EXERCISE - 01 CHECK YOUR GRASP

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Valentine Hamilton
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1 DEFNTE NTEGRATON EXERCSE - CHECK YOUR GRASP. ( ) d [ ] d [ ] d d ƒ( ) ƒ '( ) [ ] [ ] 8 5. ( cos )( c)d 8 ( cos )( c)d + 8 ( cos )( c) d 8 ( cos )( c) d sic + cos 8 is lwys posiiv f() d ( > ) ms f() is posiiv i som porio d giv i som porio from o + + c is posiiv d giv i (, ) + + c hs roo i (, ) 8. L d sc d. / / d si cos d sc / si d / cos cos d 8 log (log ) d ( ) log (log ) f ( ) f '( ) d log. F() si d 7. Now 6 si d [pu ] si d 6 [F()] F(6) F() ( ) 5 (7 ) d (5 ) (5 ) 5 (7 ) 5 dd(i) & (ii) d ( ) (7 ) 5 d d /. O diffriig oh sids d [ƒ ()] ƒ '() cos si...(i)...(ii) [ƒ (9)] ƒ '(9)...(i) Also ƒ( ) cos [ƒ()] [ƒ (9)] 7 ƒ (9)...(ii) from (i) & (ii) ƒ '(9) /9 V V cos d cos d V cos d cos d V cos d cos d / V cos d cos d / cos + si V d f() [ f() ] g(f()) g'(f())f'()

2 . g'()f'() g'() 7 f '() f '() f '() 7 lim ( r)( r) r lim r r r. Usig d ( )( ) ( ) ( ) hc d ƒ()d ƒ( )d EXERCSE - BRAN TEASERS. ( ) d ( ) [( ).] d ( ) d ( ) [ ] + + ( ) Now pu + d ( ) 8 5. L f( ) c 5 is coiuous & diffril vrywhr Now f(), f() d f( ) so f'() will hv ls o roo i (, ) ls o roo i (, ), so i will hv ls wo roos i (, ). v d 7 Pu d d. d d v 7 7 v u Hc u d d 7 u / z z dz z pu z si d / / (si )( cos )d si d si / / (cos ) cos d / / ( si ) cos d pu si cos d d/ / ( ) d Now l si d cos d / cos d 6 ( r) d K L r r k ( r) ()! d [ ] (!) ()! (!) ( + )!!.! k d d K d 9

3 . Giv d Now. d d / si d / / / si si d d / L d [Pu d d] / / si d si d / si d..d d / si cos d si d [Pu ] si d EXERCSE -. L ( ) d [( ) ] ( ) d [( ) ] ( ) ( ) d ( ) ( ) ( ) d ( ) ( )... ( )! Mch h colum :. ( A ) [ ]d [( ) ] [ ] [( ) ] [ ] [( ) ] d (i) (ii) MSCELLANEOUS TYPE QUESTONS ( D ) 55 d 55 d 55 d Assrio & Rso :. Sm- : cos d (i) + (ii) cos d. cos d...(i)...(ii) ( B ) ( C ) dd (i) & (ii) d 6 d ( )d () d Lim r r d cos d Sm- : ƒ ()d ( )ƒ ( )d (i) + (ii) (ru)...(i)...(ii) ( ) ƒ ()d {f ƒ ( + ) ƒ ()

4 ƒ ()d Hc Sm- fls u if ƒ ( + ) ƒ (), h. f() + + f'() f'() > > < f'() < < > f() ½ ƒ f() is icrsig i (, ) d dcrsig i (, ) Now g() m {f() ; } 5 g()d / ( )d 5 / d 9 / 5. (si m.si ) d if m d (si m.si ) d if m cos d cos + 6. Sm- : Pu / / d d 99 cosc d 99 cosc d Comprhsio # :. g() ƒ d g'() ƒ () From h grph i is clr h ƒ () > i [, ) d ƒ () < i (, 7) g() is icrsig i [, ] d g() is dcrsig i [, 7] mimum vlu of g() occurs g() ƒ()d.d ( )d ( 9)d g() sr dcrsig from g() ƒ d ƒ d ƒ d 9 ( 9)d Now, g() ƒ()d 9 9 ƒ d ƒ d 6 ( )d 5 g() 5 5 wich lis i [, 6]. g() coms zro 5 g() will giv i (5, 7) Comprhsio # : (,) ƒ () + (,) (,) ƒ () [, ] m ƒ () mi [, ) [, ] 9

