Page 16 - Engineering Mathematics Workbook_Final

Calculus

− + − + − +
( 2n
f
x
22. Let ( ) x = x x − x − 1 , −   1 2 3 4 5 6 .......+ − )
Lt =
2
which of the following statements is n→  n + 1 + n − 1
2
true.
1
(a) f is not differentiable at x = 0 and x = (a)  (b) 2
1
1
(b) f is differentiable at x = 0 but not (c) 0 (d) −
differentiable at x = 1 2
[MS 2007]
(c) f is not differentiable at x = 0 but
differentiable at x = 1 26. By changing order of integration
)
(d) f is differentiable at x = 0 and x = 1.   1 ex f ( ,x y dy dx can be expressed
0 1
[MS 2006] __________

23. Let 1 ln y
)
f x = 1 )( x − 2 )(x − 3 )(x − 4 )(x − ) 5 (a)   f ( , x y dx dy
( ) ( x −
0 1
x
, −  . The number of distinct

real roots of the equation (b) ∫ ∫ ( , )
d 1 1
dx ( f ( )) 0x = is exactly ______ 1 ln y
)
(c)   f ( , x y dx dy
0 0
(a) 2 (b) 3
1
(c) 4 (d) 5 (d) ∫ ∫ ( , )
1
[MS 2006] [MS 2007]

 2 1 27. Let ( )   x and
f x =


f
24. Let ( ) x =  x sin x x  0 . Then 
  x 0  x  1
 0 x = 0 




g ( ) x =  x − 1 1 x  2 for
1
x
f
(a) ( ) x is continuous at x = 0  x − 2 2   3

 0 x = 3


(b) ( ) 0f 1 exists
x    . Then ( ) x + g ( ) x is
0,3
f
(c) f 11 ( ) x is continuous at x= 0 _______
(d) f 11 ( ) 0 exists [MS 2007] (a) discontinuous at points 1 and 2
(b) continuous on [0, 3] but not derivable
25. on (0,3)

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