FIRST SEMESTER BACHELOR IN BUSINESS ADMINISTRATION COURSE: 03 Business Mathematics BLOCK-3

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1 BBA (S) 0-0 KRISHNA KANTA HANDIQUI STATE OPEN UNIVERSITY Pgon,Rnig, Guwi FIRST SEMESTER BACHELOR IN BUSINESS ADMINISTRATION COURSE: 0 Businss Mmics BLOCK- CONTENTS UNIT 6 : LOGARITHM UNIT 7 : BINOMIAL THEOREM UNIT 8 : MATRICES UNIT 9 : LIMIT AND CONTINUITY REFERENCES : FOR ALL UNITS

2 Subjc Eprs. Profssor Nripnr Nryn Srm, Mnirm Dwn Scool of Mngmn, KKHSOU. Profssor Muninr Kki, VC, ARGUCOM. Profssor Rinlini Pk Kki,Dp.of Busniss Aminisrion, GU Cours Coorinor : Hrkrisn Dk, K.K.H.S.O.U Dr, Cynik Snpi,KKHSOU Dr. Smriisik Couury,KKHSOU SLM Prprion Tm Unis Conribuor 0 & S. K. J, Luming Collg, Luming Dr. Dibyjyoi Mn, KKHSOU Mrs. Ajn Mjumr, Gui Commrc Collg Eioril Tm Conn : Prof. N.R. Ds, Gui Univrsiy Dr. Snjy Du, PETC, Gui Univrsiy Srucur Form & Grpics : Hrkrisn Dk, KKHSOU Jun 07 Tis Slf Lrning Mril (SLM) of Krisn Kn Hniqui S Opn Univrsiy is m vilbl unr Criv Commons Aribuion-Non Commrcil-Sr Alik.0 Licns (inrnionl): p://crivcommons.org/licnss/by-nc-s/.0/ Prin n publis by Rgisrr on blf of Krisn Kn Hniqui S Opn Univrsiy. Hqurrs : Pgon, Rni G, Guwi Housf Compl, Dispur, Guwi-78006; Wb: T Univrsiy cknowlgs wi nks finncil suppor provi by Disnc Eucion Buru, UGC for prprion of is suy mril.

3 BACHELOR IN BUSINESS ADMINISTRATION BUSINESS MATHEMATICS Block DETAILED SYLLABUS UNIT0: Drivivs of funcions Pg : (-6) Gomricl inrprion of riviv of funcion,driviv of vrious snr funcions, Driviv of sum n iffrnc of funcions, Driviv of prouc of funcions n quoin rul, UNIT : Applicion of Drivivs Pg : (6-7) Mimum n minimum of funcion Applicion of mim n minim, UNIT : Ingrion n Mos of Ingrion Pg : (75-) Inroucion, Ingrion of simpl funcions, propris of Ingrls, Mo of Ingrion: Ingrion by subsiuion, Ingrion by prs, Ingrion by pril frcions. UNIT : Linr progrmming problm Pg : (-) Mning,Coniions for Using LPP Tcniqu, Bsic Assumpions of LPP, Ars of pplicions of LPP, Limiions of LPP, Gnrl Linr Progrmming Problm, Formion of LPP of Two Vribl, Som Dfiniions, Grpicl Mo of Soluion of LPP.

4 BLOCK INTRODUCTION Tis is scon block of cours Businss Mmics. T block consiss of four unis. T n uni ls wi concp of rivivs n rivivs of vrious funcion. T lvn uni ls wi pplicion of rivivs suc s fining mim n minim of funcions. In wlf uni, bsics of Ingrl clculus r inrouc.in is uni,w will iscuss iffrn mos of Ingrion. T irn uni inroucs us o concp of Linr progrmming problm.in is uni,w will iscuss formion n grpicl soluion of Linr progrmming problm. Wil going roug uni, you will noic som long-so bos, wic v bn inclu o lp you know som of ifficul, unsn rms. Som ACTIVITY (s) s bn inclu o lp you pply your own ougs. Agin, w v inclu som rlvn concps in LET US KNOW long wi. An, n of c scion, you will g CHECK YOUR PROGRESS qusions. Ts v bn sign o slf-cck your progrss of suy. I will b br if you solv problms pu in s bos immily fr you go roug scions of unis n n mc your nswrs wi ANSWERS TO CHECK YOUR PROGRESS givn n of c uni.

5 UNIT 0: DERIVATIVE OF FUNCTIONS UNIT STRUCTURE 0. Lrning objcivs. 0. Inroucion 0. Gomricl inrprion of riviv of funcion. 0. Driviv of vrious snr funcions. 0.5 Driviv of sum n iffrnc of funcions. 0.6 Driviv of prouc of funcions n quoin rul. 0.7 Driviv of composi funcions. 0.0 L Us Sum Up. 0. Answrs o Cck Your Progrss 0. Furr Rings 0. Mol Qusions. 0. LEARNING OBJECTIVES Afr going roug is uni, you will b bl o : y unrsn gomricl mning of f / () or vlu riviv of lgbric, rigonomricl, ponnil n logrim funcions. rmin rivivs of composi funcions (funcion of funcions). vlu riviv of implici funcions. 0. INTRODUCTION W v lry inrouc concp of riviv in uni. In is uni, w iscuss lgbr of riviv incluing riviv of crin snr funcion. T noion of iffrniion is ssnilly concrn wi r of crg of pnn vribl wi rspc o n inpnn vribl. In is uni, w lso iscuss nur of funcion wos numror n nominor bo pproc o zro priculr poin (Inrmin form). I is cll inrmin form. Som inrmin Businss Mmics (Block )

6 Uni 0 Driviv of Funcions forms r 0 0,,, 0,, 0 o, o c. 0. GEOMETRICAL INTERPRETATION OF DERIVATIVE L yf() b funcion. L P (c, f(c)) n Q (c, f(c)) b wo nigbouring poins on grp of yf(). f(c ) Slop of Cor PQ (c ) - f(c) - c f(c )- f(c) Wn Q P long curv, cor PQ bcoms ngn PT P. Tus, slop of cor PQ 0 T Q (C, f(c)) P bcoms slop of ngn P wn Q P, i.. 0 Q (c, f(c)) X f(c ) f(c) Slop of ngn P lim 0 f '/ (c) ( by finiion) Also, if θ is ngl m by ngn P wi - is, n slop o ngn P nθ. f(c ) f(c) Tus, f '/ (c)nθ lim 0 No :- If ngn is prlll o -is, n f / (c)0. 0. DERIVATIVES OF VARIOUS STANDARD FUNCTIONS. (k) 0, k is ny consn 0. (co) cosc. (). (sc) sc n. (n ) n n. (cosc ) cosc. co Businss Mmics (Block )

7 Driviv of Funcions Uni 0. ( ) 5. (log ) 7. (sin) cos 8. (cos) sin 9. (n) sc Proof :. L y f() k f() k y By finiion, lim 0 f( ) f() k k (k) lim 0 0. L y f() f() y By finiion, lim 0 f( ) f() () lim 0 lim 0. L y f() n f () () n By finiion, y f( ) f() lim 0 n n n ( ) lim 0 Cs I : L n b posiiv ingr ( ) Businss Mmics (Block )

8 Uni 0 Driviv of Funcions lim 0 ( ) n n lim 0 lim 0 n n n n n n n c c... n n c n c n n c n... n n n Cs II : L n b ngiv ingr or frcion lim 0 ( ) n n lim 0 n n n lim n n(n ) n n....! 0 lim 0 n n(n ) n.....! n.n. n n n for ll rionl n, ( ). L y f() f() By finiion, y lim 0 f( ) f() n n ( ) lim 0 lim 0.. lim 0 Businss Mmics (Block )

9 Driviv of Funcions Uni 0 5. L y f() log f() log () By finiion, y lim 0 f( log ( log) lim 0 lim 0 ) f() log ( ) log lim 0 log lim... 0 lim L y f() sin f() sin () By finiion, y lim 0 f( ( sin) lim 0 lim 0 ) f() sin( cos ) sin sin sin lim cos. 0 lim lim. 0 0 sin Businss Mmics (Block ) 5

10 Uni 0 cos. cos 8. L y f() cos f() cos () By finiion, y lim 0 ( cos) lim 0 f( 0 ) f() cos( lim ) cos sin sin Driviv of Funcions sin lim sin 0 sin limsin lim. 0 0 sin. 9. L y f() n By finiion f() n () y lim 0 f( ( n ) lim 0 lim 0 0 ) f() n( ) n sin( ) cos( ) sin cos sin( )cos cos( )sin lim cos( )cos lim sin( ) 0 cos( )cos sin lim. lim 0 0 cos( )cos 6 Businss Mmics (Block )

11 Driviv of Funcions Uni 0. cos.cos sc 0. L y f() co f() co () By finiion, y lim 0 f( ( co ) lim 0 0 ) f() co( ) co cos( ) cos lim sin( ) sin lim 0 lim 0 sin cos( ) cos sin( ) sin( )sin sin{ ( ) } sin( )sin lim 0 sin( ) sin( )sin sin lim. lim 0 0 sin( ).sin. sin.sin cosc. L y f() sc f() sc () By finiion, y lim 0 f( ( sc ) lim 0 ) f() sc( ) sc lim 0 cos( ) cos lim 0. cos cos( ) cos( )cos lim 0. sin sin cos( ) cos Businss Mmics (Block ) 7

