Syopsis Grde 1 Mth Prt II Chpter 7: Itegrls d Itegrtio is the iverse process of differetitio. If f ( ) g ( ), the we c write g( ) = f () + C. This is clled the geerl or the idefiite itegrl d C is clled the costt of itegrtio. Some stdrd idefiite itegrls 1 C, 1 1 C si cos C cos si C sec t C cosec cot C sec t sec C cosec cot cosec C 1 1 si C or cos C 1 1 1 t C or cot C 1 1 1 sec C or cosec C 1 e e C C log 1 log C e e C Properties of idefiite itegrls d f ( ) f ( ) d f '( ) f ( ) C If the derivtive of two idefiite itegrls is the sme, the they elog to sme fmily of curves d hece they re equivlet. f ( ) g( ) f ( ) g( )
, where k is y costt kf ( ) k f ( ) Methods of itegrtio There re three importt methods of itegrtio, mely, itegrtio y sustitutio, itegrtio usig prtil frctios, d itegrtio y prts. Itegrtio y sustitutio A chge i the vrile of itegrtio ofte reduces itegrl to oe of the fudmetl itegrls, which c e esily foud out. The method i which we chge the vrile to some other vrile is clled the method of sustitutio. Usig sustitutio method of itegrtio, we oti the followig stdrd itegrls: t log cos C or log sec C cot log si C sec log (sec t ) C cosec log cosec cot C Itegrtio y prtil frctios The followig tle shows how fuctio of the form P( ), where Q() d Q( ) degree of Q() is greter th the degree of P(), is roke y the cocept of prtil frctios. After doig this, we fid the itegrtio of the give fuctio y itegrtig the right hd side (i.e., prtil frctiol form). Fuctio p q, ( )( ) p q ( ) p q r ( )( )( c) p q r ( ) ( ) p q r ( )( c) where + + c cot e fctorised, Form of prtil frctio A B A B ( ) A B C c A B C ( ) A B C c Here, A, B, C re costts tht re to e determied. Itegrls of some specil fuctios
1 1 log C 1 1 log C 1 log C 1 1 si C 1 log C 1 1 t 1 C Method of some specil types of itegrls Itegrls of the types or c : c We c reduce these types of itegrls ito stdrd form y epressig c c 4 c s d the pplyig sustitutio method y puttig s u(sy). p q p q Itegrls of the type or c : c These types of itegrls c e trsformed ito stdrd form y d epressig p q s A ( c) B A( ) B, where A d B re determied y comprig coefficiets o oth sides. Itegrtio y prts For give fuctios f() d g(), f ( ) g( ) f ( ) g( ) f ( ) g( ) I other words, the itegrl of the product of two fuctios is equl to first fuctio itegrl of the secod fuctio itegrl of {differetil of the first fuctio itegrl of the secod fuctio}. Here, the fuctios f d g hve to e tke i proper order with respect to the ILATE rule, where I, L, A, T, d E respectively represet iverse, logrithm, rithmetic, trigoometric, d epoetil fuctio. We c fid the itegrls of the type ( ) ( ) ILATE rule d otie f f e f e f f y usig the ( ) ( ) ( ) C
Usig the method of itegrtio y prts, we oti the followig stdrd itegrls: i. log C ii. log C iii. 1 si C Defiite itegrls A defiite itegrl is deoted y f ( ), where is the lower limit d is the upper limit of the itegrl. If f ( ) F( ) C, the f ( ) F( ) F( ) The defiite itegrl f ( ) represets the re fuctio A() sice f ( ) is the re ouded y the curve y = f (), [, ], the -is, d the ordites = d = The defiite itegrl f ( ) c e epressed s the sum of limits s 1 f ( ) ( ) lim f ( ) f ( h)... f ( ( 1) h),where h 0 s First fudmetl theorem of itegrl clculus Let f e cotiuous fuctio o the closed itervl [, ] d let A () e the re fuctio. The, A ( ) f ( ) [, ] Secod fudmetl theorem of itegrl clculus Let f e cotiuous fuctio o the closed itervl [, ] d let F e tiderivtive of f. The, f ( ) F( ) F( ) F( ) Some useful properties of defiite itegrls f ( ) f ( t) dt
