PROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1


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1 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses. The filter is 0 cm high nd the rdius of its bse is 0 cm. Ans: 5 cm / sec ( ) A ship trvels southwrds t the speed of km / hr nd nother ship 8 km to its south trvels estwrds t 8 km / hr. ( ) Find the rte t which the distnce between the two ships increses fter hour. ( b ) Find the rte t which their distnce increses fter two hours. ( c ) Eplin the difference in the signs of the two rtes. [ Ans: ( )  8. km / hr, ( b ) 8 km / hr, ( c ) distnce between the ships initilly decreses till the ships re closest to ech other. This hppens t time, t.8 hr when the rte of chnge of distnce between them is zero. ] ( ) The period T of simple pendulum of length l is given by the formul T the length is incresed by %, wht is the pproimte chnge in the period? l g. If [ Ans: % increse ] ( ) Find the pproimte vlue of sec  (  0 ). Ans:  00 ( 5 ) A formul for the mount of electric current pssing through the tngent glvnometer is i k tn θ, where θ is the vrible nd k is constnt. Prove tht the reltive error in i is minimum when θ. ( 6 ) Find the rdin mesure of the ngle between the tngents to y nd y t their point of intersection other thn the origin. Ans : tn  ( 7 ) Prove tht the portion of ny tngent to the curve y which lies between the coordinte es is of constnt length. ( > 0 ).
2 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( 8 ) If λ λ, then prove tht the curves y λ b λ nd other orthogonlly. y λ b λ intersect ech ( 9 ) Verify Rolle s Theorem for f ( ) sin  sin, [ 0, ]. ( 0 ) For f ( )  6 b, it is given tht f ( ) f ( ) 0. Find nd b nd (, ) such tht f ( ) 0. Ans :, b  6, ± ( ) Apply men vlue theorem to f ( ) log ( ) over the intervl [ 0, ] nd prove tht 0 <  log ( ) <, ( > 0 ). ( ) Apply men vlue theorem to f ( ) e over the intervl [ 0, ] nd prove tht e  0 < log <, ( > 0 ). ( ) Prove tht for > 0, < log ( ) <. ( ) Length of ech of the three sides of trpezium is 5. Wht should be the length of its fourth side if its re is mimum possible? [ Ans: 0 ] ( 5 ) A 8 metre long wire is to be cut into pieces. One piece is to be bent to form squre nd nother piece is to be bent to form circle. If the totl re of the squre nd the circle hs to be minimized, where should the wire be cut? 8 Ans: m for squre nd m for circle
3 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( 6 ) A window is in the shpe of semicircle over rectngle. If the totl perimeter of the window is to be kept constnt nd the mimum mount of light hs to pss through the window, then prove tht the length of the rectngle should be twice its height. ( 7 ) The illumintion due to n electric bulb t ny point vries directly s the cndlepower of the bulb nd inversely s the squre of the distnce of the point from the bulb. Two electric bulbs of cndlepowers C nd 8C re plced 6 metres prt. At wht point between them, the totl illumintion is minimum? [ Ans: m from the bulb hving cndlepower C ] ( 8 ) The perimeter of n isosceles tringle is constnt nd it is 00 cm. Its bse increses t the rte of 0. cm / sec. Find the rte t which the ltitude on the bse increses, when the length of the bse is 0 cm. Ans :  0 cm / sec ( 9 ) A boy flies kite t height of 50 m. The kite moves wy from the boy t horizontl velocity of 6 m / sec. Find the rte t which the string is relesed when the kite is 0 m wy from the boy. Ans : 7 m / sec ( 0 ) If 5 % error is committed in the mesurement of the rdius of sphere, wht percentge of error will be committed in the clcultion of its volume? [ Ans: 5 % ] ( ) If the error in mesuring the rdius of the bse of cone is δr nd if its height is constnt, wht is the error committed in clculting the totl surfce re of the cone? ( r h r rdius of bse Ans: r δr, r h h height of cone ( ) In the clcultion of the re of tringle using the formul bc sin A, A ws tken s / 6. Actully, n % error crept into this mesure of A. If b, c re constnts, wht is the percentge error in clcultion of the re? Ans: % 6
4 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Prove tht y nd 6y 7 intersect orthogonlly. ( ) Find the eqution of the tngent to 9 y 6 which is perpendiculr to the line  y 0. [ Ans: y ± 6 ] ( 5 ) Find the mesure of the ngle between y nd y. [ Ans: Angle is zero s both touch ech other t (, ) nd ( ,  ) ] ( 6 ) Prove tht 9y 5 nd  y 5 intersect orthogonlly. ( 7 ) Determine whether Rolle s theorem is pplicble in the following cses nd if so, determine c such tht f ( c ) 0. ( i ) f ( ),  < 7 , < 6 [ Ans: f is not differentible for, f ( ) 0 for ny ( , 6 ) ] ( ii ) f ( ), < 5 ,, [, 5 ] [ Ans: f is not differentible for, f ( ) 0 for ny (, 5 ) ] ( iii ) f ( ) l l, [ , ] [ Ans: f is not differentible for 0, f ( ) 0 for ny ( , ) ] ( 8 ) Apply men vlue theorem to f ( ) b c d on [ 0, ],, b > 0. Ans :  b b b ( 9 ) Find where is incresing nd where it is decresing. Ans : f increses in l 7 7 <, R, f increses in l >, R