12 Uni 0 Driviv of Funcions lim 0 sin. sin. cos( )cos sin lim sin. lim 0 0 sin.. cos.cos sin. cos cos sc. n.. L y f() cosc By finiion, f() cosc () y lim 0 f( ( cos c ) lim 0 0 ) f() cos c(. { cos( ).cos } lim 0 ) cosc lim sin ( ) sin lim 0 lim 0 sin sin( ).sin( )sin cos sin sin( ) sin.cos sin lim sin( )sin 0 8 Businss Mmics (Block )

13 Driviv of Funcions Uni 0 lim 0 cos sin. sin( ).sin cos lim sin( ).sin.lim 0 0 sin cos. sin.sin cosc co 0.5 THE DERIVATIVE OF SUM AND DIFFERENCE OF FUNCTIONS Torm.5. If f n f r wo iffrnibl funcions on inrvl I, n f b ir sum funcion so f f f, Tn f is iffrnibl n f f f Proof : L b ny poin of I n b nigbouring poin. Tn, f() f () f () n f() f () f () f()-f() f () f () f () f () f( ) f() f( ) f() f ( ) f () lim 0 f( ) f() lim 0 f ( ) Q f n f r iffrnil in I f lim 0 n f lim 0 () lim 0 f ( ) ( ) f () f f( ) f() f () f () is. lim 0 f () f() f ( ) f () () Businss Mmics (Block ) 9

14 Uni 0 Tis sows f is iffrnibl n f() f () f() i.., f f f Torm.5. If f n f r wo iffrnibl funcions on inrvl I, n f b ir iffrnc funcion so ff f n f is iffrnibl n f f f Proof : Try yourslf. ILLUSTRATIVE EXAMPLE : Driviv of Funcions Empl. Fin rivivs of following wi rspc o. Soluion : ) () b) blog log 0 ) log ( log ) ( log ) c) { } { ( )} ) ( ) ( ) () ( ) ( ) () blog b log b) ( ) ( ) ( ) log 0 b. c) ( ) log 0 log log 0 b ( log ) log0 50 Businss Mmics (Block )

15 Driviv of Funcions Uni 0 ) log 0 log log log log log () log log log 0. log log Empl : Fin rivivs of y wi rspc o for following 9 ) y co b) y n sin c) y log 7 sc Soluion : ) y y 9 co sin 9 sin co 9 ( cos c) (co ) ( ) 9( cos c.co ) ( cos c ) log 9 cos c.co cos c log b) y n ( n ) y ( n ) ( ) ( ) sc c) y log 7 sc y ( log sc ) 7 7 ( log ) ( ) ( sc ) 7 6 sc n Businss Mmics (Block ) 5

16 Uni 0 Driviv of Funcions Empl : Fin poin on curv y wic ngn is prlll o is. Soluion : Givn y ( ) y For ngns prlll o is, w v y 0 0 y T poin is 9, CHECK YOUR PROGRESS - ) Fin riviv of following funcions wi rspc o. ) b) sin c) cosc ) ) log( ) f) sin cos ) Prov no ngn o curv y is prlll o is. 0.6 Driviv of prouc of funcions n quoin rul Torm.6. : L u() n v() b wo iffrnibl funcions. Tn prouc u() v() is lso iffrnibl n / / ( u()v() ) u()v () v()u () Proof : L b ny poin in omin, n b nigbouring poin. f()u()v() f() u()v() 5 Businss Mmics (Block )

17 Driviv of Funcions Uni 0 f() f() u()v() u()v() u()v() u()v()u()v() u()v() u() {v() v()} v() {u() u()} f ( ) f ( ) v( ) v( ) u( ) u( ) u( ). v( ) lim 0 f( ) f() lim u(). lim 0 0 u(). lim 0 Qu n v r iffrnibl. u / () lim 0 n v / () lim () lim 0 0 u( ) u() f( v( v( ) v() Tus f is iffrnbl n f / ()u()v / ()v()u / () v( ) v() v() lim 0 ) v() v(). lim 0 ) f() u()v / ()v()u / () i.., (f())u()v/ ()v()u / () u( u( ) u() ) u()... () i.., u()v() u()v/ ()v()u / (). Corollry : If u, v, w r r iffrnbl funcions, n (u()v()w())u/ ()v()w()v / ()u()w()w / ()u()v() Torm.6. : L u() n v() b wo iffrnibl funcions in sm inrvl, n quoin funcion u() v() v() u() v() ( u() ) u() ( v() ) ( v() ) is lso iffrnibl n Proof : L f() u( ) v( ) v( ) u / ( ) u( ) v ( v( ) ) / ( ) u( ) f( ) v( ) Businss Mmics (Block ) 5

18 Uni 0 Driviv of Funcions u( ) f( ) f() v( ) u() v() v()u( ) u()v( ) v( )v() f( ) f() v()u( ) v()u() v()u() u()v( ).v( )v() [ ] [ ] v() u( ) u() u() v() v( ).v( )v() v( ). [ u( ) u( ) ] [ v( ) v( ) ] u( ) v( ) v( ) lim 0 f( lim u( ) u( ) lim v( ) v( ) v( ). u( ). ) f() 0 0 v( ). v( ) Q u() n v() r iffrnibl u / () lim 0 n v / () lim 0 lim () 0 u ( ) u ( ) v ( ) v( ) f( ) f() v( ) u Tus, f() is iffrnibl n v( ) u f / () / ( ) u( ) v { v( ) } / ( ) / ( ) u( ) v { v( ) } / ( )... () i.. ( f() ) i.. v( ) u u() v( ) u v() / ( ) u( ) v { v( ) } / ( ) u( ) v ( v( ) ) Illusriv mpls : Empl : Fin rivivs of following funcions wi rspc o. / ( ) / ( ) ) b) log c) sin Soluion : ) ( ) ( ) ( ).. () 5 Businss Mmics (Block )

19 Driviv of Funcions Uni 0 b) ( log) log ( ) ( log) log.. log c) ( sin ) sin ( ) sin ( ) ( sin ) sin. sin. cos (sin sin cos) Empl : Fin rivivs of following wi rspc o. ) Soluion : ) cos b) log n log log cos log c) (cos) cos ( log ) ( log ) log ( sin ) cos. sin log sin log ( log ) cos. ( log ) ( log ) log cos log log log. b) n log log n n ( ) ( log ) ( log ) log. n n ( log ) n. Businss Mmics (Block ) 55

20 Uni 0 Driviv of Funcions c) n n log ( log ) n ( ) ( ) ( ) ( ).. ( ) ( ) ( ) Empl : Fin rivivs of following funcions wi rspc o. ) log b) sc c) ) sin log Soluion : ) ( log ) ( ) ( log ) ( ) log ( ) (log) () log ( ). b) 6 log ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) 56 Businss Mmics (Block )

21 Driviv of Funcions Uni 0 ( ) (sc ) sc ( ) ( ) ( ).sc n sc ( ) ( ) sc { ( ) n } ( ) sc c) ) sin log sin ( ) log sin ( ) log sin ( ) ( ) ( log ) ( log ) log. sin..cos ( sin cos) Empl : If Soluion : W v y ( 0) ( ). ( log ) ( log) ( log ) y y, prov y( y) 5 y 5 5 ( 5) () ( 5) ( 5) ( 5). ( 0) ( 5) ( 5) 5 Businss Mmics (Block ) 57

22 Uni 0 Driviv of Funcions y 5... () ( 5) Also, y ( y) ( ) 5 From () n (), w g y y( y) () CHECK YOUR PROGRESS ) Fin rivivs of following funcions wi rspc o. n ) b) c) n ) 7 log ) log n f) ) If y sin prov cos ) If f(), fin f / (). g) log ( ) y cos 0.7 Driviv of composi funcions : If y φ () n f(), n y φ (f()) is composi funcion. Empls of composi funcions r cos, sin, log cos, sin c. Torm.7. : L y φ () wn f(). Tn y y. Proof : L k b incrmn in corrsponing o incrmn in. Q f()... () k f ( )... () () () k f() f()... () 58 Businss Mmics (Block )

23 Driviv of Funcions Uni 0 Also, 0 givs k or k 0 Now, φ (f()) φ () φ (f()) φ (k) φ (f()) φ (f()) φ (k) φ () φ ( f( ) ) φ( f() ) φ( k) φ() k. k φ( k) φ() f( ) f()., by () k Tking limi on bo sis, w g lim 0 φ ( f( ) ) φ( f() ) φ lim k 0 ( φ ( f ( )) ) ( φ( ) ). ( f ( ) ) φ( y ( ) ) ( y). ( ) y. Gnrlision of bov orm : L yf(u), ug(), () n ( k) φ( ) k f. lim 0 ( ) f( ) y y u.. u Tus, if composi funcion y is funcion of mny funcions, n y ( ( s n funcion) funcion) (. ( Abov is lso known s cin Rul. Illusriv mpls : n r funcion) funcion) (ls... funcion) Empl : Fin rivivs of following funcions wi rspc o. ) cos sin b) c) sc. Soluion : ) L ( cos ) ( cos ) Businss Mmics (Block ) cos. ( ) sin. 59

