Engineering Mathematics I (10 MAT11)

Transcription

1 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Egieeig Mthemtics I (0 MAT) LECTURE NOTES (FOR I SEMESTER B E OF VTU) VTU NOTES QUESTION PAPERS of 4

2 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 ENGNEERING MATHEMATICS I Cotet CHAPTER UNIT I DIFFERENTIAL CALCULUS I UNIT II DIFFERENTIAL CALCULUS II UNIT III DIFFERENTIAL CALCULUS III VTU NOTES QUESTION PAPERS of 4

3 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS UNIT - I DIFFERENTIAL CALCULUS I Itodctio: The mthemticl std of chge like motio, gowth o dec is clcls. The Rte of chge of give fctio is deivtive o diffeetil. 0 The cocept of deivtive is essetil i d to d life. Also pplicble i Egieeig, Sciece, Ecoomics, Medicie etc. Sccessive Diffeetitio: Let f () --() be el vled fctio. d The fist ode deivtive of deoted b o o o d The Secod ode deivtive of deoted b d o o o d Simill diffeetitig the fctio () -times, sccessivel, d the th ode deivtive of eists deoted b o o o d The pocess of fidig d d highe ode deivtives is kow s Sccessive Diffeetitio. th deivtive of some stdd fctios:. e Sol : e e Diffeetitig Sccessivel ie. e D [e ] e Fo, D [e ] e VTU NOTES QUESTION PAPERS of 4

4 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 4 of 4

5 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 VTU NOTES QUESTION PAPERS 5 of 4

6 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 6 of 4

7 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 7 VTU NOTES QUESTION PAPERS 7 of 4

8 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Leibit s Theoem : It povides sefl foml fo comptig the th deivtive of podct of two fctios. Sttemet : If d v e two fctios of with d v s thei th deivtive. The the th deivtive of v is (v) 0 v C v - C v - C - - v v 0 Note : We c itechge & v (v) (v), C, C (-) /!, C (-)(-) /!. Fid the th deivtios of e cos(b c) Soltio: e b si (b c) e cos (b c), b podct le..i.e, e [ cos( b c) b si( b c) ] Let s pt cos, d b si..ie., b d t b / b d t - (b/) Now, e [ cos cos(b c) si si(b c) ] Ie., e cos ( b c) whee we hve sed the foml cos A cos B si A si B cos (A B) Diffeetitig gi d simplifig s befoe, b c. e cos ( ) Simill e cos ( b c). Ths e cos( b c) Whee b d t - (b/). Ths D [e cos (b c)] [ [ b c ] ( b ) e cos t ( b / ) VTU NOTES QUESTION PAPERS 8 of 4

9 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS. Fid the th deivtive of log 4 8 Soltio : Let log 4 8 log (4 8 ) ½ ie., log (4 8 ) log log log { ( ) ()}, b fctoitio. {log ( ) log ( )} Now ( ) ( ) ( ) Ie., - (-) - (-)!! ( ) ( )! ( ) ( ) ( ). Fid the th deivtive of log 0 {(-) (8) 5 } Soltio : Let log 0 {-) (8) 5 } It is impott to ote tht we hve to covet the logithm to the bse e b the popet: log0 log 0 loge log 0 Ths log ( ) ( 8 ) e log 0 e e { } 5 Ie., { log( ) 5log( 8 ) } Ie., e log e. 0 ( ) (! )( ) ( ) ( ) (! ) ( ) log 0 ( ) e 5 ( ) (!8 ) ( 8 ) 5( 4) ( 8 ) VTU NOTES QUESTION PAPERS 9 of 4

10 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 4. Fid the th deivtive of e cos si Soltio : >> let e cos si e cos si ie., e (si si cos ) e si [ si si( ) ] e ( si si si ) si (-) -si 4 e (si si ) 4 Now { D ( e si ) D ( e si ) } 4 { } Ths ( 5) e si[ t ( ) ] ( ) e si[ t ( ) ] e 4 4 {( 5) si[ t ( ) ] ( ) si[ t ( ) ] } 5. Fid the th deivtive of e cos Soltio : Let e cos e. 4 ( cos cos ) Ie., 4 ( e cos e cos ) {D (e cos ) D (e cos )} 4 {( 5) e cos[ t ( ) ] ( ) e cos[ t ( ) ] } 4 { } e Ths ( 5) cos[ t ( ) ] ( ) cos[ t ( ) ] 4 VTU NOTES QUESTION PAPERS 0 of 4

11 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 6. Fid the th deivtive of ( )( ) Soltio : is impope fctio becse; the degee of the ( )( ) meto beig is eql to the degee of the deomito. Hece we mst divide d ewite the fctio fo coveiece Ie., The lgebic fctio ivolved is pope fctio. 8 Now 0 D B Let ( )( ) A Mltiplig b ( ) ( ) we hve, 8 A ( ) B ( ) B settig 0, 0 we get -/, -/. Pt -/ i (): - - A () A -/ Pt -/ i (): -9 B (-) B 9/...() 4 D 9 D VTU NOTES QUESTION PAPERS of 4

12 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ( ) ( ) ( ) ( )! ( )! 9 8 ie., ( )! 9 8 ( ) ( ) 7. Fid the th deivtive of 4 ( )( ) Soltio : 4 ( )( ) is impope fctio. (deg of. 4 > deg. of d. ) O dividig 4 b, We get ( 7 ) D ( -7)-D Bt D ( 7 ), D ( 7 ) D ( 7 ) 0... D ( 7 ) 0 if > Hece -D Now, let D 5 4 ( )( ) 5 4 A B ( ) ( ) > 5 4 A() B( ) Pt - ; - A ( ) o A - Pt - ; - 6 B ( - ) o B 6 Y D 6D VTU NOTES QUESTION PAPERS of 4

13 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ( )! ( )! 6 ( ) ( ) 6 ( )! > ( ) ( ) 8. Show tht d log ( ) d Soltio : Let We hve Leibit theoem,! log log log. d let log, v (v) v C v C v... ( ) ( )! Now, log ( )! v v Usig these i () b tkig ppopite vles fo we get, log ( )! ( ) ( )! D log... ( ) ( ). ( )... ( ) ( )!!. ( )! ( )! Ie.. log ( )! ( ) ( )!... v () ( )! log ( ) ( ) ( )... ( )! Note : (-) - ;( ) ( ) VTU NOTES QUESTION PAPERS of 4

14 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Also ( )! ( )!! ( )! d d log ( )! log If D ( log) Pove tht - (-)! d hece dedce tht log... Soltio : D ( log ) D - {D ( log } D -. log D - ( - ) D - ( - log } (-)! -. This poves the fist pt. Now Pttig the vles fo,,...we get 0! 0 log! (log ) l! l (l log ) ie., og (log /)! log! ( log ) ie., 6og ll 6 (log ll/6)! log..! log If cos (log ) b si ( log ), show tht 0. The ppl Leibit theoem to diffeetite this eslt times. o If cos (log ) b si (log ), show tht (l) l ( ) 0. [Jl-0] VTU NOTES QUESTION PAPERS 4 of 4

15 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Soltio : cos (log ) b si (log ) Diffeetite w..t - si (log ) b cos (log ). (we void qotiet le to fid ). > - si (log ) b cos (log ) Diffeetitig gi w..t we hve, - cos (log ) b si ( log ) o - [ cos (log ) b si (log ) ] - 0 Now we hve to diffeetite this eslt times. ie., D ( ) D ( ) D () 0 We hve to emplo Leibit theoeom fo the fist two tems. Hece we hve,. D ( ).. D ( ( ) ).. D. { D ( ).. D ( )} 0. ( ie., { ( ) } { } 0 ie., - 0 ie., (l) l ( l) 0 ) ). If cos - (/b ) log (/), the show tht (l) l 0 Soltio :B dt, cos - (/b) log (/) log( m ) m log > cos [ log (/ )] b o b. cos [ log (/)] Diffeetitig w..t we get, VTU NOTES QUESTION PAPERS 5 of 4

16 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS -b si [ log (/)] o - b si [ log (/ )] ( / ) Diffeetitig w..t gi we get,. -. b cos [ log (/ )]. ( / ) o ( ) b cos [ log (/) ] -, b sig (). o l 0 Diffeetitig ech tem times we hve, D( ) D ( ) D () 0 Applig Leibit theoem to the podct tems we hve,.. {.. } 0 ( )... ie 0 o ( l) 0. If si( log ( )), o If si - log ( ), show tht (l) ()() l ( 4) 0 Soltio : B dt si log ( ) cos log ( ) ( ) [Feb-0] ie., cos log ( ) ( ) ie., cos log( ) ( ) o ( ) cos log ( ) Diffeetitig w..t gi we get VTU NOTES QUESTION PAPERS 6 of 4

17 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS () - si log ( ). ( ) ( ) o ( ) () -4 o (l ) (l) 4 0, Diffeetitig ech tem times we hve, D [( ) ] D [( ) ] D [] 0 Applig Leibit theoem to the podct tems we hve, ( ) ). ( ).... ( {(l).. } 4 0 ie., (l) () - (l) l 4 0 ie., (l) ( l) ( l) ( 4) 0. If log ( ) pove tht ( ) ( ) 0 >> B dt, log ( ) ( ). Ie., ( ) o Diffeetitig w..t. gi we get.. ) 0 o ( ) 0 Now D [(l ) ] D [ ] 0 Applig Leibit theoem to ech tem we get, 7 VTU NOTES QUESTION PAPERS 7 of 4

