TOPIC 4 CONTINUITY AND DIFFRENTIABILITY SCHEMATIC DIAGRAM

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1 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM TOPIC 4 CONTINUITY AND DIFFRENTIABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of importance Refrences NCERT Tet Book XII E 007 Continuity& Differentiability Limit of a function Continuity *** E 5 QNo-, 6,30 3Differentiation * E 5 QNo- 6 E 53 QNo- 4,7,3 4Logrithmic Differentiation *** E 55 QNo- 6,9,0,5 5 Parametric Differentiation *** E 56 QNo- 7,8,0, 6 Secon orer erivatives *** E 57 QNo- 4,6,7 7 Mean Value Theorem ** E 58 QNo- 3,4 SOME IMPORTANT RESULTS/CONCEPTS * A function f is sai to be continuous at = a if Left han limit = Right han limit = value of the function at = a ie lim f () lim f () f (a) a a ie lim f (a h) lim f (a h) f (a) h0 h0 * A function is sai to be ifferentiable at = a if Lf (a) Rf (a) ie f (a h) f (a) f (a h) f (a) lim lim h0 h h0 h (i) ( n ) = n n - (ii) () = (iii) (c) = 0, c R (iv) (a ) = a log a, a > 0, a (v) (e ) = e (vi) (loga ) =, log a (vii) (log ) =, > 0 a > 0, a, 7 (iii) (cot ) = cosec, R (iv) (sec ) = sec tan, R (v) (cosec ) = cosec cot, R (vi) (sin - ) = - (vii) (cos - ) = - - (viii) (tan - ) =, R (i) (cot - ) =, R () (sec - ) =, (i) (cosec - ) = (ii) ( ) =, 0 u (iii) (ku) = k (iv) u v u v ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM

2 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM (viii) (loga ) = log a (i) (log ) =, 0 () (sin ) = cos, R, a > 0, a, 0 (v) (uv) = (vi) u v v u u v u v v u v (i) (cos ) = sin, R (ii) (tan ) = sec, R Continuity LEVEL-I Eamine the continuity of the function f()= + 5 at =- Eamine the continuity of the function f()=, R 3 3 Show that f()=4 is a continuous for all R Give an eample of a function which is continuous at =,but not ifferentiable at = k, if For what value of k,the function is continuous at = 3, if 3Fin the relationship between a an b so that the function f efine by: a + f()= b + 3 if 3 is continuous at =3 if > 3 sin3,when 0 4 If f()= Fin whether f() is continuous at =0,when 0 [CBSE 0] I cos 4, 0 For what value of k, the function f()= 8 is continuous at =0? k, 0 3sin If function f()=, for 0 is continuous at =0, then Fin f(0) 3 sin 8 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM

3 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM 3 sin, if 3cos 3Let f() = If f() be a continuous function at =, fin a an b a if b( sin ) if ( ) 4For what value of k,is the function f() = 3Differentiation sin cos k LEVEL-I, when, when Discuss the ifferentiability of the function f()=(-) /3 at = Differentiate y=tan - 3 If y= ( 3)( 4) Fin, y = cos(log ) Fin of y= tan, Fin 3If y=e a sin b, then prove that 4Fin y, if y= 3 at t, = y at t a +(a +b )y=0 I 0 continuous at = 0? 0 Fin, if y = tan Fin y = sin sin cot, 0<< sin sin a bcos 3 If y sin b a b a cos, show that = b a cos 9 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM

4 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM 4Prove that 4 4Logrithmic Differentiation Differentiate y=log 7 (log ) Differentiate, sin(log ),with respect to 3Differentiate y= ( ) If log tan 4 LEVEL-I If y =log[ -],show that ( +) +y+=0 Fin, y = cos(log ) 3 Fin if (cos) y = (cosy) [CBSE 0] I p y q y = log ( y) cos pq, prove that, fin 3 If Show that = * ( )+ [CBSE 0] y 4 Fin cot 3 when y [CBSE 0] 5 Parametric Differentiation If y = tan, prove that If = a cos log tan an y a sin fin at 4 3 If = tan /, show that ( ) ( ) [CBSE 0] 6 Secon orer erivatives 30 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM

5 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM y If y = a cos (log ) + b sin(log ), prove that y 0 If y=(sin - ), prove that (- ) y - = 3If ( ) + ( ) = c for some c>0prove that 7 Mean Value Theorem 3 y 3/ is a constant, inepenent It is given that for the function f()= 3-6 +p+q on[,3], Rolle s theorem hols with c=+ Fin the values p an q 3 Verify Rolle s theorem for the function f() = sin, in [0, ]Fin c, if verifie 3VeifyLagrange s mean Value Theorem f() = in the interval [,4] Questions for self evaluation For what value of k is the following function continuous at =? f () ; k ; 3 ; 3a b, if If f() = if, continuous at =, fin the values of a an b[cbse 0 Comptt] 5a - b, if 3 Discuss the continuity of f() = at = & = 4 If f(), efine by the following is continuous at = 0, fin the values of a, b, c sin(a ) sin f () c b 3/ b, 0, 0, 0 5If = a cos log tan an y a sin fin at 4 6 If y = log cos, fin ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM

6 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM 7 If y + y = tan + y, fin 9If 8 If y = y log = a( y), prove that, fin y = 0 Fin if (cos) y = (cosy) y If y = a cos (log ) + b sin(log ), prove that y 0 If p y q ( y) pq, prove that y 3 ISSUED BY K V - DOWNLOADED FROM WWWSTUDIESTODAYCOM

TOPIC 4 CONTINUITY AND DIFFRENTIABILITY SCHEMATIC DIAGRAM

TOPIC 4 CONTINUITY AND DIFFRENTIABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of importance Refrences NCERT Tet Book XII E 007 Continuity& Differentiability Limit of a function Continuity *** E 5 QNo-,