24 Uni 0 Driviv of Funcions b) L sin y sin By cin rul sin, wr u, wr u sin y y. u u. u ( ) (sin). ( ) u u cos. sin.cos sin cos c) L y sc sc u, wr u, wr scu By cin rul y y. u u. u ( ). ( sc u ). ( ) scu nu. 9sc u. scu nu 9sc n. Empl : Fin rivivs of following funcions wi rspc o ) y log( ) b) ylog log log c) ycos ) y co (m n b) Soluion : ) L ylog By cin rul, y y. ( log ). 60 Businss Mmics (Block )

25 Driviv of Funcions Uni 0.. ( ) b) y log log log L v log log w log y y. v v w w.. (log v). (log w). (log ). ( ) v w... v w loglog.log. (loglog )(log ) c) y cos cos u,wr u By cin rul, y y. u u ( cos u) ( ) u. u Businss Mmics (Block ) 6

26 Uni 0 Driviv of Funcions ) y co (mn b) co (m n u), wr u b, co v, wr v mn u, w, wr w co v. By cin rul, w g, y y. w w v u.. v u v u (w). ( co v ). ( mn u ). ( b) w m....b. v u mb { ( m n b) }( b ) Empl : Prov riviv of n vn funcion is lwys n o funcion. Soluion : L f b n vn funcion. Tn f() f( ) f() / / f ( ) f ( ) f( ) ( ) / f ( ).( ) f / ( ) f / ( ) f / () f / is n o funcion. sin Empl : If y,prov y ( ) y Soluion : y sin... () y. sin. ( ) 6 Businss Mmics (Block )

27 Driviv of Funcions Uni 0 sin. y y ( ) y CHECK YOUR PROGRESS ) Fin rivivs of following funcions wi rspc wi rspc o. ) y log (sc n ) b) y sin(cos ) c) y log y ) Fin of following. ) y log b c b) y c) y log (. ) ) y sin. log ) Prov riviv of n o funcion is lwys n vn funcion. 0.0 LET US SUM UP y poin on givn curv y f() rprsns grin of ngn lin poin. y For ngns prlll o is, 0, (k) 0 (k cons n) n ( ) n n Businss Mmics (Block ) 6

28 Uni 0 Driviv of Funcions ( ) (log ) (sin ) (cos ) (n ) cos sin sc (co ) cosc (sc ) (cos c) sc.n cos c.co ± (f ± f f ) u (uv) v. f v u. u v v. u. u, v 0 v v If y f(u), u g(), (), n y y u.. u If u n v r wo funcions of suc uf() n v g(), n u Driviv of u wi rspc o v Driviv of v wi rspc o 6 Businss Mmics (Block )

29 Driviv of Funcions Uni 0 0. ANSWERS TO CHECK YOU PROGRESS Cck your progrss -. ) b) sin ( ) ( ) ( ) ( ) 0 ( ) ( sin ) ( ) ( ) cos ( ) cos 5 c) ( cos c ) ( cos c) ( ) coscco log ) ( ) 0 ( ) ( ) ( ) ) { log( )} ( log log log ) ( log log log) ( log) ( log) ( ) ( log ) 0 Businss Mmics (Block ) 65

30 Uni 0 Driviv of Funcions f) ( sin cos ) ( sin ) ( cos ) ( ) ( ). Givn, y ( ) y ( ) ( ) 0 for R ( sin) log 0 sin log y Tr os no is ny vlu of for wic 0 No ngn o curv y will b prlll o sis. Cck your progrss - ) ) ( ) ( ) ( ) ( ) ( ) ( )( 0) ( )( 0) ( ) ( 6 8) ( 6 9) ( ) ( ) b) n n ( n ) ( n ) ( n ) ( n ) ( n ) 66 Businss Mmics (Block )

31 Driviv of Funcions Uni 0 ( )( ) ( )( sc ) n sc n ( n ) sc ( n n ) n ( ) c) sc ( ) n ( ) ( ) ( ) ( ) ( ). ( ) ( ) ( ) ( ) ) ( 7 log ) ( 7 log) 7 ( ) ( log) 7 ( ) ( ) log ( ) ( log) ) [ log n ] 7[.. log] log.. 7 ( log ) log ( log n ) ( ) ( ) n ( log) log ( n ) ( ) ( ) 0 Businss Mmics (Block ) 67

32 Uni 0 n. log.sc [.. ] Driviv of Funcions n log sc [ ] f) ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) 0 ( ) ( ) ( ) ( ) ( ) g) [ log( )] [ log log log ]. y sin cos y [ log log log] ( log) () log ( ) [ Q log ] log sin cos ( cos ) ( sin ) sin ( cos ) ( cos ) ( cos )( cos ) sin ( sin ) ( cos ) 68 Businss Mmics (Block )

33 Driviv of Funcions Uni 0 cos cos sin cos cos ( ) cos ( ) cos. Givn, f() f / (). f / () () ()() Cck your progrss - ) ) y log(scn) By cin rul log wr sc n y y. ( log ) ( sc n ) Businss Mmics (Block ) 69

34 Uni 0 ( sc n sc ) sc (n sc ) sc n sc. b) y sin(cos) sin,wr cos By cin rul y y. cos ( sin) cos(cos). sin ( sin ) (cos ) c) y log By cin rul, log,wr y y. ( log ). ( ). ) y log y log, wr By cin rul Driviv of Funcions 70 Businss Mmics (Block )

35 Driviv of Funcions Uni 0 y y. (log).. b c b) y y, wr bc By cin rul, y y. ( ). ( b c) (b) ( b c ).( b) c) y log (. ) log log log y ) y sin log y ( log) ( sin log) ( ).sin log log ( sin) sin ( log) sin log log.cos. sin. sin log log.cos sin Businss Mmics (Block ) 7

36 Uni 0 Driviv of Funcions. L f b n o funcion. Tn f( ) f() f ( ) { f() } f / / ( ) ( ) f ( ) f / ( ).( ) f / () f / ( ) f / () f / ( )f / () f is n vn funcion. 0. FURTHER READING ) Srm, A.K.,T Book of Diffrnil Clculus for B.A,B.Sc, B.Com,I.A.S.,P.C.S., Nw Dli,Discovry Publising Hous,00,s iion. ) Hbib, Ambr.,T clculus of finnc, Hyrb,Univrsiis Prss Ini Pv. L.0,s iion. ) Gup, Suir.,Bsics of Diffrnnil Clculus,Nw Dli,Anmol publicion Pv. L,0, s iion. ) Mron, I.A. Problms in clculus of on vribl,nw Dli,CBS publisrs & Disribuors Pv L.99,s iion. 0. MODEL QUESTIONS () Diffrni following wi rspc o () 5 (b) b (c) sc () log () Businss Mmics (Block )

37 Driviv of Funcions Uni 0 (f) sin n cos 9 (g) ( ) () Fin iffrnil cofficin of following wi rspc o () (b) ( b ) (c) log( ) () log( ) () Diffrni following wi rspc o () ( sin ) cos (b) ( n ) log sin (c) ( log ) ( sin ) sin () ( ) y () Fin if sin cos sin () ( ) ( ) y sin cos (b) y by 0 (c) y log y () y y 0 *** ***** *** Businss Mmics (Block ) 7

38 UNIT : APPLICATION OF DERIVATIVES UNIT STRUCTURE. Lrning objcivs. Inroucion. Mimum n minimum of funcion.. Mos for fining mimum n minimum.5 Applicion of mim n minim.6 L us sum up.7 Answrs o cck your progrss..8 Furr Rings.9 Mol Qusions. LEARNING OBJECTIVES Afr going roug is uni, you will b bl o fin mimum n minimum vlu of funcion rmin poins on givn curv wr rmum is, if ny. INTRODUCTION In is uni, w sll iscuss mimum n minimum of funcion. Also, w lso iscuss mos of fining mimum n minimum vlus of funcion. Lsly, w will iscuss pplicion of mim n minim.. MAXIMUM AND MINIMUM OF A FUNCTION A funcion f() is si o v mimum if f() is grr n vry or vlu ssum by f() in immi nigbouroo of. Tis mns f() > f() n f() > f( ) for sufficinly smll posiiv. (S fig..) Similrly, funcion f() is si o v minimum b if f(b) is 7 Businss Mmics (Block )

39 Applicion of Drivivs Uni smllr n vry or vlu ssum by f() in immi nigbouroo of b. Tis mns f(b) < f(b) n f(b) < f(b ) for sufficinly smll posiiv (S fig.5). Gomriclly, funcion f coninuous poin s mimum or minimum poin ccoring s grp of f s pk or cviy poin. No : i) T mimum n minimum vlus of funcion r lso known s locl (rliv) mim or locl (rliv) minm s s r grs n ls vlus of funcion rliv o som nigbouroo of poin in qusion. T rm rmum is lso us for bo mimum n minimum vlu. ii) A locl mimum vlu poin my b lss n locl minimum vlu nor poin. iii) A funcion cn v svrl mimum n minimum vlus. T mimum n minimum vlus of coninuous funcion occur lrnivly. iv) A mimum, funcion f() cngs from incrsing o crsing s n minimum funcion f()cngs from crsing o incrsing s... Mos of fining mimum or minimum Mo I : y ) Fin y ) Solv quion 0 for rl vlus of. L s vlus b, b, c c. y ) Consir. Suy sign of for vlus of sligly lss n n sligly grr n. y ) If cngs sign from posiiv o ngiv, n f() is mimum. Businss Mmics (Block ) 75

