ASSIGNMENT ON CONTINUITY AND DIFFERENTIABILITY LEVEL 1 ( CBSE AND OTHER STATE BOARDS) 1 Test the continuit of the function f() at the ; 0 origin : f () 1; 0 Show that the function f() given b sin cos, 0 f () continuous at, 0 = 0. 3 Check the continuit offunction f() e 1 if 0 f () log(1 )' at 0 7, if 0 4 Show that the function f() given b continuous at = 0 1/ e 1 1/, when 0 f () e 1 0, when 0 5 Show that the function f() = - continuous at = 0. 6 sin 3, if 0 tan 3 Show that f (), if 0 log(1 3), if 0 e 1 continuous at = 0 7 Dcuss the continuit of the function of given b: f() = 1 + - at = 1 and =. 8 Dcuss the continuit of the f () at the indicated points: f() = + - 1 at = 0, 1. 9 Find the value of the constant k so that the function given below continuous at = 0. 1 cos, 0 f () k, 0 10 the function f() defined b log(1 a) log(1 b), if 0 : f () k, if 0 continuous at = 0, find k. 11 the function f() given b a and b. 3a b, if 1 f () 11, if 1 5a b, if 1 continuous at = 1, find the values of 1 Prove that the greatest integer function [] continuous at all points ecept at integer points. 13 1 cos 4, if 0 Let f () a, if 0, if 0 16 4 Determine the value of a so that f() continuous at = 0. 14 Determine the values of a, b, c for which the 15 function: sin(a 1) sin, for 0 f () c, for 0 b, for 0 3/ b continuous at = 0. 4, if 4 4 f () a b, if 4 continuous 4 b, if 4 4
16 at = 4, find a, b. 3 1sin, if 3cos Let f () a, if. f() b(1 sin ), if ( ) continuous at find a and b. 17 Find the value of 'a' for which the function f defined b a sin ( 1), 0 f () tan sin 3, 0 continuous at = 0. DERIVATIVES DIFFERENTIABILITY 18 Show that f() = not differentiable at = 0. 19 Show that f() = - continuous but not differentiable at =. 0 Show that f() = 1/3 not differentiable at = 0. 1 Show that the function 1 sin, if 0 f () differentiable at 0, if 0 = 0 and f'(0) = 0. Dcuss the differentiabilit of 1 1 : f () e, 0 at = 0. 0, 0 3 For what choice of a and b the function, c f () differentiable at = c. a b, c 4 Show that the function 1 sin, when 0 f () continuous but 0, when 0 not differentiable at = 0. 5 Show that the function f defined as follows, continuous at =, but not differentiable 3, 0 1 thereat: f() =, 1 5 4, 6 a b, if 1 f () 1 differentiable at, if 1 7 = 1, find a, b. 3 a, for 1 f () b, for 1 everwhere differentiable, find the values of a and b. CHAIN RULE 8 Differentiate the following functions w.r.t. : (i) log sin sin(e ) (ii) sin e (iii) 9 Differentiate the following functions with respect to : (i) log (sec + tan ) (ii) 30 Differentiate the following function w.r.t. : tan 1 sin e 31 Differentiate the following functions w.r.t. :(i) e e (ii) log 7(log 7) (iii) log 3 Differentiate the following functions w.r.t. :(i) log tan 4 (ii) log sin 1 3 33 Differentiate the following functions with respect to : (i) log( a ) (ii)
a bsin log a bsin 34 Differentiate the following functions w.r.t. :(i) 35 36 37 38 sin (m sin -1 ) (ii) a 1 (sin ) (iii) 1 cos ( 1 ) e n [ a ], then prove that n d a 1 sin log 1 1 1 sin. 3/ d (1 ) a a a a a a 1 d a 3 4 4 1 tan log 1 tan, then prove that, show that, prove that sec d. DERIVATIVE OF IMPLICIT FUNCTIONS AND INVERSE TRIGO FUNCTION 39 a + h + b + g + f + c = 0, find 40 d. log( + ) = d tan, show that 1 41 1 1 0and, prove that 1 d ( 1) 4 sin = sin (a +), prove that sin (a ) d sin a 43 6 6 3 3 1 1 a( ), prove that 44 45 46 6 1, where -1 < < 1 and -1 < < 6 d 1 1. 1 4 4 1 t and t t t, then 1 prove that. 3 d. a d 1 1 b tan tan,find 1 1 a( ), prove that 1 d 1 47 = 1, prove that 0 d. 48 1 tan a, prove that (1 tan a) d (1 tan a) 49 sin (a + ) + sin cos (a + ) = 0, prove that sin (a ) d sin a 50 1 1 {log cos sin }{log sin cos } sin 1, find at d 4 LOGARITHMIC DIFFERENTIATION 51 sin Given that cos.cos.cos..., prove 4 8 that 1 1 1 sec sec... cosec 4 4 5 Differentiate the following functions w.r. to : (i) (ii) ( ) 53 tan sec (sin ) (cos ),find. d 54 Differentiate : (log ) + log w.r. to. 55 Differentiate the following function w.r. to :
log( cos ec ) 56 a e,find at a. d 57 m n mn ( ), prove that. d 58 1, prove that 59 60 61 6 d a (1 log ).log.... ( 1), find d. sin sin sin...to, prove that cos d 1 sin, prove that cos 1 sin 1 cos 1 1...to (1 )cos sin d 1 cos sin log log log...to, prove that 1 ( 1) d 65 66 67 1 1 sin t cos t a, a,a 0 and -1 < t < 1, show that d 1 t sin 1 t and 1 t tan 1 t, t > 1. Prove that 1 d. 3 3 sin t cos t,,find cos t cos t d DERIVATIVE OF ONE FUNCTION W.R.T. OTHER 68 Differentiate log sin w.r.t. cos 69 70 71 1 Differentiate sin 1 w.r.t. tan-1, -1 < < 1. 1,1, differentiate 1 1 tan with respect to 1 cos 1. Differentiate tan -1 1 1 with respect to 63 64 PARAMETRIC FUNCTIONS = a sec 3 and = a tan 3, find at d 3 Find in the following cases :(i) d 1 t a cos t log tan and a sin t 7 73 sin -1, if -1 < < 1, 0 1 Differentiate tan -1 1 a with respect to 1 a 1 a Differentiate sin 1 1 1 cot,if 0 1 1 with respect to (ii) = a( - sin ) and = a(1 cos ) 74 HIGHER ORDER DERIVATIVE = + tan, show that cos - + d
75 76 = 0 = log, show that d log 3 3 d 1 sin d, show that d (1 ) 3/ 86 = (cot -1 ), prove that ( + 1) + ( + 1) 1 = 77 = A cos (log ) + B sin (log ), prove that 78 log a d 0. d d, prove that d ( a ) 0 d d. 79 m 1, show that ( + 1) + 1 m = 0. 80 = a cos 3, = a sin 3, find d. 81 ( a) + ( b) = c, prove that of a and b. 1 d d 3/ a constant independent 8 = a (1 - cos 3 ), = a sin 3, prove that d 3 at d 7a 6 83 = a sin t - b cos t, = a cos t + b sin t, prove 84 85 that = 0 d 3 d 1 tan e, prove that (1 + ) + (- 1) 1 = [log( 1], show that d (1 ) d d