1601: ft’/seq,13. From a po int OJ twp cars leave 'at the-"lametime. -One cartravels wgpt-nnd.-nftqr '1 sac. its position in arc (2 t-{L The othercar traveirnorth ‘and it covers y 41’. 3 t z. in t s' qc."At what rateis thedistance between the two can changing after 6 age?Sol. Lot A, B be the positions of the two 9arsBat any instant t- gnd let s be thedistance between them at this instant.''.92 x2 ya 1.“ . w-.\- 'I.Sé;S 8.»,(1)J31',3-,,5*./d ‘ i!“ 1,;-za ——@We have to nwhent 5.at the instantdtxOV.We have,x t" t(3)-y 9 3: 4)Differentating (3) anti (4) w.r.t. t, we havedx'8-t' 2t 1,3,-A §, .- 3:HA- '3@3%-the dis nces of the wo cars,'ornOare'’x" 1 53 -5' 1!)'y 5? 15 404 ‘PM, Q)s2 302 -102s‘ \n \ m» 1:“.SQS ‘IS-OExercise 2.3: Page 11 of 17-Avaiilable at www.mathcity.org‘

Notes of Chapter 02: Calculus with Analytic Geometry by Ilmi Kitab Khana, Lahore.VA!»-U) ‘vv.50%soilat 23»- s . At, 3.2;)As'3?on3 ;u\ H m3-. Sb:AtV1Therefore, the distance between the two cars is changing at the'F'?'!i"f'.17 *‘ /*§ - ‘. 14. ‘Sand-falls from container at the rateof 10 its/min arid formsof the’a;)c nieaL pile whqad height is always5 it'b.ato.' How fast -is the hpe igl(lt incz;a4\qin »l\\ Q})j;lt} ,g 'g\lp',"ig;"high?Sol. Leth h be the height of the pile at any inatantt. Rhdius of the pile §.VolumeVofthe pile is‘a' ''. 1V L1-E. V2-us'\I-M3\ ‘I. . 45;, ,% 3\:.% Qt 1»‘ii1.,.'K!W “uk Eli.: \Olb '10:éh.illdb'\.a.\-4».1 'T"0“*wk“ ‘Ag/--JSVdb11g1; é \§d-t.12.L:IsaSK3.--- 13-W:0-§ \16 it high (at amo vihg?'shadbwhistip‘oftheisratespeed of 5 fl/se c.' At‘ what'At-what rate is the length ‘of his sh adew'changing?A615.\ALR.VG.“-Mb: .\ \.(1:)PMn ‘ fAt\it tall map iswalking toward a lamp postExercise 2.3: Page 12 of 17-Avaiilable at www.mathcity.org