40 Uni Applicion of Drivivs y If cngs sign from ngiv o posiiv, n f() is minimum. y If os no cng sign, n is poin nir of mimum nor of minimum. Mo II : Sps I n r sm s in mo I. y. Fin n pu vlus. Consir. y i) If <0, n f() is mimum. y ii) If >0, n f() is minimum. y y iii) If 0 bu 0, n r is nir mimum nor minimum. Illusriv Empls Empl : Fin for wic f()(5 ) is mimum. W is mimum vlu of f()? Soluion : Givn, f()(5 ) f() 5 f / () 5 For on rmum, w v f / () Also, f // () ( 5 ) f // 5 < 0 76 Businss Mmics (Block )

41 Applicion of Drivivs Uni 5 f() is mimum n mimum vlu 5 f Empl : Fin vlus of Mimum n Minimum of y 6 0. Soluion : Givn y 6 0 y ( ) 6 6 For mimum or minimum y 0 6( 6) 0 ( )( ) 0 6 or y Now, 6, ( ) A A y y, ( ) 6 0 < 0 > Tus y is mimum poin n is minimum poin. Minimum vlu of y ( ) ( ) 6( ) n Mimum vlu of y ( ) ( ) 6( ) 0 Businss Mmics (Block )

42 Uni Applicion of Drivivs Empl : Sow funcion y s mimum vlu n minimum vlu. Soluion : Givn y y 9 ( ) y For mimum n minim, 0 y Now, y A, < 0 y A, > 0 ( ) 0 ( )( ) 0, Tus y is mimum n mimum vlu is ( ) 6( ) 9( ) 8 - y is minimum n minimum vlu is ( ) 6( ) 9( ) 8 8 CHECK YOUR PROGRESS ) Fin mimum or minimum vlus of following funcions. ) 6 b) c) 5 78 Businss Mmics (Block )

43 Applicion of Drivivs Uni. APPLICATION OF MAXIMA AND MINIMA TO PROBLEMS T ory of mim n minim fins pplicions in prcicl lif. For mpl, on s o fin ou rlion bwn rius n ig of conicl n wic my v ifini cpciy n sm im us minimum moun of clo. In solving suc yp of problms w fin ou funcion wos mimum or minimum s o b foun ou. If i involvs wo vribls, n by limining on vribl from givn, funcion is ruc o on ving only on vribl. Empl : T sum of wo numbrs is consn. Prov ir prouc is mimum wn c numbr is lf of ir sum. Soluion : L wo numbrs b n y. Tn yconsn c, sy y c... () L P b prouc of numbrs. Tn P y P (c ), by () P c P c For mimum or minimum vlu of P, w v P 0 c 0 P c Now, ( c ) < 0 P is mimum wn c Also, wn c, w g from () y c Hnc P is mimum wn y c i.., prouc is mimum wn c numbr is lf of ir sum. Businss Mmics (Block ) 79

44 Uni Applicion of Drivivs CHECK YOUR PROGRESS ) Fin wo numbrs wos sum is 0 n prouc is mimum..5 LET US SUM UP y rprsns r of cng of y wi rspc o. T vlus of obin on solving f / ()0 r cll sionry or criicl poins. A funcion v locl mimum or minimum s poins ccoring f // ()<0 or f // ()>0 rspcivly..6 ANSWERS TO CHECK YOUR PROGRESS Cck your progrss -. () L f() 6 f / () 6 For mimum or minimum, w v f / ()0 60 Now, f // () ( 6) f // () 80 Businss Mmics (Block )

45 Applicion of Drivivs Uni f // > 0 f() is minimum wn n minimum vlu is f (b) L f() f ( ) / For mimum or minimum, w v f / () ()( )0, Now, f // () 0 // f ( ) < 0 f() is mimum, n mimum vlu is f( ). Agin, f // () > 0 f() is minimum n minimum vlu is f() Businss Mmics (Block ) 8

46 Uni Applicion of Drivivs. (c) L f() 5 f / () 0 For mimum or minimum, w v f / () ( 5) 0 ( )( )0, Now, f // () ( 0 ) f // < 0 f() is mimum wn n mimum vlu is f Also, f // () 0 8 > 0 f() is minimum wn n minimum vlu is f() Businss Mmics (Block )

47 Applicion of Drivivs Uni Cck your progrss -. L wo numbrs b n y. Accoring o qusion, y 0 y 0... () L P b prouc of numbrs P y P (0 ), using () P 0 P 0 For mimum or minimum of P, w v P P Now, ( 0 ) < 0 P is mimum wn 0 Also, 0 givs y0 T numbrs r 0 n 0..7 FURTHER READING ) Srm,A.K.,T Book of Diffrnil Clculus for B.A,B.Sc,B.Com,I.A.S.,P.C.S.,Nw Dli,Discovry Publising Hous,00,s iion. ) Hbib,Ambr.,T clculus of finnc,hyrb,univrsiis Prss Ini Pv.L.0,s iion. ) Gup,Suir.,Bsics of Diffrnnil Clculus,Nw Dli,Anmol publicion Pv. L,0,s iion. Businss Mmics (Block ) 8

48 Uni Applicion of Drivivs ) Mron,I.A. Problms in clculus of on vribl,nw Dli,CBS publisrs & Disribuors Pv L.99,s iion..8 MODEL QUESTIONS ) Fin poins wic following funcions v mimum n minimum vlus : () y (b) y 6 *** ***** *** 8 Businss Mmics (Block )

49 UNIT : INTEGRATION AND METHODS OF INTEGRATION UNIT STRUCTURE. Lrning Objcivs. Inroucion. Inrgrion of Simpl Funcions. Propris of Infini Ingrls.5 Mos of Ingrion.5. Ingrion by Subsiuion.5. Ingrion by Prs.6 L Us Sum Up.7 Furr Rings.8 Answrs o cck your progrss.9 Mol Qusions. LEARNING OBJECTIVES Afr going roug is uni, you will b bl o unrsn bou ingrion of simpl funcions from viw poin of invrs of iffrniion know propris of infini ingrls unrsn vrious mos of ingrions viz mo of subsiuion ingrion by prs. INTRODUCTION T vlopmn of ingrl clculus cn b viw from wo iffrn ircions wic r rl o solving of problms of following wo yps. (i) o fin funcion wn is riviv is known (ii) o fin r boun by curv (or grp) of funcion unr crin givn coniions. Businss Mmics (Block ) 85

50 Uni Ingrion n Mos of Ingrion T procss rl o firs cgory ls o vlopmn of infini ingrion n or ls o fini ingrion. T wor ingrion lirlly mns summion. I is, in fc, procss of fining limi of sum of crin numbr of lmns, s numbr of lmns ns o infiniy n c of lmns bcoms infinisimlly smll. In is uni, w sll confin ourslvs o suy of infini ingrl incluing som procss of ingrion.. INTEGRATION OF SIMPLE FUNCTIONS W v lry lrn ow o fin riviv of funcionl of fin in n inrvl. If f ( ) y is funcion n is riviv is no y by f / ( ) or. Now, r in is uni, w will lrn ow o fin f wnvr / f is givn. Ingrion is invrs procss of iffrniion, i.., o fin funcion wn iffrnil cofficin is givn. L us consir following mpls. W know (i) n (ii) ( sin ) cos Hr w sy, in cs of (i) ingrl or primiiv) of. Similrly, in (ii) cos. L us k wo or mpls : is n niriviv (or n sin is n niriviv of n ( iii) C ( iv) ( sin C) I is vin from (iii), cos Hr C is ny rl numbr C is lso n niriviv of n in (iv) sin C is lso n niriviv of cos. Trfor, w v 86 Businss Mmics (Block )

51 Ingrion n Mos of Ingrion Uni unrsoo niriviv of funcion is no uniqu. Dfiniion : T s of funcions coul possibly v givn funcion s riviv r cll nirivivs (or ingrls or primiivs) of funcion. Mor gnrlly, if r is funcion φ ( ) fin in n inrvl I φ I, n for ny rl numbr C suc [ ( ) ] f ( ), [ φ( ) C] f ( ), I. Tus φ( ) C nos fmily of nirivivs of f ( ). W inrouc symbol ( ) f wic will rprsn wol clss of ni-rivivs, i.. f ( ) φ( ) C. T symbol f ( ) is r s infini ingrl of ( ) f wi rspc o. T procss of fining ni-rivivs is known s ingrion. T sign is cll ingrl sign. T symbol is n long S wic is kn from firs lr of sum. Som Bsic Rsuls Bs upon bov finiion n lso on som snr rivivs, w obin following ingrion formul.. Drivivs n n. ( ) cos n Ingrls (or ni rivivs) n n C, n sin cos sin C. ( cos ) sin. ( ) sc sin cos C n sc n C 5. ( ) ( n ) sc sc n sc n sc C 6. ( co ) cosc cosc co C 7. ( cosc) cosc co 8. ( ) cosc co cosc C C Businss Mmics (Block ) 87

