BUSINESS MATHEMATICS

Transcription

1 BUSINESS MATHEMATICS HIGHER SECONDARY - SECOND YEAR olume- Untouchbility is sin Untouchbility is crime Untouchbility is inhumn TAMILNADU TEXTBOOK CORORATION College Rod, Chenni

2 Government of Tmilndu First Edition - Second Edition - 6 Tet Book Committee Chirperson Dr.S.ANTONY RAJ Reder in Mthemtics residency College Chenni 6. Thiru. N. RAMESH Selection Grde Lecturer Deprtment of Mthemtics Govt. Arts College (Men) Nndnm, Chenni - 6. Thiru.. RAKASH Lecturer (S.S.), Deprtment of Sttistics residency College Chenni - 6. Thiru. S.T. ADMANABHAN ost Grdute Techer The Hindu Hr. Sec. School Triplicne, Chenni - 6. Tmt. AMALI RAJA ost Grdute Techer Good Shepherd Mtricultion Hr. Sec. School, Chenni 66. rice : Rs. Reviewers - cum - Authors Authors Reviewer Dr. M.R. SRINIASAN Reder in Sttistics University of Mdrs, Chenni - 6. Thiru. R.MURTHY Selection Grde Lecturer Deprtment of Mthemtics residency College Chenni 6. Thiru. S. RAMACHANDRAN ost Grdute Techer The Chintdripet Hr. Sec. School Chintdripet, Chenni - 6. Thiru. S. RAMAN ost Grdute Techer Jigopl Grodi Ntionl Hr. Sec. School Est Tmbrm, Chenni Tmt. M.MALINI ost Grdute Techer.S. Hr. Sec. School (Min) Mylpore, Chenni 6. This book hs been prepred by the Directorte of School Eduction on behlf of the Government of Tmilndu This book hs been printed on 6 GSM pper

3 refce The most distinct nd beutiful sttement of ny truth must tlst tke the Mthemticl form -Thoreu. Among the Nobel Luretes in Economics more thn 6% were Economists who hve done pioneering work in Mthemticl Economics.These Economists not only lernt Higher Mthemtics with perfection but lso pplied it successfully in their higher pursuits of both Mcroeconomics nd Econometrics. A Mthemticl formul (involving stochstic differentil equtions) ws discovered in 97 by Stnford University rofessor of Finnce Dr.Scholes nd Economist Dr.Merton.This chievement led to their winning Nobel rize for Economics in 997.This formul tkes four input vribles-durtion of the option,prices,interest rtes nd mrket voltility-nd produces price tht should be chrged for the option.not only did the formul work,it trnsformed Americn Stock Mrket. Economics ws considered s deductive science using verbl logic grounded on few bsic ioms.but tody the trnsformtion of Economics is complete.etensive use of grphs,equtions nd Sttistics replced the verbl deductive method.mthemtics is used in Economics by beginning wth few vribles,grdully introducing other vribles nd then deriving the inter reltions nd the internl logic of n economic model.thus Economic knowledge cn be discovered nd etended by mens of mthemticl formultions. Modern Risk Mngement including Insurnce,Stock Trding nd Investment depend on Mthemtics nd it is fct tht one cn use Mthemtics dvntgeously to predict the future with more precision!not with % ccurcy, of course.but well enough so tht one cn mke wise decision s to where to invest money.the ide of using Mthemtics to predict the future goes bck to two 7 th Century French Mthemticins scl nd Fermt.They worked out probbilities of the vrious outcomes in gme where two dice re thrown fied number of times. iii

4 In view of the incresing compleity of modern economic problems,the need to lern nd eplore the possibilities of the new methods is becoming ever more pressing.if methods bsed on Mthemtics nd Sttistics re used suitbly ccording to the needs of Socil Sciences they cn prove to be compct, consistent nd powerful tools especilly in the fields of Economics, Commerce nd Industry. Further these methods not only gurntee deeper insight into the subject but lso led us towrds ect nd nlyticl solutions to problems treted. This tet book hs been designed in conformity with the revised syllbus of Business Mthemtics(XII) (to come into force from - 6)- Ech topic is developed systemticlly rigorously treted from first principles nd mny worked out emples re provided t every stge to enble the students grsp the concepts nd terminology nd equip themselves to encounter problems. Questions compiled in the Eercises will provide students sufficient prctice nd self confidence. Students re dvised to red nd simultneously dopt pen nd pper for crrying out ctul mthemticl clcultions step by step. As the Sttistics component of this Tet Book involves problems bsed on numericl clcultions,business Mthemtics students re dvised to use clcultors.those students who succeed in solving the problems on their own efforts will surely find phenomenl increse in their knowledge, understnding cpcity nd problem solving bility. They will find it effortless to reproduce the solutions in the ublic Emintion. We thnk the Almighty God for blessing our endevour nd we do hope tht the cdemic community will find this tetbook triggering their interests on the subject! The direct ppliction of Mthemticl resoning to the discovery of economic truth hs recently rendered gret services in the hnds of mster Mthemticins Alfred Mrshll. Mlini Amli Rj Rmn dmnbhn Rmchndrn rksh Murthy Rmesh Srinivsn Antony Rj iv

5 CONTENTS ge. ALICATIONS OF MATRICES AND DETERMINANTS. Inverse of Mtri Minors nd Cofctors of the elements of determinnt - Adjoint of squre mtri - Inverse of non singulr mtri.. Systems of liner equtions Submtrices nd minors of mtri - Rnk of mtri - Elementry opertions nd equivlent mtrices - Systems of liner equtions - Consistency of equtions - Testing the consistency of equtions by rnk method.. Solution of liner equtions Solution by Mtri method - Solution by Crmer s rule. Storing Informtion Reltion mtrices - Route Mtrices - Cryptogrphy. Input - Output Anlysis.6 Trnsition robbility Mtrices. ANALYTICAL GEOMETRY 66. Conics The generl eqution of conic. rbol Stndrd eqution of prbol - Trcing of the prbol. Ellipse Stndrd eqution of ellipse - Trcing of the ellipse. Hyperbol Stndrd eqution of hyperbol - Trcing of the hyperbol - Asymptotes - Rectngulr hyperbol - Stndrd eqution of rectngulr hyperbol. ALICATIONS OF DIFFERENTIATION - I 99. Functions in economics nd commerce Demnd function - Supply function - Cost function - Revenue function - rofit function - Elsticity - Elsticity of demnd - Elsticity of supply - Equilibrium price - Equilibrium quntity - Reltion between mrginl revenue nd elsticity of demnd. v

6 . Derivtive s rte of chnge Rte of chnge of quntity - Relted rtes of chnge. Derivtive s mesure of slope Slope of the tngent line - Eqution of the tngent - Eqution of the norml. ALICATIONS OF DIFFERENTIATION - II. Mim nd Minim Incresing nd decresing functions - Sign of the derivtive - Sttionry vlue of function - Mimum nd minimum vlues - Locl nd globl mim nd minim - Criteri for mim nd minim - Concvity nd conveity - Conditions for concvity nd conveity - oint of inflection - Conditions for point of inflection.. Appliction of Mim nd Minim Inventory control - Costs involved in inventory problems - Economic order quntity - Wilson s economic order quntity formul.. rtil Derivtives Definition - Successive prtil derivtives - Homogeneous functions - Euler s theorem on Homogeneous functions.. Applictions of rtil Derivtives roduction function - Mrginl productivities - rtil Elsticities of demnd.. ALICATIONS OF INTEGRATION 7. Fundmentl Theorem of Integrl Clculus roperties of definite integrls. Geometricl Interprettion of Definite Integrl s Are Under Curve. Appliction of Integrtion in Economics nd Commerce The cost function nd verge cost function from mrginl cost function - The revenue function nd demnd function from mrginl revenue function - The demnd function from elsticity of demnd.. Consumers Surplus. roducers Surplus ANSWERS 7 (...continued in olume-) vi

7 ALICATIONS OF MATRICES AND DETERMINANTS The concept of mtrices nd determinnts hs etensive pplictions in mny fields such s Economics, Commerce nd Industry. In this chpter we shll develop some new techniques bsed on mtrices nd determinnts nd discuss their pplictions.. INERSE OF A MATRIX.. Minors nd Cofctors of the elements of determinnt. The minor of n element ij of determinnt A is denoted by M i j nd is the determinnt obtined from A by deleting the row nd the column where i j occurs. The cofctor of n element ij with minor M ij is denoted by C ij nd is defined s Mi j, C ij M, i j if if i + j is even i + j is odd. Thus, cofctors re signed minors. In the cse of, we hve M, M, M, M Also C, C, C, C In the cse of M M, C, we hve, C ; ;

8 M M, C, C ; nd so on... Adjoint of squre mtri. The trnspose of the mtri got by replcing ll the elements of squre mtri A by their corresponding cofctors in A is clled the Adjoint of A or Adjugte of A nd is denoted by Adj A. Thus, Note AdjA A t c b d (i) Let A then A c d c b Adj A A t c d b c Thus the Adjoint of mtri cn be written instntly s d c b c c (ii) Adj I I, where I is the unit mtri. (iii) A(AdjA) (Adj A) A A I (iv) Adj (AB) (Adj B) (Adj A) (v) If A is squre mtri of order, then AdjA A If A is squre mtri of order, then Adj A A Emple Write the Adjoint of the mtri A Adj A Emple Find the Adjoint of the mtri A b d

9 A Now,, Adj A A t c C, C 8, C, C, C 6, C, C, C, C Hence A c Adj A t Inverse of non singulr mtri. The inverse of non singulr mtri A is the mtri B such tht AB BA I. B is then clled the inverse of A nd denoted by A. Note (i) A non squre mtri hs no inverse. (ii) The inverse of squre mtri A eists only when A tht is, if A is singulr mtri then A does not eist. (iii) If B is the inverse of A then A is the inverse of B. Tht is B A A B. (iv) A A I A - A (v) The inverse of mtri, if it eists, is unique. Tht is, no mtri cn hve more thn one inverse. (vi) The order of the mtri A will be the sme s tht of A.

10 (vii) I I (viii) (AB) B A, provided the inverses eist. (i) A I implies A A () If AB C then () A CB (b) B A C, provided the inverses eist. (i) We hve seen tht A(AdjA) (AdjA)A A I A (AdjA) (AdjA)A I (ΠA ) A A This suggests tht A A (AdjA). Tht is, A A A t c (ii) instntly s (A ) A, provided the inverse eists. b Let A Now A c Emple ~ A c d b d c d bc with A d bc, A t c d c b d c Thus the inverse of mtri d bc d c b Find the inverse of A A A b c b d provided d bc. A eists., if it eists. cn be written

11 Emple Show tht the inverses of the following do not eist : (i) A 9 6 (ii) A 6 7 (i) A 9 6 A does not eist. (ii) A 6 7 A - does not eist. Emple Find the inverse of A, if it eists. A A eists. We hve, A A A t c Now, the cofctors re C, C 7, C, C, C 8, C, C, C, C, Hence A c 8 7, A t c 8 7 A 8 7

12 6 Emple 6 Show tht A nd B re inverse of ech other. AB I Since A nd B re squre mtrices nd AB I, A nd B re inverse of ech other. EXERCISE. ) Find the Adjoint of the mtri ) Find the Adjoint of the mtri ) Show tht the Adjoint of the mtri A is A itself. ) If A, verify tht A(Adj A) (Adj A) A A I. ) Given A, B, verify tht Adj (AB) (Adj B) (Adj A)

13 7 6) In the second order mtri A ( i j ), given tht i j i+j, write out the mtri A nd verify tht Adj A A 7) Given A -, verify tht Adj A A 8) Write the inverse of A 9) Find the inverse of A ) Find the inverse of A b nd verify tht AA I. ) If A nd none of the s re zero, find A. ) If A, show tht the inverse of A is itself. ) If A, find A. ) Show tht A nd B re inverse of ech other ) If A 8, compute A nd show tht A I A 6) If A verfy tht (A ) A 7) erify (AB) B A, when A nd B 9 6 8) Find λ if the mtri ë 9 ë 7 6 hs no inverse.

14 8 9) If X 6 nd Y q p find p, q such tht Y X. ) If X 9, find the mtri X.. SYSTEMS OF LINEAR EQUATIONS.. Submtrices nd minors of mtri. Mtrices obtined from given mtri A by omitting some of its rows nd columns re clled sub mtrices of A. e.g. If A, some of the submtrices of A re :,,,,,, nd The determinnts of the squre submtrices re clled minors of the mtri. Some of the minors of A re :,,,, nd.. Rnk of mtri. A positive integer r is sid to be the rnk of non zero mtri A, denoted by ρ(a), if (i) there is t lest one minor of A of order r which is not zero nd (ii) every minor of A of order greter thn r is zero.

15 Note (i) The rnk of mtri A is the order of the lrgest non zero minor of A. (ii) If A is mtri of order m n then ρ(a) < minimum (m, n) (iii) The rnk of zero mtri is tken to be. (iv) For non zero mtrices, the lest vlue of the rnk is. (v) The rnk of non singulr mtri of order n n is n. (vi) ρ(a) ρ(a t ) (vii) ρ(i ), ρ(i ) Emple 7 Find the rnk of the mtri A Order of A is. ρ(a) < Consider the only third order minor Emple 8. 9 There is minor of order which is not zero. ρ(a) Find the rnk of the mtri A Order of A is. ρ(a) < Consider the only third order minor 6 6 The only minor of order is zero. ρ(a) < Consider the second order minors.

16 We find, Emple 9 There is minor of order which is non zero. ρ(a). Find the rnk of the mtri A 8 6 Order of A is. ρ(a) < Consider the only third order minor 6 8 (R R ) The only minor of order is zero. ρ(a) < Consider the second order minors. Obviously they re ll zero. ρ(a) < Since A is non zero mtri, ρ(a) Emple Find the rnk of the mtri A 9 Order of A is. ρ(a) < Consider the second order minors. We find, 9 8 There is minor of order which is not zero. ρ(a) 7 Emple Find the rnk of the mtri A

17 Order of A is. ρ(a) <. We find, Consider the third order minors. 8 9 There is minor of order which is not zero. ρ(a)... Elementry opertions nd equivlent mtrices. The process of finding the vlues of number of minors in our endevour to find the rnk of mtri becomes lborious unless by stroke of luck we get non zero minor t n erly stge. To get over this difficulty, we introduce mny zeros in the mtri by wht re clled elementry opertions so tht the evlution of the minors is rendered esier. It cn be proved tht the elementry opertions do not lter the rnk of mtri. (i) (ii) (iii) The following re the elementry opertions : The interchnge of two rows. The multipliction of row by non zero number. The ddition of multiple of one row to nother row. If mtri B is obtined from mtri A by finite number of elementry opertions then we sy tht the mtrices A nd B re equivlent mtrices nd we write A B. Also, while introducing mny zeros in the given mtri, it would be desirble (but not necessry) to reduce it to tringulr form. A mtri A ( i j ) is sid to be in tringulr form if i j whenever i > j. e.g., The mtri 7 9 is in tringulr form.

18 Emple Find the rnk of the mtri A - Order of A is. ρ(a) <. Let us reduce the mtri A to tringulr form. A Applying R R A ~ Applying R R R A ~ 8 Applying R R 8R A ~ This is now in tringulr form. We find, There is minor of order which is not zero. ρ(a). Emple Find the rnk of the mtri A

19 Order of A is. ρ(a) < Let us reduce the mtri A to tringulr form. A Applying R R R, R R R A ~ Applying R R + R A 8 This is now in tringulr form. We find, 8 6 There is minor of order which is not zero. ρ(a). Emple Find the rnk of the mtri A 8 6 Order of A is. ρ (A) <. A 8 6 Applying R R

20 A 6 Applying R R A 6 Applying R R R, R R R A ~ 6 6 Applying R R + R A ~ 8 6 This is in tringulr form. We find, There is minor of order which is not zero. ρ(a).. Systems of liner equtions. A system of (simultneous) equtions in which the vribles (ie. the unknowns) occur only in the first degree is sid to be liner. A system of liner equtions cn be represented in the form AX B. For emple, the equtions y+z, +yz, 67y+8z 7 cn be written in the mtri form s z y 7 A X B

21 A is clled the coefficient mtri. If the mtri A is ugmented with the column mtri B, t the end, we get the ugmented mtri, M M denoted by (A, B) M 7 A system of (simultneous) liner equtions is sid to be homogeneous if the constnt term in ech of the equtions is zero. A system of liner homogeneous equtions cn be represented in the form AX O. For emple, the equtions +yz, +y, y+z cn be written in the mtri form s y z A X O.. Consistency of equtions A system of equtions is sid to be consistent if it hs t lest one set of solution. Otherwise it is sid to be inconsistent. Consistent equtions my hve (i) unique solution (tht is, only one set of solution) or (ii) infinite sets of solution. By wy of illustrtion, consider first the cse of liner equtions in two vribles. The equtions y 8, + y represent two stright lines intersecting t (, ). They re consistent nd hve the unique solution, y. (Fig..) y -y 8 (, ) +y Consistent ; Unique solution O Fig..

22 The equtions y, y represent two coincident lines. We find tht ny point on the line is solution. The equtions re consistent nd hve infinite sets of solution such s, y - ;, y ;, y nd so on (Fig..) Such equtions re clled dependent equtions. y O Consistent ; Infinite sets of solution. The equtions y, 8 y represent two prllel stright lines. The equtions re inconsistent nd hve no solution. (Fig..) y. (, ). (, -). (, ) -y, -y Fig.. O 8 - y - y Fig.. Inconsistent ; No solution Now consider the cse of liner equtions in three vribles. The equtions + y + z, + y + z 6, + y + z 6 re consistent nd hve only one set of unique solution viz., y, z. On the other hnd, the equtions + y + z, + y + z, + y + z re consistent nd hve infinite sets of solution such s, y, z ;, y -, z ; nd so on. All these solutions re included in +k, y -k, z k where k is prmeter. 6

23 The equtions + y + z, + y z -, +y + 7z 7 do not hve even single set of solution. They re inconsistent. All homogeneous equtions do hve the trivil solution, y, z. Hence the homogeneous equtions re ll consistent nd the question of their being consistent or otherwise does not rise t ll. The homogeneous equtions my or my not hve solutions other thn the trivil solution. For emple, the equtions + y + z, y z, +y z hve only the trivil solution viz.,, y, z. On the other hnd the equtions +y -z, y +z, +6y -z hve infinite sets of solution such s, y, z ;, y 6, z 9 nd so on. All these non trivil solutions re included in t, y t, z t where t is prmeter...6 Testing the consistency of equtions by rnk method. Consider the equtions AX B in 'n' unknowns ) If ρ(a, B) ρ(a), then the equtions re consistent. ) If ρ(a, B) ρ(a), then the equtions re inconsistent. ) If ρ(a, B) ρ(a) n, then the equtions re consistent nd hve unique solution. ) If ρ(a, B) ρ(a) < n, then the equtions re consistent nd hve infinite sets of solution. Consider the equtions AX in 'n' unkowns ) If ρ(a) n then equtions hve the trivil solution only. ) If ρ(a) < n then equtions hve the non trivil solutions lso. Emple Show tht the equtions -y +z 7, +y-z, +y +z re consistent nd hve unique solution. 7

24 8 The equtions tke the mtri form s z y 7 A X B Now (A, B) 7 M M M Applying R R (A, B) 7 M M M Applying R R R, R R R (A, B) 8 M M M Applying R R R (A, B) 8 M M M Obviously, ρ(a, B), ρ(a) The number of unknowns is. Hence ρ(a, B) ρ(a) the number of unknowns. The equtions re consistent nd hve unique solution. Emple 6 Show tht the equtions + y, y - z, + y + z re consistent nd hve infinite sets of solution. The equtions tke the mtri form s

25 9 - z y A X B Now, (A, B) - M M M Applying R R - R (A, B) M M M Applying R R +R (A, B) ~ - M M M Obviously, ρ(a, B), ρ(a). The number of unknowns is. Hence ρ(a, B) ρ(a) < the number of unknowns. The equtions re consistent nd hve infinite sets of solution. Emple 7 Show tht the equtions -y +z, -y +7z 6, -8y +z re inconsistent. The equtions tke the mtri form s 8 7 z y 6 A X B

26 Now, (A, B) M M M Applying R R - R, R R -R (A, B) ~ 8 M M M Applying R R -R (A, B) ~ 8 M M M Obviously, ρ(a, B), ρ(a) Hence ρ(a, B) ρ(a) The equtions re inconsistent. Empe 8 Show tht the equtions +y +z, +y -z, -y +z hve only the trivil solution. The mtri form of the equtions is z y A X O A Applying R R R, R R R

27 A Applying R R R A ~ 9 Obviously, ρ (A) The number of unknowns is. Hence ρ (A) the number of unknowns. The equtions hve only the trivil solution. Emple 9 Show tht the equtions +y +9z, +y +z, +y +7z hve non trivil solutions lso. The mtri form of the equtions is 7 9 z y A X O A 7 9 A 7 9, ρ (A) The number of unknowns is. Hence ρ(a) < the number of unknowns. The equtions hve non trivil solutions lso.

