Page 19 - Engineering Mathematics Workbook_Final

Calculus

+
x  (a) 8 (b) 7
g
42. Let ( ) x =  sin y 2
( ) dy . Then
x 
−
(c) 10 (d) 10 − 14
g 1 ( ) x = _________
[IISC 2001]
2
(a) sin x 46. Let f be a real function defined by
 ax b if x  − 1
+

2
sin (x  + ) + sin (x  − ) 2   2

1
(b) f ( ) x =  x + 1 if − 1 x 

2  − ax b if x  1
+


2
2
(c) sin x  ( + ) − sin (x  − ) where a and b are real numbers. If f is
continuous on the real line then the
product ab is ________
2
2
(d) cos x  ( + ) − cos (x  − )
−
(a) 2 (b) 4
−
[IISC 2005] (c) 2 (d) 0
2 −
x
f
43. At x = 2, ( ) x = x e has [IISC 2001]
)
__________ 47. f ( , x y = x + 100x y + 200xy + 10y
6
7
7
5
2
2
2
(a) local minimum, but not global then x fxx + 2xy fxy + y fyy =
minimum _______
(b) local maximum, but not global (a) 42x + 4200x y + 8400xy + 420y
7
7
2
6
5
maximum
6
7
5
2
7
(b) 42x + 500x y + 200xy + 10y
(c) global minimum
5
2
6
7
7
(c) 42x + 1000x y + 1200xy + 420y
(d) global maximum
2
7
7
5
6
(d) 7x + 700x y + 1400xy + 70y
[IISC 2005]
[IISC 2002]
44. The value of lim ( 1 + 1 + ⋯ ⋯ +
→∞ +1 +2
1 ) is ___________ 48. For a real number y, let [y] denote the
+ largest integer smaller than or equal to y.
2  2
(a) 0 (b) ln2 The value of    x   dx = _________
0
2
(c) e (d) e
(a) 1 (b) 5 − 2 − 3
[IISC 2005]
8
45. The maximum value of (c) 3 − 2 (d)


10 − 3cos − 4sin + 9 for 3
 
0   2 [IISC 2004]

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