5.3 Exercise
18
CONTINUITY AND DIFFERENTIABILITY 1. Examine the continuity of the function at . Find which of the functions is continuous or discontinuous at the indicated points: 2. at 3. at 4. at 5. at 6. at 7. at 8. at 9. at 10. at .
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Transcript of 5.3 Exercise
CONTINUITY AND DIFFERENTIABILITY
1. Examine the continuity of the function at .
Find which of the functions is continuous or discontinuous at the indicated points:
2. at
3. at
4. at
5. at
6. at
7. at
8. at
9. at
10. at .
Find the value of k in each of the Exercises so that the function f is continuous at the
indicated point:
11. at
12. at
13. at
14. at
15. Prove that the function f defined by remains
discontinuous at , regardless the choice of k.
16. Find the values of a and b such that the function f defined by
is a continuous
function at .
17. Given the function . Find the points of discontinuity of the
composite function .
18. Find all points of discontinuity of the function , where .
19. Show that the function is continuous at .
Examine the differentiability of f, where f is defined by
20. at .
21. at .
22. at .
23. Show that is continuous but not differentiable at .
24. A function satisfies the equation for all
, . Suppose that the function is differentiable at and
. Prove that .
Differentiate each of the following w.r.t. x:
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37.
38.
39.
40. and
41.
42.
43. .
Find of each of the functions.
44.
45.
46. .
47. .
48. .
49. If and , prove that .
50. If and , show that .
51. If , find at .
52. Differentiate w.r.t. .
53. Differentiate w.r.t. when .
Find each of the following.
54. 55.
56. 57.
58. If , then show that .
59. If , prove that .
60. If , prove that .
61. If , show that .
62. If , prove that .
63. If , prove that .
64. If , find in terms of y alone.
Verify the Rolle’s theorem for each of the functions.
65. in [0, 1]. 66. in .
67. in . 68. in .
69. in .
70. Discuss the applicability of Rolle’s theorem on the function given by
.
71. Find the points on the curve in , where the tangent is
parallel to x-axis.
72. Using Rolle’s theorem, find the point on the curve
, where the tangent is prarallel to x-axis.
Verify mean value theorem for each of the functions.
73. in .
74. in .
75. in .
76. in .
77. Find a point on the curve , where the tangent is parallel to the chord
joining the points (3, 0) and (4, 1).
78. Using mean value theorem, prove that there is a point on the curve
between the points A (1, 0) and B (2, 1), where tangent is parallel
to the chord . Also, find that point.
79. Find the values of p and q so that is
differentiable at .
80. If , prove that
(i) and (ii) .
81. If and , prove that .
82. Find , if .
83. Find the value of the constant k so that the function f defined below is
continuous at , where .
84. Discuss the continuity of the function .
85. If is continuous at , find the
value of k.
86. Show that the function f defined by is
continuous at .
87. Given . Find the points of discontinuity of the composite function
.
88. Let , for all . Discuss the derivability of at .
89. Differentiate w.r.t. x.
90. If , find .
91. If ; prove that .
92. Find , if .
93. If and , then find .
94. If and , find at .
95. If , prove that .
96. If , prove that .
97. If , find .
98. If , find .
99. Verify Rolle’s theorem for the function, in .
100. Verify mean value theorem for the function in [3,
5].
101. If find the value of so that becomes
continuous at .
102. Show that the function f given by is discontinuous at .
103. Let . For what value of a, f is
continuous at ?
104. Examine the differentiability of the function f defined by
105. Differentiate with respect to , where .
Objective Type Questions
106. If and , then which of the following can be a
discontinuous function
(a) (b)
(c) (d)
107. The function is
(a) discontinuous at only one point
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) none of these
108. The set of points where the function f given by is
differentiable is
(a) R (b)
(c) (d) none of these
109. The function is discontinuous on the set
(a) (b)
(c) (d)
110. The function is
(a) continuous everywhere but no differentiable at
(b) continuous and differentiable everywhere
(c) not continuous at
(d) none of these.
111. If , where , then the value of the function f at , so that
the function is continuous at , is (a) 0 (b) – 1
(c) 1 (d) none of these
112. If , is continuous at , then
(a) (b)
(c) (d)
113. Let . Then
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at
(c) f is everywhere continuous but not differentiable at , .
(d) None of these.
114. If , then is equal to
(a) (b)
(c) (d)
115. If , then is equal to
(a) (b)
(c) (d)
116. The derivative of w.r.t. is
(a) 2 (b)
(c) (d)
117. If , then is
(a) (b)
(c) (d)
118. The value of c in Rolle’s theorem for the function in the interval
is
(a) 1 (b) – 1
(c) (d)
119. For the function , the value of c for mean value
theorem is
(a) 1 (b)
(c) 2 (d) none of these
120. The function is continuous at ,
then the value of k is
(a) 3 (b) 2 (c) 1 (d) 1.5
121. The function , where denotes the greatest integer function, is
continuous at
(a) 4 (b) – 2 (c) 1 (d) 1.5
122. The number of points at which the function is not continuous is
(a) 1 (b) 2 (c) 3 (d) none of these
123. The function given by is discontinuous on the set
(a) (b)
(c) (d)
124. Let . Then,
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at
(c) f is everywhere continuous but not differentiable at ,
(d) none of these.
125. The function is
(a) continuous at as well as at
(b) continuous at but not at
(c) discontinuous at as well as at
(d) continuous at but not at .
126. The value of k which makes the function defined by
, continuous at is
(a) 8 (b) 1 (c) –1 (d) none of these
127. The set of points where the functions f given by is
differentiable is
(a) R (b) (c) (d) none of these.
128. Differential coefficient of w.r.t. x is
(a) (b) (c) (d)
129. If and , then is
(a) (b) x (c) (d) 1
130. The value of c in Rolle’s Theorem for the function , is
(a) (b) (c) (d)
131. The value of c in Mean value theorem for the function ,
is
(a) (b) (c) (d)
Fill in the blanks :
132. An example of a function which is continuous everywhere but fails to be
differentiable exactly at two points is __________.
133. Derivative of w.r.t. is ________________.
134. If , then = _____________.
135. If , then = __________.
136. For the curve , at is _______________.
137. The number of points at which the function is discontinuous is
_________.
138. If is continuous, then a should be equal to
________.
139. The derivative of w.r.t. x is _____________.
140. If , then is equal to ________.
141. The derivative of w.r.t is ___________.
State True and False for the statements in each of the Exercises.
142. Rolle’s theorem is applicable for the function in [0, 2].
143. If f is continuous on its domain D, then is also continuous on D.
144. The composition of two continuous function is a continuous function.
145. Trigonometric and inverse – trigonometric functions are differentiable in their
respective domain.
146. If is continuous at , then f and g are separately continuous at .
147. For continuity, at , each of and is equal to .
148. is a continuous function.
149. A continuous function can have some points where limit does not exist.
150. is a differentiable function for every value of x.
151. is differentiable everywhere.
152. Match the following
Column-I Column-II
(A) If a function (a)
is continuous at , then k is equal to
(B) Every continuous function is differentiable (b) True
(C) An example of a function which is continuous (c) 6
Everywhere but not differentiable at exactly one point
(d) The identify function i.e. (d) False
is a continuous function
