Assignment-4
1. The domain of the function f (x, y) = √
1
16−x2 −y 2
is
(a) R2
(b) R2+
(c) open disc with radius 4 and center at (0,0).
(d) closed disc with radius 4 and centre at (0,0).
2. The domain of the function f (x, y) = sin−1 (y − x) is
(a) {(x, y) ∈ R2 : x − y ≤ 1, y − x ≤ 1}.
(b) {(x, y) ∈ R2 : x − y ≤ π2 , y − x ≤ − π2 }.
(c) {(x, y) ∈ R2 : −2π ≤ x ≤ 2π, −2π ≤ y ≤ 2π}.
(d) {(x, y) ∈ R2 : x − y ≥ 0}.
3.
cos y+1
lim
(x,y)→( π2 ,0) y−sin x
√
4.
lim
(x,y)→(4,3)
=..........
√
x− y+1
=...........
x−y−1
5. The point of discontinuities of the function f (x, y) =
(a) {(x, y) ∈ R2 : x 6= y}.
(b) {(x, y) ∈ R2 : x 6= 0, y 6= 0}.
(c) {(x, y) ∈ R2 : x = y}.
(d) {(x, y) ∈ R2 : x + y 6= 0, x − y 6= 0}.
6.
x4 −y 2
4
2=
(x,y)→(0,0) x +y
lim
(a) 1
(b) -1
(c) 0
(d) does not exist.
7. For the function f (x, y, z) = yz ln(xy)
1
x+y
x−y
are
(a)
∂f
∂y
= z ln xy.
(b)
∂f
∂y
= xz .
(c)
∂f
∂y
= z(1 + ln xy).
(d)
∂f
∂y
= ln xy.
8. For the function f (x, y) = ex + x ln y + y ln x + ey , fxy = fyx
(a) True.
(b) False.
√
9. Let w = ln(x2 + y 2 + z 2 ), x = cos t, y = sin t, z = 4 t, then
is ...........
10. Let w = z − sin xy, x = t, y = lnt, z = et−1 , then
dw
dt
dw
dt
at t=0
=
(a) et−1 + (1 + ln t) cos(t ln t).
(b) et−1 − (1 + ln t) cos(t ln t).
(c) et−1 + ln t cos(1 − ln t).
(d) et−1 − ln t cos(1 − ln t).
11. The derivative of the function f (x, y) = 2x2 +y 2 at (-1,1) in the direction
of A=3î − 4ĵ is ........
12. The direction in which the directional derivative of f (x, y) = xy + y 2 at
(3,2) is equal to zero is given by
(a) 2î + 7ĵ
(b) 2î − 7ĵ
(c) 7î − 2ĵ
(d) 7î + 2ĵ
13. For the function f (x, y) = 2xy − x2 − 2y 2 + 3x + 4,
(a) (3, 23 ) is a local maximizer.
(b) (3, 32 ) is a local minimizer.
(c) ( 23 ,3) is a local maximizer.
2
(d) ( 32 ,3) is a local maximizer.
14. The local extreme values of f (x, y) = x2 y on the the line x+y=3 are
(a) 0 and 2
(b) 0 and 4
(c) -4
(d) 2
Rb
15. The values of a,b with a ≤ b such that (6 − x − x2 )dx has its minimum
a
value are given by
(a) a=-3, b=2
(b) a=-3, b=-3
(c) a=2, b=-3
(d) a=2,b=2
Solution
1. (c).
2. (a).
3. -2
4. 0.25
5. (c).
6. (d).
7. (c).
8. (a).
9. 16
10. (b).
11. -20
3
12. (c).
13. (a).
14. (b).
15. (c).
4