f ( ) f ( ) I prticulr, f ( ) 0 c f ( ) f ( ) f ( ) c f ( ) f ( ) f ( ) f ( ) 0 0 f ( ) f ( ) f ( ) 0 0 0 f ( ), if f ( ) f ( ) f ( ) 0 f ( ) 0 0 0, if f ( ) f ( ) f ( ), if f is eve fuctio i.e., if f ( ) f ( ) 0, if f is odd fuctio i.e., if f ( ) f ( )
Chpter 8: Applictio of Itegrls Are uder simple curves Are of the regio ouded y the curve y = f(), -is, d the lies = d = ( > ) is give y A = y or A = f ( ) The re of the regio ouded y the curve = g(y), y-is, d the lies y = c d y = d is give y A = d dy or A= g( y) c Are of the regio ouded y curve d lie If lie y = m + p itersects curve y = f() t d, the the re of this curve uder the lie y = m + p or the lies = d = is A = y or A= f ( ) If lie y = m + p itersects curve = g(y) t c d d respectively, the re of this curve uder the lie y = m + p or lies y = c d y = d is give y, A = d dy = g ( y ) dy c d c d c Are etwee two curves The re of the regio eclosed etwee two curves y = f() d y = g() d the lies = d = is give y, f ( ) g( ), where f ( ) g( ) i [, ] c A = f ( ) g( ) g( ) f ( ), c where c d f ( ) g( ) i, c d f ( ) g( ) i c,
Chpter 9: Differetil Equtios A equtio is clled differetil equtio, if it ivolves vriles s well s derivtives of depedet vrile with respect to idepedet vrile. For emple: 4 3 d y d y dy y y 3 0 4 is differetil equtio. 3 4 dy d y d y d y Sometimes, we my write ' ' ' etc. s,,, 3 4 y y y y etc. respectively. Also, ote tht we cot sy tht t( y) 0is differetil equtio. Order d degree of differetil equtio Order of differetil equtio is defied s the order of the highest order derivtive of depedet vrile with respect to idepedet vrile ivolved i the give differetil equtio. For emple: The highest order derivtive preset i the differetil equtio 3 y 5 y 3 y y y 5 0 is y. Therefore, the order of this differetil equtio is 4. Degree of differetil equtio is the highest power of the highest order derivtive i it. For emple: The degree of the differetil equtio 5 ( y ) ( y ) y( y ) y 0 is, sice the highest power of the highest order derivtive, y, is. If differetil equtio is defied, the its order d degree re lwys positive itegers. Geerl d prticulr solutios of differetil equtio A fuctio tht stisfies the give differetil equtio is clled solutio of give differetil equtio. The solutio of differetil equtio, which cotis ritrry costts, is clled geerl solutio (primitive) of the differetil equtio. The solutio of differetil equtio, which is free from ritrry costts i.e., the solutio otied from the geerl solutio y givig prticulr vlues to ritrry costts is clled prticulr solutio of the differetil equtio. Formtio of differetil equtios To form differetil equtio from give fuctio, we differetite the fuctio successively s my times s the umer of costts i the give fuctio d the elimite the ritrry costts. Methods of solvig first order, first degree differetil equtios Vrile seprle method