18 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ( ) )..... ( [.. ] 0 Ie., ( ) l 0 o (l ) ( l) 0 4. If si t d cos mt, pove tht (l- ) -() l (m - ) 0. Soltio : B dt si t d cos mt si t > t si - d cos mt becomes cos [ m si - ) Diffeetitig w..t. we get - si (m si - ) m [Feb-04] o - m si (m si - ) Diffeetitig gi w..f. we get, ( ) o ( - ) - l -m o ( - ) m 0 Ths (- ) -() (m - ) 0 m cos( msi ). m 5. If t ( log ), fid the vle of (l ) (-l) (-) - [Jl-04] Soltio : B dt t(log ) > t - log o e t- Sice the desied eltio ivolves, d - we c fid d diffeetite times the eslt ssocited with d. Coside e t. e o ( ) Diffeetitig times we hve t VTU NOTES QUESTION PAPERS 8 of 4

19 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 D [(l ) ]D [] Aplig Leibit theoem oto L.H.S, we hve, {(l )D ( )..D - ( ) ( )... D ( )} Ie., ( ) (-) O (l ) (-l) (-l) VTU NOTES QUESTION PAPERS 9 of 4

20 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Cotiit & Diffeetibilit Some Fdmetl Defiitios A fctio f () is defied i the itevl I, the it is sid to be cotios t poit if lim f ( ) f ( ) f( h) f( ) A fctio f () is sid to be diffeetible t if lim f '( ) eists I h 0 h E : Coside fctio f () is defied i the itevl [-,] b f () 0 0 It is cotios t 0 Bt ot diffeetible t 0 Note : If fctio f () is diffeetible the it is cotios, bt covese eed ot be te. Geometicll : () If f () is Cotios t mes, f () hs o beks o jmps t the poit E : 0 f ( ) 0 < Is discotios t 0 () If f () is diffeetible t mes, the gph of f () hs iqe tget t the poit o gph is smooth t. Give the defiitios of Cotiit & Diffeetibilit: Soltio: A fctio f () is sid to be cotios t, if coespodig to bit positive mbe ε, howeve smll, thei eists othe positive mbe δ sch tht. f () f () < ε, whee - < δ It is cle fom the bove defiitio tht fctio f () is cotios t poit. If (i) it eists t (ii) Lt f () f () i.e, limitig vle of the fctio t is to the vle of the fctio t VTU NOTES QUESTION PAPERS 0 of 4

21 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Diffeetibilit: A fctio f () is sid to be diffeetible i the itevl (, b), if it is diffeetible t eve poit i the itevl. I Cse [,b] the fctio shold posses deivtives t eve poit d t the ed poits & b i.e., Rf () d Lf () eists.. Stte Rolle s Theoem with Geometic Itepettio. Sttemet: Let f () be fctio is defied o [,b] & it stisfies the followig Coditios. (i) (ii) f () is cotios i [,b] f () is diffeetible i (,b) (iii) f () f (b) The thee eists t lest poit C (,b), Hee < b sch tht f ( c ) 0 Poof: Geometicl Itepettio of Rolle s Theoem: Y f () P R A B A B f() Q S f() f(b) O c b c c c c 4 b Let s coside the gph of the fctio f () i ple. A (,.f()) d B (b, f( b ) ) be the two poits i the cve f () d, b e the coespodig ed poits of A & B espectivel. Now, eplied the coditios of Rolle s theoem s follows. (i) (ii) f () is cotios fctio i [,b], Becse fom fige withot beks o jmps i betwee A & B o f (). f () is diffeetible i (,b), tht mes let s joiig the poits A & B, we get lie AB. Slope of the lie AB 0 the poit C t P d lso the tget t P (o Q o R o S) is Pllel to is. Slope of the tget t P (o Q o R o S) to be Zeo eve the cve f () deceses o iceses, i.e., f () is Costt. VTU NOTES QUESTION PAPERS of 4

22 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS f () 0 f (c) 0 (iii) The Slope of the lie AB is eql to Zeo, i.e., the lie AB is pllel to is. f () f (b). Veif Rolle s Theoem fo the fctio f () 4 8 i the itel [,] Soltio: We kow tht eve Pol omil is cotios d diffeetible fo ll poits d hece f () is cotios d diffeetible i the itel [,]. Also f () 4 8 5, f () Hece f () f () Ths f () stisfies ll the coditios of the Rolle s Theoem. Now f () 4 d f () o. Clel < <. Hece thee eists t (,) sch tht f () 0. This shows tht Rolle s Theoem holds good fo the give fctio f () i the give itevl. 4. Veif Rolle s Theoem fo the fctio f () ( ) i the itevl [-, 0] Soltio: Diffeetitig the give fctio W..t we get f ( ) ( ) e ( ) e ( 6) e f () eists (i.e fiite) fo ll d hece cotios fo ll. Also f (-) 0, f (0) 0 so tht f (-) f (0) so tht f (-) f (0). Ths f () stisfies ll the coditios of the Rolle s Theoem. Now, f () 0 ( 6) e 0 Solvig this eqtio we get o - Clel < - < 0. Hece thee eists (-,0) sch tht f (-) 0 e This poves tht Rolle s Theoem is te fo the give fctio. VTU NOTES QUESTION PAPERS of 4

23 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 5. Veif the Rolle s Theoem fo the fctio Si i [-π, π] Soltio: Let f () Si Clel Si is cotios fo ll. Also f () Cos eists fo ll i (-π, π) d f (-π) Si (-π) 0; f (π) Si (π) 0 so tht f (-π) f (π) Ths f () stisfies ll the coditios of the Rolle s Theoem. Now f () 0 Cos 0 so tht X ± π Both these vles lie i (-π,π). These eists C ± Sch tht f ( c ) 0 Hece Rolle s theoem is vetified. 6. Discss the pplicbilit of Rolle s Theoem fo the fctio f () ІІ i [-,]. π Soltio: Now f () fo 0 - fo 0 f () beig lie fctio is cotios fo ll i [-, ]. f() is diffeetible fo ll i (-,) ecept t 0. Theefoe Rolle s Theoem does ot hold good fo the fctio f () i [-,]. Gph of this fctio is show i fige. Fom which we obseve tht we cot dw tget to the cve t poit i (-,) pllel to the is. Y VTU NOTES QUESTION PAPERS of 4

24 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Eecise: 7. Veif Rolle s Theoem fo the followig fctios i the give itevls. ) 6 8 i [,4] b) ( ) ( b) i [,b] b c) log i [,b] ( b) 8. Fid whethe Rolle s Theoem is pplicble to the followig fctios. Jstif o swe. ) f () i [0,] b) f () t i [0, π]. 9. Stte & pove Lgge s ( st ) Me Vle Theoem with Geometic meig. Sttemet: Let f () be fctio of sch tht (i) If is cotios i [,b] (ii) If is diffeetible i (,b) The thee eists tlest poit (o vle) C (,b) sch tht. f ( b) f ( ) f ( c) b Poof: i.e., f (b) f () (b ) f (c) [b,f(b)] [,f()] c b VTU NOTES QUESTION PAPERS 4 of 4

25 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Defie fctio g () so tht g () f () A () Whee A is Costt to be detemied. So tht g () g (b) Now, g () f () A G (b) f (b) Ab g () g (b) f () A f (b) Ab. f ( b) f ( ) i.e., A () b Now, g () is cotios i [,b] s hs of () is cotios i [,b] G() is diffeetible i (,b) s.h.s of () is diffeetible i (,b). Fthe g () g (b), becse of the choice oif A. Ths g () stisfies the coditios of the Rolle s Theoem. These eists vle c sotht < c < b t which g ( c ) 0 Diffeetite () W..t we get g () f () A g ( c ) f ( c )- A ( c) f ( c ) - A 0 ( g ( c ) 0) f ( c ) A () Fom () d () we get f ( c) f ( b) f ( ) b (o) f (b) f () (b ) f (c) Fo < c < b Cooll: Pt b h i.e., b h d c h Whee 0 < < Sbstittig i f (b) f () (b ) f ( c ) f ( h) f () h f ( h), whee 0 < <. VTU NOTES QUESTION PAPERS 5 of 4

26 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Geometicl Itepettio:- Sice f () is cotios i [,b], it hs gph s show i the fige below, At, f () At b, f (b) Y B Y B P α A C A Q 0 c b 0 Fige (ii) X Fige (i) Slope of the lie joiig the poits A (,f()) d B ( (b,f (b)) f ( b) f ( b) Is ( Slope m t ) b t α Whee α is the gle mode b the lie AB with is Slope of the tget t c f ( c ), whee < c < b Geometicll, it mes tht thee eists t lest e vle of c, whee < c < b t which the tget will be pllel to the lie joiig the ed poits t & b. Note: These c be moe th e vle t which the tgets e pllel to the lie joiig poits A & B (fom Fig (ii)). VTU NOTES QUESTION PAPERS 6 of 4

27 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0. Veif Lgge s Me vle theoem fo f() ( ) ( ) ( ) i [0,4]. Soltio: Clel give fctio is cotios i [0,4] d diffeetible i (0,4), becse f () is i polomil. f () ( ) ( ) ( ) f () 6 6 d f (0) 0 6(0) (0) 6-6 f(4) 4 6 (4) (4) 6 6 Diffeetite f () W..t, we get F () 6 Let c, f ( c) c 6c B Lgge s Me vle theoem, we hve f ( b) f ( ) f ( c) b 6 ( 6) 4 c 6c c 6c 8 0 Solvig this eqtio, we get C ± (0,4) Hece the fctio is veified. f (4) f (0) (4 0). Veif the Lgge s Me vle theoem fo f () log i [,e]. Soltio: Now Log is cotios fo ll > 0 d hece [,e]. Also f ( ) which eists fo ll i (,e) Hece f () is diffeetible i (,e) b Lgge s Me Vle theoem, we get VTU NOTES QUESTION PAPERS 7 of 4