28 Emple Find k if the equtions + y -z, -y +z, +7y -6z k re consistent. (A, B) k 6 7 M M M, A 6 7 A 6 7, Obviously ρ(a). For the equtions to be consistent, ρ(a, B) should lso be. Hence every minor of (A, B) of order should be zero. k 6 7 Epnding nd simplifying, we get k 8. Emple Find k if the equtions + y + z, +y +z 6, +y +z k re inconsistent. (A, B) k M M M 6, A We find, A, Obviously ρ(a). For the equtions to be inconsistent, ρ(a, B) should not be.

29 (A, B) k M M M 6 Applying R R R, R R R (A, B) ~ k M M M Applying R R R (A, B) ~ 9 k M M M ρ(a, B) only when k 9 The equtions re inconsistent when k ssumes ny rel vlue other thn 9. Emple Find the vlue of k for the equtions k + y + z, -y + z, k - y + z to hve non trivil solution. A k k For the homogeneous equtions to hve non trivil solution, ρ(a) should be less thn the number of unknowns viz.,. ρ(a). Hence k k Epnding nd simplifying, we get k Emple Find k if the equtions + y +z, -y -z, +y +kz hve only the trivil solution.

30 A k For the homogeneous equtions to hve only the trivil solution, ρ(a) should be equl to the number of unknowns viz.,. k, k. The equtions hve only the trivil solution when k ssumes ny rel vlue other thn. EXERCISE. ) Find the rnk of ech of the following mtrices (i) (ii) 6 6 (iii) (iv) 7 (v) 8 6 (vi) 7 (vii) 9 9 (viii) (i) ) Find the rnks of A+B nd AB where A nd B 6 6 ) rove tht the points (, y ), (, y ) nd (, y ) re colliner if the rnk of the mtri y y y is less thn. ) Show tht the equtions +8y +z, +y +z, +y z re consistent nd hve unique solution.

31 ) Show tht the equtions y 8z, +y z, +y +6z re consistent nd hve infinite sets of solution. 6) Test the system of equtions y z, y +z, + y +8z for consistency. 7) Show tht the equtions y, 6 y re inconsistent. 8) Show tht the equtions + y + z, +y z, +y +7z 7 re not consistent. 9) Show tht the equtions +y +z, y z, +y z hve no other solution other thn, y nd z. ) Show tht the equtions +y z, y +z, + 6y z hve non trivil solutions lso. ) Find k if the equtions +y z, yz, +y z k re consistent. ) Find k if the equtions +y +z, y z, +y + z k re inconsistent. ) Find the vlue of k for the equtions y +z, +y z, -y + k z to hve non trivil solutions. ) Find k for which the equtions +y +z, +y +z nd 7 +ky +9z hve no non trivil solutions.. SOLUTION OF LINEAR EQUATIONS.. Solution by Mtri method. When A, the equtions AX B hve the unique solution given by X A - B. Emple Solve using mtrices the equtions -y, +y. The equtions cn be written in mtri form s - y A X B

32 6 A 7 The unique solution is given by X A - B y 7 y y, y Emple Solve the equtions +8y +z, +y +z -, +y -z by using mtri method. The equtions cn be written in mtri form s - 8 z y A X B A - 8 The unique solution is given by X A B. We now find A. A c A t c cofctors +(--), -(--), +(-) -(-8-), +(--), -(-8) +(8-), -(-), +(-8)

33 Now A - y z A A t c y ie., y z z, y, z. Emple 6 6 A womn invested different mounts t 8%, 8 % nd 9%, ll t simple interest. Altogether she invested Rs., nd erns Rs., per yer. How much does she hve invested t ech rte if she hs Rs., more invested t 9% thn t 8%? Solve by using mtrices. Let, y, z be the mounts in Rs. invested t 8%, 8 % nd 9% respectively. According to the problem, + y + z, 8 y z z, + y + z, +y + 6z,8, z, The equtions cn be written in mtri form s, nd

34 Now, Now, 6 y,8, z, A X B A 6 The unique solution is given by X A B We now find A. A c A t c 68 A - y z y z y 68 A A t c , 8,,,,, cofctors +(--), -(--6), +(-) -(--), +(--), -(-) +(6-), -(6-), +(-) 8,,8,, z Hence the mounts invested t 8%, 8 % nd 9% re Rs.,, Rs., nd Rs., respectively.

35 .. Solution by Determinnt method (Crmer s rule) Let the equtions be + b y + c z d, + b y + c z d, + b y + c z d. Let y b b b d d d c c c c c c,, z When, the unique solution is given by, y y, z z. Emple 7 Solve the equtions +y +z, +y +z 6, 6 +y +7z 7 by determinnt method. The equtions re +y +z +y +z 6 6 +y +7z y By Crmer's rule 9 d d d 6 ; ; z b b b b b b c c c d d d

36 z z 6 ; y y 6 8 ;, y, z. 6 Emple 8 Solve the equtions -y-, +y - by Crmer s rule. The equtions re y, +y 9 ; y 9 ; By Crmer's rule 8, y y 9 9 9, y. 8 Emple 9 A slesmn hs the following record of sles during three months for three items A, B nd C which hve different rtes of commission. Months Sles of units Totl commission A B C drwn (in Rs.) Jnury 9 8 Februry 9 Mrch 6 8 Find out the rtes of commission on the items A, B nd C. Solve by Crmer s rule. Let, y nd z be the rtes of commission in Rs. per unit for A, B nd C items respectively.

37 According to the problem, 9 +y +z 8 +y +z 9 6 +y +z 8 Dividing ech of the equtions by throughout, 9 +y + z 8 + y + z y + z 8 Now, ; y By Crmer's rule -7 ; z ; y y z z 9 7 Hence the rtes of commission for A, B nd C re Rs., Rs. nd Rs. respectively. EXERCISE. ) Solve by mtri method the equtions +y 7, + y. ) Solve by mtri method the equtions y +z, y +z, +y z ) Solve by Crmer s rule the equtions 6-7y 6, 9 y. ) Solve by determinnt method the equtions +y z, + y z, +y z. ) Solve by Crmer s rule : + y, y + z 6, z +.

38 6) Two types of rdio vlves A, B re vilble nd two types of rdios nd Q re ssembled in smll fctory. The fctory uses vlves of type A nd vlves of type B for the type of rdio, nd for the rdio Q it uses vlves of type A nd vlves of type B. If the number of vlves of type A nd B used by the fctory re nd 8 respectively, find out the number of rdios ssembled. Use mtri method. 7) The cost of kg. of whet nd kg. of sugr is Rs. 7. The cost of kg. whet nd kg. of rice is Rs. 7. The cost of kg. of whet, kg. of sugr nd kg. of rice is Rs. 7. Find the cost of ech per kg., using mtri method. 8) There re three commodities X, Y nd Z which re bought nd sold by three delers A, B ndc. Deler A purchses units of X nd units of Z nd sells units of Y, deler B purchses units of X, units of Y nd sells 7 units of Z nd deler C purchses units of Y, unit of Z nd sells units of X. In the process A erns Rs. nd C Rs. but B loses Rs.. Find the price of ech of the commodities X, Y nd Z, by using determinnts. 9) A compny produces three products everydy. The totl production on certin dy is tons. It is found tht the production of the third product eceeds the production of the first product by 8 tons while the totl production of the first nd third product is twice the production of second product. Determine the production level of ech product by using Crmer s rule.. STORING INFORMATION We know tht mtri provides convenient nd compct nottion for representtion of dt which is cpble of horizontl nd verticl divisons. Now we shll study the pplictions of mtrices in the study of (i) Reltions on sets (ii) Directed routes nd (iii) Cryptogrphy. Let us first recll the concept of reltions on sets studied in erlier clsses.

39 Reltion : A reltion R from set A to set B is subset of the crtesin product A B. Thus R is set of ordered pirs where the first element comes from A nd the second element comes from B. If (, b) R we sy tht is relted to b nd write R b. If (, b) R, we sy tht is not relted to b nd write R b. If R is reltion from set A to itself then we sy tht R is reltion on A. For emple, Let A {,,, 6} nd B {, 6, 9} Let R be the reltion from A to B defined by Ry if divides y ectly. Then R {(, ), (, 6), (, 6), (, 9), (, ), (6, 6)} Inverse reltion. Let R be ny reltion from set A to set B. Then the inverse of R, denoted by R - is the reltion from B to A which consists of those ordered pirs which, when reversed, belong to R. For emple, the inverse of the reltion R {(, y) (, z) (, y)} from A {,, } to B {, y, z} is R - {(y, ) (z, ) (y, )} from B to A. Composition of reltions. Let A, B nd C be sets nd let R be reltion from A to B nd let S be reltion from B to C. i.e. R is subset of A B nd S is subset of B C. Then R nd S give rise to reltion from A to C denoted by R o S nd defined by R o S {(, c) / there eists b B for which (, b) R nd (b, c) S}. For emple, Let A {,,, }, B {, b, c, d}nd C {, y, z} nd let R {(, ), (, d), (, ), (, b), (, d)} nd S {(b, ), (b, z), (c, y), (d, z)} Then R o S {(, z), (, ), (, z)}

40 Types of reltions. A reltion R on set A is refleive if R for every A. tht is (, ) R for every A. A reltion R on set A is symmetric if whenever Rb then br tht is, whenver (, b) R then (b, ) R. A reltion R on set A is trnsitive if whenever Rb nd brc then Rc tht is, whenever (, b), (b, c) R then (, c) R. A reltion R is n equivlence reltion if R is refleive, symmetric nd trnsitive. For emple, consider the following three reltions on A {,, }. R {(, ), (, ), (, ), (, )} S {(, ), (, ), (, ), (, ), (, )} T {(, ), (, ), (, ), (, )} R is not refleive, S is refleive nd T is not refleive. R is not symmetric, S is symmetric nd T is not symmetric. R is trnsitive, S is trnsitive nd T is not trnsitive... Reltion mtrices. A mtri is convenient wy to represent reltion R from X to Y. Such reltion cn be nlysed by using computer. We lbel the rows with the elements of X (in some rbitry order) nd we lbel the columns with the elements of Y (gin in some rbitry order). We then set the entry in row nd column y to if R y nd to otherwise. The mtri so obtined is clled the reltion mtri for R. Emple Find the reltion mtri for the reltion R from {,, } to {, 6, 7, 8} where R is defined by Ry if divides y ectly. R {(, 6), (, 8), (, 6), (, 8)}

41 The reltion mri for R is R Emple Let S {,,, }. Let R be the reltion on S defined by mrn if m < n. Write out the reltion mtri for R. R {(, ), (, ), (, ), (, ), (, ), (, )} The reltion mtri for R is R Emple Given reltion mtri R y reltion R in the form of set of ordered pirs. R {(, ), (y, ), (y, )} Write down the Mtri for inverse reltion If R is reltion mtri, then its trnspose R t represents the inverse reltion R -. Emple Let A {,, } Define reltion R on A by mrn if mn m. Find the reltion mtri for R. Use it to find the reltion mtri for the inverse reltion R -. R {(, ), (, ), (, ), (, ), (, )}

42 6 Reltion mtri for R is R Reltion mtri for R - is R - R t Reltion mtri for composition of reltions. The reltion mtri for R o R is obtined by replcing ech non zero element in the mtri product R R by. Emple Let R be reltion from X {,, } to Y {, b, c} defined by R {(, ), (, b), (, ), (, b)} nd let R be the re ltion from Y to Z {, y, z} defined by R {(, ), (, y), (b, y), (b, z)}. Find the reltion mtrices for R nd R nd using them find the reltion mtri for R o R. The reltion mtri for R is R c b The reltion mtri for R is R c b z y The mtri product R R is

43 7 R R Replcing ech non zero element in R R by we get, R o R z y Type of reltion s reveled by its mtri. A reltion R is refleive if its mtri hs only s on the min digonl. A reltion R is symmetric if its mtri is symmetric Tht is, i j j i for ll i, j. A reltion R is trnsitive if whenever the entry i, j in the mtri product R is non zero, the entry i, j in the reltion mtri R is lso non zero. Emple Given reltion R {(, ), (b, b), (c, c), (d, d), (b, c), (c, b)} on A {, b, c, d}. Find the reltion mtri for R nd using it identify the type of the reltion. The reltion mtri for R is R d c b d c b The mtri hs only s on the min digonl. Hence the reltion is refleive. The mtri is symmetric. Hence the reltion is symmetric. The mtri product.

44 8 R Whenever n entry in R is non zero, the corresponding entry in R is lso non zero. Hence the reltion is trnsitive. Thus the reltion R is refleive, symmetric nd trnsitive nd hence n equivlence reltion. Emple 6 Let R be the reltion on S {,,, } defined by mrn if m-n <. Find the reltion mtri for R nd using it, identify the type of the reltion. R {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} The reltion mtri for R is R The mtri hs only s on the min digonl. Hence the reltion is refleive. The mtri is symmetric. Hence the reltion is symmetric. R Now element in R is non zero but element in R is zero. Hence the reltion is not trnsitive.

45 .. Route Mtrices. A directed route is set of points,,..., n clled vertices together with finite set of directed edges ech of which joins n ordered pir of distinct vertices. Thus the directed edge ij is different from the directed edge ji. There my be no directed edge from verte i to ny other verte nd my not be ny directed edge from ny verte to the verte i. Also there cn be no loops nd multiple directed edges joining ny two vertices. Ech edge of directed route is clled stge of length. A pth from the verte i to the verte j is sequence of directed edges from i to j. It should strt from i nd end with j, repetition of ny of the vertices including i nd j being llowed. If there is pth from i to j we sy tht j is ccessible from i or tht i hs ccess to j. A directed route is sid to be strongly connected if for ny pir of vertices i nd j, there is pth from i to j nd pth from j to i. Otherwise the route is sid to be not strongly connected. A directed route is represented by its route mtri. If G is directed route with n vertices then the n n mtri A where the (i, j) th element is if there is directed edge from i to j nd zero otherwise is clled the route mtri for the directed route G. It is to be noted tht the number of s in the route mtri will be equl to the number of edges in its route. Route mtrices re lso reltion mtrices. The route mtrices re squre mtrices wheres the reltion mtrices need not be squre mtrices. th mtri of directed route is the mtri { ij } such tht ij Emple 7, if there is pth from i to j, otherwise. Find the route mtri for ech directed route given below: 9

46 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) Emple 8 Drw the directed route for ech route mtri given below : (i) (ii) 6 6. Fig.. Fig.. Fig..6 Fig..7

47 (i) (ii) Fig..8 Fig..9 6 Theorems on directed routes. Theorem Si theorems (without proof) re stted below. If A is the route mtri then the i, j th element in A r is the number of wys in which i hs ccess to j in r stges Theorem If A is the route mtri then the sum of ll the elements in the j th column of A r is the number of wys in which j is ccessible by ll individuls in r stges. Theorem If A is the route mtri then the i, j th element in A + A + A A r is the number of wys in which i hs ccess to j in one, two,... or r stges. Theorem If A is the route mtri then the sum of ll the elements in the j th column in A + A + A A r is the number of wys in which J is ccessible by ll individuls in one, two,... or r stges. Theorem A directed route with n vertices nd hving the route mtri A is strongly connected if A + A + A A n hs no zero entries.

48 Theorem 6 If there re n vertices in route nd A is its route mtri then its pth mtri is got by replcing ech non zero element in A + A + A A n by. Now we shll hve few emples illustrting the ppliction of these theorems. Emple 9 Consider the following directed route G. 6 (i) Find the route mtri of G. (ii) (iii) (iv) (v) (vi) Find the number of wys in which hs ccess to in stges. Indicte the pths. Find the number of pths from to in stges Indicte the pths. Find the number of wys in which 6 cn be ccessed by others in stges. Find the number of wys in which hs ccess to in one, two or three stges. Find the number of wys in which 6 cn be ccessed by others in or less stges. (i) Route mtri of G is Fig..

49 A 6 6 (ii) A 6 6 A 6 6 hs ccess to in stges in wys. The pths re,, 6, 6, nd 6. (iii) The number of pths from to in stges is. The pths re, 6,, 6, 6 nd (iv) The number of wys in which 6 cn be ccessed by others in stges (v) A + A + A hs ccess to in one, two, or three stges in 7 wys.

50 (vi) The number of wys in which 6 cn be ccessed by others in or less stges Emple Show, by using the route mtri nd its powers, tht the directed route G given below is strongly connected. The route mtri of G is A Since there re vertices, let us find A + A + A +A. A, A A, A + A + A +A 6 There is no zero entry in this mtri G is strongly connected. Fig..

51 Emple Given directed route : Find the route mtri of G nd using its powers emine whether G is strongly connected. Route mtri of G is A Since there re vertices, let us find A + A + A + A. A A A A +A +A +A There is zero entry in this mtri. Hence G is not strongly connected. Fig..

52 6 Emple Show tht the directed route G with route mtri A is strongly connected. Solution: Since there re vertices, we find A + A + A. A, A. A + A + A There is no zero entry in this mtri. G is strongly connected. Emple Find the pth mtri of the directed route given below by using the powers of its route mtri. The route mtri is A Since there re vertices, let us find A + A + A + A. Fig..

53 7 A, A, A A+A +A + A Replcing ech non zero entry by, we get the pth mtri Emple The route mtri of directed route G is A Find the pth mtri of G without using the powers of A. The directed route G is Fig..

54 The pth mtri is written directly from G... Cryptogrphy Cryptogrphy is the study of coding nd decoding secret messges. A non singulr mtri cn be effectively used for this. The following emple illustrtes the method. Emple Using the substitution scheme, A B C D E F G H I J K L M N O Q R S T U W X Y Z nd the mtri A (i) Code the messge : HARD WORK nd (ii) Decode the messge : 98, 9,, 9, 8, (i) Using the substitution scheme, H A R D W O R K Grouping them, 8 8 8,,, 8

55 Applying the trnsformtion AX B, The coded messge is, 7,,, 6, 6,, 7 (ii) 98, 9,, 9, 8, Grouping them, 98 8,, 9 9 Now solve for AX B. X A B, where A Hence we hve ,,,,, 9 9 Using the substitution scheme, the messge decoded is S A E M E

56 Note When non singulr mtri is used, we group the numbers in twos in order. If number is left out without being pired, we cn include one etrneous number of our own s the lst number nd ignore its decoding, when the process is over. When mtri is used we group the numbers in threes in order. If need be one or two etrneous numbers my be included nd dispensed with when the process is over. EXERCISE. ) Find the reltion mtri for the reltion R from {,, 8, 9} to {6, 8, 9, } defined by R y if divides y ectly. ) Let S {,, 6, 9} nd R be the reltion on S defined by m R n if m > n. Write out the reltion mtri for R. l m ) Given the reltion mtri R b c Write the reltion R in the form of set of ordered pirs. ) Given the reltion mtri R b c d b Write down the mtri for the inverse reltion R. ) Let R be the reltion from X {,, 9} to Y {,, 8} defined by Ry if + y >. Let S be the reltion from Y to Z {,, } defined by ysz if y < z. Find the reltion mtrices for R, S nd Ro S. 6) Find the reltion mtri for the reltion. R {(, ), (, ), (, ), (, )} on {,,, }. Use it to identify the type of the reltion. 7) Find the reltion mtri for the reltion R {(, ), (, ), (, ), (, ), (, )} on {,,, }. Using it, decide the type of the reltion.

57 8) Find the reltion mtri for the reltion R {(, ), (, ), (, ), (, )} on {,,, }. Using it, decide the nture of the reltion. 9) Find the reltion mtri for the reltion R {(, ), (, )} on {,,, }. Using it, identify the type of the reltion. ) Find the route mtri for ech of the directed routes : (i) (ii) (iii) Fig.. Fig..6 (iv) Fig..7 Fig..8 (v) (vi) 6 Fig..9 Fig.. ) Drw the directed route for ech of the following route mtrices (i) A (ii)

58 A B C D A B ) Given the route mtri M C D for directed route G. Using the powers of M, find the number of pths from C to A with t most three stges. Indicte the pths. ) Given the directed route G : X Y (i) Find the route mtri of G. (ii) Z Fig.. Find whether G is strongly connected, by using the powers of the route mtir. (iii) Find the pth mtri of G. ) Given the following directed route G : W (i) Find the route mtri of G. (ii) Find the number of pths of length from to. Indicte them. (iii) Find the number of pths of length from to. Indicte them. (iv) Fig.. Find the number of pths from to of length or less. Indicte them.