52 Uni Ingrion n Mos of Ingrion Empls : Empl : Evlu ( i ) ( ii ) ( iii) Soluion : (i) 5 C C (using formul ) 5 ½ ½ ½ (ii) C (iii) C (using formul 9) log. PROPERTIES OF INDEFINITE INTEGRAL P Diffrniion n ingrion r invrs procsss of c or in sns of rsuls givn blow / ( ) f ( ) n ( ) f ( ) f C f C, C is consn. Proof : L φ ( ) b ny niriviv of f ( ) i.. [ φ( ) ] f ( ) C Tn f ( ) φ( ) f ( ) [ ( ) ] [ ( ) ] ( ) f ( ) φ C φ C (firs pr is prov) / Similrly, w know f ( ) f ( ) / Hnc ( ) f ( ) f C, C is ny rbirry consn. P- Two inicl ingrls wi sm riviv l o sm fmily of curvs n nc bo r quivln. Proof : L f ( ) g ( ) 0 Tn f ( ) g( ) or, [ f ( ) g ( ) ] 0 or, f ( ) g( ) or, f ( ) g( ) C C 88 Businss Mmics (Block )

53 Ingrion n Mos of Ingrion Uni I is cusomrily prss s ( ) g( ) consn of ingrion. P- [ f ( ) g( ) ] f ( ) g ( ) Proof : By propry P-, W v Also w know [ f ( ) g ( )] f ( ) g ( )... () [ f ( ) g( ) ] f ( ) g( ) f wiou mnioning ( ) g ( ) f (b) In viw of propry P-, from () n (b) w cn wri Corollry [ f ( ) g( ) ] f ( ) g( ) (A) [ f ( ) g( ) ] f ( ) g( ) (B) W cn gnrliz propry P- s blow - [ f ( ) g( ) ( )... φ( ) ] f ( ) g( ) ( )... φ( ) P- For ny rl numbr K, ( ) f ( ) K f K K, by propry P- Proof : W know f ( ) K f ( ) [ ] K f ( ) K f ( ) So by propry P- Also K f ( ) Illusriv Empls ( ) f ( ) K f K Empl : Evlu (i) (ii) ( cos ) Soluions : (i) (iii) ( sc ) (by propry P-) Businss Mmics (Block ) 89

54 Uni Ingrion n Mos of Ingrion C C C log C log C log C (ii) ( cos ) cos (Hr C C C C ) C sin C C C sin C sin C (iii) ( sc ) sc sc C n C n C C Empl : Evlu (i) 7 5 sin (ii) (iii) cos Soluion : (i) ( )( ) ( ) 90 Businss Mmics (Block )

55 Ingrion n Mos of Ingrion Uni ( ) (ii) C 7 5 (iii) C sin cos sin cos cos sc n sc C n sc C Empl : Wri n niriviv of cos. Soluion : W know ( sin ) cos or cos ( sin) sin niriviv of cos is sin C. CHECK YOUR PROGRESS - Q : Fin n niriviv of c of followings Q : Fin following ingrls (i) (i),( 0) (ii) (ii) ( cos sin ) sin (iii) ( b) Businss Mmics (Block ) 9

56 Uni Ingrion n Mos of Ingrion cos (iii) 5sin cos log log log (v) ( ) (vii) ( ) (viii) Q : Evlu (i) ( ) ( ) (iii) Q : If riviv of ( ) n fin f ( ). sin cos (iv) sin cos (vi) cos m m m m (ii) f w.r.. is n ( ) f π, Q 5: T grin poin (,y) of curv is n curv psss roug poin (, 7). Fin quion of curv..5 METHODS OF INTEGRATION In our rlir iscussion, w v consir ingrls of os funcions wic r rily obinbl from rivivs of som snr funcions. Tis mo is bs on inspcion, i.. i is pnn on src of funcion F wos riviv is f wic l us o ingrls of f. Bu ingrls of crin funcions cnno b obin ircly if y r no in on of snr forms. Hnc w n o vlop som or cniqus or mos for fining ingrls by rucing givn funcions o snr forms. Som of mos r (i) Ingrion by subsiuion, (ii) Ingrion by prs,(iii) Ingrion by pril frcions..5. Ingrion by Subsiuion T mo of vluing n ingrl by rucing i o snr form by propr subsiuion is cll ingrion by subsiuion. Suppos φ() is iffrnibl funcion. Tn o vlu / ingrls of form f ( φ( ) ) φ ( ), w subsiu φ( ) wic 9 Businss Mmics (Block )

57 Ingrion n Mos of Ingrion Uni / n givs φ ( ).. o form f ( ) If now f ( ) Tis subsiuion rucs bov ingrl is snr form n w cn fin ingrls. Finlly w rplc by wi lp of bov subsiuion. I is imporn o guss w will b usful subsiuion. Usully w mk subsiuion for funcion wos riviv lso occurs in ingrn. A fw mpls r givn blow o illusr mo. Illusriv Empls Empl : Evlu ( b) Soluion : L or n b, n n n n n n ( b) Empl : Evlu n ( ) ( b ) n C sin Soluion : L, n, or sin Now sin ( cos) C cos Empl :Evlu n C θ sc θ θ Soluion : L nθ, n sc θ θ Now 5 5 n θ sc θ θ C n θ log Empl : Evlu Soluion : L 0, n ( 0 log 0 0 ) C C Businss Mmics (Block ) 9

58 Uni Ingrion n Mos of Ingrion log 0 0 Now log C log( 0 ) C 0 0 A Fw Spcil Rsuls. f f / ( ) ( ) log f ( ) C / Proof : L f ( ) z, n f ( ) z Trfor, / ( ) ( ) f z log z C log f ( ) C f z Tis rsul cn b ppli in cs of followings () n log sc C (b) co log sin C (c) sc log sc n C () cosc log cosc co C sin sin cos cos Proof () : n Proof (c) : log cos C log sc C ( sc n ) sc sc sc n sc sc n sc n log sc n C N.B. : (b) n () r lf for lrnrs Q. Fin ingrls. 9 n CHECK YOUR PROGRESS- ( n ) sin. ( ). cos 7. ( b sin ) 0. ( )( log ) Businss Mmics (Block )

59 Ingrion n Mos of Ingrion Uni cos 5 ( ) ( ). ( ) b cos 8. sin 9. cos 0.. sin( ) sin( b). sin( ) cos( b) sin sin sin. ( ) 5. ( ).5. INTEGRATION BY PARTS An imporn ingrion formul cn b from riviv of prouc of wo funcions. W know from iffrnil clculus for wo iffrnibl funcions f ( ) n ( ) q, { f ( ). q( ) } f ( ). { q( ) } q( ). { f ( ) } Ingring bo sis w.r.. w g f ( ). q( ) f ( ). { q( ) } q( ). { f ( ) } f ( ) { q( ) } q( ). { f ( ) }....() or f ( ) { q( ) } f ( ) q( ) q( ). { f ( ) } Now l us k f ( ) u n { q( ) } v. Tn q( ) v. Businss Mmics (Block ) 95

60 Uni Ingrion n Mos of Ingrion If w subsiu r in () n w g v uv u v ( u). T bov formul cn b s s blow T ingrl of prouc of wo funcions s funcion ingrl of n funcion ingrl of [iffrnil cofficin of s funcion ingrl of n funcion] No : Wil pplying rul of ingrion by prs for prouc of wo funcions, i is vry imporn o coos firs n scon funcion vry crfully. As n i o lrnrs, w cn coos firs funcion s funcion coms firs in wor ILATE. Hr lrs sn for s givn blow I Invrs rigonomric funcion L Logrimic funcion A Algbric funcion T Trigonomic funcion E Eponnil funcion. ILLUSTRATIVE EXAMPLES Empl : Evlu Soluion : ( ) Hr w v kn s funcion n funcion ( ). {. } ( ) C ( ) C Empl : Evlu sin Soluion : sin sin ( ). sin 96 Businss Mmics (Block )

61 Ingrion n Mos of Ingrion Uni cos cos. cos cos sin cos C cos sin C 9 Empl : log Soluion : log log. Now k s funcion log n funcion n proc. Similrly in cs of sin w k sin n s scon funcion. s firs funcion ILLUSTRATIVE EXAMPLES Empl : Evlu cos Soluion :L I cos ( ) cos. cos sin sin ( ). sin sin sin sin cos cos ( sin cos ) I ( sin cos ) I I ( sin cos ) C Businss Mmics (Block ) 97

62 Uni Ingrion n Mos of Ingrion CHECK YOUR PROGRESS - Q. Evlu followings. cos. log. n. sin cos 5. sin cos. ( ). log ( ) sin sin. log( ) cos 5. ( ) cos sin 7. cos 8. sin 9. cos 0. sin. sc ( n ). ( log ) ( )( 5 ) sin. ( cos sin ) 6. sin ( ) 9. ( ) ( ).5. INTEGRATION BY PARTIAL FRACTIONS.6 LET US SUM UP Diffrniion n ingrion r invrs procsss of on nor W my lso fin ingrion s procss of fining limi of sum [T mo is iscuss in n uni]. T s of funcions coul possibly v in givn funcon s riviv r cll ni rivivs (or ingrs or primiivs) of funcion. 98 Businss Mmics (Block )

63 Ingrion n Mos of Ingrion Uni Bs on rsuls of som snr rivivs, w obin formul of som snr simpl Ingrion. Tr r iffrn mos for rminion of ingrls.som of mos r (i) Ingrion by subsiuion,(ii) Ingrion by prs,(iii)ingrion by pril frcions. A fw imporn forms of ingrls (i) / ( ) ( ) f log f ( ) C f n / n (ii) { f ( ) } f ( ) [ f ( ) ] C, n n / (iii) f ( b) f ( b) C An imporn formul for prouc of wo funcions is v uv u v ( u)..7 FURTHER READINGS ) Ingrl Clculus, Kisn,Hri,Alnic Publisrs & Disribuors.Nw Dli ) Ingrl Clculus, Dmi,H.S. Nw g Inrnionl Publisrs,Nw Dli ) T book of Ingrl Clculus,Srm,A.K. Discovry Publising Hous,Nw Dli Businss Mmics (Block ) 99

