CLASS XII CBSE MATHEMATICS CONTINUITY AND DIFFERENTIATION sin5 + cos, if 0 ) For what value of k is the function f() = { 3 k, if = 0 ) For what value of k is the following function continuous at =? ; f ( ) k; 3 ; 3) Find the value of k so that the function kcos continuous at =0 π f() = {, if π 3, if = π is continuous at = π 5, if 4) Find the value of a and b, if f() ={ a + b, if < < 0 is a continuous function at, if 0 = and = 0 5) Find the relationship between a and b so that the function f defined b is a + if 3 f() = { continuous at = 3 b + 3 if > 3 3a + b, if > 6) Find the value of a and b, if f() ={, if = 5a b, if < = 7) Find a, b, c if f() is continuous at = 0, f() = is a continuous function at sin(a+)+sin, < 0 c, = 0 +b 3, > 0 { b 8) For what value of k, the following function is continuous at = 0 cos4, 0 f() = { 8 k, = 0 Sin 9) Discuss the continuit of the function if 0 f ( ) if 0 Cosk if, 0 0) Find the value of k so that the function f ( ) Sin is continuous at = 0 if, 0 DSREENIVASULU, MSc,MPhil,BEd PGT(Mathematics), Kendria Vialaa
) Find the value of k, for which f() = { at = 0 +k k + e if < 0 if 0 < is continuous ) Show that the function f given b: f() = { e, if 0 is discontinuous at = 0, if = 0 3) Eamine the continuit of the function f() = { for 0 at = 0 for = 0 4) Show that the function f() = is continuous at = 0 5) Given f() = 6) Let f() = { cos4, if < 0 a, if = 0 6 4 sin 3, if > 0, 3cos if < π a, if = π b( sin) is continuous at = 0, find the value of a is continuous function at = π, find a and b {, if > π (π ) asin π ( + ), if 0 7) For what value of a for which the function f() = { tan sin, if > 0 is 3 continuous at = 0 DIFFERENTIATION Differentiation b using Inverse TrigonometricFunctions ) If tan, show that ) Differentiate tan ( + 3) Find + + )with respect to cos if = tan ( +sin+ sin +sin sin ) 4) Differentiate tan ( ) w rt sin ( + ) 5) If = sin [ 5+ ], find 3 6) If = cos [ 3+4 ], find 5 7) Differentiate sin ( + 3 ) with respect to +(36) 4 DSREENIVASULU, MSc,MPhil,BEd PGT(Mathematics), Kendria Vialaa
8) Find the derivative of sec ( ) w r t at = 9) Find: d cos ( + ) Differentiation b using Logarithms ) If = e, show that = log {log(e)} ) If = e ( ),then show that = ( ) (+) 3) Find, if cos + cos sin 4) If sin cos, find 5) Find when log (log ) 6) If = + (sin ) find 7) If (cos ) = (sin ), find 8) Find,when = cot + (cos) sin 9) Find if = sintan + cos sec 0) Differentiate sin + (sin ) cos with respect to ) Differentiate (sin) + sin 3 with respect to ) If = e, find 3) If + + = a b, then find p q pq 4) If, then find 5) Find the derivative of the following functions f() w r t, at = f() = cos [sin + ] + 6) Differentiate Parametric forms ) Find at t = π 3 3 4 3 45 with respect to when = 0(t sint)and = ( cost) DSREENIVASULU, MSc,MPhil,BEd PGT(Mathematics), Kendria Vialaa
) Find eθ (θ ) and = θ e θ (θ + ) θ 3) If = a (cosθ + log tan θ ) and = a sinθ, find the value of 4 4) If = a ( θ + sin θ ), = a ( cos θ ), find d at θ = π 5) If = a cos 3 θ and = asin 3 θ, then find the value of d at θ = π 6 6) Find the value of at θ = π if = 4 aeθ (sinθ cosθ) and = ae θ (sinθ + cosθ) 7) If = a(cos t + t sin t) and = a(sin t t cos t), then find d 8) If = α sin t( + cos t) and = β cos t( cos t), show that 9) If = e cos t and = e sint, prove that Implicit Functions ) Find ) Find if ( + ) =, if + = tan + 3) If sin = sin(a + ), prove that = sin (a+) sin a 4) If 0 Prove that ( ) = β α DSREENIVASULU, MSc,MPhil,BEd PGT(Mathematics), Kendria Vialaa = log log cos ( a ) 5) If cos cos( a ), with cosa, prove that sina 6) If log + = tan ( ), then show that 7) If = log a, then prove that = Second order Derivatives ) If = 3 e + e 3, prove that d = + 5 + 6 = 0 ) If = cos (log ) + 3 sin (log ), prove that d + + = 0 3) log d If, find 4) If = e asin,, then show that ( ) d a = 0 5) If = tan ( log ), show that ( + a ) d + ( a) 6) If (a + b)e / =, then show that: 3 ( d ) = ( ) 7) If log = tan, then show that( + ) d + ( ) tan t
8) If = log ( a+b ), prove that 3 = ( ) 9) If = +, prove that ( ) d + DSREENIVASULU, MSc,MPhil,BEd PGT(Mathematics), Kendria Vialaa 4 = 0 0) If = a cos θ + b sin θ, = a sin θ b cos θ, show that d + = 0 ) If = ( + + ) n, then show that ( + ) d + = n ) If = 3 log ( ), then prove that d + 3 = 0 3) If =, prove that d ( ) = 0 4) If cos(a + ) = cos then prove that = cos (a+) sin a sin a d + sin (a + ) = 0 Hence show that Rolle s Theorem and Mean Value Theorem ) Verif Mean value theorem if f( ) = - 4 3 in the interval [, 4] ) Verif Rolle s theorem for f( ) = + 8 in the interval [ 4, ] 3) Verif mean value theorem for the function f() = ( 4)( 6)( 8) on the interval [4,0] 4) Verif Mean value theorem for the function f() = sin + sin on[ 0, π] Other Problems ) If, prove that ( ) 0 ) If log, then show that 3) If f() = + ; g() = + + f [h {g ()}] 4) Let f() =, [,]find the point of discontinuit, (if an), of this function on [-,] 5) If function f() = 3 + 4,then show that f() is not differentiable at = 3 and = 4 6) Show that the functionf() = + +, for all R, is not differentiable at the points = and = 7) Find whether the following function is differentiable at = and = or not:, < f() = {, + 3 > 8) Eamine the following function f() for continuit at = and differentiabilit at = 5 4, 0 < < f() = { 4 3, < 3 + 4,