59 (v) (vi) Find the number of wys in which cn be ccessed by others in one, two or three stges. Is G strongly connected? (vii) Find the pth mtri of G. ) Given directed route G : Fig.. Find the route mtri of G nd using its powers show tht G is not strongly connected. 6) Given the route mtri for directed route G : Show tht G is strongly connected. 7) Given the directed route G : Fig.. Find the route mtri nd using its powers, find the pth mtri. 8) The route mtri of directed route G is A Find its pth mtri without using the powers of A.

60 9) Given directed route G : Fig.. Find its route nd pth mtrices. ) Using the substitution scheme, A B C D E F G H I J K L M N O Q R S T U W X Y Z nd the mtri A (i) Code the messge : CONSUMER nd (ii) Decode the messge : 68, 8, 8, 6, 6,, 8, 7.. INUT - OUTUT ANALYSIS Consider simple economic model consisting of two industries A nd A, where ech produces only one type of product. Assume tht ech industry consumes prt of its own output nd rest from the other industry for its opertion. The industries re thus interdependent. Further ssume tht whtever is produced is consumed. Tht is the totl output of ech industry must be such s to meet its own demnd, the demnd of the other industry nd the eternl demnd tht is the finl demnd. Our im is to determine the output levels of ech of the two industries in order to meet chnge in finl demnd, bsed on knowledge of the current outputs of the two industries, of course under the ssumption tht the structure of the economy does not chnge.

61 Let i j be the rupee vlue of the output of A i consumed by A j, i, j, Let nd be the rupee vlue of the current outputs of A nd A respectively. Let d nd d be the rupee vlue of the finl demnds for the outputs of A nd A respectively. These ssumptions led us to frme the two equtions Tht is + + d + + d () } Let b ij ij j, i, j, b, b, b Then equtions () tke the form b + b + d b + b + d These cn be rerrnged s (b ) b d, b, b + (b ) d This tkes the mtri form b b d b b d Tht is (I B) X D b Where B b Solving this X (I B) - D. b b d, X nd D d The mtri B is known s the technology mtri.

62 Hwkins - Simon conditions ensure the vibility of the system. If B is the technology mtri then Hwkins Simon conditions re (i) the min digonl elements in I B must be positive nd (ii) I B must be positive. Emple 6 The dt below re bout n economy of two industries nd Q. The vlues re in lkhs of rupees. roducer User Finl Demnd Totl Output Q 6 Q 8 Find the technology mtri nd test whether the system is vible s per Hwkins - Simon conditions. With the usul nottion we hve, 6,,, 8, Now b 6, b, b, b 8. The technology mtri is B I - B The min digonl elements in I B viz., nd re positive. Also 6

63 I B. I B is positive. The two Hwkins - Simon conditions re stisfied. Hence the system is vible. Emple 7 In n economy there re two industries nd Q nd the following tble gives the supply nd demnd positions in crores of rupees. roducer User Finl Demnd Totl Output Q Q 6 Determine the outputs when the finl demnd chnges to for nd for Q. With the usul nottion we hve,,, 6 Now, b, b b, b The technology mtri is B I - B I - B 7 7, 6. 6

64 Now, (I - B) - 7 X (IB) D The output of the industry should be Rs. crores nd tht of Q should be Rs. crores. EXERCISE. ) The technology mtri of n economic system of two industries is. Test whether the system is vible s per Hwkins Simon conditions. ) The technology mtri of n economic system of two industries 9 is. Test whether the system is vible s per Hwkins Simon conditions. ) The technology mtri of n economic system of two industries is 7. Find the output levels when the finl demnd chnges to nd units. ) The dt below re bout n economy of two industries nd Q. The vlues re in millions of rupees. roducer User Finl Demnd Totl Output Q Q Determine the outputs if the finl demnd chnges to for nd for Q.

65 ) Suppose the inter-reltionship between the production of two industries nd Q in yer (in lkhs of rupees) is roducer User Finl Demnd Totl Output Q Q 6 Find the outputs when the finl demnd chnges to (i) for nd 8 for Q (ii) 8 for nd for Q. 6) In n economy of two industries nd Q the following tble gives the supply nd demnd positions in millions of rupees. roducer User Finl Demnd Totl Output Q 6 Q 8 8 Find the outputs when the finl demnd chnges to 8 for nd for Q. 7) The dt below re bout n economy of two industries nd Q. The vlues re in crores of rupees. roducer User Finl Demnd Totl Output Q 7 7 Q Find the outputs when the finl demnd chnges to for nd 6 for Q. 8) The inter - reltionship between the production of two industries nd Q in crores of rupees is given below. roducer User Totl Output Q 8, Q 6, 9

66 If the level of finl demnd for the output of the two industries is, for nd, for Q, t wht level of output should the two industries operte?.6 TRANSITION ROBABILITY MATRICES These re mtrices in which the individul elements re the probbilities of trnsition from one stte to nother of n event. The probbilities of the vrious chnges pplied to the initil stte by mtri multipliction gives forecst of the succeeding stte. The following emples illustrte the method. Emple 8 Two products A nd B currently shre the mrket with shres 6% nd % ech respectively. Ech week some brnd switching tkes plce. Of those who bought A the previous week, 7% buy it gin wheres % switch over to B. Of those who bought B the previous week, 8% buy it gin wheres % switch over to A. Find their shres fter one week nd fter two weeks. If the price wr continues, when is the equilibrium reched? Trnsition robbility mtri A B A T.7. B..8 Shres fter one week A B A B A.7. A B (.6. ) (.. ) B..8 A %, B % Shres fter two weeks A B A B A.7. A (.. ). B..8 A %, B % (.) 6 B

67 Equilibrium At equilibrium we must hve (A B) T (A B) where A + B.7. (A B)..8 (A B).7 A +. B A.7 A +. (-A) A Simplifying, we get A. Equilibrium is reched when A s shre is % nd B s shre is 6% Emple 9 A new trnsit system hs just gone into opertion in city. Of those who use the trnsit system this yer, % will switch over to using their own cr net yer nd 9% will continue to use the trnsit system. Of those who use their crs this yer, 8% will continue to use their crs net yer nd % will switch over to the trnsit system. Suppose the popultion of the city remins constnt nd tht % of the commuters use the trnsit system nd % of the commuters use their own cr this yer, (i) wht percent of commuters will be using the trnsit system fter one yer? (ii) wht percent of commuters will be using the trnsit system in the long run? Trnsition robbility Mtri S C S T.9. C..8 ercentge fter one yer S C S C S.9. A (.. ). C..8 (.) 6 B

68 S %, C % Equilibrium will be reched in the long run. At equilibrium we must hve (S C) T (S C) where S + C (S C) S +.C S.9S +.(-S) S (S C) Simplifying, we get S.67 67% of the commuters will be using the trnsit system in the long run. EXERCISE.6 ) Two products nd Q shre the mrket currently with shres 7% nd % ech respectively. Ech week some brnd switching tkes plce. Of those who bought the previous week, 8% buy it gin wheres % switch over to Q. Of those who bought Q the previous week, % buy it gin wheres 6% switch over to. Find their shres fter two weeks. If the price wr continues, when is the equilibrium reched? ) The subscription deprtment of mgzine sends out letter to lrge miling list inviting subscriptions for the mgzine. Some of the people receiving this letter lredy subscribe to the mgzine while others do not. From this miling list, 6% of those who lredy subscribe will subscribe gin while % of those who do not now subscribe will subscribe. On the lst letter it ws found tht % of those receiving it ordered subscription. Wht percent of those receiving the current letter cn be epected to order subscription? ) Two newsppers A nd B re published in city. Their present mrket shres re % for A nd 8% for B. Of those who bought A the previous yer, 6% continue to buy it gin while % switch over to B. Of those who bought B the previous yer, % buy it gin nd % switch over to A. Find their mrket shres fter two yers. 6

69 6 EXERCISE.7 Choose the correct nswer ) If the minor of equls the cofctor of in i j then the minor of is () (b) (c) (d) ) The Adjoint of is () (b) (c) (d) ) The Adjoint of is () (b) (c) (d) ) If AB BA A I then the mtri B is () the inverse of A (b) the trnspose of A (c) the Adjoint of A (d) A ) If A is squre mtri of order then AdjA is () A (b) A (c) A (d) A 6) If A then AdjA is () (b) (c) (d) + 7) The inverse of is () (b) (c) (d)

70 .8.6 8) If A then A is () (b) (c) (d) ) For wht vlue of k the mtri A, k where A hs no inverse? () (b) (c) (d) ) If A then A A is 7 () (b) A (c) I (d) A. ) The rnk of n n n mtri ech of whose elements is is () (b) (c) n (d) n ) The rnk of n n n mtri ech of whose elements is is () (b) (c) n (d) n ) The rnk of zero mtri is () (b) (c) (d) ) The rnk of non singulr mtri of order n n is () n (b) n (c) (d) ) A system of liner homogeneous equtions hs t lest () one solution (b) two solutions (c) three solutions (d) four solutions 6) The equtions AX B cn be solved by Crmer s rule only when () A (b) A (c) A B (d) A B y 7) The inverse of the reltion is () y b (b) b b y 6 (c) y b (d) b y

71 b 8) The reltion R () Refleive (c) Trnsititve b is (b) Symmetric (d) Refleive nd symmetric 9) The number of Hwkins - Simon conditions for the vibility of n input - output model is () (b) (c) (d) A A.7 B. ) If T is trnsition probbility mtri, then the B.8 vlue of is (). (b). (c). (d).7 6

72 ANALYTICAL GEOMETRY. CONICS Intersections of cone by plne The prbol, ellipse nd hyperbol re ll members of clss of curves clled conics. The bove three curves cn be obtined by cutting cone with plne nd so they re clled conics (Fig..). rbol Circle Hyperbol Ellipse Fig.. A conic is the locus of point which moves in plne such tht its distnce from fied point in the plne bers constnt rtio to its distnce from fied stright line in tht plne. Focus, Directri, Eccentricity: In the bove definition of conic, the fied point is clled the focus, the fied line the directri nd the constnt rtio, the eccentricity of the conic. The eccentricity is usully denoted by the letter e. 66

73 In the Fig.., S is the focus, the line LM is the directri nd S e M M The conic is S (focus) prbol if e, L n ellipse if e < nd hyperbol if e >. directri.. The generl eqution of conic Fig.. We know tht conic is the locus of point moving such tht its distnce from the focus bers constnt rtio to its distnce from the directri. Let the focus be S(, y ) nd the directri be A + By +C Let the eccentricity of the conic be e nd (, y) be ny point on it. Then S ( ) ( ) + y y erpendiculr distnce of from A + By + C is A + By + C M + A + B S e M ( ) + ( y y) A+ By+ C e ± A + B ( A+ By + C) or ( ) + (y y ) e ( A + B ) Simplifying, we get n eqution of the second degree in nd y of the form + hy + by + g + fy + c This is the generl eqution of conic. Remrks : + hy + by + g + fy + c represents, (i) pir of stright lines if bc + fgh f bg ch 67

74 (ii) circle if b nd h If the bove two conditions re not stisfied, then + hy + by + g + fy + c represents, (iii) prbol if h b (iv) n ellipse if h b < (v) hyperbol if h b > Emple The eqution + y + y + + y + 6 represents conic. Identify the conic. Compring the given eqution, + y + y + + y + 6 with the generl second degree eqution in nd y + hy + by + g + fy + c we get, h, b h - b () () The given conic is prbol. Emple Solution: Identify the conic represented by 6 + y y -. Here, 6, h, b h b 6 - < The conic is n ellipse. EXERCISE. Identify the conics represented by the following equtions: ) 6y + 9y + 6 8y

75 { ) 7 + y y + + 6y 7 ) 7 + y + 7y 6 y +. ARABOLA.. Stndrd Eqution of prbol D y M (,y) { A O S(,) N > + Let S be the focus nd the line DD be the directri. Drw SA perpendiculr to DD cutting DD t A. Let SA. Tke AS s the is nd Oy perpendiculr to AS through the middle point O of AS s the y is. Then S is (,) nd the directri DD is the line +. Let (,y) be ny point on the prbol. Drw M DD nd N O M NA NO + OA +. S ( ) + y Then S M e [ is point on the prbol] or, S e (M) or, ( - ) + y ( + ) (Πe ) or, y D Fig.. This is the stndrd eqution of the prbol. 69

76 Note (i) (ii) In ny prbol the line which psses through the focus nd is perpendiculr to the directri is clled the is of the prbol, nd the point of intersection of the curve nd its is is clled the verte. The chord which psses through the focus nd is perpendiculr to the is is clled the ltus rectum... Trcing of the prbol y ) () utting y, the only vlue of we get is zero. The curve cuts the is t (,) only. (b) If <, y is imginry. Hence the curve does not eist for negtive vlues of. (c) The eqution of the prbol is unltered if y is replced by y. Hence the curve is symmetricl bout -is. (d) As increses, y lso increses. As, y +. Hence the curve diverges nd ssumes the form s shown in Fig... M D y L A O S N > D ) Directri : The directri is line prllel to the y -is nd its eqution is or, +. ) The -is is the is of the prbol nd the y -is is the tngent t the verte. 7 L Fig..

77 ) Ltus rectum: Through S, LSL be drwn to AS. Corresponding to, y or, y + SL SL. So, LL. LL is clled Ltus rectum of the prbol. SL (or SL ) is the Semi - ltus rectum. OS LL y y y > > > y - y -y Fig.. Note In Fig., y is prbol which lies only on the negtive side of the -is. y is prbol whose is of symmetry is the y -is nd it lies on the positive side of y -is. y lies on the negtive side of y -is. Eqution y y - y -y Focus (,) (-, ) (,) (,-) erte (,) (,) (,) (,) Directri y y Ltus rectum Ais y y Emple Find the eqution of the prbol whose focus is the point (, ) nd whose directri is the stright line + y + Let (, y) be point on the prbol. If M is drwn perpendiculr to the directri, 7

78 S where S is the focus of the prbol. M S M or, ( ) + (y ) + + y y y + + ( + y + ) + y y + +y + +y +y + y + y y +. This is the required eqution. Emple Find the focus, ltus rectum, verte nd directri of the prbol y y + 7. The given eqution cn be written s y y 8 7 y y (completing the squre) (y ) 8 ( ) Chnging the origin to the point (,) by putting y Y, X, the eqution of the prbol is Y 8X. The verte is the new origin nd ltus rectum is 8. So focus is the point (,) in the new coordintes. So, with respect to the originl es the focus is the point (,) nd the directri is the line X + or, + or The verte is (X, Y ) (, y ) (, ) Emple Find the verte, focus, is, directri nd length of semiltus rectum of the prbol y + - y + 67

79 y + y + 67 or y y 67 which cn be rerrnged nd written s (y y) 67 { y + } y 67 y 67 (y ) 67 ( + 7 ) > (y ) ( 7 ) To bring it to the form Y X, set X 7 nd Y y We now get Y X. Here nd so We now tbulte the results. Referred to (X, Y) 7 Referred to (, y) -X - 7, y Y + erte (,) 7, 7, Ais Y (X -is) y - or y Focus (, ) (, ) 7, + 7, Directri X i.e.x or Semi ltus rectum Note 7 We could lso hve tken trnsformtion X + nd Y y so tht we would hve got Y X nd compred it with y

80 Emple 6 The verge cost y of monthly output kgs. of firm producing metl is Rs. ( - + ). Show tht the verge vrible cost curve is prbol. Find the output nd verge cost t the verte of the prbol. Solution: The verge vrible cost curve is y + y + y ( ) + 7 () y 7 () (y 7.) X Y where X, Y y 7. nd. Thus we get the verge vrible cost curve s prbol whose verte is (X, Y ) or (, y 7.) or (, 7.) Hence the output nd verge cost t the verte re kgs. nd Rs. 7. respectively. Emple 7 The supply of commodity is relted to the price by the reltion p. Show tht the supply curve is prbol. Find its verte nd the price below which supply is? Solution: The supply price reltion is given by, 7

81 (p ) (p ) X where X nd p the supply curve is prbol whose verte is (X, ) (, p ) (, ) nd supply is zero below p. Emple 8 The girder of rilwy bridge is prbol with its verte t the highest point, which is metres bove the spn of length metres. Find its height metres from the mid point. Let the prbol be y. This psses through (,7) (7) () (7) 7 Hence the prbol is y 7 Now B (, ) lies on the prbol 7 or 9 7. Required height.m Emple 9 (,) B > y A h A (+, -7) (, 7) The profit Rs. y ccumulted in lkhs in months is given by y +8-. Find the best time to end the project < p O Fig..7 (,) Fig..6

82 8 y ( 7) y 9 ( 7 + ) y + 9 ( 7 ) (9y) ( 7 ) (y 9) Required time 7 months EXERCISE.. Find the equtions of the prbols with the following foci nd directrices: () (, ) ; + y (b) (, ) y (c) (, ) ; y + (d) (,), y + ) Find the verte, is, focus nd directri of the following prbol: () y (c) y 8 76 (b)y (d) 6y ) Find the foci, ltus rect, vertices nd directrices of the following prbols: () y + y + (b) y + y (c) y 8 9 (d) y + ) The verge vrible cost of monthly output of tonnes of firm producing vluble metl is Rs Show tht the verge vrible cost curve is prbol. Find lso the ouput nd the verge cost t the verte of the prbol. Note The point (, y ) lies outside, on or inside the prbol ccording s y is greter thn, equl to or less thn zero.

83 . ELLISE.. Stndrd Eqution of ellipse: D M y B L (,y) D M Z A S C S N A Z > D B L D Let S be the focus nd DD be the directri. Drw SZ perpendiculr to DD. Let A, A divide SZ internlly nd eternlly respectively in the rtio e: where e is the eccentricity. Then A, A re points on the ellipse. Let C be the mid-point of AA nd let AA. Tke CA to be the is nd Cy which is to CA to be the y is. C is the origin. SA AZ e, S A A Z e SA e (AZ) () A S e (A Z) () () + () SA + A S e (AZ + A Z) AA e (CZ CA + A C + CZ) e(cz) (since CA CA ) CZ e Fig..8 () () A S SA e (A Z AZ) A C + CS (CA CS) e (AA ) 77

84 or CS e. CS e So, S is the point (e, ). Let (, y) be ny point on the ellipse, drw M DD nd N CZ. M NZ CZ CN S M e e (since the point lies on the ellipse) S e M ( e) + y e ( e ) ( e) e + e + y e + e ( e ) + y ( e ) y + ( e ) ut b ( e ) y Hence + b ( > b) This is the equiton of ellipse in the stndrd form... y Trcing of the Ellipse + b (i) The curve does not pss through the origin. When y, +. Therefore it meets the is t the points (+, ). Similrly it meets the y is t the points (, + b). (ii) The curve is symmetricl bout both the es of coordintes since both the powers of nd y re even. If (,y) be point on the curve, so lso re (, y), (, y) nd (, y). (iii) We my put the eqution of the ellipse in the form b y + 78

85 (iv) If >. i.e. if either > or <, becomes negtive so tht is imginry. Hence there is no point of the curve lying either to the right of the line or to the left of the line. If < i.e. if < <, the epression under the rdicl sign is positive nd we get two equl nd opposite vlues of y. The curve lies entirely between these two lines. Note tht is tngent t A(, ) nd is tngent t A (, ) Similrly writing the eqution in the form + b y b we find tht the curve does not etend bove the line y b nd below the line y b. In fct the curve lies entirely between the lines y b, nd y b which re respectively tngents to the curve t B nd B. (v) If increses from to, y decreses from b to. Similrly if y increses from to b, decreses from to. (vi) Ltus rectum : Through S, LSL be drwn perpendiculr to AS. Corresponding to e, we hve e y b b + or y b (e ) b b or y + b SL SL b. So, LL rectum of the ellipse. 79 b is the ltus These informtion bout the curve re sufficient to enble us to find the shpe of the curve s given in the Fig..9. Unlike the prbol, the ellipse is closed curve.