This method is used to solve such equtio i which vriles c e seprted completely, i.e., terms cotiig y should remi with dy d terms cotiig should remi with. Homogeeous differetil equtio A differetil equtio which c e epressed s dy f (, y) or g(, y), where f (, y ) d g(, y) re homogeous dy fuctios of degree zero is clled homogeous differetil equtio. To solve such equtio, we hve to sustitute y = v i the give differetil equtio d the solve it y vrile seprle method. Lier differetil equtio ) A differetil equtio which c e epressed i the form of dy Py Q, where P d Q re costts or fuctios of oly, is clled first order lier differetil equtio. I this cse, we fid itegrtig fctor (I.F.) y usig the formul: P I.F. e The, the solutio of the differetil equtio is give y, y (I.F) = QI.F. C ) A lier differetil equtio c lso e of the form P1 Q1 dy, where P 1 d Q 1 re costts or fuctios of y oly. P 1 dy I this cse, I.F. e Ad the solutio of the differetil equtio is give y, (I.F.) = Q I.F. dy C 1
Chpter 10: Vector Alger Sclr The qutity which ivolves oly oe vlue, i.e. mgitude, is clled sclr qutity. For emple: Time, mss, distce, eergy, etc. Vector The qutity which hs oth mgitude d directio is clled vector qutity. For emple: force, mometum, ccelertio, etc. Directed lie A lie with directio is clled directed lie. Let AB e directed lie log directio B. Here, The legth of the lie segmet AB represets the mgitude of the ove directed lie. It is deoted y AB or or. AB represets the vector i the directio towrds poit B. Therefore, the vector represeted i the ove figure is AB. It c lso e deoted y. The poit A from where the vector AB strts is clled its iitil poit d the poit B where the vector AB eds is clled its termil poit. Positio vector The positio vector of poit P(, y, z) with respect to the origi (0, 0, 0) is give yop iˆ yj ˆ zkˆ. This form of y vector is kow s the compoet form. Here, iˆ, ˆj, d ˆk re clled the uit vectors log the -is, y-is, d z-is respectively., y, d z re the sclr compoets (or rectgulr compoets) log - is, y-is, d z-is respectively. iˆ, yj ˆ, zk ˆ re clled vector compoets of OP log the respective es. The mgitude of OP is give y OP y z Compoets d directio rtios The sclr compoets of vector re its directio rtios d represet its projectios log the respective is. The directio rtios of vector p iˆ j ˆ ckˆ re,, d c. Here,,, d c respectively represet projectios of p log -is, y- is, d z-is. Directio cosies
The cosies of the gle mde y the vector r iˆ j ˆ ckˆ with the positive directios of, y, d z es re its directio cosies. These re usully deoted y l, m, d. Also, l m 1 The directio cosies (l, m, ) of vector iˆj ˆckˆ re c l, m,, where r = mgitude of the vector iˆj ˆckˆ r r r Types of vectors Zero vector: A vector whose iitil d termil poits coicide is clled zero vector (or ull vector). It is deoted s 0. The vectors AA, BB represet zero vectors. Uit vector: A vector whose mgitude is uity, i.e. 1 uit, is clled uit vector. The uit vector i the directio of y give vector is deoted y 1 â d it is clculted y â Note: if l, m, d re directio cosies of vector, the liˆmj ˆ kˆ is the uit vector i the directio of tht vector. Co-iitil vectors: Two or more vectors re sid to e co-iitil vectors, if they hve the sme iitil poit. Collier vectors: Two or more vectors re sid to e collier vectors, if they re prllel to sme lie irrespective of their mgitude d directio. Equl vectors: Two vectors d re sid to equl, if they hve sme mgitude d directio regrdless of the positio of their iitil poits. They re writte s Negtive of vector: Two vectors re sid to e egtive of oe other, if they hve sme mgitude, ut their directio is opposite to oe other. For emple, the egtive of vector AB is writte s BA Additio of vectors AB
