CONTINUITY AND DIFFERENTIABILITY

Transcription

1 MCD CONTINUITY AND DIFFERENTIABILITY NCERT Solvd mpls upto th sction 5 (Introduction) nd 5 (Continuity) : Empl : Chck th continuity of th function f givn by f() = + t = Empl : Emin whthr th function f givn by f() = is continuous t = 0 Empl : Discuss th continuity of th function f givn by f() = t = 0 Empl 4 : Show tht th function f givn by f () 0 0 is not continuous t = 0 Empl 5 : Chck th points whr th constnt function f() = k is continuous Empl 6 : Prov tht th idntity function on rl numbrs givn by f() = is continuous t vry rl numbr Empl 7 : Is th function dfind by f() =, continuous function? Empl 8 : Discuss th continuity of th function f givn by f() = + Empl 9 : Discuss th continuity of th function f dfind by f(), 0 Empl 0 : Discuss th continuity of th function f dfind by f () Empl : Find ll th points of discontinuity of th function f dfind by f () 0 Empl : Discuss th continuity of th function dfind by 0 f () 0 0 Empl : Discuss th continuity of th function f givn by f() 0 Empl 4 : Show tht vry polynomil function is continuous Empl 5 : Find ll th points of discontinuity of th grtst intgr function dfind by f() = [], whr [] dnots th grtst intgr lss thn or qul to Empl 6 : Prov tht vry rtionl function is continuous Empl 7 : Discuss th continuity of sin function Empl 8 : Prov tht th function dfind by f() = tn is continuous function Empl 9 : Show tht th function dfind by f() = sin ( ) is continuous function Empl 0 : Show tht th function f dfind by f() = +, whr is ny rl numbr, is continuous function EXERCISE 5 Prov tht th function f() = 5 is continuous t = 0, t = nd t = 5 Emin th continuity of th function f() = t = Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

2 MCD Emin th following functions for continuity () f() = 5 (b) f(), (c) f(), 5 (d) f() = Prov tht th function f() = n is continuous t = n, whr n is positiv intgr 5 Is th function f dfind by f () continuous t = 0? At =? At =? 5 Find ll points of discontinuity of f, whr f is dfind by : 6 f() 7 f() f() f() f() f() 0 f() 5 Is th function dfind by f () continuous function? 5 Discuss th continuity of th function f, whr f is dfind by 4 0 f() f() f() 7 Find th rltionship btwn nd b so tht th function f dfind by continuous t = f () is b 8 For wht vlu of is th function dfind by Wht bout continuity t =? ( ) f() 4 0 continuous t = 0? 0 9 Show tht th function dfind by g() = [] is discontinuous t ll intgrl points Hr [] dnots th grtst intgr lss thn or qul to 0 Is th function dfind by f() = sin + 5 continuous t =? Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

3 MCD Discuss th continuity of th following functions : () f() = sin + cos (b) f() = sin cos (c) f() = sin cos Discuss th continuity of th cosin, coscnt, scnt nd contngnt functions Find ll points of discontinuity of f, whr sin 0 f () 0 4 Dtrmin f dfind by sin f() is continuous function? sin cos 0 5 Emin th continuity of f, whr f is dfind by f () 0 Find th vlus of k so tht th function f is continuous t th indictd point in Erciss 6 to 9 6 k cos f () t 7 k f() t = 8 k f () t = 9 cos k 5 f () t = Find th vlus of nd b such tht th function dfind by continuous function Show tht th function dfind by f() = cos ( ) is continuous function Show tht th function dfind by f() = cos is continuous function Emin tht sin is continuous function 4 Find ll th points of discontinuity of f dfind by f() = + Answrs : f is continuous t = (), (b), (c) nd (d) r ll continuous functions 5 f is continuous t = 0 nd = ; Not continuous t = 6 Discontinuous t = 7 Discontinuous t = 8 Discontinuous t = 0 9 No point of discontinuity 0 No point of discontiniuty No point of discontinuity f is discontinuity t = f is not continuous t = 5 f () b 0 is 0 4 f is not continuous t = nd = 5 = is th only point of discontinuity 6 Continuous 7 b 8 For no vlu of, f is continuous t = 0 but f is continuous t = for ny vlu of 0 f is continuous t = (), (b) nd (c) r ll continuous Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

4 MCD 4 Cosin function is continuous for ll R; coscnt is continuous cpt for = n, n Z; scnt is continuous cpt for (n ), n Z nd cotngnt function is continuous cpt for = n, n Z Thr is no point of discontinuity 4 Ys, f is continuous for ll R 5 f is continuous for ll R 6 k = 6 7 k 8 4 k 9 9 k 0 =, b = 5 4 Thr is no point of discontinuity NCERT Solvd mpls upto th sction 5 (Dfrntibility) : Empl : Find th drivtiv of th function givn by f() = sin ( ) cos Empl : Find th drivtiv of tn ( + ) sc ( + ) Empl : Dfrntit sin (cos ( )) with rspct to sin cos (cos ) EXERCISE 5 Dfrntit th functions with rspct to in Ercis to 8 sin ( + 5) cos (sin ) sin ( + b) 4 sc (tn ()) 5 sin( b) cos(c d) 6 cos sin ( 5 ) 7 cot( ) 8 cos () 9 Prov tht th function f givn by f() =, R 0 Prov tht th grtst intgr function dfind by f() = [], 0 < < is not dfrntibl t = nd = Answrs : cos( + 5) cos sin (sin ) cos ( + b) 4 sc(tn )tn(tn )sc 5 cos ( + b) sc (c + d) + c sin ( + b) tn (c + d) sc (c + d) sin 5 cos 5 cos sin sin 5 7 sin sin 8 sin Empl 4 : Find y = d Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