28 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Loge Log e c e c C e < e - < < e Sice e (,) So tht c e lies betwee & e Hece the Theoem.. Fid fo f () L m b Lgge s Me Vle theoem. Soltio: f () L m f () L m We hve f ( h) f () hf ( h) O f ( h) f () hf ( h) i.e., { ( h) m ( h) } { m } h { ( h) m} Compig the Co-efficiet of h, we get (0,) Eecise:. Veif the Lgge s Me Vle theoem fo f () Si i 0, π 4. b b Pove tht, < t - b t - < b if 0 < < b d edce tht π 4 π < t < Si Show tht < < i 0, π b b 6. Pove tht < Si b Si < Whee < b. Hece edce b π π < Si < VTU NOTES QUESTION PAPERS 8 of 4

29 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 7. Stte & pove Cch s Me Vle Theoem with Geometic meig. Poof: The tio of the icemets of two fctios clled Cch s Theoem. Sttemet: Let g () d f () be two fctios of sch tht, (i) Both f () d g () e cotios i [,b] (ii) Both f () d g () e diffeetible i (,b) (iii) g () 0 fo ε (,b) These thee eists t lest e vle c ε (,b) t which f ( c) g ( c) f ( b) f ( ) g( b) g( ) Poof: Defie fctio, φ () f () A. g () () So tht φ () φ (b) d A is Costt to be detemied. Now, φ () f () A g () φ (b) f (b) A. g (b) f () A g () f (b) A. g (b) f ( b) f ( b) A () g( b) g( ) Now, φ is cotios i [,b] s.h.s of () is cotios i [,b] d φ () is diffeetible i (,b) s.h.s of () is diffeetible i [,b]. Also φ () φ (b) Hece ll the coditios of Rolle s Theoem e stisfied the thee eists vle c (,b) sch tht φ ( c ) 0. Now, Diffeetitig () W..t, we get φ () f () A.g () t c (,b) VTU NOTES QUESTION PAPERS 9 of 4

30 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS φ ( c ) f ( c ) A g (c) 0 f ( c) A g ( c ) ( g () 0) f ( c) A g ( c) Sbstittig () i (), we get f ( c) f ( b) f ( ) g ( c) g( b) g( ) Hece the poof (), whee < c < b 8. Veif Cch s Me Vle Theoem fo the fctio f (), g () i [,] Soltio: Hee f (), g () Both f () d g () e Polomil i. Hece the e cotios d diffeetible fo ll d i pticl i [,] Now, f (), g () Also g () 0 fo ll (,) Hece f () d g () stisf ll the coditios of the cch s me vle theoem. Theefoe f () f () g() g() f ( c), fo some c : < c < g ( c) i.e., c i.e., C c 6 6 Clel C lies betwee d. 6 Hece Cch s theoem holds good fo the give fctio. VTU NOTES QUESTION PAPERS 0 of 4

31 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 9. Veif Cch s Me Vle Theoem fo the fctios f () Si, g () Cos i 0, π Soltio: Hee f () Si, g () Cos so tht f () Cos,g () - Si Clel both f () d g () e cotios i Also g () -Si 0 fo ll ( 0, π ),, d diffeetible i 0 π ( 0, π ) Fom cch s me vle theoem we obti π f f (0) π g g(0) f ( c) g ( c) fo some C : 0 < C < π 0 Cosc i.e., i.e., - - Cot c (o) Cot c 0 Sic π π π C, clel C lies betwee 0 d 4 4 Ths Cch s Theoem is veified. Eecises: 0. Fid C b Cch s Me Vle Theoem fo ) f () e, g () e - i [0,] b) f (), g () i [,]. Veif Cch s Me Vle theoem fo ) f () t -, g () i, b) f() log, g () i [,e] VTU NOTES QUESTION PAPERS of 4

32 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Geelied Me Vle Theoem:. Stte Tlo s Theoem d hece obti Mcli s epsio (seies) Sttemet: If f () d its fist ( ) deivtives e cotios i [,b] d its th deivtive eists i (,b) the ( b ) ( b ) ( b ) f(b) f() (b ) f () f () f - ()! ( )!! f ( c) Whee < c < b Remide i Tlo s Theoem: We hve ( ) ( ) ( ) f () f () ( ) f () f () f - ()! ( )! f [ ( ) } f () S () R () Whee R () Whee 0, R () f (), 0 < <! Tlo s d Mcli s Seies: We hve f () S () R () Lim[ f ( ) S ( )] Lim R ( ) If ( )! f [ ( ) ] is clled the Lgge s fom of the Remide. Lim R ( ) 0thef ( ) Lim S ( ) Ths Lim S () coveges d its sm is f (). This implies tht f () c be epessed s ifiite seies. i.e., f () f () ( ) f () This is clled Tlo s Seies. ( )! f () to VTU NOTES QUESTION PAPERS of 4

33 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Pttig 0, i the bove seies, we get ( ) F () f (0) f (0) f (0) to! This is clled Mcli s Seies. This c lso deoted s ( ) ( ) Y (0) (0) (0) (0) to!! Whee f (), f (), f (). B sig Tlo s Theoem epd the fctio e i scedig powes of ( ) Soltio: The Tlo s Theoem fo the fctio f () is scedig powes of ( ) is ( ) f () f () ( ) f () f () () Hee f () e d f () e f () e f () e f () e () becomes ( ) e e ( ) e e ( ) e { ( ) } 4. B sig Tlo s Theoem epd log si i scedig powes of ( ) Soltio: f () Log Si, d f () log si Now f () Cos Si Cot, f () Cot f () - Cosec, f () - Cosec f () - Cosec (-Cosec Cot) Cosec Cot f () Cosec Cot ( ) ( ) f () f () ( ) f () f () f () !! ( ) ( ) Log Si f () ( ) f () f () f () !! ( ) ( ) logsi ( ) Cot (-Cosec ) Cosec Cot ---!! VTU NOTES QUESTION PAPERS of 4

34 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Eecise: 5. π Epd Si is scedig powes of 6. Epess t i powes of ( ) p to the tem cotiig ( ) 7. Appl Tlo s Theoem to pove e h e h h h!! Poblems o Mcli s Epsio: 8. Epd the log ( ) s powe seies b sig Mcli s theoem. Soltio: Hee f () log ( ), Hece f (0) log 0 We kow tht f d d ) { log( ) } d d ( ( ) ( )!,,, ( ) Hece f (0) (-) - ( )! f (0), f (0) -, f (0)!, f v (0) -! Sbstittig these vles i f () f (0) f (0) f (0) f (0) !! 4 log () 0. (-)! -! !! 4! This seies is clled Logithmic Seies. VTU NOTES QUESTION PAPERS 4 of 4

35 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 9. Epd t b sig Mcli s Theoem p to the tem cotiig 5 Soltio: let t, Hece (0) 0 We fid tht which gives (0) Fthe ( ), Diffeetitig we get Y. ( ) 0 (o) ( ) 0 Hece (0) 0 Tkig th deivtive both sides b sig Leibi s Theoem, we get ( ) ( ) i.e., ( ) ( ) ( ) 0 Sbstittig 0, we get, (0) - ( ) (0) Fo, we get (0) - (0) - Fo, we get 4 (0) -.. (0) 0 Fo, we get 5 (0) -.4. (0) 4 Usig the foml Y (0) (0) (0) (0) !! 5 We get t Eecise: 0. Usig Mcli s Theoem pove the followig: 5 4 ) Sec ! 4! 5 b) Si c) e Cos d) Epd e Cos b b Mcli s Theoem s f s the tem cotiig VTU NOTES QUESTION PAPERS 5 of 4

36 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Eecise : Veif Rolle s Theoem fo π 5π (i) f ( ) e (si cos ) i,, 4 4 (ii) / f ( ) ( ) e i [ 0,] si π 5π (iii) f( ) i., e 4 4 Eecise : Veif the Lgge s Me Vle Theoem fo (i) f( ) ( )( ) i 0, (ii) f ( ) T i [ 0,] Eecise : Veif the Cch s Me Vle Theoem fo (i) f ( ) d g ( ) i, 4 (ii) f( ) d g ( ) i [ b, ] [ b, ] (iii) f( ) Si d g( ) Cos i VTU NOTES QUESTION PAPERS 6 of 4

37 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 UNIT II DIFFERENTIAL CALCULUS-II Give diffeet tpes of Idetemite Foms. If f () d g () be two fctios sch tht Lim f () d Lim g() both eists, the f ( ) Lim g( ) Lim f ( ) Lim g( ) If Lim f () 0 d Lim g() 0 the f ( ) Lim g( ) 0 0 Which do ot hve defiite vle, sch epessio is clled idetemite fom. The othe idetemite foms e,0,, 0 0, 0 d. Stte & pove L Hospitl s Theoem (le) fo Idetemite Foms. L Hospitl le is pplicble whe the give epessio is of the fom 0 0 o Sttemet: Let f () d g () be two fctios sch tht () Lim f () 0 d Lim g() 0 () f () d g () eist d g () 0 The f ( ) Lim g( ) Lim f ( ) Lim g ( ) f ( ) Poof: Now g 0 Lim Lim, which tkes the idetemite fom. Hece pplig the g( ) 0 f ( ) L Hospitls theoem, we get VTU NOTES QUESTION PAPERS 7 of 4