Trigle lw of vector dditio: If two vectors re represeted y two sides of trigle i order, the the third closig side of the trigle i the opposite directio of the order represets the sum of the two vectors. AC AB BC Note: The vector sum of the three sides of trigle tke i order is 0 Prllelogrm lw of vector dditio: If two vectors re represeted y two djcet sides of prllelogrm i order, the the digol closig side of the trigle of the prllelogrm i the opposite directio of the order represets the sum of two vectors. c Properties of vector dditio Commuttive property: Associtive property: ( c) ( ) c Eistece of dditive idetity: The vector 0 is dditive idetity of vector, sice 0 0 Eistece of dditive iverse: The vector is clled dditive iverse of, sice ( ) ( ) 0 Opertios o vectors The multiplictio of vector ˆ ˆ ˆ 1i j 3k y y sclr is give y, ( ) iˆ ( ) ˆj ( ) kˆ 1 3 The mgitude of the vector is give y The sum of two vectors ˆ ˆ ˆ 1i j 3k d ˆ ˆ ˆ 1i j 3k is give y, ( ) iˆ ( ) ˆj ( ) kˆ 1 1 3 3 The differece of two vectors ˆ ˆ ˆ 1i j 3k d ˆ ˆ ˆ 1i j 3k is give y ( ˆ ˆ ˆ 1 1 ) i ( ) j ( 3 3 ) k
Equlity of vectors The vectors ˆ ˆ ˆ 1i j 3k d ˆ ˆ ˆ 1i j 3k re equl, if d oly if 1 = 1, =, d 3 = 3 Distriutive lw for vectors Let 1 d e two vectors, d k 1 d k e y sclrs, the the followig re the distriutive lws of dditio d multiplictio of vector y sclr: k k ( k k ) 1 1 1 1 1 k k k k 1 1 1 1 k k k 1 1 1 1 1 Collier vectors Two vectors d re collier, if d oly if there eists o-zero sclr such tht Two vectors ˆ ˆ ˆ 1i j 3k d ˆ ˆ ˆ 1i j 3k re collier, if d 1 3 oly if 1 3 Vector joiig two poits The mgitude of the vector joiig the two poits P 1 ( 1, y 1, z 1 ) d P (, y, z ) is give y PP ( ) ( y y ) ( z z ) 1 1 1 1 Sectio formul The positio vector of poit R dividig lie segmet joiig the poits P d Q, whose positio vectors re d respectively, i the rtio m : m iterlly, is give y m m eterlly, is give y m Sclr product of vectors The sclr product of two o-zero vectors d is deoted y d it is give y the formul cos, where is the gle etwee d such tht 0 If either 0 or 0, the i this cse, is ot defied d 0 The followig re the oservtios relted to the sclr product of two vectors: is rel umer. The gle etwee vectors d is give y, 1 cos cos
0, if d oly if If = 0, the If =, the iˆ iˆ ˆj ˆj kˆ kˆ 1, iˆ ˆj ˆj kˆ kˆ iˆ 0 If ˆ ˆ ˆ 1i j 3k d ˆ ˆ ˆ 1i j 3k, the 1 1 33 Properties of sclr product Commuttive property: Distriutivity of sclr product over dditio: ( c) c Projectio of vector If ˆp is the uit vector log lie l, the the projectio of vector o the lie l is give y p.. ˆ Projectio of vector o other vector is give y or. Vector product of vectors The vector product (or cross product) of two o-zero vectors d is deoted y d is defied y siˆ, where is the gle etwee d, 0, d ˆ is uit vector perpediculr to oth d. If ˆ ˆ ˆ 1i j 3k d ˆ ˆ ˆ 1i j 3k re two vectors, the their cross iˆ ˆj kˆ product, is defied y 1 3 1 3 The followig re the oservtios mde y the vector product of two vectors: 0, if d oly if iˆ iˆ ˆj ˆj kˆ kˆ 0 iˆ ˆj kˆ, ˆj kˆ iˆ, kˆ iˆ ˆj ˆj iˆ kˆ, kˆ ˆj iˆ, iˆ kˆ ˆj I terms of vector product, the gle etwee two vectors d is 1 give y si or si If d represet the djcet sides of trigle, the its re is give s 1. If d represet the djcet sides of prllelogrm, the its re is give s.