86 y M B M Z A S (-e,) C S (e,) A Z > M B Fig..9 An importnt property If is ny point on the ellipse whose foci re S nd S, then S + S where is the length of the mjor is... Centre, vertices, foci, es nd directrices for the y ellipse + ( > b) b (i) Centre We hve seen tht if (, y) be point on the curve, then (, y) is lso point on the curve. Agin if (, y) is point on the curve, (, y) is lso point on the curve. This shows tht every line through C meets the curve t two points equidistnt from C. Thus every chord through C is bisected t C. The point C is therefore clled the centre of the ellipse. C(,) is the middle point of AA. (ii) ertices The points A nd A where the line joining the foci S nd S meets the curve re clled the vertices of the ellipse. A is the point (, ) nd A is the point (, ). (iii) Foci The points S(e, ) nd S (e, ) re the foci of the ellipse. (iv) Aes The two lines AA nd BB with respect to which the curve is symmetricl re clled respectively the mjor is nd the minor is of the ellipse. 8 M

87 Since e <, e is lso less thn unity. Therefore, b ( e ) is less thn so tht b <. Thus BB < AA. AA is clled the mjor is nd BB the minor is. The segment CA is clled the semi-mjor is nd the segment CB b the semi minor is. (v) Directrices Eqution to the directri MZ in the Fig..9 is e Eqution to the directri M Z in the Fig..9 is e Due to symmetry, there re two directrices. (vi) b (e ) \ e b Emple Find the eqution of the ellipse whose eccentricity is, one of the foci is (-, ) nd the corresponding directri is - y +. Given the focus is S(,), directri is y + nd e Let (, y ) be ny point on the ellipse. Then S e M where M is the perpendiculr distnce of y + from. ( + ) + (y ) y + + 8( + ) + 8 (y ) ( y + ) 7 + y + 7y + y + 7 Locus of (, y ), i.e., the eqution of the ellipse is 7 + y + 7y + y + 7 8

88 Emple Find the eqution of the ellipse whose foci re (,) nd (-, ) nd eccentricity is. Solution: We know tht S (e, ) nd S (e,) re the foci of the y ellipse + b, ( > b) Now the foci re (,) nd (-, ) nd e e nd e or 6 The centre C is the mid-point of SS nd hence C is (,). S nd S re on the is. So the eqution of the ellipse is of the y form + b b ( e ) 6 ( ) the equiton of the ellipse is y + 6 Emple Find the eccentricity, foci nd ltus rectum of the ellipse 9 + 6y. The eqution of the ellipse is y 9 + 6y y The given eqution is of the form + b. Then nd b e b 9 7 6

89 The foci re S (e, ) nd S (e, ). or, S( 7, ) nd S (- 7, ) Emple The ltus rectum b ( ) 9 Find the centre, eccentricity, foci nd directrices of the ellipse + y y - The given eqution cn be written s ( 6) + (y + 8y) ( ) + (y + y) ( + ) + (y + y + ) () + (y + ) + + ( ) ( y + ) + If we put X nd Y y + in the bove eqution X we get, Y + y which cn be compred with the stndrd eqution + b. b ( e ) gives ( e ) or e e Now let us tbulte the results: Referred to (X, Y) Referred to (, y) X+, y Y- Centre (,) (+, ) (, ) Foci Directrices (+e,) (,) nd (, ) (, ) nd (, ) X + e or X + + or nd 8

90 EXERCISES. ) Find the eqution of the ellipse whose (i) focus is (, ) directri is y + 6 nd eccentricity is (ii) focus is (, ) directri is + y - nd eccentricity is 6 (iii) focus is (, ) directri is y + nd e ) Find the eqution of the ellipse whose (i) foci re (, ) nd (, ) nd e (ii) foci re (, ) nd (, ) nd e 8 (iii) the vertices re (, + ) nd foci re (, + ). ) Find the centre, vertices, eccentricity, foci nd ltus rectum nd directrices of the ellipse. (i) 9 + y 6 (ii) 7 + y - + y + 79 (iii) 9 + 6y + 6 y 9. HYERBOLA.. Stndrd Eqution of Hyperbol y B D M (,y) S A Z C Z A S N > B Fig.. Let S be the focus nd DD be the directri. Drw SZ DD. Let A, A divide SZ internlly nd eternlly respectively 8 D

91 in the rtio e: where e is the eccentricity; then A, A re points on the hyperbol. Tke C, the mid-point of AA s origin, CZ s -is nd Cy perpendiculr to CZ s y-is. Let SA S A AA. Now AZ e, A Z e. So SA e (AZ) () nd SA e (A Z) () () + () SA + SA e (AZ + A Z) or, CS - CA + CS + CA e. AA CS e. CS e () () SA SA e (A Z AZ) or, AA e (CZ + CA CA + CZ) e. CZ So, CZ e Let (, y) be ny point on the hyperbol, nd let M DD nd N CA. S Then M e or, S e M ( e) + (y ) e [CN CZ] e ( e ) (e ) (e ) y e (e ) y (e ) y ( e ) ut b (e ) Hence y b This is the stndrd eqution of hyperbol. The line AA is the trnsverse is nd the line through C which is prependiculr to AA is the conjugte is of the hyperbol. 8

92 y.. Trcing of the hyperbol - b (i) The curve does not pss through the origin. When y, +. Therefore it meets the is t the points (+, ). Thus the points A nd A in which the curve cuts the is re equidistnt from the centre. (ii) (iii) We hve CA CA nd AA. Similrly when, y becomes imginry. Therefore the curve does not cut the y-is. Let us tke two points B nd B on the y-is such tht CB CB b. Then BB b. The curve is symmetricl bout both is nd y is since both the powers of nd y re even. If (, y) be point on the curve, so lso re the points (,y), (, y) nd (, y). We write the eqution of the hyperbol in the form b y + (iv) If > i.e. if either > or <, > nd we get two equl nd opposite vlues of y. In this cse, s the vlue of numericlly increses, the corresponding two vlues of y increse numericlly. The curve therefore consists of two brnches ech etending to infinity in two directions s shown in the Fig... If < or, < <, then is negtive quntity. Therefore y imginry nd there is no point of the curve between the lines nd. The curve lies to the left of the line nd to the right of. Similrly by writing the eqution in the form + b y + b 86

93 (v) we find tht y cn hve ny rel vlue without limittion nd tht for ech vlue of y we get two equl nd opposite vlues of. These informtion bout the curve re sufficient to enble us to find the shpe of the curve. The curve drwn is therefore s shown in the Fig... Ltus rectum : Through S, LSL be drwn perpendiculr to AS. Corresponding to e, we hve e y b or y b b (e ) b b or y + b SL SL b. So, LL rectum of the hyperbol. b is the ltus Note tht the hyperbol is not closed curve. The curve consists of two prts detched from ech other. y A A S (-e, ) Z C Z S(e, ) > An importnt property: The difference between the focl distnces of ny point on hyperbol is constnt nd equl to the length of the trnsverse is of the hyperbol. i.e., S S... Asymptote of curve Fig.. A stright line which touches curve t infinity but does not lie ltogether t infinity is clled n symptote of tht curve. 87

94 Note + b + c hs both roots equl to zero if b c nd both roots infinite if b. y The symptotes of the hyperbol - b The coordintes of the point of intersection of the line y m + c ( m + c) nd the hyperbol re given by - b m mc c b b b (b m ) m c c b Now if y m + c is n symptote, both the roots of this eqution re infinite. Coefficient of nd coefficient of. m c nd b m. c, m + b Hence there re two symptotes, nmely y b nd y b or, y nd y + b b Their combined eqution is Note (i) (ii) ( y ) ( y + ) or, b b 88 y b - The symptotes evidently pss through the centre C(,) of the hyperbol (Fig.). The slopes of the symptotes re b nd b. Hence the symptotes re eqully inclined to the trnsverse is. Tht is, the trnsverse nd conjugte es bisect the ngles between the symptotes (Fig.).

95 (iii) (iv) If α is the ngle between the symptotes then tn α b Angle between the symptotes tn ( b ) The combined eqution of the symptotes differs from tht of the hyperbol by constnt only. y B S A O α A S B Emple Find the eqution of the hyperbol in stndrd form whose eccentricity is nd the distnce between the foci is 6 Given e Let S nd S be the foci, then S S 6 But S S e e 6 Thus we hve ()( ) 6, Also b (e ) ( ) ( ) The eqution of the hyperbol is y - y - b y Fig... 89

96 .. Rectngulr Hyperbol (R.H.) A hyperbol is sid to be rectngulr if the symptotes re t right ngles. If α is the ngle between the symptotes, then tn α b. If the hyperbol is R.H., then α 9 o α b. Eqution of the rectngulr hyperbol is y. A hyperbol is lso sid to be rectngulr when its trnsverse nd conjugte es re equl in length. i.e. b b (e ) gives (e ). or e.. Stndrd eqution of rectngulr hyperbol. Let the symptotes of rectngulr hyperbol be tken s the coordinte es. The equtions of the symptotes re nd y. Their combined eqution is y. Since the eqution of the hyperbol differs from the eqution of symptotes by constnt, the eqution of the hyperbol is y k () where k is consnt. Let the trnsverse is, AA. Drw AM perpendiculr to one symptote, the -is. ACM o where C is the centre So, CM CA cos o nd MA CA sin o The coordintes of A re, It lies on the rectngulr hyperbol. Therefore, 9

97 k or, k The eqution for the rectngulr hyperbol is y Let c Hence y c This is the stndrd form of the eqution of the rectngulr hyperbol. Emple Find the eqution of the hyperbol whose eccentricity is, focus is (, ) nd the corresponding directri is + y. Focus S is (, ), directri is + y nd e If (, y ) is ny point on the hyperbol, then S e M, where M is the perpendiculr to the directri + y. ( ) + (y ) ( ) + y ( + + y y + ) ( + y ) 7 + y y + y Locus of (, y ) is 7 + y y + y Emple 6 Find the equtions of the symptotes of the hyperbol + y + y - - 7y - The combined eqution of the symptotes differs from the eqution of the hyperbol only by constnt. 9 A y C Fig... M A (, )

98 So, the combined eqution of the symptotes is + y + y 7y + k...() which is pir of stright lines stisfying the condition bc + fgh f bg ch...() In the given eqution, h 7, b, f, g, c k Substituting in () we get k. So the combined eqution of the symptotes is + y + y 7y + > ( + y + y ) 7y + > ( + y ) ( + y) 7y + > ( + y + l) ( + y + m) > l + m (compring the coefficients of ) nd l + m 7 (compring the coefficents of y) l, m The equtions of the symptotes re + y nd + y. Emple 7 Find the centre, eccentricity, foci nd ltus rectum of the hyperbol 9-6y - 8-6y - 99 The given eqution cn be written s 9( ) 6 (y + y) 99 Completing squres in nd y, we get 9( ) 6(y + ) ( ) ( y + ) 6 9 To bring this to the stndrd form, set X nd Y y + ; 9

99 we get, e 6 9 X Y 6 9 b (e ) + 6 or,, e Now we cn tbulte the results: Referred to (X, Y) 9 Referred to (, y) X+, y Y- Centre (,) ( +, ) (, ) Foci (+e,) (+, ) nd (+, ) Ltus rectum (,) nd (-,) (6, ) nd (, ) b 9 9 Emple 8 Find the eqution to the hyperbol which hs the lines + y nd - y + for its symptotes nd which psses through the point (,). The combined eqution of the symptotes is ( + y - ) ( - y +) The eqution of the hyperbol differs from this combined eqution of symptotes only by constnt. Thus the hyperbol is ( + y -) ( - y + ) k, where k is constnt Since the hyperbol psses through (, ) [ + () - ] [() - () + ] k i.e. k The eqution of the hyperbol is ( + y ) ( y + ) or + y y 9 + 9y

100 Emple 9 The cost of production of commodity is Rs. less per unit t plce A thn it is t plce B nd distnce between A nd B is km. Assuming tht the route of delivery of the commodity is long stright line nd tht the delivery cost is pise per unit per km, find the curve, t ny point of which the commodity cn be supplied from either A or B t the sme totl cost. Solution: Choose the midpoint of AB s the origin O(,). Let be point on the required curve so tht the commodity supplied from either A or B t the sme totl cost. Let the cost per unit t B C the cost per unit t A C Delivery cost per unit from A to A y (,y) A (-,) O (,) B (,) Fig... Derlivery cost per unit from B to B Totl cost is sme whether the commodity is delivered from either A or B. (C) + (A) C + (B) 9

101 A B i.e. A B 6 ( + y + ) ( + y ) 6 + y y + 6 Simplifying we get, 6 6y y y y 9 6 () () Thus we get the required curve s hyperbol. 9 Emple A mchine sells t Rs.p nd the demnd, (in hundreds) 9 mchines per yer is given by p Wht type of demnd curve corresponds to the bove demnd s lw? At wht price does the demnd tend to vnish? Solution: The demnd curve is p + ( + 6) (p + ) 9 X 9 where X +6, p + The demnd curve is rectngulr hyperbol. Demnd 6(p+) 9 p Rs.. EXERCISE. ) Find the eqution of the hyperbol with () focus (, ), eccentricity nd directri - y. (b) focus (, ), eccentricity nd directri cos α + y sin α p.

102 ) Find the eqution of the hyperbol whose foci re (6, ) nd (, ) nd eccentricity. ) Find the eqution of the hyperbol whose () centre is (, ), one focus is (6, ) nd trnsverse is 6. (b) centre is (, ), one focus is (, ) nd one verte is (, ). (c) centre is (6, ), one focus is (, ) nd e. ) Find the centre, eccentricity, foci nd directrices for the following hyperbols: () 9 6y ( + ) ( y + ) (b) 9 7 (c) y + y 7 ) Find the eqution to the symptotes of the hyperbol () y y y + (b) 8 + y y + y 6) Find the eqution to the hyperbol which psses through (,) nd hs for its symptotes the lines + y 7 nd y. 7) Find the eqution to the hyperbol which hs y + 7 nd + y + for symptotes nd which psses through the origin. EXERCISE. Choose the correct nswer ) The eccentricity of prbol is () (b) (c) (d) ) The eccentricity of conic is. The conic is () prbol (b) n ellipse (c) circle (d) hyperbol ) Ltus rectum of y is () (b) (c) (d) ) Focus of y - is () (, ) (b) (, ) (c) (, ) (d) (, ) 96

103 ) Eqution of the directri of y is () + (b) (c) y+ (d) y y 6) + b represents n ellipse ( > b) if () b ( e ) (b) b (e ) (c) b (d) b e e y 7) Ltus rectum of n ellipse + b ( > b) is () b (b) (c) b b (d) b 8) Focus of y 6 is () (, ) (b) (, ) (c) (8, ) (d) (, ) 9) Eqution of the directri of y 8 is () + (b) (c) y+ (d) y ) The length of the ltus rctum of + 8y, is () 8 (b) (c) 8 (d) 8 ) The prbol + 6y is completely () bove -is (b) below -is (c) left of y-is (d) right of y-is ) The semi mjor nd semi minor es of 6 + y 97 is () (, ) (b) (8, ) (c) (, ) (d) (, 8) ) The length of ltus rectum of + 9y 6 is () (b) 8 (c) 9 ) In n ellipse e, the length of semi minor is is. The length of mjor is is () (b) (c) 8 (d) ) Eccentricity of the hyperbol y () (b) 9 (c) (d) is (d) 8 9

104 6) The sum of focl distnces of ny point on the ellipse is equl to length of its () minor is (b) semi minor is (c) mjor is (d) semi mjor is 7 The difference between the focl distnces of ny point on the hyperbol is equl to length of its () trnsverse is (b) semi trnsverse is (c) conjugte is (d) semi conjugte is. 8) Asymptotes of hyperbol pss through () one of the foci (b) one of the vertices (c) the centre of the hyperbol (d) one end of its ltus rectum. 9) Eccentricity of the rectngulr hyperbol is () (b) (c) (d) ) If is the length of the semi trnsverse is of rectngulr hyperbol y c then the vlue of c is () (b) (c) (d) 98

105 ALICATIONS OF DIFFERENTIATION - I Differentition plys vitl role in Economics nd Commerce. Before we embrk on demonstrting pplictions of differentition in these fields, we introduce few Economic terminologies with usul nottions.. FUNCTIONS IN ECONOMICS AND COMMERCE.. Demnd Function Let q be the demnd (quntity) of commodity nd p the price of tht commodity. The demnd function is defined s q f(p) where p nd q re positive. Generlly, p nd q re inversely relted. Observe the grph of the demnd function q f(p) y q f(p) Quntity Demnd curve y O Fig.. Following observtions cn be mde from the grph (Fig.) (i) only the first qudrnt portion of the grph of the demnd function is shown since p nd q re positive. (ii) slope of the demnd curve is negtive... Supply Function Let denotes mount of prticulr commodity tht sellers offer in the mrket t vrious price p, then the supply function is given by f(p) where nd p re positive. 99 rice

106 Generlly nd p re directly relted Observe the grph of the supply function, f (p) y f(p) Unit price Supply curve O y Fig.. Supply Following observtions cn be mde from the grph (Fig.) (i) only the first qudrnt portion of the grph of the supply function is shown since the function hs mening only for nonnegtive vlues of q nd p. (ii) slope of the supply function is positive... Cost Function Normlly totl cost consists of two prts. (i) rible cost nd (ii) fied cost. rible cost is single - vlued function of output, but fied cost is independent of the level of output. Let f() be the vrible cost nd k be the fied cost when the output is units. The totl cost function is defined s C() f() + k, where is positive. Note tht f() does not contin constnt term. We define Averge Cost (AC), Averge rible Cost (AC), Averge Fied Cost (AFC), Mrginl Cost (MC), nd Mrginl Averge Cost (MAC) s follows. (i) Averge Cost (AC) f ( ) + k Totl Cost Output

107 f ( ) (ii) Averge rible Cost (AC) rible Cost Output (iii) Averge Fied Cost (AFC) k Fied Cost Output (iv) Mrginl Cost (MC) d d C() C () (v) Note Mrginl Averge Cost (MAC) d d (AC) If C() is the totl cost of producing units of some product then its derivtive C () is the mrginl cost which is the pproimte cost of producing more unit when the production level is units. The grphicl representtion is shown here (Fig.). Cost y C(+) C() A T c() B l y l O + roduction level Fig.. A C( + ) C() B C () Mrginl Cost.. Revenue Function Let units be sold t Rs. p per unit. Then the totl revenue R() is defined s R() p, where p nd re positive. Averge revenue (AR) Totl revenue p quntity sold (i.e. Averge revenue nd price re the sme) Mrginl revenue (MR) d (R) R () d p.

108 Note If R() be the totl revenue gined from selling units of some product, then its derivtive, R () is the mrginl revenue, which is pproimte revenue gined from selling more unit when the sles level is units. The grphicl represention is shown here (Fig.) y R(+) T R() Revenue R() A B A R(+) R() B R () mrginl revenue.. rofit Function The profit function () is defined s the difference between the totl revenue nd the totl cost. i.e. () R() C()...6. Elsticity The elsticity of function y f(), with respect to, is defined s η O + y Sles level Ey E Ä y y Lt y dy d Thus the elsticity of y with respect to is the limit of the rtio of the reltive increment in y to the reltive increment in, s the increment in tends to zero. The elsticity is pure number, independent of the units in nd y. Fig..

109 ..7 Elsticity of Demnd Let q f(p) be the demnd function, where q is the demnd nd p is the price. Then the elsticity of demnd is η d (Fig.) y q f(p) p q dq dp Quntity q Demnd curve p y O Elsticity of demnd p Lt q q p p q p Since the slope of the demnd curve is negtive nd elsticity is positive quntity the elsticity of demnd is given by p dq η d - q dp p p+ p rice Fig Elsticity of Supply Let f(p) be the supply function, where is the supply nd p is the price. The elsticity of supply is defined s p η s d dp..9 Equilibrium rice The price t which quntity demnded is equl to quntity supplied is clled equilibrium price... Equilibrium Quntity The quntity obtined by substituting the vlue of equilibrium price in ny one of the given demnd or supply functions is clled equilibrium quntity. dq dp

110 .. Reltion between Mrginl Revenue nd Elsticity of Demnd Let q units be demnded t unit price p so tht p f(q) where f is differentible. The revenue is given by R(q) qp R(q) q f(q) [ p f(q)] Mrginl revenue is obtined by differentiting R(q) with respect to q. R (q) q f (q) + f(q) dp dp q + p [Œ f (q)] dq dq q dp R (q) p( + ) p dq p + dq q p dp p + p dq q dp p dq Since η d, q dp Mrginl Revenue R (q) p η d Emple A firm produces tonnes of output t totl cost C() Find (i) Averge cost (ii) Averge rible Cost (iii) Averge Fied Cost (iv) Mrginl Cost nd (v) Mrginl Averge Cost. C() - + +

111 Totl cost (i) Averge Cost output ( + + ) rible cost (ii) Averge rible Cost output + (iii) Averge Fied Cost Fied cost output (iv) Mrginl Cost d C() d d ( + +) d ( 8 + ) (v) Mrginl Averge Cost d (AC) d d ( + + ) d ( ) Emple The totl cost C of mking units of product is C ,. Find the mrginl cost t units of output. C , Mrginl Cost dc d (.) ( ) (.6) when dc d (.)() (.)() + + At units, Mrginl Cost is Rs.