38 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 f ( ) Lim g( ) g Lim f ( ) [ g( )] ( ) [ f ( )] g Lim f ( ) f ( ) ( ) ( ) g g Lim f ( ) f ( ) Lim ( ) ( ) g If f ( ) Lim g( ) 0d the g Lim f ( ) f ( ) Lim ( ) g( ) i.e If f ( ) Lim g( ) Lim f ( ) g( ) f ( ) Lim g ( ) 0 o the bove theoem still holds good. Si 0. Evlte Lim fom 0 Soltio: Appl L Hospitl le, we get Cos Lim Cos Lim Si log Si. Evlte Lim Cot log Si log Si0 log 0 Soltio: Lim fom Cot Cot0 Appl L Hospitl le Cosce Lim Cosec Cosec Cot VTU NOTES QUESTION PAPERS 8 of 4

39 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Lim Cot 0 log Si Lim Cot 0 Eecise: Evlte ) b) c) d ) t Lim 0 Lim( ) 0 Lim Lim 0 4. Epli - d 0 Foms: Soltio: Sppose Lim f () 0 d Lim g() i this cse 0 Lim f () - g() 0, edce this to o fom 0 f ( ) 0 Let Lim[ f ( ). g( ) ] Lim fom 0 ( ) g g( ) O Lim[ f ( ). g( ) ] Lim fom f ( ) L Hospitls le c be pplied i eithe cse to get the limit. Sppose Lim f () d Lim g() i this cse Lim f ( ). g( ) fom, edce 0 this o fom d the ppl L Hospitls le to get the limit 0 [ ] VTU NOTES QUESTION PAPERS 9 of 4

40 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 5. Evlte log( ) Lim 0 Soltio: Give log( ) Lim 0 - fom Reqied limit log( Lim fom Appl L Hospitl le. Lim 0 Lim 0 Lim 0 ( ) 6. Evlte Lim Cot 0 Soltio: Give limit is - fom t 0. Hece we hve Reqied limit Cos Lim 0 Si Si Cos 0 Lim fom 0 Si 0 Appl L Hospitl s le Cos Cos Si Lim 0 Cos Si Si 0 Lim fom 0 Cos Si 0 Appl L Hospitls le VTU NOTES QUESTION PAPERS 40 of 4

41 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Cos Si Lim 0 Cos Si Cos Evlte Lim t log 0 Soltio: Give limit is (0 - ) fom t 0 Reqied limit Appl L Hospitls le log Lim 0 Cot fom Lim 0 Co sec Si 0 Lim fom 0 0 Appl L Hospitls le SiCos Lim 0 0 π 8. Evlte Lim Sec.log Soltio: Give limit is ( 0) fom t Reqied limit log Lim π Cos 0 0 fom Appl L Hospitls le Lim π π π Si. VTU NOTES QUESTION PAPERS 4 of 4

42 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Eecise: Evlte ) b) Lim log Lim Cot 0 c) Lim Sec π Si d) Lim Epli Idetemite Foms 0,,, 0 Soltio: At, g ( ) [ f ( ) ] tkes the idetemite fom 0 (i) 0 if Lim f() 0 d Lim g () 0 (ii) if Lim f() d Lim g () 0 (iii) if Lim f() d Lim g() 0 d (iv) 0 if Lim f() 0 d Lim f () I ll these cses the followig method is dopted to evlte Lim g ( ) [ f ( ) ] Let L Lim g ( ) [ f ( ) ] so tht Log L Lim [g() log f ()] 0 Redcig this to L e 0 o 0 d pplig L Hospitls le, we get Log L O VTU NOTES QUESTION PAPERS 4 of 4

43 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 0. Evlte Lim Si 0 Soltio: let L Lim Si 0 0 fom t 0 0 Hece Log L Lim 0 Si log 0 fom log log LogL Lim Lim fom 0 0 Cosce Si Appl L Hospitl le, Si.t Lim Lim ( 0 ) fom 0 CosceCot 0 0 Appl L Hospitls le we get si Sec Lim 0 Cos t LogL L e e 0 L 0. Evlte Lim( ) Soltio: let L Lim( ) is fom 0 LogL Lim log 0 fom Appl L Hospitls le Lim Lim VTU NOTES QUESTION PAPERS 4 of 4

44 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Log L - L e e. Evlte t Lim 0 Soltio: let L t Lim 0 fom t LogL Lim log ( 0) 0 log LogL Lim ( t ) fom fom Appl L Hospitls le Sec Lim 0 t t LogL Sec Lim t fom Appl L Hospitl le, we get Log L Sec Sec t Sec Lim 0 Lim( Sec 0 L e t ) VTU NOTES QUESTION PAPERS 44 of 4

45 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Eecise: Evlte the followig limits. Cot ) Lim ( Sec) b) 0 Lim π t ( ) e c) Lim d) 0 Lim( Cos) 0 e e log( ) log( ) ii 0 0 si si () i lim ( ) lim si cos log( ) logsi ( iii) lim ( iv) lim 0 t π π ( ) cosh log( ) si si () v lim ( vi) lim 0 π e ( ) ( vii) lim 0 log( ) Evlte the followig limits. ( i) lim cot( ) ( ii) lim coseccot 0 iii iv π 0 ( ) π ( ) lim t sec ( ) lim cot () v lim ( vi) lim t sec 0 t π b () i lim(cos ) ( ii) lim 0 0 cos [ π ] VTU NOTES QUESTION PAPERS 45 of 4

46 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 log( ) 0 si ( iii) lim( ) ( iv) lim Evlte the followig limits. ( ) cot ( ) t lim(si ) ( ) lim si 0 0 v iv 0 ( ) cosec t ( vii) lim(cos ) ( viii) lim t π 4 0 b c ( i) lim( ) ( ) lim b () i lim(cos ) ( ii) lim 0 0 cos log( ) si ( iii) lim( ) ( iv) lim 0 ( ) cot ( ) t lim(si ) ( ) lim si 0 0 v iv 0 ( ) cosec t ( vii) lim(cos ) ( viii) lim t π 4 0 b c ( i) lim( ) ( ) lim VTU NOTES QUESTION PAPERS 46 of 4

47 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Pol Cves If we tvese i hill sectio whee the od is ot stight, we ofte see ctio bods hipi bed hed, shp bed hed etc. This gives idictio of the diffeece i the mot of bedig of od t vios poits which is the cvte t vios poits. I this chpte we discss bot the cvte, dis of cvte etc. Coside poit P i the -Ple. legth of OP dil distce Pol gle (, ) Pol co-odites Let f () be the pol cve (, t () ) Cos Si Reltio () ebles s to fid the pol co-odites (, ) whe the Ctesi co-odites (, ) e kow. Epessio fo c legth i Ctesi fom. Poof: Let P (,) d Q ( δ, Y δ) be two eighboig poits o the gph of the fctio f (). So tht the e t legth S d S δs mesed fom fied poit A o the cve. Y f () A δs δ Q δ T (Tget) N O δ X Fom fige, PQ δs, AP S TPR ψ d PR δ, RQ δ VTU NOTES QUESTION PAPERS 47 of 4

48 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Ac PQ δs Fom le PQR, we hve [Chod PQ] PR QR [Chod PQ] (δ) (δ) ( fom fige) Whe Q is ve close to poit P, the legth of c PQ is eql to the legth of Chod PQ. i.e c PQ Chod PQ δs (δs) (δ) (δ) () (δ), we get δs δ δ δ Whe Q P log the cve, δ 0, δs 0 δs Lim δ 0 δ δ Lim δ 0 δ i.e., ds d d d ds d () d d Simill, dividig () b δ d tkig the limit s δ 0, we get ds d d d () VTU NOTES QUESTION PAPERS 48 of 4

49 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Epessios fo ds d & ds d Tce tget to the cve t the poit P, it mkes gle ψ with the is. Fom le PRT, we hve t ψ d d Eqtio () becomes, Y Q Tget T A P ψ R ψ P R T ψ O M N X ds d t ψ Sec ψ secψ ds d Sec ψ ( O) Cos ψ d ds d eqtio () becomes ds d Cot ψ Cosec ψ Cosec ψ ds d Cosecψ o Si ψ d ds Deive epessio fo c legth i pmetic fom. Soltio: Let the eqtio of the cve i Pmetic fom be f (t) d g (t). We hve, VTU NOTES QUESTION PAPERS 49 of 4

50 0 (δs) (δ) (δ) b (δt), we get Tkig the limit s δt 0 o both sides, we get (O) (4) Deive epessio fo c legth i Ctesi fom. Soltio: Let P (,) d Q ( δ, δ) be two eighboig poits o the gph of the fctio f (). So tht the e t legths S d s δs fom fied poit A o the cve. PQ (S δs) s δs Dw PN OQ Fom le OPN, Q Tget N δ P φ δ X O t t t s δ δ δ δ δ δ t Lim t Lim t s Lim t t t δ δ δ δ δ δ δ δ δ dt d dt d dt ds dt d dt d dt ds VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 50 of 4

51 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 PN Siδ OP i.e., PN Si δ (o) PN siδ d ON OP Cos δ i.e ON Cos δ (o) ON Cos δ PQ. Whe Q is ve close to P, the legth of c PQ s eql to δs, whee δs s the legth of chod I le PQN, (PQ) (PN) (QN) bt PN si δ ( Si δ δ) Ad QN OQ ON ( δ) Cos δ δ ( Cos δ ) QN δ Ad (PQ) (PN) (QN) 0 (δs) (δ) (δ) (5) b (δ) we get δs δ δ δ Whe Q P log the cve δ 0 s δs 0 d δ 0 VTU NOTES QUESTION PAPERS 5 of 4

