Page 20 - Engineering Mathematics Workbook_Final

Calculus
(
f
49. Let : 0, ) → R be the function 52. Find ‘C’ of Rolle’s Theorem for
f ( ) x = e x (sin x − cos ) x in
e x
defined by ( ): . Then   
f x
x x  / 4,5 / 4
Lt f ( ) x = ________  
x→  (a) / 2 (b) 3 / 4
(c)  (d) does not exist
(a) does not exist (b) 0
53. The value of  of
(c) 1 (d) e
1
f ( ) b − f a = − ) ( )
( ) (b a f  for the
[IISC 2003] function ( ) x = Ax + Bx c in the
+
f
2
50. Let  ,  be two real numbers and interval [a, b] is ________
→
  0. The function : f R R
+
−
(a) b a (b) b a
defined by
−
+
b a b a
 0 if x  0 (c) (d)
 2 2

f ( ) x =   1 
 x  sin   if x  0
      54. If the Rolle’s Theorem holds for the
  x  function ( ) 2f x = x + ax + bx in the
3
2
is differentiable at 0 iff __________ 1

−
(a)   = (b)    interval  1,1 for the point c = 2 then
(c)    (d)   1 value of a & b are
1
2
[IISC 2003] (a) a = , b = −
2
5 − 3 x
x
51. The value of the limit Lt is 1
x
x→ 0 3 − 2 x (b) a = − 2 , b = 2
___________

10 ( ) (c) a = 1 , b = 2
5
(a) log e 9 (b) log 3 3 2
2
1
log 5 (d) a = − , b = −
2
(c) 2 (d) log 5 2
log 3 2
2
1 1
f
g
[IISC 2004] 55. If ( ) x = , ( ) x = in [1, 2] then
x x 2
Mean Value Theorems the mean value C of Cauchy’s mean
value theorem is

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