Properties of vector product Not commuttive: However, Distriutivity of vector product over dditio: c c
Chpter 11: Three Dimesiol Geometry Directio cosies (d.c. s) of lie D.c. s of lie re the cosies of gles mde y the lie with the positive directio of the coordite es. If l, m, d re the d.c. s of lie, the l + m + = 1 D.c. s of lie joiig two poits P ( 1, y 1, z 1 ) d Q (, y, z ) re 1 1 1, y y, z z, where PQ = ( 1 ) ( y y1) ( z z1) PQ PQ PQ Directio rtios (d.r. s) of lie D.r. s of lie re the umers which re proportiol to the d.c. s of the lie. D.r. s of lie joiig two poits P ( 1, y 1, z 1 ) d Q (, y, z ) re give y 1, y 1 y, z 1 z or 1, y y 1, z z 1. If,, d c re the d.r. s of lie d l, m, d re its d.c. s, the l m c c l, m, d c c c Equtio of lie through give poit d prllel to give vector Vector form Equtio of lie tht psses through the give poit whose positio vector is d which is prllel to give vector is r, where is costt. Crtesi form o Equtio of lie tht psses through poit ( 1, y 1, z 1 ) hvig d.r. s 1 y y1 z z1 s,, c is give y c o Equtio of lie tht psses through poit ( 1, y 1, z 1 ) hvig d.c. s 1 y y1 z z1 s l, m, is give y l m Equtio of lie pssig through two give poits Vector form: Equtio of lie pssig through two poits whose positio vectors re d is give y r ( ), where R Crtesi form: Equtio of lie tht psses through two give poits ( 1, y 1, z 1 ) d (, y, z ) is give y, 1 y y1 z z1 y y z z 1 1 1 Skew lies d gle etwee them
Two lies i spce re sid to e skew lies, if they re either prllel or itersectig. They lie i differet ples. Agle etwee two skew lies is the gle etwee two itersectig lies drw from y poit (preferly from the origi) prllel to ech of the skew lies. Agle etwee two o-skew lies Crtesi form o If l 1, m 1, 1, d l, m, re the d.c. s of two lies d is the cute gle etwee them, the cos l1l m1m 1 o If 1, 1, c 1 d,, c re the d.r. s of two lies d is the cute 1 1 c1c gle etwee them, the cos c. c 1 1 1 Vector form If is the cute gle etwee the lies r 1 1 d r 1, the cos 1 1 Two lies with d.r. s 1, 1, c 1 d,, c re perpediculr, if 1 1 c 1 c 0 1 1 c1 prllel, if c Shortest distce etwee two skew lies: The shortest distce is the lie segmet perpediculr to oth the lies. Vector form: Distce etwee two skew lies r 1 1 d d r is give y, d ( ) ( ) 1 1 1 Crtesi form: The shortest distce etwee two lies 1 y y1 z z1 y y z z d is give y, c c 1 1 1 y y z z 1 1 1 c 1 1 1 c ( c c ) ( c c ) ( ) 1 1 1 1 1 1
The shortest distce etwee two prllel lies r 1 d r is give y, d ( ) 1 Equtio of ple i orml form Vector form: Equtio of ple which is t distce of d from the origi, d the uit vector ˆ orml to the ple through the origi is r.ˆ d, where r is the positio vector of poit i the ple Crtesi form: Equtio of ple which is t distce d from the origi d the d.c. s of the orml to the ple s l, m, is l + my + z = d Equtio of ple perpediculr to give vector d pssig through give poit Vector form: Equtio of ple through poit whose positio vector is d perpediculr to the vector N is ( r ).N = 0, where r is the positio vector of poit i the ple Crtesi form: Equtio of ple pssig through the poit ( 1, y 1, z 1 ) d perpediculr to give lie whose d.r. s re A, B, C is A( ) B( y y ) C( z z ) 0 1 1 1 Equtio of ple pssig through three o-collier poits Crtesi form: Equtio of ple pssig through three o-collier poits ( 1, y 1, z 1 ), (, y, z ), d ( 3, y 3, z 3 ) is y y z z 1 1 1 y y z z 0 1 1 1 y y z z 3 1 3 1 3 1 Vector form: Equtio of ple tht cotis three o-collier poits hvig positio vectors,, d c is ( r ) ( ) ( c ) 0, where r is the positio vector of poit i the ple. Itercept form of the equtio of ple Equtio of ple hvig, y, d z itercepts s,, d c respectively i.e., the equtio of the ple tht cuts the coordite es t (, 0, 0), (0,, 0), d (0, 0, c) is give y, y z 1 c Ples pssig through the itersectio of two ples Vector form: Equtio of the ple pssig through itersectio of two ples r. 1 d1 d r. dis give y, r.( ) d d, where is o-zero costt 1 1