52 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS δs Lim δ δ 0 δ Lim δ 0 δ ds d i.e d d ds d i.e., (6) d d Simill eqtio (5) b (δ) d tkig the limits s δ 0 we get δs Lim δ δ 0 Lim δ 0 δ δ ds d i.e d d ds d (7) d d Note: Agle betwee Tget d Rdis Vecto:- We hve, T φ d d i.e., Siφ Cosφ d d d. ds Siφ Cosφ ds d d ds d ds d Siφ d Cos φ ds d ds VTU NOTES QUESTION PAPERS 5 of 4

53 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ds ds Fid d fo the followig cves:- d d ) C Cos h c Soltio: C Cos h c Diffeetitig w..t. we get d d Si h c ds d d d sih c ( ) Cosh ( ) c ds d Cosh c Agi Diffeetitig w..t we get C Si h c C d d i.e ) d d Soltio : Cosech c ds Co sec h d ( ) c Diffeetitig w..t d septel we get VTU NOTES QUESTION PAPERS 5 of 4

54 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 d d d d d d d i.e., d d d We kow tht ds d d d ds d ds d 9 d 4 d d ds d ( ). log cos Soltio. log cos Diffeetitig w..t d septel, we get d d ( Si ) - t Cos i.e., d d - t d (- Si ) Cos d d VTU NOTES QUESTION PAPERS 54 of 4

55 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 d d i.e., - t (O) - Cot d d We hve ds d d d d ds d d d ds t d ds & d Cot Sec ds ds Sec d ( Cot ) d d ds Fid fo the followig Cves:- dt. (Cos t t Si t), (Si t t Cos t). Sec t, b t t ( ). Cost log t t, Si t Soltio of Give (Cos t t Si t), (Si t t Cos t ) d dt Diffeetitig & W..t t, we get d dt (-Si t Si t t Cos t) t cos t d d (Cos t Cos t t Si t) dt d dt t Si t ds dt d dt d dt VTU NOTES QUESTION PAPERS 55 of 4

56 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 t Cos t t Si t ds t dt Soltio of Sec t, b t t d d Sec t t t, b Sec t dt dt We hve ds dt d dt d dt Soltio of ( Sec t t t b Sec 4 t) [ Sec t (Sec t ) b Sec 4 t] [ Sec 4 t Sec t b Sec 4 t] ds dt [( b ) Sec 4 t Sec t] (Cos t log t t ), Si t Diffeetitig d w..t t we get d dt it Sec t d S., Cos t t t dt ds dt d dt d dt Sec t S it Cos t, t t Cot t S it Sec t t t Sec 4 t 4 t t VTU NOTES QUESTION PAPERS 56 of 4

57 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ds d Fid d fo the followig cves d d 0. ( Cos ). Cos. e Cot α Soltio of ( Cos ) Hece Diffeetitig w..t we get d d Si ds d d d ds d { ( Cos ) Si } { ( Cos Cos ) Si } { Cos } { Cos } { ( Cos )} { ( Si )} Si ds Ad d d d ( Cos ) Si VTU NOTES QUESTION PAPERS 57 of 4

58 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Si ( Cos ) Si Soltio of Cos Diffeetitig W..t we get d - Si. d ds d { ( Cos )} d d Si Cos - Si Si Si Si Si Cos Hece d d ds d Siβ d d Si 4 Si 4 Si Cos Cos 4 Cos 4 Si Cos VTU NOTES QUESTION PAPERS 58 of 4

59 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 4 Cos Cos 4 ds O ( Cos ) d ds Ad d d d Si 4 4 Si 4 Si 4 4 Cos si Soltio Hece 4 Cosec 4 si Si e Cot α, hee α is costt Diffeetitig w..t we get d d e Cot α. Cot α ds d d d { e Cot α e Cot α cot α} e Cot α { Cot α} e Cot α {Cosce α} VTU NOTES QUESTION PAPERS 59 of 4

60 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ds d e Cot α Cosce α 0 ds d d d d e Cotα e CotαCot α Eecises: { t α} {Sec α} Sec α ds ds Fid d to the followig cves. d d. Cos. ( Cos ). Note: We hve Si φ Cos φ d ds d ds d d Cosφ ( Si φ) ds p Sice P Si φ. d ds p VTU NOTES QUESTION PAPERS 60 of 4

61 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 ds d p Q O δ P φ P(,) R Si φ X OR P P Si φ OR OP p Pove tht with sl ottios d t φ d Let P (, ) be poit o the cve f () X O P d OP Let PL be the tget to the cve t P sbtedig gle ψ with the positive diectio of the iitil lie ( is) d φ be the gle betwee the dis vecto OP d the tget PL. Tht is O P L φ Fom the fige we hve VTU NOTES QUESTION PAPERS 6 of 4

62 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ψ φ 0 (Recll fom geomet tht eteio gle is eql to the sm of the iteio opposite gles) t ψ t ( φ ) t φ t φ o t ψ () - t φ t Let (, ) be the Ctesi coodites of P so tht we hve, X cos, si Sice is fctio of, we c s well egd these s pmetic eqtios i tems of. We lso kow fom the geometicl meig of the deivtive tht d t ψ d ie., ie., slope of the tget PL d d t ψ sice d e fctio of d d d t ψ d d d ( si ) cos si whee d d ( cos ) - si cos Dividig both the meto d deomito b O t ψ cos si cos cos si cos cos cos cos we hve, t t ψ () -. t Compig eqtios () d () we get t φ d d d o tφ d VTU NOTES QUESTION PAPERS 6 of 4

63 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Pove with sl ottios p d o 4 d p whee d d Poof : Let O be the pole d OL be the iitil lie. Let P (, ) be poit o the cve d hece we hve OP d L O P Dw ON p (s) pepedicl fom the pole o the tget t P d let φ be the gle mde b the dis vecto with the tget. Fom the fige O N P 90 L O P Now fom the ight gled tigle ONP ON si φ OP P ie., si φ o we hve p si φ p si φ () d d cot φ () d Sqig eqtio () d tkig the ecipocl we get, VTU NOTES QUESTION PAPERS 6 of 4

64 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS. ie., cosec p si φ p O ( cot φ) p Now sig () we get, p d d φ 0 O p 4 d d () Fthe, let Diffeetitig w..t. we get, d d d d d d 4 d Ths () ow becomes p d d, b sqig d (4). Fid the gle of itesectio of the cves: ( cos ) & b( - ) cos Soltio : ( cos ) : b( cos ) log log log ( cos ) : log log b log ( cos ) Diffeetitig these w..t. we get d d - si 0 cos ( /) cos ( /) ( /) si cot φ : cos : d d si 0 cos si cot φ si ( /) cos ( /) ( /) ie., cot φ - t ( /) cot ( π/ /) : cot φ cot ( /) φ π/ / : φ / VTU NOTES QUESTION PAPERS 64 of 4

65 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 gle of itesectio φ φ π/ / -/ / Hece the cves itesect othogoll. - π. S.T. the cves ( si ) & ( - si) ct ech othe othogoll Soltio : log log log ( si ) : log log log ( si ) ie., Diffeetitig these w..t we get d cos d si cos cot φ : si We hve : d - cos d si cot φ si - si t φ d t φ cos - cos - cos si t -si cos φ. t φ cos - cos Hece the cves itesect othogoll. -. Fid the gle of itesectio of the cves: si cos, si Soltio : log log (si cos ) : si ( si cos ) log log : log log log (si ) Diffeetitig these w..t we get d cos - si d si cos ie., : ( - t ) ( t ) d d cos si cos cot φ : cot φ cot φ cos ie., φ cot ( π/4 ) φ π/4 cot φ - φ π/4 - π/4 The gle of itesectio is π / 4 VTU NOTES QUESTION PAPERS 65 of 4

66 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 4. Fid the gle of the cves: log d / log Soltio : log : / log log log log ( log ) : log log - log ( log ) Diffeetitig these w..t q, we get, d d log. : d d log. ie., cot φ : log cot φ - log Note : We c ot fid φ d φ eplicitl. t φ log : t φ log Now coside, t φ log : t φ - log Now coside, t ( φ -φ ) log t φ t φ We hve to fid b solvig the give pi of eqtios : log d /log log... () ( log ) Eqtig the R.H.S we hve log log ie., ( log ) o log e Sbstittig e i () we get e t - e ( φ φ ) ( log e ) gle of itesectio φ e - e - - φ t t 5. Fid the gle of itesectio of the cves: ( cos ) d cos Soltio : ( cos ) : cos Tkig logithms we hve, Log log log ( cos ) : log log log (cos ) Diffeetitig these w..t, we get, - e VTU NOTES QUESTION PAPERS 66 of 4

67 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS d d si - cos ( ) ( ) ( ) si / cos / ie., cot φ :cot φ - t si / : d -si d cos ie., cot φ cot ( /) : φ cot ( π/ ) cot 0 φ / : φ π/ φ φ / -π/- π / () Now coside ( - cos ) cos - O cos o cos ( /) Sbstittig this vle i () we get, - The gle of itesectio π / /. cos ( /) 6. Fid the gle of itesectio of the cves : d / Soltio : : / log log log : log log - log Diffeetitig these w..t, we get, d d : - d d ie., cot φ : cot φ - o t φ : t φ - Also b eqtig the R.H.S of the give eqtios we hve / o ± Whe, t φ, t φ - d Whe, t φ -, t φ. - t φ. t φ - φ - φ The cves itesect t ight gles. π / VTU NOTES QUESTION PAPERS 67 of 4