5 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge 5 ( 0 ) Prove tht e >, R  { 0 }. ( ) Prove tht tn is incresing over 0,. ( ) If > 0, prove tht log ( ) > . Check for globl or locl etreme vlues: ( to 7 ) ( ) f ( ) , [ , ]. Ans : globl mm. f (  ) 57, globl min. f ( ) , locl min. f ( 0 ) 5 nd f ( ) , locl mm. f 87 6 ( ) f ( )  , [ , ]. Ans : f (  ) 9 6 is locl nd globl mimum, f ( ) f ' ( )  0, but ( , ), is globl minimum. ( 5 ) f ( ) 500, [ 0, ]. Ans : globl mimum locl minimum is f ( 0 ) f ( ) 0. 5 For , ( 6 ) f ( ) sin cos, R. Ans: M. f k, 5 Min. f k 
6 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge 6 ( 7 ) f ( ) sin, [ 0, ]. Ans: locl m. f, locl min. f,  globl min. f ( 0 ) 0, globl m. f ( ). ( 8 ) If the length of the hypotenuse of right tringle is given, prove tht its re is mimum when it is n isosceles right tringle. ( 9 ) If the volume of right circulr cone with given oblique height is mimum, prove tht the rdin mesure of its semi verticl ngle is tn. ( 0 ) Find pproimte vlue of log 999. [ Ans: ] 0 ( ) Prove tht the length of the tngent is constnt for the curve t cos t log tn, y sin t. ( ) Prove tht for the curve y, the midpoint of the segment of tngent intercepted between the two es is precisely the point of contct ( i.e., the point of tngency ). ( ) Using men vlue theorem, prove tht if > 0, then < tn  <. ( ) Prove tht if > 0, then log ( ) is decresing function. ( 5 ) By cutting equl squres from the four corners of 6 0 tin sheet, bo is to be constructed. Wht should be the length of ech squre if the volume of the bo is to be mimum? [ Ans: ]
7 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge 7 ( 6 ) Prove tht is minimum when e. ( 7 ) The cost of mnufcturing TV sets per dy is 5 5 nd the sle price of one TV set then is How mny TV sets should be mnufctured to mimize profit? Prove tht then the cost of mking one TV set is minimum. ( 8 ) Find the minimum distnce of (, ) from y 8. [ Ans: ] ( 9 ) Which line pssing through (, ) mkes tringle of minimum re with the two es in the first qudrnt? [ Ans: y ] ( 50 ) The eqution of motion of prticle moving on line is s t t bt c, where displcement s is in m, time t is in sec., nd, b, c, re constnts. t s 7 nd velocity, v 7 m/s ccelertion, m / s. Find, b, c. [ Ans:, b , c 5 ] ( 5 ) Find the rte of increse of the volume, V, of sphere with respect to its surfce dv re, S. Ans: S / ds ( 5 ) Find the re of n equilterl tringle with respect to its perimeter, p. Ans: p / 6 ( 5 ) A m tll mn wlks towrds source of light situted t the top of pole 8 m high, t the rte of.5 m / s. How fst is the length of his shdow chnging when he is 5 m wy from the pole? [ Ans:  m / s ]
8 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge 8 ( 5 ) Two sides of tringle re 5 nd 0. How fst is the third side incresing when the ngle between the given sides is 60 nd is incresing t the rte of per sec.? Ans: 9 unit / s ( 55 ) A metl bll of rdius 90 cm. is coted with uniformly thick lyer of ice, which is melting t the rte of 8 cc / min. Find the rte t which the thickness of the ice is decresing when the ice is 0 cm thick. [ Ans: cm / min ] ( 56 ) Obtin the equtions of the tngent nd the norml to the curve cos θ nd y b sin θ t θpoint. y by Ans: cos θ sin θ, b   b cos θ sin θ ( 57 ) Obtin the equtions of the tngent nd the norml to the curve n n y b t point where. y Ans:, by b   b ( 58 ) Obtin the equtions of the tngent nd the norml to the curve e t, y e t t t 0. [ Ans:  y 0, y ] ( 59 ) Find the lengths of the subtngent, subnorml, tngent nd norml t θ curve ( θ sin θ ) nd y (  cos θ ). to the Ans: l l,, l l,
9 PROBLEMS  APPLICATIONS OF DERIVATIVES Pge 9 ( 60 ) For f ( ) (  ) m (  b ) n, [, b ] nd m, n N, determine whether Rolles theorem is pplicble nd if so, determine c (, b ) such tht f ( c ) 0. Ans: Rolles' theorem is pplicble nd c mb m n n ( 6 ) Apply the men vlue theorem to f ( ) cos, 0, tn <. nd prove tht ( 6 ) Show tht 0 < < <  sin <  sin. ( 6 ) For > 0, prove tht  < tn  <. ( 6 ) Using f ( ), > 0, which is greter, e or e? [ Ans: e ] ( 65 ) Apply Men vlue theorem to f ( ) log ( ) over the intervl [, b ], where b  b b 0 < < b nd prove tht log  < < b. ( 66 ) Using f ( ) tn sin , 0 < < tn, prove tht <. sin ( 67 ) Prove tht the rectngle inscribed in given semicircle hs mimum re, if its length is double its bredth. ( 68 ) Fuel cost of running trin is proportionl to squre of its speed in km / hr. nd is Rs. 00 / hr t 0 km / hr. Wht is its most economicl speed, if the fied chrges re Rs. 00 per hour? [ Ans: 0 km / hr ]
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