5 MCD 5 Empl 5 : Find, y + sin y = cos d y (n+ ) Empl 6 : Find th drivtiv of f givn by f() = sin ssuming it ists Empl 7 : Find th drivtiv of f givn by f() = tn ssuming it ists EXERCISE 5 Find in th following : d + y = sin + y = sin y + by = cos y 4 y + y = tn + y 5 + y + y = y + y + y = 8 7 sin y + cos y = 8 sin + cos y = 9 y sin 0 y tn, y cos, 0 < < y sin, 0 < < y cos, < < 4 y sin, 5 y sc,0 Answrs : cos cosy by sin y 4 sc y y 5 ( y) 6 ( y) ( y y ) ( y y ) 7 y sin y sin y sin y 8 sin sin y Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

6 MCD 6 NCERT Solvd mpls upto th sction 54 (Eponntil nd Logrithmic Functions) : Empl 8 : Is it tru tht = log for ll rl? Empl 9 : Dfrntit th following wrt : (i) (ii) sin (log ), > 0 (iii) cos ( ) (iv) cos (i) (ii) cos(log ) (iii) (iv) (sin ) cos EXERCISE 54 Dfrntit th following wrt : sin sin 4 sin (tn ) 5 log (cos ) 6 5 cos 7, > 0 8 log (log ), > 9, 0 log 0 cos (log + ), > 0 Answrs : (sin cos ), n,n Z sin sin, (,) 4 cos(tn ) 5 tn, (n ),n N , 0 8, log 4 (sin log cos ) 9, 0 0 sin(log ), 0 (log ) NCERT Solvd mpls upto th sction 55 (Logrithmic Dfrntition) : Empl 0 : Dfrntit ( )( 4) 4 5 wrt ( )( 4) 6 4 ( ) Empl : Dfrntit wrt, whr is positiv constnt log log = log Empl : Dfrntit sin, > 0 wrt sin sin + sin cos log Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

7 MCD 7 Empl : Find, y + y + = b d [y d log y y y y ( log )] y log EXERCISE 55 Dfrntit th functions givn in Ercis to wrt cos cos cos ( )( ) ( )( 4)( 5) (log ) cos 4 sin 5 ( + ) ( + 4) ( + 5) (log ) + log 8 (sin ) + sin 9 sin + (sin ) cos 0 cos ( cos ) (sin ) Find of th functions givn in Ercis to 5 d y + y = y = y 4 (cos ) y = (cos y) ( y) 5 y = 6 Find th drivtiv of th function givn by f() = ( + ) ( + ) ( + 4 ) ( + 8 ) nd hnc find f () 7 Dfrntit ( 5 + 8) ( ) in thr wys mntiond blow : (i) (ii) (iii) by using produc rul by pnding th product to obtin singl polynomil by logrithmic dfrntition Do thy ll giv th sm nswr? 8 If u, v nd w r functions of, thn show tht d (u, v, w) d du dv dw vw u w u v d d d in two wys-first by rptd ppliction of product rul, scond by logrithmic dfrntition Answrs : cos cos cos [tn + tn + tn ] ( )( ) ( )( 4)( 5) 4 5 cos cos (log ) sin log(log ) log 4 ( + log ) sin cos log Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

8 MCD 8 5 ( + ) ( + 4) ( + 5) ( ) 6 log log 7 (log ) [ + log log (log )] + log log 8 (sin ) ( cot + log sin ) + sin 9 sin cos log + (sin ) cos [cos cot sin log sin ] 0 cos [cos ( + log ) sin log ] ( 4 ) cot log(sin ) ( cos ) [ tn + log ( cos )] + ( sin ) y y y log y y y log log y y y log y 4 y tn logcosy tny logcos 5 y( ) (y ) ( )( )( )( ) ;f () NCERT Solvd mpls upto th sction 56 (Drivtivs of Functions in Prmtric Forms) : Empl 4 : Find, = cos, y = sin d cot Empl 5 : Find, = t, y = t d /t Empl 6 : Find, = ( + sin ), y = ( cos ) d tn Empl 7 : Find, y d y Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