1 1 1 1 Crtesi form: Equtio of ple pssig through the itersectio of two ples A1 B1 y C1z D1 0 d A B y Cz D 0 is give y, (A B y C z D ) (A B y C z D ) 0, where is ozero costt Co-plrity of two lies Vector form: Two lies r 1 1 d r re co-plr, if ( ) ( ) 0 1 1 1 y y1 z z1 Crtesi form: Two lies d 1 1 c1 y y z z re co-plr, if c y y z z 1 1 1 c 1 1 1 c 0 Agle etwee two ples: The gle etwee two ples is defied s the gle etwee their ormls. Vector form: If is the gle etwee the two ples r 1 d1d 1 r d, the cos 1 Note tht if two ples re perpediculr to ech other, the 1. 0; d if they re prllel to ech other, the 1 is prllel to. Crtesi form: If is the gle etwee the two ples A1 B1 y C1z D1 0 d A B y Cz D 0, the cos A A B B C C 1 1 1 A B C A B C 1 1 1 Note tht if two ples re perpediculr to ech other, the A1A B1B C1C 0 ; d if they re prllel to ech other, the A1 B1 C1 A B C Distce of poit from ple Vector form: The distce of poit, whose positio vector is, from the ple r ˆ d is d ˆ. Note:
o If the equtio of the ple is i the form of rn d, where N is the N d orml to the ple, the the perpediculr distce is. N o Legth of the perpediculr from origi to the ple d rn dis. N Crtesi form: The distce from poit ( 1, y 1, z 1 ) to the ple A + By A1 By1 Cz1 D + Cz + D = 0 is. A B C Agle etwee lie d ple: The gle etwee lie r d the ple r d is the complemet of the gle etwee the lie d the orml to the ple d is give y si 1
Chpter1: Lier Progrmmig Prolems which seek mimise (or miimise) of lier fuctio (sy, of two vriles d y) suject to certi costrits s determied y set of lier iequlities re clled optimistio prolems. A Lier Progrmmig Prolem (L.P.P.) is the oe tht is cocered with fidig the optiml vlue (mimum or miimum vlue) of lier fuctio of severl vriles (clled ojective fuctio), suject to the coditios tht the vriles re o-egtive d stisfy set of lier iequlities (clled costrits). The vriles re sometimes clled the decisio vriles. For emple: The followig is L.P.P. Mimize Z = 10 + 1y Suject to the followig costrits: 5 + 3y 30... (1) + y... () 0, y 0... (3) I this L.P.P, the ojective fuctio is Z = 10 + 1y The iequlities (1), (), d (3) re clled costrits. The commo regio determied y ll the costrits icludig the oegtive costrits 0, y 0 of lier progrmmig prolem is clled the fesile regio (or solutio regio) for the prolem. The regio outside this fesile regio is clled ifesile regio. Poits withi d o the oudry of the fesile regio represet fesile solutios of the costrits. Ay poit outside the fesile regio is ifesile solutio. Ay poit i the fesile regio tht gives the optiml vlue (mimum or miimum) of the ojective fuctio is clled optiml solutio. Fudmetl theorems for solvig lier progrmmig prolems Theorem 1: Let R e the fesile regio for lier progrmmig prolem d let Z = + y e the ojective fuctio. Whe Z hs optiml vlue, where the vriles d y re suject to costrits descried y lier iequlities, this optiml vlue must occur t corer poit of the fesile regio. Theorem : Let R e the fesile regio for lier progrmmig prolem, d let Z = + y e the ojective fuctio. If R is ouded, the the ojective fuctio Z hs oth mimum d miimum vlue o R d ech of these occurs t corer poit of R. If the fesile regio is uouded, the mimum or miimum my ot eist. However, if it eists, the it must occur t corer poit of R.
