Continuity and Differentiability A)NCERT Problems: 1) 2) 3) 4) 5) 6) 7) |π₯| + 3, ππ π₯ ≤ −3 Find all points of discontinuity of π is defined by π(π₯) = {−2π₯, ππ − 3 < π₯ < 3 6π₯ + 2, ππ π₯ ≥ 3 Find the relationship between π and π so that the function π defined by π(π₯) = ππ₯ + 1, ππ π₯ ≤ 3 { is continuous at π₯ = 3 ππ₯ + 3, ππ π₯ > 3 π(π₯ 2 − 2π₯), ππ π₯ ≤ 0 For what value of π is the function defined by π(π₯) = { continuous at π₯ = 0 ? 4π₯ + 1, ππ π₯ > 0 What about continuity at π₯ = 1? Show that the function defined by π(π₯) = π₯ − [π₯] is discontinuous at all integral points. Here [π₯] denotes the greatest integer less than or equal to π₯ sin π₯ , ππ π₯ < 0 Find all points of discontinuity of π, where π(π₯) = { π₯ π₯ + 1, ππ π₯ ≥ 0 1 2 π₯ π ππ π₯ , ππ π₯ ≠ 0 Determine if π defined by π(π₯) = { is a continuous function? 0 ,π₯ = 0 π cos π₯ π , ππ π₯ ≠ 2 π π−2 π₯ Find the values of π so that the function π(π₯) = { is continuous at π₯ = 2 π 3, ππ π₯ = 2 5, ππ π₯ ≤ 2 8) Find the values of π and π such that the function defined by π(π₯) = {ππ₯ + π, ππ 2 < π₯ < 10 is a 21, ππ π₯ ≥ 10 continuous function. 9) Show that the function defined by π(π₯) = πππ (π₯ 2 ) is a continuous function. 10) Prove that the function π given by π(π₯) = |π₯ − 1|, π₯ ∈ π is not differentiable at π₯ = 1 11) Prove that the greatest integer function defined by π(π₯) = [π₯], 0 < π₯ < 3 is not differentiable at π₯ = 1 and π₯ = 2 (π₯−3)(π₯ 2 +4) 12) Differentiate √ ππ¦ 3π₯ 2 +4π₯+5 π¦ π₯ with respect to π₯ 13) Find ππ₯ , if π¦ π₯ +π₯ + π₯ = ππ 14) Differentiate (log π₯)π₯ + π₯ log π₯ with respect to π₯ 15) Differentiate (sin π₯)π₯ + sin−1 √π₯ with respect to π₯ 1 16) Differentiate (π₯ cos π₯)π₯ + (π₯ sin π₯)π₯ with respect to π₯ 17) Differentiate π₯ sin π₯ + (sin π₯)cos π₯ with respect to π₯ ππ¦ 18) Find ππ₯ of (cos π₯)π¦ = (cos π¦)π₯ ππ¦ 19) Find ππ₯ of π₯π¦ = π (π₯−π¦) 20) Find 21) Find ππ¦ ππ₯ ππ¦ ππ₯ and and π2 π¦ ππ₯ 2 π2 π¦ ππ₯ 2 π‘ , π₯ = π (cos π‘ + log π‘ππ 2) and π¦ = π sin π‘ , π₯ = π(cos π + π sin π) , π¦ = π(sin π − π cos π) ππ¦ −1 −1 22) If π₯ = √πsin π‘ , π¦ = √π cos π‘ , show that ππ₯ = − π¦ π₯ 23) If π¦ = cos −1 π₯. Find π2 π¦ ππ₯ 2 in terms of π¦ 24) If π¦ = 3π 2π₯ + 2π 2π₯ , prove that π2 π¦ ππ¦ - 5 ππ₯ + 6π¦ = 0 ππ₯ 2 π2 π¦ ππ¦ 25) If π¦ = sin−1 π₯, show that (1 − π₯ 2 ) ππ₯ 2 − π₯ ππ₯ = 0 26) If π¦ = 3 πππ (ππππ₯) + 4 π ππ(ππππ₯), show that π₯ 2 π¦2 + π₯π¦1 +π¦ = 0 π2 π¦ ππ¦ 2 27) If π π¦ (π₯ + 1) = 1, show that ππ₯ 2 = (ππ₯ ) 28) If π¦ = (tan−1 π₯)2 , show that (π₯ 2 + 1)2 π¦2 + 2π₯(π₯ 2 + 1)π¦1 = 2 29) State Rolle’s theorem. Write the geometrical interpretation of Rolle’s theorem. 