9 EXERCISE 56 MCD 9 If n r connctd prmtriclly by th qution givn in Erciss to 0, without liminting th prmtr, find d = t, y = t 4 = cos, y = b cos = sin t, y = cos t 4 = 4t, y = t 4 5 = cos cos, y = sin sin 6 = ( sin ), y = ( + cos ) 7 sin t cos t t, y 8 cost log tn y sin t cost cost 9 = sc, y = b tn 0 = (cos + sin ), y = (sin cos ) If Answrs : sin t, y cos t, show tht d y t b 4 sin t 4 t 5 cos cos sin sin 6 cot 7 cot t 8 tn t b 9 cosc 0 tn NCERT Solvd mpls upto th sction 57 (Scond Ordr Drivtiv) : Empl 8 : Find, y = + tn d 6 + sc + tn Empl 9 : If y = A sin + B cos, thn prov tht y 0 d Empl 40 : If y = +, prov tht 5 6y 0 d d Empl 4 : If y = sin, show tht ( ) 0 d d EXERCISE 57 Find th scond ordr drivtivs of th functions givn in Erciss to cos 4 log 5 log 6 sin cos 8 tn 9 log (log ) 0 sin (log ) If y = 5 cos sin, prov tht y 0 d Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

10 MCD 0 If y = cos, find d in trms of y lon If y = cos (log ) + 4 sin (log ), show tht y + y + y = 0 4 If y = A m + B n, Show tht (m n) mny 0 d d 5 If y = , show tht 49y d 6 If y ( + ) =, show tht d d 7 If y = (tn ), show tht ( + ) y + ( + ) y = Answrs : 80 8 cos sin 4 5 (5 + 6 log ) 6 (5 cos 5 sin 5) ( cos 4 sin ) 8 9 ( ) ( log ) (log ) 0 sin(log ) cos(log ) cot y cosc y NCERT Solvd mpls upto th sction 58 (Mn Vlu Thorm) : Empl 4 : Vry Roll s thorm for th function y = +, = nd b = Empl 4 : Vry Mn Vlu Thorm for th function f() = in th intrvl [, 4] EXERCISE 58 Vry Roll s thorm for th function f() = + 8, [ 4, ] Emin Roll s thorm is pplicbl to ny of th following functions Cn you sy som thing bout th convrs of Roll s thorm from ths mpl? (i) f() = [] for [5, 9] (ii) f() = [] for [, ] (iii) f() = for [, ] If f : [ 5, 5] R is dfrntibl function nd f () dos not vnish nywhr, thn prov tht f( 5) f(5) 4 Vry Mn Vlu Thorm, f() = 4 in th intrvl [, b], whr = nd b = 4 5 Vry Mn Vlu Thorm, f() = 5 in th intrvl [, b], whr = nd b = Find ll c (, ) for which f (c) 0 6 Emin th pplicbility of Mn Vlu Thorm for ll thr functions givn in th bov rcis MISCELLANEOUS EXAMPLES Empl 44 : Dfrntit wrt, th following function : (i) sc (ii) cos (iii) log 7 (log ) 4 Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

11 MCD (i) (ii) sc tn sc (iii) log 7log Empl 45 : Dfrntit th following wrt (i) cos sin (sin ) (ii) tn cos (iii) sin 4 (i) f () (ii) f () (iii) 4 log Empl 46 : Find f () f() = (sin ) sin for ll 0 < < ( + log (sin )) (sin ) sin cos Empl 47 : For positiv constnt find, whr d y t t, nd t t t t log t t Empl 48 : Dfrntit sin wrt cos cos cos MISCELLANEOUS EXERCISE ON CHAPTER 5 Dfrntit wrt th function in Erciss to ( 9 + 5) 9 sin + cos 6 (5) cos 4 sin (), 0 cos 5, 7 6 cot sin sin sin,0 sin 7 (log ) log, > 8 cos ( cos + b sin ), for som constnt nd b 9 (sin cos ) (sin cos ), , for som fid > 0 nd > 0 ( ), for > Find, y = ( cos t), = 0 (t sin t), t d Find, y = sin + sin, d 4 If y y 0, for < <, prov tht d ( ) Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857

12 MCD 5 If ( ) + (y b) = c, for som c > 0, prov tht nd b d d is constnt indpndnt of 6 If cos y = cos ( + y), with cos ±, prov tht 7 If = (cos t + t sin t) n = (sin t t cos t), find cos ( y) d sin d 8 If f() =, show tht f () ists for ll rl nd find it 9 Using mthmticl induction prov tht d ( d n n ) n for ll positiv intgrs n 0 Using th fct tht sin (A + B) = sin A cos B + cos A sin B nd th dfrntition, obtin th sum formul for cosins Dos thr ist function which is continuous vrywhr but not dfrntibl t ctly two points? Justy your nswr If f() y l g() m b h() n c, prov tht d f () l m n g() b h () c If cos y,, show tht ( ) y 0 d d Answrs : 7 ( 9 + 5) 8 ( ) sin cos (sin cos 4 ) cos cos (5) 6sin log cos ( 7) 6 log log(log ) 7 (log ), 8 ( sin b cos ) sin ( cos + b sin ) 9 (sin cos) sin cos (cos + sin) ( + log (sin cos )), sin > cos 0 ( + log ) + + log log ( ) log( ) 7 6 cot 5 sc t t t,0 t 0 Einstin Clsss, Unit No 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Etn, Outr Ring Rod Nw Dlhi 0 08, Ph : 96905, 857