64 Uni Ingrion n Mos of Ingrion.8 ANSWERS TO CHECK YOUR PROGRESS CHECK YOUR PROGRESS Ans o Q No : (i) W know ( log ), > 0 n ( log ( ) ) ( ), < 0 Combining bov,w g ( log ), 0 Trfor, log is ni-riviv of. (ii) W know ( cos) sin sin ( cos ) sin cos Trfor, ni -riviv of sin is cos. (iii) W know ( b) ( b ) ( b) ( b) ( b) ( b) Trfor, ni -riviv of ( b) is ( b) Ans o Q No : (i). ( 5 ). 00 Businss Mmics (Block )

65 Ingrion n Mos of Ingrion Uni C C. (ii) ( cos sin ) ( cos sin sin cos ) ( sin ) cos C. cos (iii) co cosc sc 5sin cos 5 co cosc sc 5 n 5 ( cosc ) C sin cos sin cos (iv) sin cos sin cos sin cos ( sc cosc ) n co C log log log log log log (v) ( ) ( ) ( ). C. log (vi) cos cos, cos cos. (vii) ( ) cos sin C. Businss Mmics (Block ) 0

66 C 5 5. C 5 5 (viii) m m m m m m m m C m m m m m m log log. Ans o Q No : (i) L.Tn ( ) ( ) ( ) C C 6 6 (ii) ( ) ( )( ) ( ) C. (iii) Ingrion n Mos of Ingrion Uni 0 Businss Mmics (Block )

67 Ingrion n Mos of Ingrion Uni 6 6 C C C. 7 f / Ans o Q No : W v ( ) f ( ) Wn, w v f ( ) π n log C π π C C 0 ( ) log n f. y Ans o Q No 5: Givn, f ( ) / π ( ) f / ( ) ( ) f. C ( ) C f... () () psss roug poin (,-7). 7. C log n C Businss Mmics (Block ) 0

68 Uni C 9 Rquir quion of curv is ( ) 9 CHECK YOUR PROGRESS Ans o Q No : L 9 8 Ans o Q No : L 9 8 log C 8 f. log 9 C 8 ( ) Ingrion n Mos of Ingrion ( ) C ( ) C Ans o Q No : L n Ans o Q No : W pu n C scθ nθθ n C scθ scθ nθθ scθ sc θ scθ nθθ scθ n θ θ scθ θ sc 0 Businss Mmics (Block )

69 Ingrion n Mos of Ingrion Uni Ans o Q No 5: L θ C sc 5 C 5 C C Ans o Q No 6: L Ans o Q No 7: L cos ( b sin ) C ( 5) C sinθ cosθθ cosθ θ sin θ sin θ cosθ θ sin θ cosθ cosc θθ coθ C co ( sin ) C bsin b cos cos b b b Businss Mmics (Block ) 05

70 Uni Ingrion n Mos of Ingrion. C b. C b. b b sin C Ans o Q No 8: Diviing numror n nominor by L ( ) log C, w v log C Ans o Q No 9: L ( n ) sin Ans o Q No 0: n ( )( log ) sin co C co ( n ) C L log ( log ) ( ) log ( log ) 06 Businss Mmics (Block )

71 Ingrion n Mos of Ingrion Uni C ( log ) C Ans o Q No : L I sin I C In I,w subsiu I I sin C. Ans o Q No : L 7 ( 6 ) 7 ( ) ( ) C C C 7 C Ans o Q No : L (. ) ( ) Businss Mmics (Block ) 07

72 Uni ( ) sc cos n C n ( ) C ( ) cos Ans o Q No : Muliplying n iviing by,w v. ( ) ( ) ( ). 5 (. ) Ingrion n Mos of Ingrion 5 5 L 5 5 ( ) 5 ( ) ( ) C C C 08 Businss Mmics (Block )

73 Ans o Q No 5: 5 ( )( ) ( ) C Ans o Q No 6: ( ) ( )( ) b b b b ( ) ( ) ( ) b b ( ) ( ) ( ) b b b b { } b b b { } v v uu b wr u n v b u n v C v u b C v u b ( ) ( ) ( ) C b b Ans o Q No 7: ( ) cos cos cos [ ] cos cos cos cos Ingrion n Mos of Ingrion Uni 09 Businss Mmics (Block )

74 Uni Ingrion n Mos of Ingrion cos cos cos cos cos cos cos 8 8 sin sin C 8 Ans o Q No 8: From iniy sin sin sin sin sin sin sin sin sin cos cos C Ans o Q No 9: From iniy, cos cos cos cos cos Ans o Q No 0: L Ans o Q No : ( cos ) cos sin C ( ) 5 log C 5 7 ( 5 ) C log 7 W v sin ( ) sin ( b) sin ( b) sin sin {( b) ( ) } ( ) sin ( b) 0 Businss Mmics (Block )

75 Ingrion n Mos of Ingrion Uni Ans o Q No : sin sin co ( b) { co( ) ( b) } sin sin { log sin ( ) log ( b) } C ( b) ( ) ( b) sin cos c ( b) log C. sin cos( b) ( ) cos( b) cos( b) sin ( ) cos( b) cos{ ( b) ( ) } cos( b) sin ( ) cos( b) cos( ) cos( b) sin ( ) sin ( b) cos( b) sin ( ) cos( b) { co( ) n( b) } cos ( b) [ log sin ( ) log ( b) ] C ( b) cos cos cos ( b) sin log cos ( ) ( b) C. Ans o Q No : W v sin sin sin ( sin sin ) Ans o Q No : ( ) sin ( cos cos) sin ( sin cos sin cos) ( sin sin sin 6) cos cos cos6 C. 6 ( ) ( ) Businss Mmics (Block )

76 Uni Ingrion n Mos of Ingrion Ans o Q No 5: L ( ) ( ) z z pu z z z log z C log z C z log C ( 6 ) ( ) C z z. C. C. ( ) CHECK YOUR PROGRESS Ans o Q No : ( ) C. I n C 9 () 6 Ans o Q No 7: W v 6 5 ( ) 6 5 L,Tn 6 5 ( ) Businss Mmics (Block )

77 C log C 5 log. Ans o Q No 8: W v ( ) ( ) () L 5 5 From (), ( ) ( ) C log. 5 C log 5 C log Ans o Q No 9: W v ( ) { } Ingrion n Mos of Ingrion Uni Businss Mmics (Block )

78 5 C 5 sin C 5 sin. Ans o Q No 7: W v ( ) Now ( ) ( ) C sin C sin. Ingrion n Mos of Ingrion Uni Businss Mmics (Block )

79 Ingrion n Mos of Ingrion Uni Ans o Q No 8: L A ( 5 6) B ( ) B A 5 Compring cofficins of lik powrs of, w g A n 5 A B A n B 5 6 ( 5) 5 6 ( 5) wr log 5 6 C log 5 6 C Ans o Q No 9: W pu, so ( ) ( ). n n C [ ] C Ans o Q No 0: W pu, so. Also, ( ) Businss Mmics (Block ) 5

80 Uni Ingrion n Mos of Ingrion [ ( ) ] ( ) ( ). C C CHECK YOUR PROGRESS Ans o Q No : cos C sin sin [ ( cos ) ( cos ) ] sin sin cos sin ( ) sin cos Ans o Q No : log log log. log log log... C log C. 6 Ans o Q No : n n n. 6 Businss Mmics (Block )

81 Ingrion n Mos of Ingrion Uni n n Ans o Q No 6: ( ) n ( ) ( ) [ ( ). ( ) ] ( ) Ans o Q No 9: L sin,tn sin cos sin cos sin. ( sin ) sin cos C sin sin C sin C Ans o Q No : W subsiu,w g cos cos Ans o Q No 7: W pu cos [ sin sin ] [ sin ] C cos [ sin ] C cos nθ,so sc θθ θ sc θ n cos cos θ sc n ( cosθ ) sc θθ θθ θθ [ θ nθ.nθ ] [ θ nθ logsc ] C θ θ [ n log ] C. Businss Mmics (Block ) 7

82 Uni Ingrion n Mos of Ingrion I sin cos cos Ans o Q No 8: L ( ) I cos ( sin cos ) I I ( sin cos ) cos sin sin sin I Ans o Q No 0: L I sin sin ( cos cos) cos cos ( cos sin ) ( cos sin ) 0 C ( cos sin ) ( cos sin ) C sin cos sin cos Ans o Q No : W v ( ) sin cos cos sin C Ans o Q No : L I sc ( n ) ( sc sc n ) / Also, l sc f ( ) sc n f ( ) W know, { f ( ) f ( ) } f ( ) C / I sc C. Ans o Q No : L log,n ( sin ) cos ( cos) I cos cos I cos sin I I cos sin I [ sin sin ] cos ( sin cos) C Hnc, sin(log ) { sin( log) cos( log) } C 8 Businss Mmics (Block )

83 Ans o Q No : L ( ) B A, A n B r consns. ( ) B A On compring cofficins,w g A A Also, B A B 7 B ( ) ( ) 7 ( ) ( ) 7 ( ) ( ) 7 ( ) sin. 7. ( ) C sin MODEL QUESTIONS A. Evlu.. 5. cos sin n. 5 Ingrion n Mos of Ingrion Uni 9 Businss Mmics (Block )