Corer poit method: This method is used for solvig lier progrmmig prolem d it comprises of the followig steps: Step 1) Fid the fesile regio of the L.P.P. d determie its corer poits. Step ) Evlute the ojective fuctio Z = + y t ech corer poit. Let M d m respectively e the lrgest d smllest vlues t these poits. Step 3) If the fesile regio is ouded, the M d m respectively re the mimum d miimum vlues of the ojective fuctio. If the fesile regio is uouded If the ope hlf ple determied y + y > M hs o poit i commo with the fesile regio, the M is the mimum vlue of the ojective fuctio. Otherwise, the ojective fuctio hs o mimum vlue. If the ope hlf ple determied y + y < m hs o poit i commo with the fesile regio, the m is the miimum vlue of the ojective fuctio. Otherwise, the ojective fuctio hs o miimum vlue. If two corer poits of the fesile regio re oth optiml solutios of the sme type, i.e. oth produce the sme mimum or miimum, the y poit o the lie segmet joiig these two poits is lso optiml solutio of the sme type. A few importt lier progrmmig prolems re: diet prolems, mufcturig prolems, trsporttio prolems, d lloctio prolems. Emple 1: A firm is egged i reedig gots. The gots re fed o vrious products grow i the frm. They require certi utriets, med A, B, d C. The gots re fed o two products P d Q. Oe uit of product P cotis 1 uits of A, 18 uits of B, d 5 uits of C, while oe uit of product Q cotis 4 uits of A, 9 uits of B, 5 uits of C. The miimum requiremet of A d B re 144 uits d 108 uits respectively wheres the mimum requiremet of C is 50 uits. Product P costs Rs 35 per uit wheres product Q costs Rs 45 per uit. Formulte this s lier progrmmig prolem. How my uits of ech product my e tke to miimise the cost? Also fid the miimum cost. Solutio: Let d y e the umer of uits tke from products P d Q respectively to miimise the cost. Mthemticl formultio of the give L.P.P. is s follows: Miimise Z = 35 + 45y Suject to costrits 1 + 4y 144 (costrits o A) + y 1... (1) 18 + 9y 108 (costrits o B) + y 1... () 5 + 5y 50 (costrits o C) + y 10... (3) 0, y 0... (4) The fesile regio determied y the system of costrits is s follows:
The shded regio is the fesile regio. The corer poits re L (4, 4), M (, 8), d N (8, ). The vlue of Z t these corer poits re s follows: Corer poit Z = 35 + 45y L (4, 4) 30 Miimum M (, 8) 430 N (8, ) 370 It c e oserved tht the vlue Z is miimum t the corer poit L (4, 4) d the miimum vlue is 30. Therefore, 4 uits of ech of the products P d Q re tke to miimise the cost d the miimum cost is Rs 30.
Chpter 13: Proility Coditiol proility If E d F re two evets ssocited with the smple spce of rdom eperimet, the the coditiol proility of evet E, give tht F hs lredy occurred, is deoted y P(E/F) d is give y the formul: P(E/F) = P( E F ), where P (F) 0 P( F) Properties of coditiol proility If E d F re two evets of smple spce S of eperimet, the the followig re the properties of coditiol proility: 0 P(E/F) 1 P(F/F) = 1 P(S/F) = 1 P(E /F) = 1 P(E/F) If A d B re two evets of smple spce S d F is evet of S such tht P(F) 0, the o P((A B)/F) = P(A/F) + P(B/F) P((A B)/F) o P((A B)/F) = P(A/F) + P(B/F), if the evets A d B re disjoit. Multiplictio theorem of proility If E, F, d G re evets of smple spce S of eperimet, the P(E F) = P(E). P(F/E), if P(E) 0 P(E F) = P(F). P(E/F), if P(F) 0 P(E F G) = P(E). P(F/E). P(G/(E F)) = P(E). P(F/E). P(G/EF) Idepedet evets Two evets E d F re sid to e idepedet evets, if the proility of occurrece of oe of them is ot ffected y the occurrece of the other. If E d F re two idepedet evets, the o P(F/E) = P(F), provided P(E) 0 o P(E/F) = P(E), provided P(F) 0 If three evets A, B, d C re idepedet evets, the P(A B C) = P(A). P(B). P(C) If the evets E d F re idepedet evets, the o E d F re idepedet o E d F re idepedet Prtitio of smple spce A set of evets E 1, E, E is sid to represet prtitio of the smple spce S, if E i E j =, i j, i, j = 1,, 3, E 1 E E = S P(E i ) > 0, i = 1,, 3, Theorem of totl proility
Let {E 1, E, E } e prtitio of the smple spce S, d suppose P(E i ) > 0, i = 1,,. Let A e y evet ssocited with S, the P(A) = P(E 1 ). P(A/E 1 ) + P(E ). P(A/E ) + + P(E ). P(A/E ) = j1 P( E )P( A/ E ) j j Byes theorem If E 1, E,.. E re o-empty evets, which costitute prtitio of smple spce S, the P( Ei)P( A/ Ei) P( Ei / A), i 1,,3..., P( E )P( A/ E ) j1 j j Rdom vriles d their proility distriutio A rdom vrile is rel-vlued fuctio whose domi is the smple spce of rdom eperimet. The proility distriutio of rdom vrile X is the system of umers: X: 1 P(X): p 1 p p Where, p 0 p 1, i 1,,... i i1 i Here, the rel umers 1,,, re the possile vlues of the rdom vrile X d p i (i = 1,,, ) is the proility of the rdom vrile X tkig the vlue of i i.e., P(X = i ) = p i Me/epecttio of rdom vrile Let X e rdom vrile whose possile vlues 1,, 3, occur with proilities p 1, p, p 3, p respectively. The me of X (deoted y ) or the epecttio of X (deoted y E(X)) is the umer p i i. Tht is, E( X ) p 1 p1 p... p i1 i i Vrice of rdom vrile Let X e rdom vrile whose possile vlues 1,, occur with proilities p( 1 ), p( ), p( ) respectively. Let = E(X) e the me of X. The vrice of X deoted y Vr (X) or is clculted y y of the followig formule: i p i i1 E( X ) ( ) ( ) i1
( ) ( ) i p i i p i i1 i1 E( ) E( ), where X X E( X ) i p( i) i1 It is dvisle to studets to use the fourth formul. Stdrd devitio: The o-egtive umer Vr ( X ) is clled the stdrd devitio of the rdom vrile X. X X E( ) E( ) Beroulli trils: Trils of rdom eperimet re clled Beroulli trils, if they stisfy the followig coditios: There should e fiite umer of trils. The trils should e idepedet. Ech tril hs ectly two outcomes: success or filure. The proility of success remis the sme i ech tril. A iomil distriutio with -Beroulli trils d proility of success i ech tril s p is deoted y B(, p). Biomil distriutio: For iomil distriutio B(, p), the proility of successes is deoted y P(X = ) or P(X) d is give y P( X ) C q p, 0,1,,..., q 1 p Here, P(X) is clled the proility fuctio of the iomil distriutio.