112 Emple Find the elsticity of demnd for the function - p - p when p. p p d dp p. Emple Elsticity of demnd η d When p, η d p d dp p( p) p p p + p p p Find the elsticity of supply for the supply function p +8p+ p +8p+ d p+8 dp Elsticity of supply η s Emple p d dp p + 8 p p + 8 p+ p + p p + p+ For the function y -8 find the elsticity nd lso obtin the vlue when 6. y 8 dy d

113 Elsticity η y dy d η () 8 when 6, η 6 6 Emple 6 E y If y + find E. Obtin the vlues of h when nd. We hve y + Differntiting with respect to, we get dy ( + )( ) ( )() d ( + ) ( + ) ( + ) Ey dy η E y d ( + ) 7 ( ) ( + ) 7 η ( )( + ) when, η when, η 7 Emple 7 A demnd function is given by p n k, where n nd k re constnts. Clculte price elsticity of demnd. Given p n k k p n 7

114 d dp nk p n Elsticity of demnd η d Emple 8 p d dp p n (nk p n ) kp n, which is constnt. Show tht the elsticity of demnd t ll points on the curve y c (c is constnt), where y represents price will be numericlly equl to. We hve y c c y Differentiting with respect to y, d c dy y y d y Elsticity of demnd η d dy c c y y Emple 9 The demnd curve for monopolist is given by -p (i) Find the totl revenue, verge revenue nd mrginl revenue. (ii) At wht vlue of, the mrginl revenue is equl to zero? We hve p p Totl revenue R p 8

115 Averge revenue p Mrginl revenue d (R) d d d [] (ii) Mrginl revenue is zero implies Mrginl revenue is zero when. Emple If AR nd MR denote the verge nd mrginl revenue t ny output level, show tht elsticity of demnd is equl to AR. erify this for the liner demnd lw p + b. AR MR where p is price nd is the quntity. Totl Revenue R p Averge Revenue AR p Mrginl Revenue MR d (R) d (p) d d dp p + d Now, AR p (ARMR) dp p( p+ ) d p - d dp AR (ARMR) Elsticity of demnd η d η d 9

116 Given p + b Differentiting with respect to, dp b d R p + b AR +b MR d ( + b ) d + b. ( AR price) AR + b ( + b) (ARMR) + b b b -----() η d p d dp ( + b) ( + b) b b () from () nd () we get tht AR (ARMR) η d Emple Find the equilibrium price nd equilibrium quntity for the following demnd nd supply functions, Q d -.6p nd Q s.6+.p At the equilibrium price Q d Q s -.6p.6 +.p.7p. p..7 p when p, Q d (.6)()..8 Equilibrium price nd Equilibrium quntity.8

117 Emple p The demnd for given commodity is given by q p (p>), where p is the unit price. Find the elsticity of demnd when p 7. Interpret the result. p Demnd function q p Differentiting with respect to p, we get dq (p ) ( ) p( ) dp (p- ) (p ) p dq p(p ) Elsticity of demnd η d q dp p (p) p when p 7, η d. 7 This mens tht if the price increses by % when p 7, the quntity demnded will decrese by pproimtely.%. Also if the price decreses by % when p 7, the quntity demnded will increse by pproimtely.%. Emple { } The demnd for given commodity is q - 6p + 8, ( < p < 7) where p is the price. Find the elsticity of demnd nd mrginl revenue when p 6. Demnd function q 6p + 8 Differentiting with respect to p, we get dq 6 dp p dq p Elsticity of demnd η d - (6) q dp 6 p+ 8 when p 6, η d p p8

118 Mrginl revenue p (- η d ) 6( ) Mrginl revenue Rs. EXERCISE. ) A firm produces tonnes of output t totl cost C() Rs. ( - ++8). Find (i) Averge Cost (ii) Averge rible Cost nd (iii) Averge Fied Cost. Also find the vlue of ech of the bove when the output level is tonnes. ) The totl cost C of mking units of product is C() + + units.. Find the mrginl cost t output level of ) The totl cost of mking units is given by C() + +. Wht is the mrginl cost t units of output? ) If the cost of mking units is C the mrginl cost t output of 96 units.. Find ) If the totl cost C of mking tonnes of product is C +. Find the mrginl cost t tonnes output nd find the level of output t which the mrginl cost is Rs.. per ton. 6) The cost function for the production of units of n item is given by C Find (i) the verge cost (ii) the mrginl cost nd (iii) the mrginl verge cost. 7) If the totl cost C of mking units is C + +. Find the verge cost nd mrginl cost when.. 8) The totl cost C of producing units is C ,. Find the mrginl cost of units output. 9) Show tht the elsticity of demnd t ll points on the curve y c (y being price, nd c is the constnt) will be numericlly equl to one.

119 ) Find the elsticity of demnd when the demnd is q p+ nd p. Interpret the result. ) Given the demnd function q 6 p p, find the elsticity of demnd t the price p. Interpret the result. ) Show tht the elsticity of demnd function p for every vlue of q. q is unity ) Find the elsticity of demnd with respect to the price for the following demnd functions. (i) p b, nd b re constnts (ii) p / ) A demnd curve is p m b where m nd b re constnts. Clculte the price elsticity of demnd. ) Find the elsticity of supply for the supply function p +. 6) The supply of certin items is given by the supply function p b, where p is the price, nd b re positive constnts. (p>b). Find n epression for elsticity of supply η s. Show tht it becomes unity when the price is b. 7) For the demnd function p 6 where is the quntity demnded nd p is the price per unit, find the verge revenue nd mrginl revenue. 8) The sles S, for the product with price is given by S, e.6. Find (i) totl sles revenue R, where R S (ii) Mrginl revenue 9) The price nd quntity of commodity re relted by the eqution p p. Find the elsticity of demnd nd mrginl revenue. ) Find the equilibrium price nd equilibrium quntity for the following demnd nd supply functions. q d.p nd q s.8 +.p ) Find the mrginl revenue for the revenue function R() +, where. 8

120 ) The price nd quntity q of commodity re relted by the eqution q p p. Find the elsticity of demnd nd mrginl revenue when p.. DERIATIE AS A RATE OF CHANGE Let reltion between two vribles nd y be denoted by y f(). Let be the smll chnge in nd y be the corresponding chnge in y. Then we define verge rte of chnge of y with respect to is y. where y f( + ) f() nd Lt y dy d Instntneous rte of chnge of y with respect to... Rte of chnge of quntity Let the two quntities nd y be connected by the reltion y f(). Then f ( o ) represents the rte of chnge of y with respect to t... Relted rtes of chnge We will find the solution to the problems which involve equtions with two or more vribles tht re implicit functions of time. Since such vribles will not usully be defined eplicitly in terms of time, we will hve to differentite implicitly with respect to time to determine the reltion between their time-rtes of chnge. Emple If y, find the verge rte of chnge of y with respect to when increses from to.. Find lso the instntneous rte of chnge of y t. (i) Averge rte of chnge of y with respect to t is y f ( f + ) ( ) Here f(), nd.

121 Averge rte of chnge of y with respect to is f (.) f () units per unit chnge in. The negtive sign indictes tht y decreses per unit increse in. (ii) The instntneous rte of chnge of y is dy d y dy d dy At, d - () The instntneous rte of chnge t is units per unit chnge in. The negtive sign indictes the decrese rte of chnge with respect to. Emple A point moves long the grph of y in such wy tht its bsciss is incresing t the rte of units per second when the point is (, 7). Find the rte of chnge of y - coordinte t tht moment. Here nd y re functions of time t. y Differentiting with respect to t we get dy + y dt dt d (y) d () dt dt d dy dt We need the vlue of dy dt y d dt when, y 7 nd d dt

122 dy 7 dt. units per second. i.e. y co-ordinte is decresing t the rte of. units per second. Emple 6 The unit price, p of product is relted to the number of units sold,, by the demnd eqution p -. The cost of producing units is given by C() + 6,. The number of units produced nd sold, is incresing t rte of units per week. When the number of units produced nd sold is,, determine the instntneous rte of chnge with respect to time, t (in weeks) of (i) Revenue (ii) Cost (iii) rofit. (i) Revenue R p ( ) R d (R) d () d ( ) dt dt dt dr ( ) d dt dt when,, nd d dt dr (, )() dt Rs. 76, per week. i.e. Revenue is incresing t rte of Rs.76, per week. (ii) C() + 6,. d (C) dt d () + d (6,) dt dt 6

123 when i.e. (iii) i.e. dc dt d dt d + dt d dt, dc dt Rs., per week. Cost is incresing t rte of Rs., per week. rofit R C d dr dc dt dt dt 76,, Rs.66, per week. rofit is incresing t rte of Rs.66, per week. Emple 7 If the perimeter of circle increses t constnt rte, prove tht the rte of increse of the re vries s the rdius of the circle. Let be the perimeter nd A be the re of the circle of rdius r. Then πr nd A πr d dt da dt using () nd () we get, da dt π dr () dt πr dr () dt r d dt Since perimeter increses t constnt rte d is constnt. dt da r i.e. the rte of increse of A is proportionl dt to the rdius. 7

124 Emple 8 A metl cylinder is heted nd epnds so tht its rdius increses t rte of. cm per minute nd its height increses t rte of. cm per minute retining its shpe. Determine the rte of chnge of the surfce re of the cylinder when its rdius is cms. nd height is cms. The surfce re of the cylinder is A πrh. Differentiting both sides with respect to t da π dt r dh + h dr dt dt Given r, h, dr., dt da π [. +.] dt π[6 + 6] Emple 9 π cm / minute. dh. dt For the function y +, wht re the vlues of, when y increses 7 times s fst s? y + Differentiting both sides with respect to t dy d + dt dt dy d dt dt Given dy 7 d dt dt d 7 d dt dt 8

125 7 +. Emple The demnd y for commodity is y, where is the price. Find the rte t which the demnd chnges when the price is Rs.. The rte of chnge of the demnd y with respect to the price is dy. d We hve y Differentiting with respect to, we get dy d The rte of chnge of demnd with respect to price is When the price is Rs. the rte of chnge of demnd is 6 This mens tht when the price is Rs., n increse in price by % will result in the fll of demnd by.7%. EXERCISE. ) If y find the verge rte of chnge of y with respect to, when increses from to. units. Find lso the instntneous rte of chnge of y t. ) A point moves on the grph of y 8 in such mnner tht its y - coordinte is incresing t rte of units per second, when the point is t (, ). Find the rte of chnge of the - coordinte t tht moment. ) A point moves on the curve + y 8 in such wy tht its co-ordinte is decresing t rte of units per second when the point is t (,). Find the rte of chnge of the y - coordinte t tht moment. 9

126 ) A point moves long the curve y in such wy tht its - coordinte is incresing t the rte of units per second when the point is t (, 6). Show tht the y - coordinte increses t the sme rte s tht of - coordinte. ) Given re the following revenue, cost nd profit equtions R 8, C +,, R C, where denotes the number of units produced nd sold (per month). When the production is t units nd incresing t the rte of units per month, determine the instntneous rte of chnge with respect to time, t (in months), of (i) Revenue (ii) Cost nd (iii) rofit 6) The unit price, p of some product is relted to the number of units sold,, by the demnd function p. The cost of producing units of this product is given by C +,. The number of units produced nd sold is incresing t the rte of units per week. When the number of units produced nd sold is, determine the instntneous rte of chnge with respect to time, t (in weeks) of (i) Revenue (ii) Cost (iii) rofit. 7) Using derivtive s rte mesure prove the following sttement : If the re of circle increses t uniform rte, then the rte of increse of the perimeter vries inversely s the rdius of the circle. 8) The rdius of circulr plte is incresing t the rte of. cm per second. At wht rte is the re incresing when the rdius of the plte is cm.? 9) A metl cylinder when heted, epnds in such wy tht its rdius r, increses t the rte of. cm. per minute nd its height h increses t rte of. cm per minute. retining its shpe. Determine the rte of chnge of the volume of the cylinder when its rdius is cms nd its height is cms. ) For wht vlues of, is the rte of increse of ++8 is twice the rte of increse of?

127 . DERIATIE AS MEASURE OF SLOE.. Slope of the Tngent Line Geometriclly dy represents the slope or grdient of the d tngent line to the curve y f() t the point (, y). If θ is the inclintion of the tngent line with the positive direction of - is, then slope of the line (Fig..6). dy m tnθ d t (, y). y N yf() (, y) Note l y l T Fig..6 (i) If the tngent to the curve is prllel to the -is, then θ which implies tn θ dy t tht point d (ii) If the tngent to the curve is prllel to the y - is, then θ 9 which implies tn θ dy d or d dy t tht point... Eqution of the Tngent θ From Anlyticl Geometry, the eqution of the tngent to the curve y f() t (, y ) is y y dy ( - ) where dy is the slope of the tngent t.. d d

128 dy or y y m ( - ) where m t. d The point is clled the oint of Contct. Note (i) Two tngents to the curve y f() will be prllel if the slopes re equl nd (ii) perpendiculr to ech other if the product of their slopes is... Eqution of the Norml The line which is perpendiculr to the tngent t the point of contct (, y) is clled norml. The eqution of the norml t (, y ) is y y dy ( ), provided dy t (, y d ) d or y y dy ( ) where m ( ) t (, y m d ). Emple Find the slope of the curve y, ( )t the point (, ) nd determine the points where the tngent is prllel to the is of. We hve y Differentiting with respect to, we get dy d ( )() ( )() ( ) 8 + ( ) dy The slope of the curve t (, ) t (, ) d

129 The points t which tngents re prllel to the -is re dy given by d 8 + ( ) ( 6),6 when, y when 6, y The points t which the tngents re prllel to the - is re (, ) nd (6, ). Emple Determine the vlues of l nd m so tht the curve, y l + + m my pss through the point (, ) nd hve its tngent prllel to the -is t.7. We hve y l + + m. Differentiting with respect to, we get dy l +. d dy At.7, l (.7) + d.l +. The tngent t.7 is prllel to the -is dy At.7, d.l + l.. Since the curve is pssing through the point (, ) we get tht l() + () + m m. l nd m.

130 Emple For the cost function y + +, prove tht + the mrginl cost flls continuously s the output increses. We hve y y () + dy Mrginl cost is d Differentiting () with respect to, we get dy d ( + )(+ 8) ( + 8)() + ( + ) ( ) ( ) ( + ) ( + ) ( + ) + ( + ) + ( + ) dy This shows tht s increses, the mrginl cost decreses. d Emple rove tht for the cost function C + +, where is the output, the slope of AC curve (MC-AC). (MC is the mrginl cost nd AC is the verge cost) Cost function is C + + Averge cost (AC)

131 Slope of AC Mrginl cost MC d (AC) d d ( ++) d d (C) d () d (++ ) + d MC AC (+) ( ++) + (MC AC) ( +) () From () nd () we get Slope of AC Emple (MC-AC) Find the equtions of the tngent nd norml t the point ( cosq, b sinq) on the ellipse y + b. We hve y + b Differentiting with respect to, we get () + b y dy d dy b - d y dy At ( cosθ, b sinθ) b cos θ m. d sin θ Eqution of the tngent is y y m ( )

132 y b sinθ b cos θ ( cosθ) sin θ y sinθ b sin θ b cosθ + b cos θ b cosθ + y sinθ b (sin θ + cos θ) b Dividing both sides by b, we get y cosθ + sinθ b Eqution of the tngent is cosθ + b y sinθ Eqution of the norml is y y m ( ) > yb sinθ sin θ ( cosθ) bcos θ > by cosθ - b sinθ cosθ sinθ sinθ cosθ > sinθ + by cosθ sinθ cosθ ( b ) When sinθ cosθ, dividing by sinθ cosθ we get by cosθ sin θ b The eqution of the norml is by cosθ sin θ b Emple 6 Find the eqution of the tngent nd norml to the demnd curve y - t (, 7). Demnd curve y Differentiting both sides with respect, we hve dy 6 d dy At (, 7) 6 m. d Eqution of the tngent is y y m ( ) 6

133 y 7 6 ( ) 6 + y. Eqution of the norml is y y m ( ) y 7 ( ) 6 y 7 ( ) 6 6y 6y +. Emple 7 Find the points on the curve y (-) (-) t which the tngent mkes n ngle o with the positive direction of the -is. We hve y () () Differentiting with respect to, we get dy () () + () () d () Also the tngent is mking o with the -is. dy m tnθ d tn o tn (8 o - o ) tn o tn o -----() Equting () nd () we get or When, y () (). The required point is (, ). 7

134 EXERCISE. ) Find the slope of the tngent line t the point (, ) of the curve y ( + ). At wht point of the curve the slope of the tngent line is 8? ) Determine the coefficients nd b so tht the curve y 6 +b my pss through the point (, ) nd hve its tngent prllel to the -is t.. ) For the cost function y + 7 +, prove tht the mrginl + cost flls continuously s the output increses. ) Find the equtions of the tngents nd normls to the following curves (i) y t (, ) (ii) y sin t π 6 (iii) + y t (-, -) (iv) y 9 t. (v) y log t e (vi) cosθ, y b sinθ t θ π ) Find the eqution of the tngent nd norml to the supply curve y + + when 6. 6) Find the eqution of the tngent nd norml to the demnd curve y 6 when y. 7) At wht points on the curve y the tngents re inclined t o to the -is. 8) rove tht y + touches the curve y b b point where the curve cuts the y-is. 8 e / t the 9) Find the eqution of the tngent nd norml to the curve y() () + 7 t the point where it cuts the is. ) rove tht the curves y + nd (y+) intersect t right ngles t the point (, ). ) Find the eqution of the tngent nd norml t the point ( secθ, b tnθ) on the hyperbol y b.

135 ) At wht points on the circle + y - - y +, the tngent is prllel to (i) -is (ii) y-is. Choose the correct nswer EXERCISE. ) The verge fied cost of the function C is () (b) (c) 9 (d) 8 ) If 6 units of some product cost Rs. nd units cost Rs. to mnufcture, then the vrible cost per unit is () Rs. (b) Rs. 6 (c) Rs. (d) Rs. ) If units of some product cost Rs. nd units cost Rs. to produce, the liner cost function is () y +9 (b) y + 9 (c) y + (d) y + 9 ) rible cost per unit is Rs., fied cost is Rs. 9 nd unit selling price is Rs. 7. Then the profit eqution is () 9 (b) - 7 (c) 9 (d) ) For the cost function c e, the mrginl cost is () (b) e (c) e (d) e 6) Given the demnd eqution p + ; ( < < ) where p denotes the unit selling price nd denotes the number of units demnded of some product. Then the mrginl revenue t units is () Rs. (b) Rs. (c) Rs. (d) Rs. 7) The demnd for some commodity is given by q p + ( < p < ) where p is the unit price. The elsticity of demnd is () 9 p + p (b) 9 p p (c) p 9 p (d) p p +

136 8) For the function y + the verge rte of chnge of y when increses from. to.6 is () (b). (c).6 (d). 9) If y +, the instntneous rte of chnge of y t is () 6 (b) 9 (c) (d). ) If the rte of chnge of y with respect to is 6 nd is chnging t units/sec, then the rte of chnge of y per sec is () units/sec (c) units/sec (b) units/sec (d) units/sec ) The weekly profit, in rupees of corportion is determined by the number of shirts produced per week ccording to the formul. -. Find the rte t which the profit is chnging when the production level is shirts per week. () Rs. (b) Rs. (c) Rs. (d) Rs. 9 ) The bottom of rectngulr swimming tnk is m by m. Wter is pumped into the tnk t the rte of m /min. Find the rte t which the level of the wter in the tnk is rising? ().m/min (c).m/min (b).m/min (d).m/min ) The slope of the tngent t (, 8) on the curve y is () (b) (c) 6 (d) 8 ) The slope of the norml to the curve + y t (9, ) is () (b) - (c) (d) ) For the curve y + - the tngent t (, ) is prllel to -is. The vlue of is () (b) (c) (d)

137 6) The slope of the tngent to the curve y cost, sin t t π t is () (b) (c) (d) 7) The point t which the tngent to the curve y mkes n π ngle with the -is is ()(, ) (b) (, ) (c) (, ) (d) (, -) 8) The tngent to the curve y - + t (, ) is prllel to the line () y (b) y + (c) + y + 7 (d) y 7 9) The slope of the tngent to the curve y log t is () 7 (b) 7 (c) 7 (d) 7 ) The slope of the curve y 6y t the point where it crosses the y is is () (b) (c) 6 (d) 6

138 The concept of mim nd minim is pplied in Economics to study profit mimistion, inventory control nd economic order quntity. We lso lern wht prtil derivtive is nd how to clculte it. Appliction of prtil derivtives re lso discussed with the production function, mrginl productivities of lbour nd cpitl nd with prtil elsticities of demnd.. MAXIMA AND MINIMA.. Incresing nd Decresing Functions A function y f() is sid to be n incresing function of in n intervl, sy < < b, if y increses s increses. i.e. if < < < b, then f( ) < f( ). A function y f() is sid to be decresing function of in n intervl, sy < < b, if y decreses s increses. i.e. if < < < b, then f( ) > f( ).... Sign of the derivtive Let f be n incresing function defined in closed intervl [,b]. Then for ny two vlues nd in [, b] with <, we hve f( ) < f( ). f( ) < f( ) nd > f ( ) f ( ) > Lt ALICATIONS OF DIFFERENTIATION-II f ( ) f ( ) f () > for ll [,b]. >, if this limit eists. Similrly, if f is decresing on [,b] then f () <, if the derivtive eists.