84 Uni Ingrion n Mos of Ingrion B. Ingr following cos. sin. ( ) sin.. cos 5 ( log ) 5. cos 6. sin cos n ( ) C. Evlu. n 7. 5 cos. sin sin. sin 0 5. log 6. log 7. cos cos 0.. ( ). sin. 5 8 D. Evlu cos ( )( ) ( ) ( )( ) 8.. ( )( ) ( )( ) *** ***** *** 0 Businss Mmics (Block )

85 UNIT : LINEAR PROGRAMMING PROBLEM (LPP) UNIT STRUCTURE. Lrning Objcivs. Inroucion. Mning. Coniions for Using LPP Tcniqu.5 Bsic Assumpions of LPP.6 Ars of pplicions of LPP.7 Limiions of LPP.8 Gnrl Linr Progrmming Problm.9 Formion of LPP of Two Vribl.0 Som Dfiniions. Grpicl Mo of Soluion of LPP. L Us Sum Up. Furr Rings. Answrs To Cck Your Progrss.5 Mol Qusions. LEARNING OBJECTIVES Afr going roug is uni, you will bl o- unrsn mning of linr progrmming problm (LPP) know vrious rms ssoci wi LPP know bou vrious fil of pplicion of LPP rw grp of linr inquliis lrn ow o fin fsibl rgion of LPP of wo vribl from grp obin opiml soluion of n LPP of wo vribl from grp.. INTRODUCTION T cnrl m of conomic ory n mngmn scinc is Businss Mmics (Block )

86 o opimis us of scrc rsourcs ir isposl, wic inclu mcin, mn-powr, mony, rw mrils, wrous spc c. Tr r svrl oricl ools o ccomplis is purpos in bo srm. Bu suc ools r no qu for ring compl conomic problm wi svrl lrnivs c wi is own rsricions n limiions. I is for ckling suc problms us of linr progrmming s bn foun o b mos usful.. MEANING T wor progrmming mns plnning n rfrs o procss of rmining priculr pln of cion from mongs svrl lrniv. T wor linr mns rlions involv mong wo or mor vribl in priculr problm r linr (i.. inics of ll vribls involv mus v uni powr only). Linr Progrmming is mmicl cniqu for rmining opiml llocion of rsourcs n obining priculr objciv (i..) cos minimizion or profi mzimizion) wn r r lrniv uss of rsourcs lik ln, lbour, cpil, mrils, mcins c. No : (i) T vribls involv in LPP r cll cision vribls. (ii) (iii) T linr funcions wic is o b opimiz is cll Objciv Funcion. T inquliis or quions involving cision vribls of n LPP wic scrib crin rsricions (or coniions) unr wic objciv funcion is o b opimiz r cll consrins. Signs,, r us in consrins.. CONDITIONS FOR USING LPP TECHNIQUE LPP cniqu my b us for opimizion problms subjc o fulfillmn of following coniions :. Tr mus b wll-fin objciv funcion rling o profi, cos or quniy prouc.. Tr mus b s of linr consrins rling o inmn of Businss Mmics (Block )

87 objciv.. All cision vribls mus ssum non-ngiv vlus.. Tr mus b n lrniv cours of cion ling o lrniv soluions for purpos of coosing bs on..5 BASIC ASSUMPTIONS OF LPP T following four ssumpions r m in LPP :. Proporionliy : I is ssum proporionliy iss in objciv funcion n in consrins.. Criniy : All co-fficins in objciv funcion n consrins r complly known wi criny n o no cng uring prio bing s.. Divisibiliy : Tis ssumpion ss cision vribls cn k ny non-ngiv vlus i.. frcionl vlus of cision vribls r prmi.. Aiiviy : W ssum iiviy of ll civiis wic mns ol civiis qul sum of ll civiis..6 AREAS OF APPLICATIONS OF LPP Hisoriclly, fil of linr progrmming ws vlop by Gorg B. Dnzig n is ssocis uring Worl Wr II n ws firs us by U.S. Air Forc s n i in cision mking. Toy i s bn ppli o wi vriy of problms. Som of s r : (i) Mnufcuring Problm : To fin numbr of ims of c yp soul b mnufcur so s o mimiz profi subjc o proucion rsricions impos by limiion on us of mcinry, lw mril n lbour. (ii) Trnsporion Problm : To fin ls cosly wy of rnsporion from wrouss o cusomrs. (iii) Invsmn Problm : To fin moun o b invs in vriy of srs n bnurs so s o mimiz rrn on invsmn. (iv) Avrising Mi Problm : To fin opimum llocion of Businss Mmics (Block )

88 vrismns in vrious mi subjc o ol vrising bug o mimiz ffciv numbr of cusomrs. (v) Di Problm : To fin minimum rquirmn of nurins subjc o vilbiliy of foos n ir prics. (vi) Proucion Problm : To ci proucion scul coul sisfy fuur mns (ssonl or orwis) n minimiz cos in fc of flucuing rs n sorg (invnory) cos. (vii) Job Assigning Problm : To ssign job o workrs for mzimum ffcivnss n opimum rsuls subjc o rsricions of wgs n or cos..7 LIMITATIONS OF LPP Inspi of ving wi pplicbiliy of LPP, i s crin limiions lso. Som of wic r s follows : (i) (ii) (iii) In rl lif siuion, mny objciv funcion n consrins cn no b prss linrly. In prcicl siuions, i my no b possibl o s ll cofficins in objciv funcion n consrins wi criniy. Linr progrmming will fil o giv soluion if mngmn v mulipl gols ins of on gol..8 GENERAL LINEAR PROGRAMMING PROBLEM T mmicl smn (mol) of gnrl form of LPP my b wrin s follows : Opimiz (Mimiz or Minimiz) Z C C C n n Subjc o n n ( ) b n n ( ) b } () m m mn n ( ) b m Businss Mmics (Block )

89 n,, n 0 () Wr (i),, n r vribls wos vlus r o b rmin r cll cision vribl. (ii) T linr funcion z wic is o b mzimiz or minimiz is cll objciv funcion. (iii) T inquliis () r cll consrins. (iv) T s of inquliis () r known s non-ngiv rsricions. (v) T consns ij (i,, m, j,, n) r cll cnologicl cofficins. (vi) T consns bi (i,,..... m) r cll vilbiliy prmrs. (vii) T prssion (,, ) mns only on of rlionsip woul ol for priculr consrin. No : Hr w will rsric our iscussion on LPP of wo vribls only..9 FORMATION OF A LPP OF TWO VARIABLE T following sps r o b follow in formulion of LPP involving wo vribls : Sp I : Inify wo cision vribls n no m by n y. W v o rmin vlu of n y. Sp II : Inify objciv funcion n prss i s linr funcion of cision vribls n y i.. s z c c y wr c n c r consns. W v o mimiz z s profi (or proucion) or o minimiz z s cos. Sp III : Inify ll consrins n prss m s linr inquliy/ quions in rms of n y by using givn coniions. Sp IV : Impos non-ngiviy rsricions on n y i.. k () s cision vribls cn no b ngiv. Empl : A compny mnufcurs wo yps of sowpic m of plywoo. SHow pic of yp A rquirs 5 minus c for cuing n 0 minus c for ssmbling. Sowpic of yp B rquirs 8 minus for cuing n 8 minus c for ssmbling. Tr r ours n 0 minus vilbl for cuing n ours vilbl for ssmbling. T Businss Mmics (Block ) 5

90 profi is Rs. 0 for c sowpic of yp A n Rs. for of yp B. Giv mmicl formulion of bov LPP in rms of mzimizing profi. Soluion : L Numbr of sowpic of yp A y numbr of sowpic of yp B T informion suppli in givn problm my b summris in bulr form s follows : Typ of Numbr of Mnufcuring Tim (In minu) Profi Sowpic Sowpic Cuing Assmbling pr uni 6 Businss Mmics (Block ) (Rs.) A B y 8 8 Tol ours 0 ours 0 vilbl minus00 minus im profi, minus From bov bl, w obsrv s our objciv is o mimiz T objciv funcion is Z 0 y. W obrv ol im (in minus) rquir in cuing mcin for numbrs of sowpic of yp A n y numbrs of sowpic of yp B is 5 8y. Tus limiion of ours 0 minus ( 00 minus) mos on cuing mcin is prss by consrin. 5 8y 00 Similrly for ssmbling consrin w g 0 8y 0 Agin, numbr of sowpic mnufcur cnno b ngiv 0, y 0 Tus mmicl formulion of givn LPP is s blow: Mzimiz Subjc o consrins 5 8y y 0, y 0. Z 0 y

91 Empl : A i is o conin l 00 unis of crboyrs, 500 unis of f n 00 unis of proin. Two foos r vilbl : F wic coss Rs. pr uni, F wic coss Rs. pr uni. A uni of foo F conins 0 unis of crboyr, 0 unis of f n 5 unis of proin. A uni of foo F conins 5 unis of crboyr, 0 unis of f n 0 unis of proin. Fin minimum cos for i consiss of miur of s wo foos n lso minimum nuriion rquirmn formul problm s LPP. Soluion : L Numbr of unis of foo F o b purcs y Numbr of unis of foo F o b purcs. T informion suppli in givn problm my b summris in bulr form s follows : Typ of Numbr of Nurins Cos foo unis of Crboyr F Proin pr uni foo (Rs.) F F y Minimum rquirmn Hr w obsru our objciv is o minimiz cos T objciv funcion is Z y. T ol moun of crboyrs from foos F n F is 0 5y n is is rquir o b ls minimum moun of crboyrs rquir, nmly 00 unis. Trfor, w mus v 0 5y 00 Similrly, for f consrin, w g 0 0y 500 For proin consrin, w g 5 0y 00 Agin, unis of foo mnufcur cn no b ngiv. 0, y 0 Ts mmicl formulion of givn LPP is s blow. Minimiz Z y Subjc o consrins Businss Mmics (Block ) 7