68 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 7. Fid the pedl eqtio of the cve: ( - cos ) Soltio : ( - cos ) ( - cos ) log log log Diffeetitig w..t, we get d d si - cos 0 o d - si d - cos ( /) cos ( /) ( /) - si cot φ - cot si ie., cot φ cot (- /) φ - ( /) Coside p si p si φ (- /) o p - si ( /) Now we hve, ( - cos ) ( /) () ( /) We hve to elimite fom () d () () c be pt i fom. si ( / ) ie., si ( /) Bt p/- si (/), fom () p o p Ths p is the eqied pedl eqtio. p - si () Soltio : sec 8. Fid the pedl eqtio of the cve: sec log log log (sec ) Diffeetitig w..t, we get, d sec t d sec ie., d t d ie., cot φ cot ( π/ - ) φ π/ - VTU NOTES QUESTION PAPERS 68 of 4

69 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Coside p si φ p si ( π/ - ) ie., p cos Now we hve, sec () p cos () Fom () p/ cos o /p sec Sbstittig i () we get, ( /p) Ths p is the eqied pedl eqtio. o p 9. Fid the pedl eqtio of the cve: cos Soltio : cos log log log Diffeetitig w..t q we get d - si ie., d cos cot φ cot Coside p si ( cos ) d - t d ( π/ ) φ π/ φ p si ( π/ ) ie., p cos Now we hve, cos () s coseqece of () is () p cos () ( p/) Ths p is the eqied pedl eqtio. Soltio : 0. Fid the pedl eqtio of the cve: m m (cos m si m) m m (cos m si m) Diffeetitig w..t, we get, m ie., d - m si m m cos m d cos m si m d cos m -si m cos m d cos m si m cos m cot φ cot ( π/4 m) φ π/4 m ( - t m) ( t m) VTU NOTES QUESTION PAPERS 69 of 4

70 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Coside p p si si φ ( π/4 m) ie., p [ si ( π/4) cos m cos ( π/4) si m] ie., p ( cos m si m) (we hve sed the foml of si (A B) d lso the vles si ( π /4) cos ( π/4) / ) m m Now we hve, ( cos m si m ) Usig () i () we get, () p ( cos m si m) () 0 m m. p o m m p Ths m m p is the eqied pedl eqtio.. Estblish the pedl eqtio of the cve: si b cos i the fom ( ) p b Soltio : We hve log log Diffeetitig w..t we get d d si b cos ( si b cos ) cos - b si si b cos Dividig b, cot φ Coside p si φ cos - b si b si cos Sice φ cot be fod, sqig d tkig the ecipocl we get, p cosec p ( cot φ) φ o p ( cos - b si ) ( si b cos VTU NOTES QUESTION PAPERS 70 of 4

71 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS i e., i e., p p ( si b cos ) ( cos - b si ) ( si b cos ) ( si cos ) b ( cos si ) ( si b cos ) (podct tems ccels ot i the meto) 0 ie., p. b ( si b cos ) b o., b sig the give eqtio. p ( ) ( b ) p is the eqied pedl eqtio.. Defie Cvte d Rdis of cvte Soltio: A Cve Cts t eve poit o it. Which is detemied b the tget dw. Y f () Tget A s P (,) O ψ X Let P be poit o the cve f () t the legth s fom fied poit A o it. Let the tget t P mkes e gle ψ with positive diectio of is. As the poit P moves log cve, both s d ψ v. dψ The te of chge ψ w..t s, i.e., s clled the Cvte of the cve t P. ds The ecipocl of the Cvte t P is clled the dis of cvte t P d is deoted b ρ. ρ dψ ds ds dψ VTU NOTES QUESTION PAPERS 7 of 4

72 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ds dψ ρ (o) dψ ρ ds dψ Also deoted ρ K K ds K is ed it s Kpp. 0. Deive epessio fo dis of cvte i Ctesi fom. Soltio :() Ctesi Fom: Y f () O P A S ψ T ψ ψ P d R X d Let f () be the cve i Ctesi fom. d We kow tht, t ψ (Fom Fige) () d Whee ψ is the gle mde b the tget t P with is. Diffeetitig () W..t, we get dψ d Sec ψ. d d d i.e., Sec ψ d dψ ( t ψ) d dψ ( t ψ). d ( t ψ) ρ dψ d ds d d d d d.. fom eq () d ρ d VTU NOTES QUESTION PAPERS 7 of 4

73 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 ρ d d d ρ d d d ρ ( ) / () d d Whee, d d This is the foml fo Rdis of Cvte i Ctesi Fom. 4. Show tht the Cvte of Cicle t poit o it, is Costt Tget O PP P A ψ X T Soltio: Coside Cicle of dis. Let A be fied poit d P be give poit o the cicle sch tht c AP S. Let the gle betwee the tget to the Cicle t A d P be ψ. The clel AOP ψ. AP ψ i.e., S ψ This is the itisic eqtio of the cicle. Diffeetitig w..t S we get dψ dψ O K ds ds K which is Costt Hece the Cvte of the Cicle t poit o it is costt. VTU NOTES QUESTION PAPERS 7 of 4

74 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 5. Deive epessio fo dis of cvte i pmetic fom. Soltio: We hve ρ ( ) / Let f (t), g (t) be the cve i Pmetic Fom. d d The dt Y d d dt X d d d d Y d Y Y d d d d X dt X d Y. d Y. dt X d dt X X dt ( ) ( X ) X Y Sbstittig Y d Y i eqtio. dt d ρ ρ ( Y ) ( ( Y ) ( ) [( ) ( ) ], we get ρ () Eqtio () is clled the Rdis of Cvte i Pmetic Fom. VTU NOTES QUESTION PAPERS 74 of 4

75 0 6. Deive epessio fo dis of cvte i pol fom. Soltio: Let f () be the cve i the Pol Fom. We kow tht, Agle betwee the tget d dis vecto, T φ. i.e., t φ Diffeetite w..t we get Sec φ. Fom fige ψ φ Diffeetitig w..t, we get d d d d ( ) d d φ d d. d d d d d d d d ( ) ( ) φ φ d d d d d d Sec d d ( ) ( ) t φ d d d d d d d d d d d d d d φ d d d d d d d d VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 75 of 4

76 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS dψ d dφ d 0 dψ d dψ d ds ds Now, ρ. dψ d d d d d d d d d d d ds Also we kow tht d ρ () d d d. d ( ) d d d dψ d d d d d d d Whee, d d d Eqtio () s clled the dis of cvte i Pol fom. 7. Deive epessio fo dis of cvte i pedl fom. Soltio: Let p Si φ be the cve i Pol Fom. We hve p Si φ Diffeetitig p W..t, we get dp d Si φ Cos φ dφ d VTU NOTES QUESTION PAPERS 76 of 4

77 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 d Bt Si φ, Cos φ ds d ds dp d d ds d ds dφ d d dφ. ds d d ds d ds dφ ds d ( φ) ds dψ ds dp d dψ ds dp d dψ. ρ ds ρ d ρ (4) dp Eqtio (4) is clled Rdis of Cvte of the Cve i Pol Fom. 8. Fid the dis of cvte t (, ) fo the cve. Soltio: Give () is i Ctesi fom. We hve, Rdis of cvte i Ctesi fom. ρ ( ) () Diffeetitig () w..t we get, VTU NOTES QUESTION PAPERS 77 of 4

78 0 Diffeetitig w..t we get Sbstitte Y d Y i (), we get. 9. Fid the dis of cvte t (,) fo the cve c log Sec Soltio: Give c log Sec () Diffeetitig () w..t, we get c t. t Diffeetitig W..t we get Sbstitte d i ρ d d d d d d 4 ( ) ρ c c c c Sec c Sec c t.. c c c c Sec c c c Sec ( ) δ ( ) [ ] ( ) ( ) [ ] ( ) c Sec c c Sec c Sec c c t δ ( ) c csec VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 78 of 4

79 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 0. Fid the dis of cvte t the poit t o the cve (t Sit), ( Cost). Soltio: Give Cves e i Pmetic Fom ( ) ( ) ) Rdis of Cvte, ρ () Diffeetitig the give Cves W..t t, we get d d ( Cost) Sit dt dt Diffeetitig W..t t we get - Sit, Cost Sbstitte,, d i (), we get ρ Cost Cos t Si t { Cost Cos t Si t} { ( ) } { }. t 8. t Cost Cos Cos ( Cost) Cos t Cos t ρ 4Cos t. Fid the Rdis of Cvte to { Cost log t( t )}, Sit t t. Soltio: Hee Cost log t t, Si t d S it. Sec t. dt t t { -sit } si t / cost / { -sit } si t / { (-si t) / sit } d dt Cos t / sit ( Cost) S it) ( ( Cost) Cost S it( S it) { } { ( )} VTU NOTES QUESTION PAPERS 79 of 4

80 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS d d d Cost 0 d d d / dt d / dt Cost tt cost cost / si t d t t d Diffeetitig W..t we get d d d d Sec t dt d 4 Sec Sec ts it t d dt Sec t. Cos t S it d Sbstitte & i ρ, we get d d d ( ) ( ) t t i.e., ρ 4 Sec ts it ρ Cot t. Sec t 4 Sec ts it Cost S it Cott. Fid the Rdis of Cvte to t, 4 4 Soltio: Give () Diffeetitig () w..t to we get d d 0 ( ) d i.e., (Fom ()) d () VTU NOTES QUESTION PAPERS 80 of 4

81 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 d Also () d At the give poit, 4 The 4 - & 4 ( ) 4 Sbstitte d i ρ ρ ( ( ) ) 4 4 ρ 4 ( ). Show tht fo the Cdioids ( Cos), ρ / is costt Soltio: ( Cos) We hve, d d - Si P d 4 d, is Pedl Eqtio. Si 4 Si 4 ( Cos ) 4 Si ( Cos ) 4 VTU NOTES QUESTION PAPERS 8 of 4