30) Verify Rolle’s theorem for the function π(π₯) = π₯ 2 + 2π₯ − 8, π₯ ∈ [ −4, 2] 31) State Mean value theorem. Write the geometrical interpretation of Mean value theorem. 32) Verify Mean value theorem, if π(π₯) = π₯ 2 − 4π₯ − 3 in the interval [π, π], where π = 1 and π = 3. 33) If π: [ −5, 5] → π is a differentiable function and π ′ (π₯) does not vanish anywhere , then prove that π(−5) ≠ π(5) 2π₯+1 34) Differentiate πππ7 (log π₯) and sin−1 (1+ 4π₯ ) with respect to π₯ 1 π 1 ππ¦ 35) For a positive constant π find ππ₯ , where π¦ = ππ‘+ π‘ and π₯ = (π‘ + π‘ ) 36) Differentiate cot −1 [ ππ¦ √1+sin π₯+√1−sin π₯ π √1+sin 2 ] ,0 < π₯ < π₯ −√1−sin π₯ with respect to π₯ 37) Find ππ₯ , if π¦ = sin−1 π₯ + sin−1 √1 − π₯ 2 , −1 ≤ π₯ ≤ 1 ππ¦ 1 38) If π₯√1 + π¦ + π¦√1 + π₯ = 0, for – 1< π₯ < 1, prove that ππ₯ = − (π₯+1)2 39) If (π₯ − π)2 + (π¦ − π)2 = π 2, for some π > 0, prove that 3 ππ¦ 2 2 [1+( ) ] ππ₯ π2 π¦ ππ₯2 is a constant independent of π and π πππ 2 (π+π¦) ππ¦ 40) If cos π¦ = π₯ cos(π + π¦) , with cos π ≠ ±1, prove that = ππ₯ 41) If π(π₯) = |π₯|3, show that π ′′ (π₯) exists for all real π₯ and find it. 42) If π¦ = π π cos −1 π₯ , −≤ π₯ ≤ 1, show that (1 − π₯ 2 )2 π2 π¦ ππ₯ 2 -π₯ ππ¦ ππ₯ sin π − π2 π¦ = 0 a) extra problems: sin 3π₯+π sin 2π₯+ππππ π₯ 1) If f(x) = 2) A function f is defined as f(x) = π₯3 is continuous at x = 0, find the value of a and b π₯ 2 − 4π₯+3 π₯ 2 + 2π₯−3 =3) Let f(x) = π₯ 3 + π₯ 2 −16π₯+20 ( π₯−2)2 1 2 for x ≠ 1 for x = 1. Show that f(x) is differentiable at x = 1 and find its value if x ≠ 2 = k, if x = 2. If f(x) is continuous for all x, then find the value of k. 4) Let f(x) be a function of x defined as f(x) = = Discuss the continuity of function at x = 1 5) π₯2 − 1 π₯ 2 − |π₯|− 1 1 2 ,x≠1 , x = 1. Determine the values of a, b, c for which the function f(x) = sin( π+1)π₯+π πππ₯ π₯ , for x < 0 = c , for x = 0 1 = 6) f(x) = , for x > 0 3 ππ₯ 2 1−πππ 4π₯ , π₯2 = a, = 1 ( π₯+ππ₯ 2 )2 − π₯ 2 x<0 x= 0 √π₯ √16+ √π₯ − 4 , x >0 is continuous at x = 0 π₯ + π√2 π πππ₯, 7) 0≤π₯< π 4 Find the value of a and b so that the function π(π₯) = 2π₯ cot π₯ + π, ≤ π₯≤ { π cos π₯ − π sin π₯ π ,2 π 2 π 4 is continuous at 0 ≤ <π₯ ≤ π x≤π π+3 cos π₯ 8) Let π(π₯) = { π π‘ππ ( π₯2 π , π₯<0 ), π₯ ≥ 0 where [ ] represents the greatest integer function. If f(x) is [ π₯+3] 1 continuous at x = 0, then prove that a = -3 and b = - √3 2 9) ππ₯ 2 − π, |π₯| < 1 1 π(π₯) = { . The above function is continuous and differentiable , then prove that a − , |π₯| ≥ 1 |π₯| 1 3 2 2 = ,b= 10) Discuss the continuity and differentiability of the function f(x) = π₯ 1+ |π₯| = 11) 12) 3 πππ 2 π₯ , ππ π₯ < π, Let π(π₯) = π ( 1−sin π₯) { 14) (π−2π₯)2 ππ π₯ = π 2 π , ππ π₯ > Prove that function π(π₯) = { π . If f(x) be a continuous at x = 2 , find a and b. 2 π 2 π₯ |π₯|+ 2π₯ 2 , ππ π₯ ≠ 0 π, is discontinuous at x = 0, regardless of the value of k. ππ π₯ = 0 π cos π₯ , ππ π−2π₯ 15) Find the value of k, such that function ‘f’ defined by π(π₯) = { 16) Find the set of all points where the function π (π₯) = { (cos π₯ − sin π₯) 17) , |π₯| ≤ 1 2 + √( 1 − π₯ 2 ), |π₯| ≤ 1 Discuss the continuity and differentiability of the function π(π₯) = { 2 2π ( 1−π₯) , |π₯| > 1 ππ₯ 2 + π, ππ π₯ > 2 Determine the constants a and b, such that the function π(π₯) = { 2, ππ π₯ = 2 is continuous 2ππ₯ − π, ππ π₯ < 2 1− π ππ3 π₯ 13) , |π₯| ≥ 1 π₯ 1− |π₯| 3, is continuous at x = 1+ π π₯ πππ ππ π₯ π , ππ − < π₯ < 0 2 ππ π₯ = 0 1 2 3 π π₯+ π π₯+ π π₯ { ππ π₯ = π 2 π 2 2 3 π ππ₯ + π ππ₯ π 2 0 , ππ π₯ = 0 π₯ is differentiable. 1 , ππ π₯ ≠ 0 π, Let f(x) be defined as follows π(π₯) = π₯ ≠ , ππ π<π₯< π 2 . If f(x) is continuous at x = 0, then find the value of and b . 1 π₯ (3π π₯ +4 ) 18) , ππ π₯ ≠ 0 1 Discuss the continuity and differentiability of the function π (π₯) = { ( 2− π π₯ ) 0 , 19) 1 −1 π₯ (π π₯ −π π₯ ) 1 −1 ( ππ₯ + π π₯ ) Let π (π₯) = { 0 , , ππ π₯ ≠ 0 ππ π₯ = 0 . prove that f(x) is not differentiable at x = 0 ππ π₯ = 0 y = (sinx) x + ( cosx ) tanx 20) Differentiate w.r.t. x- 21) Differentiate w.r.t. x- 22) If xy = e x-y, prove that y = (logx) x + x logx ππ¦ ππ₯ = log π₯ (1+log π₯)2 1 23) at x = 0 Differentiate w.r.t. x - y = tan −1 1 π₯ 3+ π3 ( 1 1 ) 1− π3 π₯ 3 cos π₯−sin π₯ 24) Differentiate w.r.t. x- y = tan−1 (cos π₯+ π πππ₯) 25) If y = √π₯ + √π₯ + √π₯ + β― … … … , prove that 26) Differentiate y = π₯ π₯ , w.r.t. x 27) If y √1 − π₯ 2 + x √1 − π¦ 2 = 1 , prove that ππ¦ ππ₯ π₯ π₯2 ππ₯ 1− π₯ 2 ππ¦ π₯π ππ₯ π! 29) If √1 − π₯ 2 + √1 − π¦ 2 = a( x- y) , prove that 30) If sin−1(π₯ 2 √1 − π₯ 2 + π₯ √1 − π₯ 4 ) , then prove that π! , show that 1− π¦ 2 If y = 1 + x + 3! + ……… + π₯π =-√ 2π¦−1 28) 2! + π₯3 ππ¦ 1 = ππ¦ ππ₯ Differentiate w.r.t.x : tan−1 32) If √1 − x 6 + √1 − y 6 = a3 ( x3 – y3), show that 33) If x = 34) √cos cos3 θ ,y= 2t 1− π¦ 2 1− π₯ 2 ππ¦ ππ₯ =√ 2π₯ 1−π₯4 +√ 1 1− π₯2 √1+ x2 − √1−x2 , find √cos 2t =0 √1+ x2 +√1−x2 31) sin3 θ =√ -y + dy dx at t = x2 dy 1− y6 = √ 1−x6 dx y2 π 6 a−x dy a−x If y = √( a − x)(x − b) - ( a – b) tan−1 √x−b , then prove that dx = √x−b 1 35) If y = log [ 36) If y = 1 + { c1 ( c1 − x) √1+x+ √1−x 2 dy √1+x− √1−x c1 x c2 dx ] , then x− c1 + 37) If y = 2 tan x 2n ( x x 2 22 = c2 x− c1 )( x− c2 ) c3 ( c2 – x) −1 x √2 If cos cos tan +( 1 2n + 1− x2 x cos cot x 2n + 1 2x √1− x2 x2 c3 x− c1 )( x− c2 )( x− c3 ) c4 1−x √2+ x2 x …………… cos – cot x +( x3 c4 x− c1 )( x− c2 )( x− c3 )(x− c4 ) , show that dy dx = y x } ( c3 − x) (c4 − x) 1+x √2+ x2 ) + log 23 +( =- 2n , then = dy dx sin x x = 2n sin( n ) 2 4√2 1+ x4 , prove that that 1 2 tan x 2 + 1 22 tan x 22 + ………. + 1 2n x n y 38) If cos −1 (b) = log ( ) , prove that x 2 π¦2 + x y1 n2 y = 0 n 39) If y = (sin−1 x)2 + ( cos−1 π₯)2 , then prove that ( 1- x2 )π¦2 - x y1 = 4 40) If y = tan−1 ( a1 x − α a2 − a1 a1 α+x 1+ a1 a2 ) + tan−1 ( tan−1(ππ ) , then find π πππ₯ 41) If y = dy ) + tan−1 ( a3 − a2 1+ a2 a3 ) + ……………. + tan−1 ( ππ − ππ−1 1+ ππ−1 ππ )- dx cos π₯ 1+ sin π₯ 1+ 1+cos π₯…………∞ , then prove that 1 dy dx = ( 1+π¦) cos π₯+sin π₯ 1+2π¦+cos π₯−sin π₯ 1 1 42) If y = tan−1 ( π₯ 2 + π₯+1) + tan−1 (π₯ 2 + 3π₯ + 3) + tan−1 (π₯ 2 + 5π₯ +7) + …………………. To n terms, show that dy dx = 1 (π₯+π)2 + 1 - 1 π₯2+ 1 ( 1 + x)a 43) If f( x) = | 1 (1 + 2x)b (1 + 2x)b ( 1 + x)a 1 1 (1 + 2x)b | , then find i) constant term ii) coefficient of x ( 1 + x)a sin( π+1)π₯+π πππ₯ 44) Determine the values of a, b, c for which the function f(x) = π₯ = c , for x = 0 1 = , for x < 0 1 ( π₯+ππ₯ 2 )2 − π₯ 2 3 , for x > 0 ππ₯ 2 ππ₯ 2 + π, ππ π₯ > 2 Determine the constants a and b, such that the function π(π₯) = { 2, ππ π₯ = 2 is continuous 2ππ₯ − π, ππ π₯ < 2 45) 46) 47) If y √1 − π₯ 2 + x √1 − π¦ 2 = 1 , prove that 1 ππ¦ ππ₯ 1 1− π¦ 2 = - √1− π₯2 1 If y = tan−1 ( π₯ 2 + π₯+1) + tan−1 (π₯ 2 + 3π₯ + 3) + tan−1 (π₯ 2 + 5π₯ +7) + …………………. To n terms, show that 1 1 (π₯+π)2 + 1 π₯ 2 + 1 b) extra problems( Advanced) 1) Let π(π₯) = π₯ π π₯ π πππ(1+ )−πππ(1− ) π₯ , π₯ ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0 2) If y = √( a − x)(x − b) - ( a – b) tan−1 √ 3) If x = sin3 θ √cos 2t ,y= cos3 θ , √cos 2t dy a−x x−b , then prove that dy dx =√ a−x x−b π find dx at t = 6 4) Find the values of a and b such that the function f(x) is defined by π π₯ + π√2π πππ₯, 0 ≤ π₯ < 4 π 2π₯ πππ‘π₯ + π, 4 ≤ π₯ ≤ {π πππ 2π₯ − π π π πππ₯, 2 π 2 <π₯≤π is continuous for all values of x and 0≤ π₯ ≤ π π(π₯) = dy dx = (4π₯ −1)3 5) π₯ π₯2 π ππ( )πππ(1+ ) π 3 ,π₯ ≠ 0 Find the value of a for which the function π(π₯) = { may be continuous at x = 0 3 12(πππ4) , π₯ = 0 6) If y = √π₯ + √π₯ + √π₯ + β― … … … , prove that 7) Differentiate tan−1 ( 8) If π¦ = (cos π₯)(cos π₯) 9) If π¦ = tan−1 ( 10) If π¦ = 11) If π¦ = (cos π₯)…∞ Differentiate 13) If π¦ = to tan−1 π₯ , when π₯ ≠ 0 π¦ 2 tan π₯ ππ¦ , show that ππ₯= π¦πππ(cos π₯)−1 π₯+ π₯+β―.∞ √1+π₯ 2 +√1−π₯ 2 w.r.t. √1 − π₯ 4 √1+π₯ 2 −√1−π₯ 2 2 [log(π₯ + √π₯ 2 + 1)] , show that (1 + π₯ 2 ) 1−cos 4π₯ , { 8π₯ 2 Find the value of π for which π(π₯) = π2 π¦ ππ₯ 2 +π₯ π₯≠0 π, π₯ = 0 π cos π₯ π−2π₯ 15) 1 = 2π¦−1 ππ¦ √1+π₯−√1−π₯ ), find ππ₯ √1+π₯+√1−π₯ 2π₯−3√1−π₯ 2 ππ¦ cos−1 ( ), find ππ₯ √13 1 ππ¦ π₯+ , then find ππ₯ 1 π₯+ 1 12) 14) √1+π₯ 2 −1 ) w.r. π₯ ππ¦ ππ₯ Determine the values of π and π, π(π₯) = 3, ππ‘ππ 2π₯ { (2π₯−π) ππ¦ ππ₯ =2 is continuous at π₯ = 0 π , ππ π₯ < 2 ππ π₯ = , ππ π₯ > π 2 π 2 is continuous at π₯ = π 2 16) Show that the function π defined as follows, is continuous at π₯ = 2, but not differentiable at π₯ = 2. π(π₯) = 3π₯ − 2, ππ 0 < π₯ ≤ 1 { 2π₯2 − π₯, ππ1 < π₯ ≤ 2 5π₯ − 4 , ππ π₯ > 2 π(1−sin ππ₯) 1 , ππ π₯ < 2 (1+cos 2ππ₯) 17) Let π(π₯) = π, ππ π₯ = √2π₯−1 1 {√4+√2π₯−1−2 1 2 , ππ π₯ > Determine the sum of π + π so that π(π₯) is continuous at π₯ = 1 2 . 2 18) 19) Letπ(π₯) = 1−ππ₯ +π₯ππ₯ log π , π π₯ π₯2 0, ππ π₯ < 0 ππ π₯ = 0 2π₯ ππ₯ −π₯ log 2−π₯ log π−1 π₯2 { is continuous at π₯ = 0, find the value of π , ππ π₯ > 0