92 0 5y y y 00 y 0.0 SOME DEFINITIONS I is ppropri is sg o giv som finiions prining o soluion of LPP. Dfiniion : A s of vlus of cision vribls wic sisfy consrins of gnrl LPP is cll soluion o LPP. Dfiniion : Any soluion o gnrl LPP wic lso sisfis nonngiv rsricions of problm is cll fsibl soluion. T s of ll fsibl soluions consius w is cll fsibl rgion. Dfiniions : Any fsibl soluion wic opimizs objciv funcion of gnrl LPP is cll n opimum soluion.. GRAPHICAL METHOD OF SOLUTION OF LPP Now, w r going o scuss grpicl mo for solving LPP involv only wo vribls. Tis mo is bs on orm. cll rm poin orm, wic ss s follows : Erm Poin Torm : n opimum soluion o LPP, if i iss, occurs on of cornr (or rm) poins of r of fsibl rgion. Sps for Grpicl Mo of Soluion. Sp I : In is sp, w v o rw grp of c linr consrin (gnrlly prss s linr quliy). T grp of lin inquliy in wo vribls n y is s of ll poins (, y) for wic inquliy ols. In is sp w v o rmin r of fsibl rgion. Fsibl rgion is r wic conins s of ll poins simulnously sisfy ll consrins. (incluing non-ngiv consrins) No : T procur for grping linr inquliy is s follows. 8 Businss Mmics (Block )

93 Sp (i): Grp corrsponing linr quion, lin. Tis is on by fining ny wo poins on lin n rwing srig lin roug m. For convninc, w coos wo poins so c is on zis (i.. ir of wo coorins is zro) Sp (ii): Coos poin P no on lin. Normlly, is poin is kn s origin s long s i os no li on lin. Ts P o s wr P sisfis inquliy. Sp (iii): I coorin of cosn poin P sisfy inquliy, n ll poins on sm si of lin s poin P sisfy inquliy. I coorin of poin P o no sisfy inquliy, n ll poins on opposi si of lin from P sisfy inquliy. Sp II: No coorin of c cornr poin of fsibl rgion n subsiu coorins of cornr poins ino objciv funcion. Sp III: T opimum soluion occurs in mzimizion cs cornr poin yiling lrgs vlu of objciv funcion n in minimizion cs cornr poin yiling smlls vlu of objciv funcion. No : T mo of fining fsibl rgion is illusr by n mpl givn blow : Grp sysm y 8 () y () 0, y 0 Firs l us rw grp of corrsponing quions. y 8 () y () For () : Wn 0, y 6 Wn y 0, 9 Grp of () is lin PQ obin by joining (0, 6) n (9, 0). To s wic si of lin sisfis inquliy, cooins of origin r subsii ino inquliy. Businss Mmics (Block ) 9

94 (0) (0) 0 8. Wic is ru. Trfor ll poins on sm si of lin s origin sisfy inquliy. T grp of inquliy is s rgion sown by Similrly, grp of inquliy y is s rgion sown by 0 Businss Mmics (Block )

95 T inquliis 0 n y 0 r sisfi by fc w confin grp o firs qurn only. Hnc fsibl rgion of bov sysm will b OABCO sown by Empl : Mimiz Z y Subjc o consrins y 50 y n y 0 Soluion : Firs, w fin fsibl rgion wic consiss of s of ll poins wos coorins simulnously sisfy ll consrins incluing nonngiv rsricions. Applying cniqus illusr bov, fsibl rgion for sysm of inquliis is OABCO s sown in igrm blow Businss Mmics (Block )

96 T cornr poins of fsibl rgion r 0(0, 0), A(00, 0), B(50, 00) n C(0, 50). No coorins of poin B v bn obin by solving y 50 n y 600 simulnously for n y. Now vlus of objciv funcion s cornr poins r summriz blow Cornr Coorins Vlu of objciv funcion poins z y O (0, 0) Z A (00, 0) Z B (50, 00) Z C (0, 50) Z Hnc mimum vlu of z is 800 n i is in vrs C(0, 50) i.. Mimum vlu of objciv funcion is 800 wn 0 n y 50 Empl : Minimiz z y Subjc o 6 6y 08 y 6 0 0y 00, y 0 Businss Mmics (Block )

97 Soluion : Firs, w fin fsibl rgion wic consiss of s of ll poins wos coorins simulnously sisfy ll consrins incluing non-ngiv rsricions. T fsibl rgion for sysm of inquliis is YDCBAX s sown in igrm blow. T cornr poins of fsibl rgion r D(0, 8), C(, 6), B(, ) n A(, 0). No coorin of C is obin by solving 6 6y 08 n 0 0y 00 n of B by solving 0 0y 00 n y 6 simulnously for n y. Now, vlus of objciv funcion s cornr poins r summriz blow. Businss Mmics (Block )

98 Cornr Coorins Vlu of objciv funcion poins z y D (0, 8) Z C (, 6) Z 6 B (, ) Z A (, 0) Z 0 So, minimum vlu of Z is n i is in vrs B(v, ) i.. minimum vlu of Z is wn n y. Empl 5 : Mimiz Z 0 5 Subjc o , 0. Subjc : Firs, w fin fsibl rgion wic consiss of s of ll poins wos coorins simulnously sisfy ll cosrins incluing non-ngiv rsricions. T fsibl rgion for sysm of inquliis is OABCDO s sown in igrm blow. T cornr poins of fsibl rgion r A(, 0), B(8, 0), C(6, ) n D(0, 5). No coorins of B is obin by solving Businss Mmics (Block )

99 6 n 56 n of C is obin by solving 56 n 5 simulnously for n. Now vlus of objciv funcion s cornr poins r sumriz blow. Poin Coorins Vlu of objciv funcion z 0 5 A (, 0) Z B (8, 0) Z C (6, ) Z D (0, 5) Z So, mimum vlu of z is 0 n i is in vr B(8, 0) i.. mimum vlus of z is 0 wn 8 n 0. CHECK YOUR PROGRESS Q : W o you mn by linr progrmming problm? Q : Dfin fsibl soluion, opimum soluion n fsibl rgion. Q : Sy wr following smns r ru (T) or fsl (F) (i) In LPP, ll prmrs r ssum o b consn. (ii) S vribls in soluion of LPP r ir posiiv or ngiv. (iii) Proucion plnning is on of pplicion rs of LPP. (iv) Linr progrmming is licniqu for fining bs uss of n orgnizion s mnpowr, mony n mcinry. Q : Fill in blnks. (i) T vribls involv in LPP r cll... vribl. (ii) In Linr Progrmming, linr funcion o b opimiz is cll... funcion. (iii) In Linr Progrmming, ll rlionsip mong cision vribls mus b... (iv) Mos of consrins in n LPP r prss s... Businss Mmics (Block ) 5

100 (v)... of cision vribls is on of ssumpions of LPP. (vi) Linr Progrmming is us o lloc o civiis so s o opimiz vlu of objciv funcion.. LET US SUM UP In is uni, w v lrn T mning of LPP. Formulion of LPP. Vrious rms ssoci wi LPP. Abou bsic ssumpions, limiions n impornc of LPP. Grpicl soluion of LPP involving wo vribls.. FURTHER READINGS ) Tukrl, Dr. J. K. Mmics for Businss Suis, Myoor Pprbcks. ) Bowl M. K. Funmnls of Businss Mmics, Asin Books Priv Limi. ) Gos, R. K. n S, S. Businss Mmics n Sisics, Nw Cnrl Books Agncy (P) L. ) Srm, J. K. Qunivs Mos, Tory n Applicion, Mcmillon Publisrs Ini L. 6 Businss Mmics (Block )

101 . ANSWERS TO CHECK YOUR PROGRESS Ans o Q No : (i) T (ii) F (iii) T (iv) T. Ans o Q No : (i) cision (ii) objciv (iii) linr (iv) inquliy (v) criniy (vi) scrc rsourc..5 MODEL QUESTIONS Q : W o you unrsn by Linr Progrmming? W r is mjor ssumpions? Q : W r rquirmns of n LPP? Q : Wri no on limiions of linr progrmming? Q : Discuss scop of linr progrmming in solving Businss Problm. Q 5: Coos corrc lrnivs () A consrin in n LP mol rsrics. (i) vlu of objciv funcion. (ii) vlu of cision vribls. (iii) us of vilbl rsourcs. (iv) ll of bov. (b) T isingusing fur of LPP is (i) rlionsip mong ll vribls is linr. (ii) vlu of cision vribls is non-ngiv. (iii) i s singl objciv funcion n consrins. (iv) ll of bov. (c) Wic of following is n ssumpion of LPP. (i) ivisibiliy (ii) criniy (iii) iiviy (iv) ll bov. Businss Mmics (Block ) 7