139 The converse holds with the dditionl condition, tht f is continuous on [, b]. Note Let f be continuous on [,b] nd hs derivtive t ech point of the open intervl (,b), then (i) (ii) (iii) (iv) (v) If f () > for every (,b), then f is strictly incresing on [,b] If f () < for every (,b), then f is strictly decresing on [,b] If f () for every (,b), then f is constnt function on [,b] If f () > for every (,b), then f is incresing on [,b] If f () < for every (,b), then f is decresing on [,b] The bove results re used to test whether given function is incresing or decresing... Sttionry lue of Function A function y f() my neither be n incresing function nor be decresing function of t some point of the intervl [,b]. In such cse, y f() is clled sttionry t tht point. At sttionry point f () nd the tngent is prllel to the - is. Emple If y -, prove tht y is strictly incresing function for ll rel vlues of. ( ) We hve y Differentiting with respect to, we get dy + > for ll vlues of, ecept d y is strictly incresing function for ll rel vlues of. ( )

140 Emple If y +, show tht y is strictly decresing function for ll rel vlues of. ( ) We hve y + dy d < for ll vlues of. ( ) y is strictly decresing function for ll rel vlues of. ( ) Emple Find the rnges of vlues of in which is strictly incresing nd strictly decresing. Let y dy d 6( + ) 6( ) ( ) dy > when < or > d lies outside the intervl (, ). dy < when < < d The function is strictly incresing outside the intervl [, ] nd strictly decresing in the intervl (, ) Emple Find the sttionry points nd the sttionry vlues of the function f() Let y 9 +

141 dy 6 9 d dy At sttionry points, d 6 9 ( + ) ( ) The sttionry points re obtined when - nd when -, y () () 9() + when, y () () 9() + - The sttionry vlues re nd The sttionry points re (, ) nd (, ) Emple For the cost function C find when the totl cost (C) is incresing nd when it is decresing. Also discuss the behviour of the mrginl cost (MC) Cost function C dc 8 + d dc d ( ) ( ) or (i) For, < <, dc > (i) then dc s6 > d d

142 (ii) < <, dc < (ii) then dc 7< d d (iii) > ; dc > (iii) then dc 6 > d d C is incresing for < < nd for >. C is decresing for < < MC d (C) d MC 8 + d (MC) + 6 d d (MC) 6 d. 6 For, d d (i) < <, (MC) < (i) then (MC)9< d d d d (ii) >, (MC) > (ii) thend (MC) > d MC is decresing for < nd incresing for >... Mimum nd Minimum lues Let f be function defined on [,b] nd c n interior point of [,b] (i.e.) c is in the open intervl (,b). Then (i) (ii) f(c) is sid to be mimum or reltive mimum of the function f t c if there is neighbourhood (cδ, c + δ) of c such tht for ll (c δ, c + δ) other thn c, f(c) > f() f(c) is sid to be minimum or reltive minimum of the function f t c if there is neighbourhood (c δ, c + δ) of c such tht for ll (c δ, c + δ) other thn c, f(c) < f().

143 (iii) f(c) is sid to be n etreme vlue of f or etremum t c if it is either mimum or minimum... Locl nd Globl Mim nd Minim Consider the grph (Fig..) of the function y f(). y 7 B 8 O 6 A Fig.. The function y f() hs severl mimum nd minimum dy points. At the points,,... 8,. In fct the function hs d mim t,,, 7 nd minim t,, 6, 8. Note tht mimum vlue t is less thn the minimum vlue t 8. These mim nd minim re clled locl or reltive mim nd minim. If we consider the prt of the curve between A nd B then the function hs bsolute mimum or globl mimum t 7 nd bsolute minimum or globl minimum t. Note By the terminology mimum or minimum we men locl mimum or locl minimum respectively...6. Criteri for Mim nd Minim. Mimum Minimum Necessry condition Sufficient condition dy dy d d dy d y ; < d d dy d y ; > d d 7

144 ..7 Concvity nd Conveity Consider the grph (Fig..) of the function y f(). Let T be the tngent to the curve y f() t the point. The curve (or n rc of the curve) which lies bove the tngent line T is sid to be concve upwrd or conve downwrd. y y f() O T Fig.. The curve (or n rc of the curve) which lies below the tngent line T (Fig..) is sid to be conve upwrd or concve downwrd. y y f() O T Fig....8 Conditions for Concvity nd Conveity. Let f() be twice differentible. Then the curve y f() is (i) concve upwrd on ny intervl if f () > (ii) conve upwrd on ny intervl if f () <..9 oint of Inflection A point on curve y f(), where the concvity chnges from upto down or vice vers is clled oint of Inflection. 8

145 For emple, in y (Fig..) hs point of inflection t y y O.. Conditions for point of inflection A point (c, f(c)) on curve y f() is point of inflection (i) if f (c) or f (c) is not defined nd (ii) if f () chnges sign s increses through c i.e. f (c) when f () eists Emple 6 Fig.. Investigte the mim nd minim of the function Let y Differentiting with respect to, we get dy 6 d () dy d + 6 ( + ) ( ), Agin differentiting () with respet to, we get d y + 6 d d y when, () + 6 < d 9

146 It ttins mimum t Mimum vlue is y () + () 6() + 9 d y when, () + 6 > d It ttins minimum t Emple 7 Minimum vlue is y () + () 6() + Find the bsolute (globl) mimum nd minimum vlues of the function f() in the intervl [-, ] Given f() f () The necessry condition for mimum nd minimum is f () ( ) ( ) +, -, ( [, ]) f () 6 f () 6() () 8 < f() is mimum. f () 6() (-) 9 > f() is minimum. f () 6() () 9 < f() is mimum. The mimum vlue when is f() () () + 6() + -

147 The minimum vlue when is f() () () + 6() + -7 The mimum vlue when is f() () () + 6() + 9 Absolute mimum vlue 9. nd Absolute minimum vlue 7 Emple 8 Wht is the mimum slope of the tngent to the curve y nd t wht point is it? We hve y Differentiting with respect to, we get dy d Slope of the tngent is Let M Differenting with respect to, we get dm () d Slope is mimum when dm d M nd < d d dm d Agin differentiting () with respect to, we get d M 6 <, M is mimum t d Mimum vlue of M when is M () + 6()+9 When ; y () +() +9()7-6 Mimum slope The required point is (, -6)

148 Emple 9 Find the points of inflection of the curve y - +. We hve y + Differentite with respect to, we get dy d 8 d y d d y ( ) d, d y 8 d when, d y d. points of inflection eist. when, y () () + when, y () () + The points of inflection re (, ) nd (, ) Emple Find the intervls on which the curve f() is conve upwrd nd conve downwrd. We hve f() Differentiting with respect o, f () + 9 f () 6 f () 6( ) -

149 For (i) - < <, f () < (i) then f () < (ii) < <, f () > (ii) then f () 6 > The curve is conve upwrd in the intervl (, ) The curve is conve downwrd in the intervl (, ) Eercise. ) Show tht the function is n incresing function for ll rel vlues of. ) rove tht lwys decreses s increses. ) Seprte the intervls in which the function is incresing or decresing. ) Find the sttionry points nd the sttionry vlues of the function f() ) For the following totl revenue functions, find when the totl revenue (R) is incresing nd when it is decresing. Also discuss the behviour of mrginl revenue (MR). (i) R 9 +6 (ii) R +6 6) For the following cost functions, find when the totl cost (C) is incresing nd when it is decresing. Also discuss the behviour of mrginl cost (MC). (i) C (ii) C +. 7) Find the mimum nd minimum vlues of the function (i) (ii) + (iii) 6 + (iv) ) Find the bsolute (globl) mimum nd minimum vlues of the function f() in the intervl [, ]. 9) Find the points of inflection of the curve y + +. ) Show tht the mimum vlue of the function f() is 8 more thn the minimum vlue.

150 ) Find the intervls in which the curve y is conve upwrd nd conve downwrd. ) Determine the vlue of output q t which the cost function C q 6q + is minimum. ) Find the mimum nd minimum vlues of the function +. Discuss its nture t. ) Show tht the function f() + hs minimum vlue t. ) The totl revenue (TR) for commodity is TR +. Show tht t the highest point of verge revenue (AR), AR MR (where MR Mrginl Revenue).. ALICATION OF MAXIMA AND MINIMA The concept of zero slope helps us to determine the mimum vlue of profit functions nd the minimum vlue of cost functions. In this section we will nlyse the prcticl ppliction of Mim nd Minim in commerce. Emple A firm produces tonnes of output t totl cost C ( - + +). At wht level of output will the mrginl cost nd the verge vrible cost ttin their respective minimum? Cost C() Rs.( + +) Mrginl Cost d (C) d MC + rible cost Averge vrible cost AC ( + )

151 (i) Let y MC + Differentiting with respect to, we get dy d dy d y Mrginl cost is minimum when nd d > d dy or d d y when, > MC is minimum. d Mrginl cost ttins its minimum t units. (ii) Let z AC + Differentiting with respect to, we get dz d dz d z AC is minimum when, nd d > d dz. d d z when, > AC is minimum t units. d Averge vrible cost ttins minimum t units. Emple A certin mnufcturing concern hs totl cost function C Find, when the totl cost is minimum. Cost C Differentiting with respect to, we get dc () d

152 Cost is minimum when dc d C nd d > d dc + 9 d +, Differentiting () with respect to we get d C d + 6 when ; d C d < C is mimum when, d C d > C is minimum when, the totl cost is minimum Emple The reltionship between profit nd dvertising cost is given by -. Find which mimises.. + rofit + Differentiting with respect to we get d (+ ) ()() d ( + ) (+ ) () rofit is mimum when d d nd < d d d d (+ ) ( + ) + 6

153 Differentiting () with respect to we get d d ( + ) d when 9 ; < rofit is mimum. d Emple The totl cost nd totl revenue of firm re given by C nd R Find the output (i) when the revenue is mimum (ii) when profit is mimum. (i) Revenue R 8 Differentiting with respect to, dr 8 8 d d R 8 d dr d R Revenue is mimum when nd d d dr d 8 d R Also 8 <. R is mimum d (ii) 7 < When the output 8 8 units, revenue is mimum rofit R C (8 ) ( ) Differentiting with respect to, d d

154 d d rofit is mimum when d d nd d d d > d > 6 > ( + ) ( 7) > d when, d d when 7, d or 7-6( 8 < ) > is minimum -6(7) < is mimum when 7 units, profit is mimum. Emple A telephone compny hs profit of Rs. per telephone when the number of telephones in the echnge is not over,. The profit per telephone decreses by. pis for ech telephone over,. Wht is the mimum profit? Let be the number of telephones. The decrese in the profit per telephone (,) (.), >,. (. ) The profit per telephone (. ) (.) The totl profit for telephones (.).

155 Let the totl profit. Differentiting with respect to, we get d () d Conditions for the mimum profit re d d nd < d d d. d.,. Differentiting () with respect to we get d. < is mimum d when,, the mimum profit (,) (.) (,) pise Rs. (,,) Rs., Mimum profit is Rs.,. Emple 6 The totl cost function of firm is C where is the output. A t t Rs. per unit of output is imposed nd the producer dds it to his cost. If the mrket demnd function is given by p -, where Rs. p is the price per unit of output, find the profit mimising output nd price. Totl Revenue (R) p ( ) Totl cost fter the imposition of t is C

156 rofit Revenue Cost ( ) ( + + ) + Differentiting with respect to, d () d Conditions for mimum profit re d d nd d d < d d or Differentiting () with respect to d d When, d < is mimum d rofit mimising output is units When, price p ( ) Rs. 8.. Inventory Control Inventory is defined s the stock of goods. In prctice rw mterils re stored upto cpcity for smooth nd efficient running of business... Costs Involved in Inventory roblems (i) Holding cost or storge cost or inventory crrying cost. (C ) The cost ssocited with crrying or holding the goods in stock is known s holding cost per unit per unit time. (ii) Shortge cost (C ) The penlty costs tht re incurred s result of running out of stock re known s shortge cost.

157 (iii) Set up cost or ordering cost or procurement cost : (C ) This is the cost incurred with the plcement of order or with the initil preprtion of production fcility such s resetting the equipment for production... Economic Order Quntity (EOQ) Economic order quntity is tht size of order which minimises totl nnul cost of crrying inventory nd the cost of ordering under the ssumed conditions of certinty with the nnul demnds known. Economic order quntity is lso clled Economic lot size formul... Wilson s Economic Order Quntity Formul The formul is to determine the optimum quntity ordered (or produced) nd the optimum intervl between successive orders, if the demnd is known nd uniform with no shortges. (i) (ii) (iii) (iv) (v) Let us hve the following ssumptions. Let R be the uniform demnd per unit time. Supply or production of items to the inventory is instntneous. Holding cost is Rs. C per unit per unit time. Let there be n orders (cycles) per yer, ech time q units re ordered (produced). Let Rs C be the ordering (set up) cost per order (cycle). Let t be the time tken between ech order. Digrmtic representtion of this model is given below : y q q q Rt O A t t t Fig..

158 If production run is mde t intervls t, quntity q Rt must be produced in ech run. Since the stock in smll time dt is Rt dt, the stock in period t is t Rt dt Rt qt (ΠRt q) Are of the inventory tringle OA (Fig..). Cost of holding inventory per production run C Rt. Set up cost per production run C. Totl cost per production run C Rt + C Averge totl cost per unit time C C(t) C Rt () t d d C(t) is minimum if dt C(t) nd C(t) > dt Differentiting () with respect to t we get d dt C(t) C C R () t d dt C(t) C C R t t C C R Differentiting () with respect to t, we get d C(t) dt C >, when t t C C R Thus C(t) is minimum for optimum time intervl t o C C R Optimum quntity q to be produced during ech production run,

159 CR EOQ q Rt C This is known s the Optiml Lot - size formul due to Wilson. Note : (i) Optimum number of orders per yer demnd C RC n EOQ R C R C t (ii) Minimum verge cost per unit time, C C C R q (iii) Crrying cost C, Ordering cost (iv) At EOQ, Ordering cost Crrying cost. R q C Emple 7 A mnufcturer hs to supply, units of product per yer to his customer. The demnd is fied nd known nd no shortges re llowed. The inventory holding cost is pise per unit per month nd the set up cost per run is Rs.. Determine (i) the optimum run size q (ii) optimum scheduling period t (iii) minimum totl vrible yerly cost. Supply rte R,, units / month. C pise per unit per month Rs. per run. C C R (i) q C.,87 units / run. t C C R. 6 dys (iii) C C CR. ( ) Rs.,9 per yer.

160 Emple 8 A compny uses nnully, units of rw mterils which costs Rs.. per unit, plcing ech order costs Rs.. nd the holding cost is.% per yer of the verge inventory. Find the EOQ, time between ech order, totl number of orders per yer. Also verify tht t EOQ crrying cost is equl to ordering cost Requirement, units / yer Ordering Cost (C ) Rs.. Holding cost (C ).% of the vlue of ech unit.., Re..67 per unit per yer. EOQ RC C..67 q Time between ech order t o R R Number of order per yer q 6 q At EOQ crrying cost Ordering cost Emple 9 R q C C units. yer 6.67 Rs.. Rs. A mnufcturing compny purchses 9 prts of mchine for its nnul requirements. Ech prt costs Rs.. The ordering cost per order is Rs. nd crrying chrges re % of the verge inventory per yer. Find (i) economic order quntity (ii) time between ech order (iii) minimum verge cost

161 Requirement R 9 prts per yer C % unit cost Rs. ech prt per yer. C Rs. per order EOQ CR C units. q t R 9 9 yer 6 dys (pproimtely). Minimum Averge cost C C R 9 Rs.9 EXERCISE. ) A certin mnufcturing concern hs the totl cost function C 6 +. Find when the totl cost is minimum. ) A firm produces n output of tons of certin product t totl cost given by C +. Find the output t which the verge cost is lest nd the corresponding vlue of the vege cost. ) The cost function, when the output is, is given by C (e +e` ). Show tht the minimum verge cost is. ) A firm produces tons of vluble metl per month t totl cost C given by C Rs.( ). Find t wht level of output, the mrginl cost ttins its minimum.

162 ) A firm produces units of output per week t totl cost of Rs. ( + + ). Find the level t which the mrginl cost nd the verge vrible cost ttin their respective minimum. 6) It is known tht in mill the number of lbourers nd the totl cost C re relted by C ( - ) +. Wht vlue of will minimise the cost? 7) R - nd C re respectively the sles revenue nd cost function of units sold. Find (i) At wht output the revenue is mimum? Wht is the totl revenue t this point? (ii) Wht is the mrginl cost t minimum? (iii) Wht output will mimise the profit? 8) A firm hs revenue function R 8 nd production cost function C + 6. Find the totl profit function 9 nd the number of units to be sold to get the mimum profit. 9) A rdio mnufcturer finds tht he cn sell rdios per week t Rs p ech, where p ( ). His cost of production of rdios per week is Rs. (+ 6 ). Show tht his profit is mimum when the production is rdios per week. Find lso his mimum profit per week. ) A mnufcturer cn sell items per week t price of p 6 rupees. roduction cost of items works out to Rs. C where C +. How much production will yield mimum profit? ) Find the optimum output of firm whose totl revenue nd totl cost functions re given by R nd C +, being the output of the firm. ) Find EOQ for the dt given below. Also verify tht crrying costs is equl to ordering costs t EOQ.