5 Now, 6 g() h() g()d h'() g( ). g( ) 6 g() h'() < i ( 6, 7] d hc h() is dcrsig g() g() lim (cos( )) 5 from 6 7. g() g() d 5 g '() lim si cos g '() lim from g ''() lim sc g ''() EXERCSE - [A] CONCEPTUAL SUBJECTVE EXERCSE 5. ( ) ( ) [ ]d.d d.d d 5 cos cos cos cos cos cos [cos ]d d d d.d cos + cos + cos + ( ) d [( ) ] 6 5 d ( )d d d d ( )( ) d / si cos d si / si cos d...() si cos / cos si...() cos si. ( ). dd.() & () / ( ) ( ) d d. ( ) ( ) d Pu + + d d d d ( ) si d si ( ) ( ) Add. (i) d (ii) d si si d d ( / ) ( / ) [ ]...(i) si d...(ii) 6

6 6. 8. si si cos d...(i) h, ( ) si ( ) si cos( ) d...(ii) ( ) ( ) si si cos d ( ) si si cos d dd quio (i) & (ii) si si cos d si cos si cos d Pu cos / si d / / / si d d / / si d si d cos si ( )d L d d/ ( )d d L z 5 / 5 / zdz z dz ( )d d d zdz U () 5 V 8 U 8 V / / 9. J m. ( c ). () m d mj m [ m m ] m. d d d d...() d...() 6 dd () & () 5 6 si cos f() si d cos d Pu si i s igrl d cos i h scod igrl h / / f() si d si d si d si d / 7. ( c ) L si d P lim /... P lim... / P lim... / P lim log log...log lim r r log ( ) lim () + lim / d [ ] lim p P / / lim ( / )

7 EXERCSE - [B] BRAN STORMNG SUBJECTVE EXERCSE. { ( ) ( ) } d ( ) ( ) k f k < {( )( ) ( )} d ( ) ( ) ( + ) k + ( k)d ( ) d ( )( ) ( ) ( ) d d d ( )( ) d ( + ) / so hr will hv wo rl d disic roos for k < Th quio will hv wo rl d disic roos for k R, ( )( ) ( ) 8..( ) d. 5 ( 5 ) d / / 9 d.( ) ( ) d L 5 ( 5 ) d ( 5 + ) {usig propry ( ) / / d 9 ( / ) (( 5 )5 ) d f()d ( ) f(( ) )d } d 9 ( ) d whr + + ( /) k + + ( + ) (k + ) + f k R.H.S. d k d k d so hr will wo rl d disic roos for. ( ) ( ) +, ( )d ( ) d ( ) 5.( 5) 6 6 so f() y f(y)dy yf(y)dy ( y f(y)dy) ( yf(y)dy) f() is qudric prssio of h form + whr y f(y)dy y (y y )dy 5 5 d yf(y)dy y(y y ) dy...(i)

8 9...(ii) from (i) d (ii) 8 8, 9 9 so 8 8 f() 9. u { ( )}. ( ) C C + C...+( ) C (-) + (C C +C ( ) C )( ) (C C + C ( ) C ) (C C + + C ( ) C ) ( ) d du d du d d u d {( )} { }.u u ( )u { } {u +. ( )u { }} C C C ( ) C... C C C ( ) C... ( )u (-)u.u ( )(-)u (-)u ( )u {+ } u ( )u ( )u ( ) u ( )u u { } v.u d & pply y prs wic C C C ( ) C... C C C C...( ) C C C ( ) ( )( ) ( )( ) upo ( + ) rms.... () m ( ) d ( ) d m ( ) m m ( ) d m m ( ) d pu si d si cos d / ( ) d si cos ( si cos )d / si cos d m(m ) ( )( ) m ( ) d m(m )... m ( )( )...( m ) m () ( ) ( )

9 EXERCSE - 5 [A] JEE-[MAN] : PREVOUS YEAR QUESTONS. si d si d si d r r r lim. sc Pu d ; r si d si d si d si d 9 si d 9 8. [ ]d [ ]d [ ]d d d []. f(y) y, g(y) y ; y > d F() 7. f() 8. f( y)g(y)dy. y y ydy ydy [ y y y ] [ + ] ( + ) f() f( ) f( ) g[( )]d, f( ) f() + f( ) f( ) f( ) f( ) f( ) f( ) f( ) g[( )]d g{( )}d {f() f( ) }g( )()d f( ) g{( )}d f( ) f() + f( ) r r sc r lim lowr limi r r / r Pu sc d ; d d ; d d,, sc d ( ) 9. for < <, > d for < <, > for, < <, > d for < <, < d d d d d > d <. Puig for + cos ( d) cos d cos cos d cos d ( cos )d si d

10 [] 5..f '()d.f '()d... []f '()d [f() f()] + [f() f()] [] [f()] [] f() {f() + f() f[]} 6. F() f()+f(/) pu F() l z / log log d d d dz z l l / z d dz ( ) ( / z) z y propry 8. Now f( )d f( )d l l d d ( ) ( ) si < si si d < < / d < l d.5. [ ]d.5 d d d.5 (.5 ). g() cos d g( + ) cos d cos d cos d cos d cos d g() + g() Bcus g() so g() g() is lso corrc As.. Sm- : / / 6 d < cos < / / 6 cos d si cos...() 9. cos < cos d < J < [co ]d... () [co( )]d dd () & () d < < [ co ]d... () [co ] [ co ]d [] + [ ] d us ƒ( )d ƒ( )d / / 6 () + () / d / si d cos si...() So Sm- is fls. d sm- is ru s i is propry.