82 0 ( ( Cos) Diffeetitig w..t P we get P Now, Ad 4. Fid the Rdis of Cvte of the Cve Soltio: Give Diffeetitig w..t to P, we get P 4 Cos P P dp d 4 p dp d p dp d 4 ρ p ρ b b P b b P dp d b P p b dp d.. p b p b dp d ρ VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 8 of 4

83 0 5. Fid the Rdis of Cvte t (,) o Soltio: Give Diffeetitig w..t to we get We hve Diffeetitig bove eslt w..t to P we get d d. d d 4 d d P 4 P P.. P ( ).. dp d dp d / / dp d. dp d ). ( VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 8 of 4

84 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS ( ) ( ) d dp ( ) d. Ths, ρ. dp ρ ( ) d dp Eecises: π ψ () Fid the Rdis of the Cvte t the poit (s, ψ) o S log t 4 () Fid the Rdis of the Cvte of C t (,) () Fid the Rdis of the Cvte of 4 t (,) (4) Fid the Rdis of Cvte t the poit o C Si ( Cos ), C Cos ( Cos ) (5) If ρ d ρ e the dii of cvte t the etemities of chod of the cdiode 6 ( Cos) which posses thogh the Pole pove tht ρ ρ 9 VTU NOTES QUESTION PAPERS 84 of 4

85 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS DIFFERENTIAL CALCULUS III 0 PARTIAL DIFFERENTIATION Itodctio :- Ptil diffeetil eqtios bod i ll bches of sciece d egieeig d m es of bsiess. The mbe of pplictios is edless. Ptil deivtives hve m impott ses i mth d sciece. We shll see tht ptil deivtive is ot mch moe o less th pticl sot of diectiol deivtive. The ol tick is to hve elible w of specifig diectios... so most of this ote is coceed with fomliig the ide of diectio So f, we hd bee delig with fctios of sigle idepedet vible. We will ow coside fctios which deped o moe th oe idepedet vible; Sch fctios e clled fctios of sevel vibles. Geometicl Meig Sppose the gph of f (,) is the sfce show. Coside the ptil deivtive of f with espect to t poit (0, 0). Holdig costt d vig, we tce ot cve tht is the itesectio of the sfce with the veticl ple 0. The ptil deivtive f(0,0). meses the chge i pe it icese i log this cve. Tht is, f(0, 0) is jst the slope of the cve t (0, 0). The geometicl itepettio of f(0, 0). is logos. VTU NOTES QUESTION PAPERS 85 of 4

86 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Rel-Wold Applictios: Rtes of Chge: I the Jv pplet we sw how the cocept of ptil deivtive cold be pplied geometicll to fid the slope of the sfce i the d diectios. I the followig two emples we peset ptil deivtives s tes of chge. Specificll we eploe pplictio to tempete fctio ( this emple does hve geometic spect i tems of the phsicl model itself) d secod pplictio to electicl cicits, whee o geomet is ivolved. I. Tempete o Metl Plte The scee cpte below shows cet website illsttig theml flow fo chemicl egieeig. O fist pplictio will del with simil flt plte whee tempete vies with positio. * The emple followig the picte below is tke fom the cet tet i SM,: Mltivible Clcls b Jmes Stewt. 0 VTU NOTES QUESTION PAPERS 86 of 4

87 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Sppose we hve flt metl plte whee the tempete t poit (,) vies ccodig to positio. I pticl, let the tempete t poit (,) be give b, 0 T(, ) 60/ whee T is mesed i o C d d i metes. Qestio: wht is the te of chge of tempete with espect to distce t the poit (,) i () the -diectio? d (b) i the -diectio? Let's tke () fist. Wht is the te of chge of tempete with espect to distce t the poit (,) i () the -diectio? Wht obsevtios d tsltios c we mke hee? Rte of chge of tempete idictes tht we will be comptig tpe of deivtive. Sice the tempete fctio is defied o two vibles we will be comptig ptil deivtive. Sice the qestio sks fo the te of chge i the -diectio, we will be holdig costt. Ths, o qestio ow becomes: Wht is dt d t the poit (,)? T(, ) 60 / 60( ) T 60( )( ) T (,) 60(4)( 4 ) 0 Coclsio : The te of chge of tempete i the -diectio t (,) is 0 degees pe mete; ote this mes tht the tempete is decesig! VTU NOTES QUESTION PAPERS 87 of 4

88 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Pt (b): The te of chge of tempete i the -diectio t (,) is compted i simil me. Coclsio : T(, ) 60 / 60( ) T 60( )( ) T (,) 60()( 4 ) 0 The te of chge of tempete i the -diectio t (,) is 0 degees pe mete; ote this mes tht the tempete is decesig! II. Electicl Cicits: Chges i Cet The followig is dpted fom emple i fome tet fo SM, Mltivible Clcls b Bdle d Smith. * I electicl cicit with electomotive foce (EMF) of E volts d esistce R ohms, the cet, I, is IE/R mpees. Qestio: () At the istt whe E0 d R5, wht is the te of chge of cet with espect to voltge. (b) Wht is the te of chge of cet with espect to esistce? () Eve thogh o geomet is ivolved i this emple, the te of chge qestios c be sweed with ptil deivtives. we fist ote tht I is fctio of E d R, mel, I(E,R) ER - VTU NOTES QUESTION PAPERS 88 of 4

89 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS The te of chge of cet with espect to voltge the ptil deivtive of I with espect to voltge, holdig esistce costt is I R E whe E0 d R5, we hve I E vebl coclsio : If the esistce is fied t 5 ohms, the cet is icesig with espect to voltge t the te of mpees pe volt whe the EMF is 0 volts. Pt (b): Wht is the te of chge of cet with espect to esistce? Usig simil obsevtios to pt () we coclde: The ptil deivtive of I with espect to esistce, holdig voltge costt whe E0 d R5, we hve I E (0,5) 0(5) 0.5 I ER E Coclsio : If the EMF is fied t 0 volts, the cet is decesig with espect to esistce t the te of 0.5 mpees pe ohm whe the esistce is 5 ohms. Ke Wods :- The the ptil deivtive of w..t is give b f( δ, ) f(, ) lim δ 0 δ The ptil deivtive of w..t is give b f(, δ ) f(, ) lim δ δ 0 VTU NOTES QUESTION PAPERS 89 of 4

90 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS show tht b - b -b. If e si ( b) -b Soltio : e si ( b) 0 e ie., -b ( b)..e si ( b) - b e cos - b cos ( b) - b -b Also e cos ( b). b (- b) e si ( b) ie., b e - b cos ( b) b () () Now b b sig () d () becomes Ths be b b b - b cos -b ( b) b - be cos ( b) b b. If e f ( - b), pove tht b b b Soltio : e f ( - b), b dt e. f b ( - b) e f ( - b) b b O e. f ( - b) b b Net, e f ( - b). (- b) b e f ( - b) b O b e f ( - b) b () () Now coside L.H.S b VTU NOTES QUESTION PAPERS 90 of 4

91 0 ( ) { } ( ) { } b b - f - be b - f e b b b ( ) ( ) b b - f b - b e b - f b e b b b R.H.S Ths b b. If log, show tht ( ) Soltio : B dt ( ) log log The give is smmetic fctio of,,, (It is eogh if we compte ol oe of the eqied ptil deivtive).. ie., ( ) ( ) ( ). - ( ) () Simill ( ) () ( ) () Addig (), () d () we get, ( ) Ths ( ) VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 9 of 4

92 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 4. If log (t t t ), show tht, si si si Soltio : log (t t t ) is smmetic fctio. 0 si sec t t t ( si cos ) sec. t t t O Simill t si () t t t t si () t t t t si () t t t Addig (), () d () we get, si ( t t t ) ( t t t ) si si Ths si si si 5. If log ( ) the pove tht d hece show tht - 9 ( ) Soltio : log ( ) is smmetic fctio () () () Addig (), () d () we get, VTU NOTES QUESTION PAPERS 9 of 4

93 0 ( ) ( ) Recllig stdd elemet eslt, ( )( ) c bc b c b c b bc c b We hve, ( ) ( )( ) Ths Fthe, b sig the elie eslt. ( ) ( ) ( ) ( ) 9 Ths ( ) -9 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 9 of 4

94 0 6. If () si, cos d f, pove tht () () f f Soltio :Obsevig the eqied ptil deivtive we coclde tht mst be fctio of,. Bt f( ) b dt d hece we eed to hve s fctio of,. Sice cos, si we hve. () whee f hve we () ( ) ( ) d. - f f () ( ) ( ). - f f Addig these eslts we get, () ( ) { } ( ) ( ) f - f () ( ) () ().. f f f f Ths () () f f 7. Pove tht Poof : Sice f (, ) is homogeeos fctio of degee we hve b the defiitio, ( ) / g () Let s diffeetite this w..t d lso w..t. ( ) ( ) / g -. / g. - ie., ( ) ( ) / g / g - - () Also ( ). / g. VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 94 of 4

95 0 ie., ( ) /.g - () Now coside s coseqece of () d () ( ) ( ) [ ] ( ) [ ] / g / g / g ( ) ( ) ( ) / g / g / g - ( ) / g., b sig () Ths we hve poved Ele s theoem ; 8. Pove tht ( ) - Poof : Sice f (, ) is homogeeos fctio of degee, we hve Ele s theoem () Diffeetitig () ptill w..t. d lso w..t we get,. () Also,. () We shll ow mltipl () b d () b. d Addig these sig the fct tht we get, VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 95 of 4