163 Item Monthly Ordering cost Crrying cost Requirements per order er unit. A 9 Rs. Rs..6 B Rs. 68 Rs.. C 8 Rs. Rs..6 ) Clculte the EOQ in units nd totl vrible cost for the following items, ssuming n ordering cost of Rs. nd holding cost of % Item Annul demnd Unit price (Rs.) A 6 Units. B 9 Units 8.6 C 8 Units. D Units. ) A mnufcturer hs to supply his customer with 6 units of his products per yer. Shortges re not llowed nd storge cost mounts to 6 pise per unit per yer. When the set up cost is Rs. 8 find, (i) the economic order quntity. (ii) the minimum verge yerly cost (iii) the optimum number of orders per yer (iv) the optimum period of supply per optimum order. ) The nnul demnd for n item is units. The unit cost is Rs.6 nd inventory crrying chrges % per nnum. If the cost of one procurement is Rs., determine (i) Economic order quntity. (ii) Time between two consecutive orders (iii) Number of orders per yer (iv) minimum verge yerly cost.. ARTIAL DERIATIES In differentil clculus, so fr we hve discussed functions of one vrible of the form y f(). Further one vrible my be epressed s function of severl vribles. For emple, production my be treted s function of lbour nd cpitl nd 7

164 price my be function of supply nd demnd. In generl, the cost or profit depends upon number of independent vribles, for emple, prices of rw mterils, wges on lbour, mrket conditions nd so on. Thus dependent vrible y depends on number of independent vribles,,.. n. It is denoted by y f(,,.. n ) nd is clled function of n vribles. In this section, we will restrict the study to functions of two or three vribles nd their derivtives only.... Definition Let u f(, y) be function of two independent vribles nd y. The derivtive of f(, y) with respect to, keeping y constnt, is clled prtil derivtive of u with respect to nd is denoted by u f or or f or u. Similrly we cn define prtil derivtive of f with respect to y. Thus we hve f Lt f( +, y) f (, y) provided the limit eists. (Here y is fied nd is the increment of ) Also f f(, y + y) f (, y) y Lt y y provided the limit eists. (Here is fied nd y is the increment of y)... Successive rtil Derivtives. f f The prtil derivtives nd y re in generl functions f f of nd y. So we cn differentite functions nd y prtilly with respect to nd y. These derivtives re clled second order prtil derivtives of f(, y). Second order prtil derivtives re denoted 8

165 by Note y y f f y f y f f f f y f yy f f y y f y f y If f, f, f y re continuous then f y f y.. Homogeneous Function A function f(, y) of two independent vrible nd y is sid to be homogeneous in nd y of degree n if f(t, ty)t n f(, y) for t >... Euler s Theorem on Homogeneous Function Theorem : Let f be homogeneous function in nd y of degree n, then f f + y y n f. Corrollry : In generl if f(,,... m ) is homogeneous function of degree n in vribles,,... m, then, f Emple f + f + f m m 9 n f. It u(, y) - - y + y 6 + 8y, find ech of the following. (i) u (ii) u u (iii) (iv) u (v) u y y y u(, y) y + y 6 + 8y (vi) u y

166 (i) (ii) u u y u (iii) u (iv) y (v) (vi) ( y + y 6 + 8y) + ( )y y 6. y ( y + y 6 + 8y) y + (6y ) + 8 y + y + 8 u y u ( + y 6 ) 6 +()y y 6. y y u y (y + y + 8) + (y ) y u y (y + y + 8) + ( )y + 7 y. u y y u y ( + y 6 ) + (6y ) 7 y 6

167 Emple If f(, y) + y + 6y - y + find (i) f (, -) (ii) f yy (, ) (iii) f y (, ) (i) f(, y) + y + 6y y + (ii) f (iii) f y (f) ( + y + 6y y + ) ()y ()y y y. f (, ) 6() + 6() ()() f y y (f) f yy f y (, ) y ( + y + 6y y + ) y + 6 y y y f yy (, ) 8 Emple u f y (y + 6 y ) y 6 y f y (y + 6 y ) 6 6y 6 If u log + y + z, then prove tht u y + u z + + y + z We hve u log ( + y + z ) () 6

168 6 Differentiting () prtilly with respect to, u z y + + z y + + u u + + z y ) ( ) ( )() ( z y z y ) ( z y z y ) ( z y z y Differentiting () prtilly with respect to y we get, y u z y + + y u ) ( ) ( )() ( z y y y z y ) ( z y z y Differentiting () prtilly with respect to z we get, z u z y z + + z u ) ( ) ( )() ( z y z z z y ) ( z y y z u + y u + z u ) ( z y y z z y z y ) ( z y z y z y + + Emple erify Euler s theorem for the function u(, y) + y + y. We hve u(, y) + y + y () u(t, ty) t + t y + t (ty)

169 t ( + y + y) t u(, y) u is homogeneous function of degree in nd y. We hve to verify tht u u +y y u. Differentiting () prtilly with respect to, we get u + y u + y Differentiting () prtilly with respect to y, we get u y y + u u y y u + y y y + y + y + y + y ( + y + y ) u Thus Euler s Theorem is verified, for the given function. Emple Using Euler s theorem if u log show tht u y u log y + y e u y + u + y y. + y y This is homogeneous function of degree in nd y By Euler s theorem, (eu ) + y y (eu ) e u 6

170 e u u u + ye u y dividing by e u we get e u u u + y y 6 Emple Without using Euler s theorem prove tht u u + y u y + z u, if u z yz + y z + y We hve u yz + y z + y () Differentiting prtilly with respect to, we get u ()yz + ()y z + 6yz + y z Differentiting () prtilly with respect to y, we get u y ()z + (y)z + y z + 8yz + y Differentiting () prtilly with respect to z, we get u z y() + y () + y + y u u + y y + z u z 6 yz + y z + yz + 8y z + y + yz + y z yz + 6y z + y ( yz + y z+ y ) u. Emple 6 The revenue derived from selling clcultors nd y dding mchines is given by R(, y) - +8-y +6y+y+. If clcultors nd dding mchines re sold, find the mrginl revenue of selling (i) one more clcultor (ii) one more dding mchine.

171 (i) The mrginl revenue of selling one more clcultor is R. R (R) ( + 8 y + 6y + y + ) ()(y) R (, ) () () 6 At (, ), revenue is incresing t the rte of Rs.6 per ccultor sold. (ii) Mrginl revenue is Rs. 6. Mrginl Revenue of selling one more dding mchine is R y R y y (R) ( + 8 y + 6y + y + ) y + y () y R y (, ) -() () Thus t (, ) revenue is incresing t the rte of pproimtely Rs. per dding mchine. Hence Mrginl revenue is Rs.. EXERCISE. ) If u y + 6y, find u u nd. y ) If u + y + z yz, prove tht u +y u y 6 + z u z u ) If z y + 8y + y, find ech of the follwoing (i) u (ii) u z z (iii) (iv) (v) y y z y (vi) y z ) If f(, y) 8y + 6 y + + 6y + 9, evlute the following. (i) f (ii) f (, ) (iii) f y (iv) f y (, ) (v) f (vi) f (, ) (vii) f yy (viii) f yy (, ) (i) f y () f y (, ) (i) f y (, )

172 ) If u y + y z + z, show tht 6) If u log + y, show tht u u + y 7) If u + y + y, prove tht u y y u 8) If z y e y, prove tht y z + z y 9) erify tht u y y u 66 u + u + u ( + y + z). y z y. + y for the function u y + siny. ) If u log ( + y + z ) prove tht y u z y z u z u y ) erify Euler s theorem for ech of the following functions. (i) u y (ii) f + y y y (iii) z (iv) u + y + y + y y (v) u (vi) u log + y ) Use Euler s theorem to prove the following + y u (i) If u then prove tht + y u + y y u (ii) If z e + y z z then prove tht + y z logz y + y (iii) If f log f f + y then show tht + y y (iv) If u tn - + y then prove tht y u u + y y sin u. ) Without using Euler s theorem prove the following (i) If u z y + z + y, then prove tht u + y u y + u z z

173 (ii) If u log + y + y, then prove tht u + y u y ) The cost of producing wshers nd y dryers is given by C(, y) + y + y +. resently, wshers nd 9 dryers re being produced. Find the mrginl cost of producing (i) one more wsher (ii) one more dryer. ) The revenue derived from selling pens nd y note books is given by R(, y) + y + + y + 8 At present, the retiler is selling pens nd notebooks. Which of these two product lines should be epnded in order to yield the greter increse in revenue? 6) The nnul profit of certin hotel is given by (, y) + y + + y +. Where is the number of rooms vilble for rent nd y is the monthly dvertising ependitures. resently, the hotel hs 9 rooms vilble nd is spending Rs. per month on dvertising. (i) If n dditionl room is constructed, how will this ffect nnul profit? (ii) If n dditionl rupee is spent on monthly dvertising ependitures, how will this ffect nnul profit?. ALICATIONS OF ARTIAL DERIATIES In this section we lern how the concept of prtil derivtives re used in the field of Commerce nd Economics... roduction Function roduction of firm depends upon severl economic fctors like investment or cpitl (K), lbour (L), rw mteril (R), etc. Thus f (K, L, R,...). If depends only on lbour (L) nd cpitl (K), then we write f(l, K)... Mrginl roductivities Let f(l, K) represent production function of two vribles L nd K. is clled the Mrginl roductivity of Lbour nd L is the Mrginl roductivity of Cpitl. 67 K

174 .. rtil Elsticities of Demnd Let q f(p, p ) be the demnd for commodity A which depends upon the prices p nd p of commodities A nd B respectively. The prtil elsticity of demnd q with respect to p is defined s p q Eq q p Ep Similrly the prtil elsticity of demnd of q with respect to p price p is. q Eq q p Ep Emple 7 Find the mrginl productivities of cpitl (K) nd lbour (L), if K - K + KL, when K nd L 6 We hve K K + KL () The mrginl productivity of cpitl is K Differentiting () prtilly with respect to K we get K K + () L K + L when K, nd L 6, () + 6 K The mrginl productitivy of lbour is L Differentiting () prtilly with respect to L we get K L when K, nd L 6. L Mrginl productivity of cpitl units Mrginl productivity of lbour units 68

175 Emple 8 For some firm, the number of units produced when using units of lbour nd y units of cpitl is given by the production function f(, y) 8 y. Find (i) the equtions for both mrginl productivities. (ii) Evlute nd interpret the results when 6 units of lbour nd 8 units of cpitl re used. Given f(, y) 8 y () Mrginl producitivity of lbour is f (, y). Differentiting () prtilly with respect to, we get f 8 y y Mrginl productivity of cpitl is f y (, y) Differentiting () prtilly with resepect to y we get f y 8 y 6 y (ii) f (6, 8) ( 6 ) ( 8 ) (7). i.e. when 6 units of lbour nd 8 units of cpitl re used, one more unit of lbour results in. more units of production. f y (6, 8) 6 ( 6 ) ( ) 8 6() (i.e.) when 6 units of lbour nd 8 units of cpitl re used, one more unit of cpitl results in more units of production. 69

176 Emple 9 The demnd for commodity A is q -p +6p -p p. Eq Eq Find the prtil Elsticities nd Ep Ep when p nd p. Given q p + 6p p p (i) (ii) q p q p Eq Ep p p 6 p p q - q p p - p + 6p p p when p nd p Eq Ep Eq Ep ( )( ) p q - p q p p ( 6 p ) + 6 p when p nd p Eq Ep ( ) p p (p p ) EXERCISE. ) The production function of commodity is L + K L K + KL. Find (i) the mrginl productivity of lbour (ii) the mrginl productivity of cpitl (iii) the two mrginl productivities when L nd K.

177 ) If the production of firm is given by K L L K, prove tht L + K. L K ) If the production function is Z y y + where is the lbour nd y is the cpitl find the mrginl productivities of nd y when nd y. ) For some firm, the number of units produced when using units of lbour nd y units of cpitl is given by the production function f(,y) y. Find (i) both mrginl productivities. (ii) interpret the results when units of lbour nd units of cpitl re used. ) For the production function p (L).7 (K). find the mrginl productivities of lbour (L) nd cpitl (K) when L nd K. 6) For the production function C(L) α (K) β where C is positive constnt nd if α +β show tht K K + L L. 7) The demnd for quntity A is q 6 p p. Find Eq Eq (i) the prtil elsticities, (ii) the prtil elsticities Ep Ep for p nd p. 8) The demnd for commodity A is q p p. Find the prtil elsticities when p p. 9) The demnd for commodity X is q p p. Find the prtil elsticities when p nd p. ) The demnd function for commodity Y is q - p + p p. Find the prtil elsticities when p nd p. 7

178 EXERCISE. Choose the correct nswer ) The sttionry vlue of for f() (-)(-) is () (b) (c) - (d) ) The mimum vlue of f() cos is () (b) (c) (d) ) y is lwys () n incresing function of (b) decresing function of (c) constnt function (d) none of these. ) The curve y is () concve upwrd (c) stright line ) If u y e +, then u is equl to 7 (b) concve downwrd (d) none of these. () y u (b) u (c) u (d) yu 6) If u log (e + e y ) then u u + y is equl to () y (b) e (c) (d) e + e y y e + e e + e 7) If u y u ( > ) then y is equl to () y log (b) log (c) y log (d) log y 8) f(, y) () + y is homogeneous function of degree + y (b) (c) (d) 6 9) If f(, y) + ye -, then f y (, ) is equl to () e (b) (c) e (d) e e ) If f(, y) + y + y then f y is () 6 (b) 6y (c) (d)

179 ) If mrginl revenue is Rs. nd the elsticitiy of demnd with respect to price is, then verge revenue is () Rs. (b) Rs. (c) Rs.7 (d) Rs.. ) The elsticity of demnd when mrginl revenue is zero, is () (b) (c) (d) ) The mrginl revenue is Rs. nd the verge revenue is Rs.6. The elsticity of demnd with respect to price is () (b) (c) (d) ) If u y + y u then y () (b) y (c) y (d) y ) If z + y + y then the mrginl productivity of is () + y (b) 6y + y (c) ( + y ) (d) ( + y ) 6) If q + 8p p then is q p () 8 (b) (c) (d) 7) The mrginl productivity of lbour (L) for the production function K L + KL when L nd K is () (b) (c) (d) 8) The production function for firm is L KL + k. The mrginl productivity of cpitl (K) when L nd K is () (b) (c) 6 (d) 9) The cost function y + is minimum when () (b) (c) (d) ) If R units / yer, C pise, C Rs. then EOQ is () (b) (c) (d) is 7

180 In the present chpter we give some properties of definite integrl, geometricl interprettion of definite integrl nd pplictions of integrtion in finding totl nd verge functions from the given mrginl functions. We further find demnd function when the price nd elsticity of demnd re known. Finlly we discuss few problems under consumers surplus nd producers surplus.. FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS We stte below n importnt theorem which enbles us to evlute definite integrls by mking use of ntiderivtive. Theorem : Let f be continuous function defined on the closed intervl [, b]. Let F be n ntiderivtive of f. Then b f () d F(b) F().. roperties of definite integrls b ) f () d f () d roof : b Let F() be the ntiderivtive of f(). Then we hve, b b f () d [ F ( )] F(b)F() [F() F(b)] b 7 f () d ) f () d f () d + f () d for < c < b. ALICATIONS OF INTEGRATION c b c b

181 roof : Let, b, c be three rel numbers such tht < c < b. b L.H.S. f () d F(b) F() -----() c R.H.S. f () d + f () d F(c) F() + F(b) F(c) b c b 7 F(b) F() -----() From () nd (), f () d f () d + f () d b ) f () d f ( + b ) d roof : b Let + b t d dt t b b t Thus when vries from to b, t vries from b to. b f () d - f ( + b t) dt b b c f ( + b t) dt [by property ()] b f ( + b ) d [since f () d f (t) dt] ) f ( ) d f ( ) d roof : Let t d dt t t b b c b

182 f ( ) d f ( t) (dt) f ( t) dt 76 f ( ) d ) (i) f () d f ( ) d if f() is n even function. (ii) f () d roof : (i) if f() is n odd function. If f() is n even function, then f() f(). ut f () d f ( ) d + f ( ) d [by property ()] t in the first integrl then, dt d t t f () d f ( t ) dt + f ( ) d (ii) f ( ) d + f ( ) d f ( ) d + f ( ) d (f() is n even function) f ( ) d If f() is n odd function then f() f() f () d f ( ) d + f ( ) d ut t in the first integrl. Then dt d t t

183 f () d f ( t ) dt + f ( ) d f ( ) d + f ( ) d f ( ) d + f ( ) d (since f() is n odd function) Emple Evlute ( + ) d f() + is n odd function. ( + ) d [by property (ii)] Emple Evlute ( + ) d f() + is n even function ( + Emple ) d ( + ) d [by property (i)] [ + ] + ] ð [ sin sin cos 77 7 Evlute d +

184 Let π I sin + sin cos d () By property (), f ( ) d f ( ) d π Agin I π sin ( sin ( π ) + π cos cos + sin Adding () nd () we get, ð sin + I sin + ð I ð Emple π Evlute d [ ] ) cos( cos cos ð sin sin + cos ð (-) d π 78 ) d d () d d π By property (), f ( ) d f ( ) d () d ) (+) ( d ( ) d

185 () d Emple ð d Evlute ð + tn ð d Let I ð + tn 6 ð 6 ( ) d cos d ð sin + cos () By property (), b ð I sin( + f ( ) d b f (+b) d cos( π 6 + ) + π 79 ) cos( π π π π 6 ð 6 6 ð ð 6 ð cos( ) ð d sin( ) + cos( ) ð ð + d ) sin d () ð cos + sin Adding () nd () we get 6 ð cos + I ð cos + 6 sin sin d ð ð 6 ð d [ ] ð 6 ð 6

186 I ð ð d ð + tn 6 ð EXERCISE. Evlute the following using the properties of definite integrl: ) ( ) d ) ( + ) d ) ð ð sin d ) - ð ð ð 6) ( ) d 7) d ð + cot cos d ) d - 6 ð 9) ð sin d ) sin + bcos d sin + cos 8) d +. GEOMETRICAL INTERRETATION OF DEFINITE INTEGRAL AS AREA UNDER A CURE The re A of the region bounded by the curve y f(), the - is nd the ordintes t nd b is given by, Are, A b y d b f () d y y f() A O b 8 Fig.

187 Note The grph of y f() must not cross the -is between nd b. Similrly the re A of the region bounded by the curve g(y), the y - is nd the bscisse y c nd y d is given by Note Are, A d dy c d g(y) dy The grph of g(y) must not cross the is of y between y c nd y d. Emple 6 c Find the re enclosed by the prbol y,, nd the - is. A The re under the curve is b y d d y yd yc O y Fig.. g(y) A y d ( ) 8 sq. units. 8 O Fig..

188 Emple 7 Find the re of the region bounded by the prbol y, y, y nd the y - is The re under the curve is, A d dy c Emple 8 y dy y dy y ) 8 sq. units. ( Find the re under the curve y bounded by the y - is - is nd the ordinte t. The y - is is the ordinte t. The re bounded by the ordintes t, nd the given curve is A b y d ( 8 + 6) d y y y O Fig.. y y y [ 6] () () + 6() - sq. units. 8 O Fig..

189 Emple 9 Find the re bounded by the semi cubicl prbol y nd the lines, y nd y. Are, A d dy c y dy y sq. units. Emple Find the re bounded by one rch of the curve y sin nd the - is. The limits for one rch of the curve y sin re π nd y y y y O fig.6 y Are, A b y d π sin d π cos [cosπ cos] O y sin π fig.7 sq. units 8

190 Emple Find the re of one loop of the curve y (- ) between nd. y Eqution of the curve is y ( ) y + Are, A b y d O (, ) Are in the I qudrnt A t d (Œ y > in the I qudrnt) ( dt ) t t dt 6 sq. units. EXERCISE. ut t - dt - d dt fig.8 d. when, t when, t ) Find the re under the curve y included between, nd the - is. ) Find the re of the region bounded by the curve y +, the - is nd the lines nd. ) Find the re under the curve y, -is,, nd +. ) Find the re contined between the - is nd one rch of the curve y cos bounded between π nd π 8

191 ) Find the re of one loop of the curve y ( ) between nd. 6) Find the re under the demnd curve y bounded by the ordintes, 9 nd -is. 7) Find the re cut off from the prbol y by its ltus rectum. 8) Find the re bounded by the curve y 9 nd the lines, y nd y. 9) Find the re bove the is of bounded by y, nd. ) Find the re of the circle of rdius using integrtion. ) Find the re of the ellipse y + b 8.. ALICATIONS OF INTEGRATION IN ECONOMICS AND COMMERCE We lernt lredy tht the mrginl function is obtined by differentiting the totl function. We were given the totl cost, totl revenue or demnd function nd we obtined the mrginl cost, mrginl revenue or elsticity of demnd. Now we shll obtin the totl function when mrginl function is given... The cost function nd verge cost function from mrginl cost function : If C is the cost of producing n output then mrginl cost function, MC dc. Using integrtion s reverse process of d differentition we obtin, Cost function, C (MC) d + k where k is the constnt of integrtion which cn be evluted if the fied cost is known. If the fied cost is not known, then k. C Averge cost fucntion, AC,

192 Emple The mrginl cost function of mnufcturing units of commodity is Find the totl cost nd verge cost, given tht the totl cost of producing unit is. Given tht, MC C (MC) d + k ( ) d + k k k () Given, when, C () k k 6 Totl Cost function, C Averge Cost function, AC C, Emple The mrginl cost function of mnufcturing units of commodity is If there is no fied cost find the totl cost nd verge cost functions. Given tht, MC + 8 C (MC)d + k ( + 8) d + k k 86

193 No fied cost k Totl cost, C + 8 Averge Cost, AC C, Emple + 8. The mrginl cost function of mnufcturing units of commodity is - -. If the fied cost is, find the totl cost nd verge cost functions. Given tht, MC C (MC) d + k ( ) d + k + k () Given tht fied cost C () k C + C AC, +.. The revenue function nd demnd function from mrginl revenue function If R is the totl revenue function when the output is, then mrginl revenue MR dr. d Integrting with respect to we get 87

194 Revenue function, R (MR) d + k where k is the constnt of integrtion which cn be evluted under given conditions. If the totl revenue R, when, Demnd function, p Emple R, If the mrginl revenue for commodity is MR 9-6 +, find the totl revenue nd demnd function. Given tht, MR R (MR) d +k ( ) d +k k Since R when, k R 9 + Emple 6 p R, p 9 + For the mrginl revenue function MR - -, find the revenue function nd demnd function. Given tht MR R (MR) d + k ( ) d + k 88

195 + k since R, when, k R p R, p Emple 7 Given tht, If the mrginl revenue for commodity is e MR + +, find the revenue function. MR e + + R (MR) d + k ( e e ) d + k + + k when no product is sold, revenue is zero. when, R. e Revenue, R + + +k k e The demnd function when the elsticity of demnd is given We know tht, Elsticity of demnd η d p d dp

196 Integrting both sides dp p - dp p η d d This eqution yields the demnd function p s function of. The revenue function cn be found out by using the reltion, R p. Emple 8 The elsticity of demnd with respect to price p for commodity is, > when the demnd is. Find the demnd function if the price is when demnd is 7. Also find the revenue function. Given tht, Elsticity of demnd, η d p i.e. d dp d dp - - p Integrting both sides, d 9 η d d - - p + log k log ( ) log p + log k log ( )+ log p log k log p ( ) log k p ( ) k () when p, 7, k The demnd function is,

197 p Revenue, R p or R Emple 9 9, > The elsticity of demnd with respect to price for commodity is constnt nd is equl to. Find the demnd function nd hence the totl revenue function, given tht when the price is, the demnd is. Given tht, Elsticity of demnd, η d p d dp d dp p Integrting both sides, d dp p + log k log log p + log k log + log p log k p k () Given, when, p From () we get k () p or p Demnd function p Emple ; Revenue R p The mrginl cost nd mrginl revenue with respect to commodity of firm re given by C () +.8 nd R (). Find the totl profit, given tht the totl cost t zero output is zero.