11 EXERCSE - 5 [B] JEE-[ADVANCED] : PREVOUS YEAR QUESTONS 6. Giv h f() is v fucio, h o prov / / f(cos ) cos d f(si ) cos d L / f(cos ) cos d...() / ƒ cos cos d Usig f( )d f( )d / / f( cos ) si d f(cos ) si d...() [As ƒ() is v fucio] ddig wo vlus of i () d () w g / f(cos )(si cos )d / f(cos ) si cos d / f(cos ) cos( / )d L / d d / f[cos ( / )]cos d / / / / f[ si ]cos d f (si )cos d / [ f is v fucio] / f(si ) cos d [ f is v fucio] 8. () / f(si ) cos d R.H.S. [ ( ) cos( )]d ( ) si( ) cos( ) cos 9. L si cos cos cos si d cos si cos si d cos cos cos si d + Now usig h propry h f()d W g, d if f( ) f() f()d if f( ) f() / cos cos cos si d / cos 6 cos cos si d Pu cos si d d, w g or 6 cos d 6[( cos ) si d] 6 cos( / ) ( si / ) cos / d. 6 cos si( / ). 6 6 cos( / ) si( / ) cos( / ) si 5 / si d si si si ( ) 8

12 . f''() <, (, ), for c (, ) c c F(c) (f() f(c)) (f() f(c)) c c f(c) f() f() F'(c) f '(c) f() f() [( )f '(c) f() f()] F ''(c) ( )f ''(c). L Th, 5 ( ) d d 5 ' ( ) d '.( ) d (( ) ) ( ) d ( ) d 5 5 ' 55 ( ) d ' 55 ( 5 ) d. [ f ''(), (, )d ] F(c) is m. h poi (c, f(c)) whr F'(c) f'(c) lim f() f() f()d (f() f()) ( ) h h f( )d (f( h) f()) lim h h h f( h) [f() f( h)] (f '( h)) lim h h [Usig L'Hospil rul] ( ) d 55 ( 5 ) d 55 ' ' ' 7. S k k k d S ( h fucio is dcrsig) S S d h f( h) f() f '( h) lim h h h f '( h) f '( h) f ''( h) lim h 6h S 6 S Now T S T S > [Usig L' Hospil rul] f ''( h) lim f''(), R h f() mus of m. dgr T > S + s S so T

13 8. (ƒ '()) d ƒ()d, diffriig oh h sids & squrig ƒ '() (ƒ ' ()) ƒ () ƒ () si ƒ () + c ƒ () ƒ () si si for [, ] ƒ 9. < d ƒ <. si si d si si d + (ii) (i) si d si..(i) si d...(i) si ( + ) cos m m + m Pu i quio (i) m m si d si. f() f d...(i) si d si f'() f() f() k. From (i) f() f() k. k f(). Applyig L-Hospil rul, ( ) d ( ) lim lim ( ) lim ( ).. d 5 d 6 d d d d. d d d {} wh 9 < 8; 7 6,... f() {} wh 9; 8 7,... Sic f() & cos oh r priodic fucios hvig priod. ( {}) cos d {}cos d ( ) cos d ( ) cos d cos d cos d cos d cos d. f() + d f '() f() f'() f() dy y d (sy)...(i) cosidrig y f(). so h f'(y) d f '() dy y... (ii) for f() i.. y f () dy d from (), f ' ()

14 5. si d ; pu si si( 6 ) d d si d...(i) si si( 6 ) si( 6 ) d si( 6 ) si Addig quio (i) & (ii) d 6. Ar (OABC) y C(, ) (, /) Shdd r is S. Clrly S < d d > S > B(, ) (, /) A d ((B) is corrc Agi S Ar (rpzium ACDO) y A B C E D F / S S C is wrog y...(ii) Also S Sum of rs of rcgls ABDO & CEFD S S ( (D) is corrc) + cos d cosd / / / / / / cos d cos d / / / ( si ) si d / / ( cos ) cos d / cos d L lim lim lim ims r r r r r r lim d & will rjcd s d is o dfid.

(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is

(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is [STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs