continuity and derivatives

1. CHAPTER – 5
CONTINUITY AND DIFFERENTABILITY
1. A real valuedfunctioniscontinuousata pointa in itsdomainif LHL= RHL = f(a). A functionis
continuousif itiscontinuousonwhole domain.
2. Sum,difference,productandquotientof continuousfunctions are continuous.
3. A functionisdifferentiable atapointif LHD= RHD
4. Everydifferentiable functioniscontinuousbutconverse isnottrue.
5. Rolle’s Theorem: if f isa continuousfunctionon[a,b] anddifferentiableon(a,b) suchthatf(a)=f(b),
thenthere existsome cin(a,b) such that f’(c)=0.
6. Mean value Theorem: if f is a continuousfunctionon[a,b] anddifferentiable on(a,b),thenthere
existsome cin (a,b) suchthat f’(c)=
𝑓( 𝑏)−𝑓(𝑎)
𝑏−𝑎
4 Marks Questions
1. Examine the continuity of following functions at indicated points:
(i) f(x) = {
2𝑥 − 1, 𝑥 < 0
2𝑥 + 1, 𝑥 ≥ 0
(ii)f(x) = {
5𝑥 − 4, 𝑥 < 1
4𝑥2 − 3𝑥, 𝑥 ≥ 1
(iii) f(x) ={
𝑥3 − 3, 𝑥 ≤ 0
𝑥2 + 1, 𝑥 > 0
(iv)f(x) = {
𝑥
| 𝑥|
, 𝑥 ≠ 0
0, 𝑥 = 0
2. find all points of discontinuity of f , where f is defined by
(i)f(x) = {
2𝑥 + 3, 𝑥 ≤ 2
2𝑥 − 3, 𝑥 > 2
(ii)f(x) ={
𝑥3 − 3, 𝑥 ≤ 2
𝑥2 + 1, 𝑥 > 2
(iii) f(x) ={
𝑠𝑖𝑛 𝑥
𝑥
, 𝑥 < 0
𝑥 + 1, 𝑥 ≥ 0
(iv) f(x) = {
| 𝑥| + 3 𝑥 ≤ −3
−2𝑥 −3 < 𝑥 < 3
6𝑥 + 2 𝑥 ≥ 3
2. For what values of a and b the function defined is continuous at x = 1,
f(x) = {
3𝑎𝑥 + 𝑏 𝑖𝑓 𝑥 < 1
11 𝑖𝑓 𝑥 = 1
5𝑎𝑥 − 2𝑏 𝑖𝑓 𝑥 > 1
4. Discuss the continuity of the function f defined by f(x) = {
𝑥 + 2 𝑖𝑓 𝑥 ≤ 1
𝑥 − 2 𝑖𝑓 𝑥 > 1
5. Show that the function f(x) = {
𝑥 𝑠𝑖𝑛
1
𝑥
𝑖𝑓 𝑥 ≠ 1
0 𝑖𝑓 𝑥 = 1
is continuous at x = 0.
6. For what value of 𝛾 is the function defined by f(x) ={
𝛾(𝑥2 − 2𝑥) 𝑖𝑓 𝑥 ≤ 0
4𝑥 + 1 𝑖𝑓 𝑥 > 0
is continuous at x = 0.
7. Examine the continuity of f , where f is defined by f(x) = {
𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥 𝑖𝑓 𝑥 ≠ 0
−1 𝑖𝑓 𝑥 = 0
.

2. 8. Find the values of K so that the function f is continuous at indicated points.
(i) f(x) = {
𝐾𝑥 + 1 𝑖𝑓 𝑥 ≤ 5
3𝑥 − 5 𝑖𝑓 𝑥 > 5
𝑎𝑡 𝑥 = 5 (ii) f(x) = {
𝐾𝑥 + 1 𝑖𝑓 𝑥 ≤ 𝜋
𝑐𝑜𝑠 𝑥 𝑖𝑓 𝑥 > 𝜋
𝑎𝑡 𝑥 = 𝜋
(iii) f(x) = {
𝐾 cos𝑥
𝜋 −2𝑥
𝑖𝑓 𝑥 ≠
𝜋
2
3 𝑖𝑓 𝑥 =
𝜋
2
𝑎𝑡 𝑥 =
𝜋
2
(iv) f(x) =
{
(1−𝑐𝑜𝑠4𝑥)
𝑥2 𝑖𝑓 𝑥 < 0
𝐾 𝑖𝑓 𝑥 = 0
√ 𝑥
√16+√ 𝑥− 4
𝑖𝑓 𝑥 > 0
𝑎𝑡 𝑥 = 0
9. Determine if f defined by f(x) = {
𝑥2
𝑠𝑖𝑛
1
𝑥
𝑖𝑓 𝑥 ≠ 0
0 𝑖𝑓 𝑥 = 0
is a continuous function?
10. Examine the continuity of the function f(x) = {
| 𝑠𝑖𝑛 𝑥|
𝑥
𝑖𝑓 𝑥 ≠ 0
1 𝑖𝑓 𝑥 = 0
at x = 0
.11 Find the value of k so that f(x) = {
𝑥2
+3𝑥−10
(𝑥−2)
, 𝑖𝑓 𝑥 ≠ 0
𝑘 , 𝑖𝑓 𝑥 = 0
is continuous at x = 0 .
12 Discuss the continuity of f(x) = |x – 1| +|x| + |x - 1| at – 1, 0 and 1
13. if f(x) = {
√1+𝑘𝑥 − √1−𝑘 𝑥
𝑥
− 1 ≤ 𝑥 < 0
2𝑥+1
𝑥−2
0 ≤ 𝑥 ≤ 1
is continuous at x = 0. Find the value of k
14. Find the value of a and b such that the following function f(x) is a continuous function :
f(x) = {
5; 𝑥 ≤ 2
𝑎𝑥 + 𝑏; 2 < 𝑥 < 10
21; 𝑥 ≥ 10
15.For what value of k, the following function is continuous at x = 0 :
f(x) = {
1−𝑐𝑜𝑠 4𝑥
8𝑥2 , 𝑥 ≠ 0
𝑘 , 𝑥 = 0
16. Find the value of ‘ a’ for which the function f defined as
f(x) = {
𝑎 𝑠𝑖𝑛
𝜋
2
( 𝑥 + 1) , 𝑥 ≤ 0
𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛 𝑥
𝑥3 , 𝑥 > 0
, is continuous at x = 0.
17. If the function f , as defined below is continuous at x = 0, find the values of a, b and c.
f(x) =
{
𝑠𝑖𝑛( 𝑎+1) 𝑥+𝑠𝑖𝑛𝑥
𝑥
, 𝑥 < 0
𝑐 , 𝑥 = 0
√𝑥+𝑏𝑥2 − 𝑥
𝑏𝑥3/2 , 𝑥 > 0

3. 18. Show that the function ‘f ‘ defined by f(x) = {
3𝑥 − 2 , 0 < 𝑥 ≤ 1
2𝑥2
− 𝑥 , 1 < 𝑥 ≤ 2
5𝑥 − 4 , 𝑥 > 2
is continuous at x = 2, but
not differentiable.
19. Prove that the greatest integer function defined by f(x) = [x], 0<x<3 ,
is not differentiable at x = 1 and x = 2.
20. Prove that the function f given by f(x) = | 𝑥 − 1|, 𝑥 ∈ 𝑅 is not differentiable at x = 1.
21 If log ( 𝑥2
+ 𝑦2) = 2 tan -1(
𝑦
𝑥⁄ ) , then Show that
𝑑𝑦
𝑑𝑥
=
𝑥+𝑦
𝑥−𝑦
22 Differentiate sin -1(
3𝑥+4√1−𝑥2
5
) 𝑤. 𝑟. 𝑡. 𝑥
23 Differentiate tan –1 (
√1+𝑥2
−√1−𝑥2
√1+𝑥2+√1−𝑥2 )w.r.t cos – 1 x2
24 IF x = a sec 3 , y = a tan3 𝜃,Prove that
𝑑2
𝑦
𝑑𝑥2 at 𝜃 =
𝜋
4
is 1/12a
25 IF x = tan (
1
𝑎
log 𝑦 ) 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 that (1 + 𝑥2)
𝑑2
𝑦
𝑑𝑥2 + (2x-a )
𝑑𝑦
𝑑𝑥
= 0
26 If Y = 𝑙𝑜𝑔(𝑥 + √𝑥2 + 1)2 then show that (1+𝑥2
)
𝑑2
𝑦
𝑑𝑥2 +x
𝑑𝑦
𝑑𝑥
= 2
27 If y = sin ( log x) then prove that 𝑥2 𝑑2
𝑦
𝑑𝑥2 + x
𝑑𝑦
𝑑𝑥
+ y = 0
28 Differentiate sin -1(
(2 𝑥+1
3 𝑥)
1+ (36) 𝑥 )w.r.t x
29 Differentiate the following functions w.r.t.x (chain rule)
(a) log (sin x) (b) sin (ex2) (c) sin3x
(d)
1
√𝑎2−𝑥2 (e) log √
𝑥
𝑥−1
(f) log(x+ √ 𝑎2 + 𝑥2 )
(g) sin √ 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠 𝑥 (h) cos (x3) sin2 (x5) (i) sinmx cosnx
30. If y = (x + √ 𝑥2 + 𝑎2 )n , Prove that
𝑑𝑦
𝑑𝑥
=
𝑛𝑦
√𝑥2 +𝑎2
31. If y = (x + √ 𝑥2 − 1)m , Prove that (x2 – 1) (y1)2 = m2y2
32. If x√1 + 𝑦 + y√1 + 𝑥 = 0, find
𝑑𝑦
𝑑𝑥

4. 33. If y =
𝑥 sin−1
𝑥
√1−𝑥2 + log √1 − 𝑥2 , Prove
𝑑𝑦
𝑑𝑥
=
sin−1
𝑥
(1−𝑥2)
3
2
34. Differentiate the following functions w.r.t.x :
(i) sin−1
(
2𝑥
1+𝑥2) (ii) tan−1
(
1−𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 𝑥
) (iii) tan−1
(
𝑐𝑜𝑠 𝑥
1+𝑠𝑖𝑛 𝑥
)
(iv) tan−1
(
5 𝑥
1−6𝑥2) (v) tan-1[
√1+𝑥2
√1+𝑥2
+√1−𝑥2
−√1−𝑥2] (vi) tan-1[
√1+𝑠𝑖𝑛 𝑥
√1+𝑠𝑖𝑛 𝑥
+√1−𝑠𝑖𝑛 𝑥
−√1−𝑠𝑖𝑛 𝑥
]
(vii) cot-1(
1−𝑥
1+𝑥
)
35. Find
𝑑𝑦
𝑑𝑥
in the following: (implicit functions)
(a) x3 +y3 = 3axy (b) tan-1 (x2 + y2) = a (c) ax2 + 2hxy + by2 = c2
(d) ex-y log (
𝑥
𝑦
) (e) x2/3 + y2/3 = a2/3
36. If √1 − 𝑥2 + √1 − 𝑦2 = a (x – y), Prove
𝑑𝑦
𝑑𝑥
= √
1−𝑦2
1−𝑥2
37. If y =√ 𝑥 + √ 𝑥 + √ 𝑥 − ∞, Prove
𝑑𝑦
𝑑𝑥
=
1
2𝑦−1
38. Differentiate the following functions w.r.t.x :
(a) xsin x (b) (sin x) logx (c) 𝑥cos−1
𝑥
(d) cos (xx)
(e)𝑥sin−1
𝑥
(f)( 𝑠𝑖𝑛 𝑥)cos−1
𝑥
(g) x logx + (log x)x (h) (sin x)cosx + x sinx
(i) x cotx +
2𝑥2
−3
𝑥2 +𝑥+2
(j) x sinx-cos x +
𝑥2
−1
𝑥2+1
(k) xx cos x + (x cos x)x
39. If xm . yn = (x + y)m+n , Prove that
𝑑𝑦
𝑑𝑥
=
𝑦
𝑥
40. If xy = e x-y , Prove that
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔 𝑥
(1+𝑙𝑜𝑔𝑥)2
41. If xy = y x , Prove that
𝑑𝑦
𝑑𝑥
=
𝑦(𝑥 𝑙𝑜𝑔 𝑦−𝑦)
𝑥(𝑦𝑙𝑜𝑔𝑥−𝑥)
42. If (cos x)y = (siny)x, find
𝑑𝑦
𝑑𝑥
.
43. If xy + yx = ab , find
𝑑𝑦
𝑑𝑥
.
44. If x = a ( t + sin t), y = a(1 – cos t), find (
𝑑𝑦
𝑑𝑥
)t =
𝜋
2
.
45. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that
𝑑2
𝑦
𝑑𝑥2 =
𝑏 𝑠𝑒𝑐3
𝑡
𝑎2 𝑡
.

5. 46. If y = ( x + √𝑥2 − 1 )m, prove that (x2 – 1) y2 + xy1 – m2 y = 0.
47. Find
𝑑𝑦
𝑑𝑥
in the following:
(i) x = a [cos t + log tan t/2], y = a sin t
(ii) x = at2 , y = 2at
(iii) x = a(𝜃 + 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(1 + 𝑐𝑜𝑠𝜃)
(iv) x = a(cos 𝜃 + 𝜃𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(𝑠𝑖𝑛𝜃 − 𝜃𝑐𝑜𝑠𝜃)
(v) x = a sec3 𝜃 , 𝑦 = 𝑎 tan3 𝜃
48. If y = 3 cos(logx) + 4 sin (logx), then show that x2 𝑑2
𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑦 = 0.
49. If y = 3e2x + 2e3x , Prove
𝑑2
𝑦
𝑑𝑥2 − 5
𝑑𝑦
𝑑𝑥
+ 6𝑦 = 0.
50. If y = (tan-1 x)2 , Show that (x2 + 1)2 y2 + 2x(x2 + 1)y1 = 2
51. If y =𝑒 𝑎 cos−1
𝑥
, Show (1 – x2) y2 – xy1 = a2y.
52. If x = a cos3 𝜃 , 𝑦 = 𝑎 𝑠𝑖𝑛3
𝜃, 𝑓𝑖𝑛𝑑
𝑑2
𝑦
𝑑𝑥2 .
53. If y = cosec x + cot x. Show that sin x
𝑑2
𝑦
𝑑𝑥2 = y2 .
54. If x = a(𝜃 − 𝑠𝑖𝑛 𝜃), 𝑦 = 𝑎 ( 1 − 𝑐𝑜𝑠𝜃), Find (
𝑑2
𝑦
𝑑𝑥2 )
𝜃=
𝜋
2
.
55. Differentiate sin-1(
2 𝑥+1
1+4 𝑥) w.r.t.x
56. If x = sin t, y = sin mt, Prove that (1 – x2)
𝑑2
𝑦
𝑑𝑥2 - x
𝑑𝑦
𝑑𝑥
+ 𝑚2
𝑦 = 0.
57. Find
𝑑𝑦
𝑑𝑥
, if yx + xy + xx = ab
58. If y = sin -1 x, Show y2 =
𝑥
(1−𝑥2)
3
2
.
59. Differentiate log sin x w.r.t. √ 𝑐𝑜𝑠 𝑥 .
60. Differentiate sin-1(
2𝑥
1+𝑥2) w.r.t. tan-1 x .
64. Find
𝑑𝑦
𝑑𝑥
, if y = sin -1 x + sin -1√1 − 𝑥2 .
65. If xy = ex-y , show that
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔 𝑥
{ 𝑙𝑜𝑔 (𝑥𝑒)}2 .
66. Prove that :
𝑑
𝑑𝑥
[
𝑥
2
√𝑎2 − 𝑥2 +
𝑎2
2
sin−1
(
𝑥
𝑎
)] = √𝑎2 − 𝑥2 .
67. Differentiate tan-1[
√1+𝑥2
−1
𝑥
] with respect to x.
68. If y = log tan (
𝜋
4
+
𝑥
2
), show that
𝑑𝑦
𝑑𝑥
- sec x = 0 .

6. 69. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos2t), show that (
𝑑𝑦
𝑑𝑥
)at t =
𝜋
4
=
𝑏
𝑎
.
70. If y = cos-1(
2 𝑥+1
1+4 𝑥) , find
𝑑𝑦
𝑑𝑥
.
71. Find
𝑑𝑦
𝑑𝑥
, if y = sin-1 [x √1 − 𝑥 − √ 𝑥√1 − 𝑥2] .
72. If y = log [x + √𝑥2 + 1], prove that (𝑥2
+ 1)
𝑑2
𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
= 0 .
73. If y =
sin−1
𝑥
√1−𝑥2 , show that (1-𝑥2
)
𝑑2
𝑦
𝑑𝑥2 − 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦 = 0 .
74 if x = a cos t + b sin t , y = a sin t – b cos t , show that y2 𝑑2
𝑦
𝑑𝑥2 – x
𝑑𝑦
𝑑𝑥
+ y = 0