198 Given tht, MC +.8 C () (MC) d + k ( +.8) d + k k +. + k () But given when,c () + + k k C() () Given tht, MR. R() MR d + k d + k + k Revenue when. k R () () Totl profit function, () R() C(). 8.. Emple The mrginl revenue function (in thousnds of rupees) of commodity is 7 + e -. where is the number of units sold. Find the totl revenue from the sle of units ( e -.67) 9

199 Given tht, Mrginl revenue, R () 7 + e -. Totl revenue from sle of units is R ( 7 + e -. ) d. [ 7 + e ]. 7 (e - -) 7 (.67 - ) (7.) thousnds Revenue, R Rs.7,9,866. Emple The mrginl cost C () nd mrginl revenue R () re given by C () + nd R () The fied cost is Rs.. Determine the mimum profit. Given C () + C() C ()d + k ( + ) d + k + + k () When quntity produced is zero, the fied cost is Rs.. i.e. when, C, k Cost function is C() + + 9

200 The revenue, R () R() R () d + k d + k + k When no product is sold, revenue () i.e., when, R Revenue, R() rofit, Totl revenue Totl cost d d d < d ; d d rofit is mimum when 9 Mimum profit is rofit Rs. 8. Emple A compny determines tht the mrginl cost of producing units is C ().6. The fied cost is Rs.. The selling price per unit is Rs.. Find (i) Totl cost function (ii) Totl revenue function (iii) rofit function. Given, C ().6 C() C () d + k.6 d + k.6 + k

201 . + k () Given fied cost Rs. (i.e.) when, C k Hence Cost function, C. + (ii) Totl revenue number of units sold price per unit Let be the number of units sold. Given tht selling price per unit is Rs.. (iii) Revenue R(). rofit, Totl revenue Totl cost Emple - (. + ).. Determine the cost of producing units of commodity if the mrginl cost in rupees per unit is C () +. Given, Mrginl cost, C () +. C () C () d + k ( +.) d + k +. + k. 6 When, C k. C() +. 6 When, Cost of production, C() Rs.9 Emple The mrginl cost t production level of units is given 7 by C () 8 +. Find the cost of producing incrementl units fter units hve been produced. 9

202 7 Given, C () 8 + C() C () d + k The cost of producing incrementl units fter units hve been produced C () d Rs. 86. Required cost Rs EXERCISE. ) The mrginl cost function of production units, is MC + nd the totl cost of producing one unit is Rs.. Find the totl cost function nd the verge cost function. ) The mrginl cost funciton is MC. Find the cost function C() if C(6). Also find the verge cost function. ) The mrginl cost of mnufcturing units of product is MC +. The totl cost of producing one unit of the product is Rs.7. Find the totl cost nd verge cost function. ) For the mrginl cost function MC 6 +, is the output. If the cost of producing items is Rs.8, find the totl cost nd verge cost function. ) The mrginl cost function is MC. +. where is the number of units produced. The fied cost of production is Rs. 7,. Find the totl cost nd the verge cost. 6) If the mrginl revenue function is R () 9, find the revenue function nd verge revenue function. 7) If the mrginl revenue of commodity is given by MR 9 +, find the demnd function nd revenue function. d

203 8) Find the totl revenue function nd the demnd function for the mrginl revenue function MR 9. 9) Find the revenue function nd the demnd function if the mrginl revenue for units is MR +. ) The mrginl revenue of commodity is given by MR. Find the revenue function nd the demnd function. ) The elsticity of demnd with respect to price p is, <. Find the demnd function nd the revenue function when the price is nd the demnd is. ) The elsticity of demnd with respect to price p for p commodity is, when the demnd is. Find the demnd function nd revenue function if the demnd is when the price is. ) Find the demnd function for which the elsticity of demnd is. ) The mrginl cost function of commodity in firm is + e where is the output. Find the totl cost nd verge cost function if the fied cost is Rs.. ) The mrginl revenue function is given by R (). Find the revenue function nd demnd function if R() 6. 6) The mrginl revenue is R () 6. Find the revenue nd demnd function. 7) The mrginl cost of production of firm is given by C () +.. The mrginl revenue is given by R () 8. The fied cost is Rs.. Find the profit function. 8) The mrginl revenue (in thousnds of rupees) of commodity is R () + e -. where denotes the number of units sold. Determine the totl revenue from the sle of units of the commodity (e -.). CONSUMERS SURLUS 97 A demnd curve for commodity shows the mount of the commodity tht will be bought by people t ny given price p.

204 Suppose tht the previling mrket price is p. At this price n mount of the commodity determined by the demnd curve will be sold. However there re buyers who would be willing to py price higher thn p. All such buyers will gin from the fct tht the previling mrket price is only p. This gin is clled Consumers Surplus. It is represented by the re below the demnd curve p f() nd bove the line p p. Thus Consumers Surplus, CS [Totl re under the demnd function bounded by, nd -is Are of the rectngle OAB] CS f ( ) d p Emple 6 Find the consumers surplus for the demnd function p - - when p 9. Given tht, The demnd function is p p ( + ) ( ) (or) [demnd cnnot be negtive] p 9 8 CS f ( ) d p 98 rice B O y CS p pf() A Quntity Fig..9

205 ( )d 8 Emple 7 8 [ ] 8 [() ] 8 units The demnd of commodity is p 8 - Find the consumers surplus when demnd Given tht, The demnd function, p 8 when p 8 p CS f ( ) d p Emple 8 (8 )d [ 8 ] [8 ] units The demnd function for commodity is p. Find + the consumers surplus when the previling mrket price is. 99

206 Given tht, Demnd function, p + p + or + 6 or p 6 CS f ( ) d p d 6 + [log( +)] 6. [log 6 log ] 6 log 6 6 log 6. RODUCERS SURLUS A supply curve for commodity shows the mount of the commodity tht will be brought into the mrket t ny given price p. Suppose the previling mrket price is p. At this price n mount of the commodity, determined by the supply curve, will be offered to buyers. However, there re producers who re willing to supply the commodity t price lower thn p. All such producers will gin from the fct tht the previling mrket price is only p. This gin is clled roducers Surplus. It is represented by the re bove the supply curve p g() nd below the line p p. y Thus roducers Surplus, S S [ Are of the whole rectngle B OAB Are under the supply curve bounded by, p nd - is] S p - g( ) d Emple 9 The supply function for commodity is p + + where denotes supply. Find the producers surplus when the price is. rice O A Quntity Fig. pg()

207 Given tht, Supply function, p + + For p, ( + ) ( ) or Since supply cnnot be negtive, is not possible. p nd p roducers Surplus, Emple S p g( ) d ( + + ) d + [ + ] [ ++] 8 units. Find the producers surplus for the supply function p + + when. Given tht, supply function p + + when, p + + p 9. roducers Surplus S p - g( ) d 9 - ( + + ) d

208 + 9 - [ + ] 9 - [ ] units. Emple Find the producers surplus for the supply function p + when the price is. Given tht, supply function, p +. When p, + or 9 or + Since supply cnnot be negtive,. i.e., p 6. roducers Surplus, S p g( ) d ( + ) d 6 [ ] [9+ 7 ] 8 units. Emple The demnd nd supply functions under pure competition re p d 6 - nd p s +. Find the consumers surplus nd producers surplus t the mrket equilibrium price. For mrket equilibrium, Quntity demnded Quntity supplied But is indmissible. (i.e.)

209 p 6 () p. Consumers Surplus, roducers Surplus CS f ( ) d p ( 6 - ) d units. [ ] S p g( ) d ( + ) d [ ] 8 8 units. EXERCISE. + ) If the demnd function is p nd the demnd is, find the consumers surplus..) If the demnd function for commodity is p 6 find the consumers surplus for p. ) The demnd function for commodity is p. Find the consumers surplus for (i) p (ii) p 6. ) The demnd function for commodity is p 8. Find the consumers surplus for p. ) If the supply function is p + nd, find the producers surplus. 6) If the supply lw is p +, find the producers surplus when the price is 6. 7) The supply function for commodity is p +. Find the producers surplus when (i). (ii) 6.

210 8) For commodity, the supply lw is p +. Find the producers surplus when the price is. 9) The demnd nd supply function for commodity re p d 6 nd p s +. Find the consumers surplus nd producers surplus t the mrket equilibrium price. ) The demnd nd supply lw under pure competion re given by p d nd p s. Find the consumers surplus nd producers surplus t the mrket equilibrium price. ) Under pure competition the demnd nd supply lws for commodity nd p d 6 nd p s 8 +. Find the consumers surplus nd producers surplus t the equilibrium price. ) Find the consumers surplus nd the producers surplus under mrket equilibrium if the demnd function is p d nd the supply function is p s. ) In perfect competition the demnd nd supply curves of commodity re given by p d nd p s Find the consumers surplus nd producers surplus t the mrket equilibrium price. ) The demnd nd supply function for commodity re given by p d nd p s. +. Find the consumers surplus nd producers surplus t the mrket equlibrium price. ) The demnd nd supply curves re given by p d 6 nd + p s. Find the consumers surplus nd producers surplus t the mrket equilibrium price. EXERCISE. Choose the correct nswer ) If f() is n odd function then f ( ) d is () (b) (c) (d)

211 ) If f() is n even function then f ( ) d is () ) d is f ( ) d (b) f ( ) d (c) (d) () (b) (c) (d) ) d is () (b) 6 (c) 6 (d) 8 π ) sin d is π () (b) (c) (d) π π 6) cos d is π () (b) (c) (d) 7) The re under the curve y f(), the -is nd the ordintes t nd b is b () y d b (b) y dy b (c) b dy (d) d 8) The re under the curve g(y), the y - is nd the lines y c nd y d is d () y dy c d (b) dy c d (c) y d c d (d) d c 9) The re bounded by the curve y e, the - is nd the lines nd is () e - (b) e + (c) e (d) e - ) The re bounded by y, y - is nd y is () (b) (c) log (d)

212 ) The re of the region bounded by y + the - is nd the lines nd is () (b) (c) (d) ) The re bounded by the demnd curve y, the - is, nd is () log (b) log 6 (c) log (d) log ) If the mrginl cost function MC e, then the cost function is () e (b) e +k (c) 9e (d) e ) If the mrginl cost function MC -, then the cost function is () +k (b) (c) (d) ) The mrginl revenue of firm is MR 8. Then the revenue function is () +k (b) 8 (c) 8 (d) 8 6) The mrginl revenue R () then the revenue function is + () log + + k (b) - ( +) (c) (d) log ( + ) + 7) The consumers surplus for the demnd function p f() for the quntity nd price p is () f ( ) d p (b) f ( ) d (c) p f ( ) d (d) f ( ) d 8) The producers surplus for the supply function p g() for the quntity nd price p is () g ( ) d p (b) p - g ( ) d (c) g ( ) d (d) g( ) d p p

213 7 ANSWERS. ALICATIONS OF MATRICES AND DETERMINANTS Eercise. ) ) 6 8) 6 9) ) b ) ) 7 8), 9), ) Eercise. ) (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (i) ),. 6) inconsistent ) k ) k ssumes ny rel vlue other thn ) k ) k ssumes ny rel vlue other thn 8 Eercise. ),. ),,. ),. ), -,. ),,. 6),. 7) Rs., Rs., Rs.. 8) Re., Rs., Rs.. 9) tons, tons, 9 tons. Eecrise. ) )

214 8 ) {(, l), (b, m), (c, m)} ) b d c b ) 9 8, 8, 9 6) ; Equivlence reltion. 7) ; Refleive, Not symmetric, Trnsitive. 8) ; Not refleive, Not symmetric, Trnsitive. 9) ; Not refleive, Not symmetric, Not trnsitive. ) (i) (ii) (iii) (iv)

215 9 (v) 6 6 (vi) ) i) ii) ), CBA, CDA, CDBA ) (i) W Z Y X W Z Y X (ii) Not strongly connected. (iii) W Z Y X W Z Y X ) (i) (ii),,. (iii),,,,,. (iv),,,,. (v) (vi) Not strongly connected. (vii)

216 ) 7) ; 8) 9) ; ) (i),, 6, 7, 76,,, 8 (ii) THURSDAY Eercise. ) The system is vible. ) The system is not vible. ) units, units. ) Rs.7 millions, Rs.96 millions. ) (i) Rs lkhs, Rs. 78 lkhs (ii) Rs.8 lkhs, Rs. lkhs. 6) Rs. 8 millions, Rs. millions. 7) Rs. crores, Rs. 6 crores. 8) Rs. 7 crores, Rs. 68 crores. Eercise.6 ) 7.8%,.% ; 7%, % ) 9% ).6%,.% Eercise.7 ) c ) b ) c ) c ) 6) 7) b 8) b 9) b ) c ) ) ) ) ) 6) b 7) 8) b 9) d ) b ANALYTICAL GEOMETRY Eercise. ) prbol ) hyperbol ) n ellipse

217 Eercise. ) () + y y y + 6 (b) + y + y + y + (c) + y +y + 8y (d) + y +y 6y + ) () (, ), (, ),, y + (b) (, ), (, ), y, + (c) (, ), (7, ), y, 7 (d) (, ), (, ),, y ) () (-, ), (-, ),, (b) (, ), (, ), +, (c) ( 8 9 -, ), ( 8 7, ), 8 +, 8 (d) (, ), (, 7 ), y, ) Output tons. nd cost Rs. Eercise. ) (i) + 8y + 8y y + (ii) 7 + y y y (iii) 7 + y + y 8 + 8y + 9 ) (i) + y 8 (ii) + y (iii) + y 9 ) (i) (, ), (, + ); ; (, + ); 8 (ii) (, ), (, + 7 ); ; (, + 7 ); y 7 -, y 7 -, (iii) (, ), (, ) (6, ); 6 6, ; (+ 7 -, ); 9,

218 Eercise. ) () 9 + 6y y 6 7y + 79 (b) 6( + y ) ( cosα + y sinα p) ) y + y 7 ) () 6 9y 8 (b) y 8 + y + (c) y 6 + y + ) () (, ); ; (+, ); +6 (b) (, ); ; (, ) (-6, );, + 7 (c) (, ); ; (6, ) (-, ); 9, + ) () + y + nd + ; (b) y + nd + y 6) y 6y + y + 7 7) 7y y + + 7y Eercise. ) ) b ) c ) d ) c 6) 7) c 8) b 9) b ) ) b ) c ) b ) b ) 6) c 7) 8) c 9) c ) c Eercise. ALICATIONS OF DIFFERENTIATION-I ) (i) (ii) + (iii) 8. AC Rs..8, AC Rs., AFC Rs..8 ) Rs.6. ) Rs.. ) Rs..8 ) Rs.., Rs.6. 6) (i) (ii) (iii) 7) Rs., Rs. 8) Rs.9 ).7 ).

219 ( b) p p ) (i) (ii) ) m ) b 6) p + ( p b) 7) AR p, MR 6 8 8) (i) R, e.6 p + p (ii) MR, e -.6 [.6] 9), p p p + 8p ( p + ) Eercise. ), ) Rs. ), Rs..9 ).,. ) - unit / sec ) units / sec. ) (i) revenue is incresing t the rte of Rs., per month (ii) cost is incresing t the rte of Rs., per month (iii) profit is incresing t the rte of Rs.6, per month 6) (i) revenue is incresing t the rte of Rs.8, per week (ii) cost is incresing t the rte of Rs., per week (iii) profit is incresing t the rte of Rs.6, per week. 8) π cm / sec 9) π cm / minute ),. Eercise. ), ), b ) (i) y +, + y (ii) y + π ; + y π (iii) + y + ; y (iv) 9 + 6y 7 ; 6 6y 7 (v) e y e ; + ey e e (vi) b + y b ; by + b ) y ; + y 78 6) + y 6 ; y +

220 7) (, ), (-, ) 9) y 7 ; + y y by ) secθ tnθ ; + b sec θ tn θ + b. ) (i) (, ) nd (, ) (ii) (, ) nd (-, ) Eercise. ) d ) c ) ) ) b 6) c 7) d 8) d 9) b ) ) d ) ) b ) c ) b 6) d 7) c 8) 9) ) c ALICATIONS OF DIFFERENTIATION-II Eercise. ) incresing in (-,) nd (, ) decresing in (, ) ) (, 7), (, ) ) (i) R is incresing for < < decresing for >, MR is incresing for < < nd decresing for >. (ii) R is incresing for < < 7, decresing for < < nd > 7. MR is incresing for < < nd decresing for >. 6) (i) TC is incresing for < < nd for > nd decresing for < <. MC is decresing for < < nd incresing for >. (ii) TC is incresing for < < nd decresing for >. MC is lwys decresing. 7) (i) m. vlue 7, min. vlue (ii) m. vlue, mini. vlue (iii) min. vlue (iv) m. vlue 9, mini. vlue 8) m. vlue, min. vlue. 9) (, ), (, -9) ) conve up for < < conve down for - < < nd < <.

221 ) q. ) m. vlue mini. vlue 8 point of infleion eists. Eercise. ) ), ) ), 6) 8 7) (i)., Rs.. (ii), (iii) 6 8) 6 9) Rs.6 ) 7 ) ) A :, B : 8, C : 6 ) A :.76, Rs.. B : 67. Rs.8.6 C :, Rs., D : 7.8, Rs.7.9 ) (i) (ii) Rs. (iii) orders / yer (iv) of yer ) (i) 8 (ii) of yer (iii) (iv) Rs.. Eercise. ) 8 + 6y; 6 6y ) (i) + y + 6y 7 (ii) y (iii) 8 + 6y (iv) y (v) y + 6 (vi) y + 6 ) (i) y (ii) (iii) y y + 6 (iv) 9 (v) y + 8 (vi) 968 (vii) 8y (viii) (i) 6 y () 88 (i) 88

222 ) (i) 9 (ii) 7 ) Note Book (6) (i) Rs.8, (ii) Rs.8 Eecrise. ) (i) L + K, (ii) K + L (iii), ), ).9, ).8,.8 7) (i) (ii) Eercise. 8), 9) 6, ), 6 ) b ) d ) ) b ) c 6) c 7) 8) c 9) b ) d ) ) ) d ) ) c 6) 7) c 8) d 9) ) Eercise. ) ) 8 ) π 8) 9) Eercise. ALICATIONS OF INTEGRATION π ) ( + b) π Answers re in squre units. ) 9 ) 6 ) π 9) log ) π ) πb Eercise. ) ) 6 ) ) 6 6) log 7) 6) 7) π 8 8) 8 ) C + +, AC + + ) C (log 6 +), AC (log 6 +) ) C + + 8, AC + + ) C + +, AC + + 8

223 ) C AC ) R 9, AR 7) R 9 +, p 9 + 8) R, p 7 9 9) R +, p + ) R, p ) p, R ) p, R ) p k, k is constnt. ) C + e e +, AC + ) R log + 9, p - 6) R 6, p 6 8) R Rs.,,667 Eercise. ) CS 7 units ) CS units ) (i) CS 6 units (ii) CSunits - + log + 9 7).6 ) CS 6 units ) S 8 units 6) S units 7) (i) S 9 units (ii) S 8 units 8) S 8 units 9) CS 9 units ; S 8 units ) CS8 units ; S 6units ) CS units S 8 units ) CS ) CS 6 units S 9 units 6 units S units

224 ) CS units S units ) CS 6 log -8 units S units Eercise. ) (c) ) () ) () ) (b) ) () 6) () 7) () 8) (b) 9) () ) (b) ) (c) ) () ) (b) ) () ) () 6) () 7) () 8) (b) 8