96 0 ie., ( ) ( ) b sig, o ( ) ( ) - - Ths ( ) - ie., ( ) - 9. If show tht 0 Soltio : (Obseve tht the degee is 0 i eve tem) We shll divide both meto d deomito of eve tem b. ( ) { } /, / g / / / / / is homogeeos of degee 0. 0 We hve Ele s theoem, Pttig 0 we get, 0 0. If show tht log 4 4 Soltio : we cot pt the give i the fom g (/) ( ) ( ) ( ) ( ) / / / / e ie., e g (/) e is homogeeos of degee Now pplig Ele s theoem fo the homogeeos fctio e VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 96 of 4

97 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 e ( ) ( ) We hve e e ie., e e e Dividig b e we get -. If t show tht Soltio : (i) (i) si (ii) si 4 si t - - t - b dt ( / ) ( - / ) ( / ) ( / ) ie., t /g (/) t is homogeeos of degee. Applig Ele s theoem fo the fctio t we hve, ( t ) ( t ). t ; ie., sec sec t o si t si cos cos si sec cos si (ii) We hve si () Diffeetitig () w..t d lso w..t ptill we get. cos. () Ad. cos. Mltiplig () b d () b we get, () cos. VTU NOTES QUESTION PAPERS 97 of 4

98 0. cos Addig these b sig the fct tht, we get ( ) ( ) cos B sig () we hve, si si - cos (sice si cos si, fist tem i the R.H.S becomes si 4) Ths si 4 si. If,, f Pove tht 0 >> hee we eed to covet the give fctio ito composite fctio. Let ( ), q, whee p q, p, f ie., ( ) ( ) { },,,, q, p, q q p p ie., q. p - p () Simill b smmet we c wite, p - q () q - () Addig (), () d () we get 0 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 98 of 4

99 0. If f(,, ) show tht 0 >> Let f(p, q, ) whee p, q, q q p p ie., ( ) -.0 q. p - p () Simill we hve b smmet p - q () q - () Addig (), () d () we get, 0 4. If f(, ) whee cos d si Show tht Soltio : ( ) ( ) { } ( ),,, ; ie., si cos () d ( ) ( ) cos si - cos si - o cos si sqig d ddig (), () d collectig sitble tems hve, VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 99 of 4

100 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS [ cos si ] 0 [ si cos ] cos si - si cos ie., R.H.S L.H.S v - v 5. If ( ) f, whee e e, e e Pove tht v Soltio : { (, ) (, v) } (, v) ie.,. e ; v v v - (- e ) -v -v (- e ) (- e ) v () () Coside R.H.S d () () ields v -v - v ( e e )- ( e e ). -. Ths 6. Fid - ie., R.H.S L.H.S (, v, w) (,, ) whee, v, w Soltio : The defiitio of J (, v, w) (,, ) v v v w w w Bt, v, w VTU NOTES QUESTION PAPERS 00 of 4

101 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Sbstittig fo the ptil deivtives we get J Epdig b the fist ow, J {( ) ( )} - {( ) ( )} {( ) ( )} (-) (-) (-) ( ) 0 Ths J 0 7. If, v, w, show tht (, v, w) (,, ) 4 Soltio : b dt, v, w (, v, w) (,, ) v w v w v w - 04 Ths (, v, w) (,, ) 4 VTU NOTES QUESTION PAPERS 0 of 4

102 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 8. If v e cos d v e si fid the jcobi of the fctios d v w..t d. Soltio : we hve to fid (, v) (, ) v v Usig the give dt we hve to solve fo d v i tems of d. B dt v e cos v e si () () () () gives : e (cos si ) () () gives : v e (cos si ) Ie., e e (cos si ) ; v (cos si ) e (cos si ), v e (- si - cos ) Now (, v) (, ) e e (cos si ) (cos si ) e ( si cos ) e (si cos ) e. e { - ( cos si ) (cos si ) } e {si ) ( si )} 4 Ths (, v) (, ) e e VTU NOTES QUESTION PAPERS 0 of 4

103 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 9. () If cos, si fid the vle of (b) Fthe veif tht (, ) (, ). (, ) (, ) (, ) (, ) () Soltio : We shll fist epess, i tems of d. We hve cos, si b dt. d t o t - (/) Coside Diffeetitig ptill w..t d lso w..t we get, d d Also coside t ( / ) ( / ). d ( / ). i.e., d Now (, ) (, ) d d d i e., ( ) ( ) ( ) ( ). (, ) (, ) VTU NOTES QUESTION PAPERS 0 of 4

104 Soltio (b) : Coside si, cos d ) si (cos cos si si cos ), ( ), ( ), ( ), ( Fom () d () : ), ( ), ( ), ( ), ( 0. If ( ) v v, the show tht / JJ Soltio : v v ); ( v v v v, ), ( v v v v v J ) ( ), ( ), ( J v v ) ( Net we shll epess d v i tems of d. B dt - v d v Hece. Also v Now we hve,,, ; v VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 04 of 4

105 0 ) ( ) ( 0 ) ( v / ) ( ) ( ), ( ), ( v v v J ) ( ) ( ) ( ) ( Ths J / Hece J J / Ths / JJ. Stte Tlo s Theoem fo Fctios of Two Vibles. Sttemet: Cosideig f ( h, k) s fctio of sigle vible, we hve b Tlo s Theoem f ( h, k) f (, k) h () Now epdig f (, k) s fctio of ol, f (, k) f (, ) k (i) tkes the fom f ( h, k) f (,) k h Hece f ( h, k) f (, ) h I smbol we wite it s ), (! ), ( k f h k f ), (! ), ( f k f ), (! ), ( f k f ), (! ), ( ), ( f k f k f k f h ), ( ), (! ()! f k f hk f h f k f VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 05 of 4

106 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS F ( h, k) f (,) h k f h k! f Tkig d b, () becomes f ( h, b k) f (,b) [h f (,b) kf (,b)] [h f (,b) hkf (,b)! Pttig h d b k so tht h, k b, we get F (,) f (,b) [( ) f (,b) ( b) f (,b)] k f (,b)] [( ) f (,b) ( ) ( b) f (,b) ( b) f (,b)] ()! This is Tlo s epsio of f (,) i powes of ( ) d ( b). It is sed to epd f (,) i the eighbohood of (,b) cooll, pttig 0, b 0 i (), we get f (,) f (0,0) [ f (0,0) f (0,0)] [ f (0,0) f (0,0)! f (0,0) ] () This is Mcli s Epsio of f (,). Epd e log ( ) i powes of d p to tems of thid degee. Soltio: Hee f (,) e log ( ) f (0,0) 0 f (,) e log ( ) f (0,0) 0 f (,) e f (0,0) f (,) e log ( ) f (0,0) 0 f (,) e f (0,0) VTU NOTES QUESTION PAPERS 06 of 4

107 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 f (,) - e ( ) - f (0,0) - f (,) e log ( ) f (0,0) 0 f (,) e f (0,0) f (,) -e ( ) f (0,0) - f (,) e ( ) - f (0,0) Now, Mcli s epsio of f (,) gives f (,) f (0,0) (f (0,0) f (0,0) { f (0,0) f (0,0)! f (0,0)} { f (0,0) f (0,0) f (0,0) f (0,0)} ! e log ( ) 0.0 () {.0 () (-)}! {.0 () (-) ()} ! - ( ) Epd f (,) e Cos b Tlo s Theoem bot the poit, π p to the Secod 4 degee tems. π Soltio: f (,) e Cos d, b f, π e 4 4 f (,) e Cos f, π e 4 f (,) -e Si f -, π e 4 f (,) e Cos f, π e 4 f (,) -e Si f -, π e 4 f (,) - e Cos f -, π e 4 Hece b Tlo s Theoem, we obti VTU NOTES QUESTION PAPERS 07 of 4

108 0 f (,) f i.e., e Cos e Cos } Eecise: ) Epd e p to Secod degee tems b sig Mcli s theoem ) Epd Log ( ) p to Thid degee tems b sig Mcli s theoem ) Epd bot the poit (,-) b Tlo s epsio 4) Obti the Tlo s epsio of e Si bot the poit p to Secod degee tems 5) Epd e si p to the tem cotiig 4 4, π f f 4 ) ( π! f f 4 ) ( ) ( π e 4 ) ( e e π! 4 4 ) ( ) ( e e e π π e 4 4 ) ( ) (! 4 ) ( π π π ( ), 0 π VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS VTU NOTES QUESTION PAPERS 08 of 4

109 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 Mim d Miim:- I mthemtics, the mimm d miimm (pll: mim d miim) of fctio, kow collectivel s etem (sigl: etemm), e the lgest d smllest vle tht the fctio tkes t poit withi give eighbohood. A fctio f (, ) is sid to hve Mimm vle t (,b) if thei eists eighbohood poit of (,b) (s (h, bk)) sch tht f (, b) > f (h, bk). Simill, Miimm vle t (,b) if thee eists eighbohood poit of (,b) (s (h, bk)) sch tht f (, b) < f (h, bk). A Miimm poit o the gph (i ed) f (, ) ( ) A Mimm poit o the gph is t the top (i ed) VTU NOTES QUESTION PAPERS 09 of 4

110 VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS 0 A sddle poit o the gph of (i ed) Sddle poit betwee two hills. Necess d Sfficiet Coditio:- If f 0 d f 0 (Necess Coditio) Fctio will be miimmm if AC-B > 0 d A > 0 Fctio will be mimm if AC-B > 0 d A < 0 Fctio will be eithe mim o miim if AC-B < 0 If AC-B 0 we cot mke coclsio withot fthe lsis whee A f, B f, C f A VTU NOTES QUESTION PAPERS 0 of 4