15.
16.

20.

21.

23,

25;

. Use a polar double integral to find the area enclose

24.

Evaluate limgy)4@,0) [sin (ಗಣಿ + y2)]/(x? + )2) = 09 converting to polar coordinates,

Show that the functions f(x, y) = 3x’y? and f(x, y) = sin(3x’y°) are continuous everywhere.

Answer any 2 complete questions each having 7 marks

. Let L(x, y) denote the local linear approximation to f(x, y) = 22 +y? at the point (3, 4).

Compare the error in approximating f (3.04, 3.98) = J(3.04 (2 + (3.98 2 (2+ (3.98 )2 ४७५ 1, (3.04,
3.98) with the distance between the points (3,4) and (3.04, 3.98).

. Suppose that w = x? + y? - 22 and x =p sin ¢ cos 6, y=p sin फ sin 6, z= 0 cos फ. Use

⋅ ∙ ow
appropriate form of the chain rule to find 2p and 30

. Logate the relative extrema and saddle points of f(x,y) = 3൧ - 2xy + y? - ‏برع‎

Answer any 2 complete questions each having 7 marks

Let f(x, y) = xe”. Find the maximum value of a directional derivative at (-2,0) and find the

unit vector in the direction in which the maximum value occur. ۱

Find the angle between the tangent lines to the graphs of 7,(t)=tan™'ti+sin¢ ‏رز‎ (3%

१ (४) = (13 -ഥ7 )2- ‏زر‎ + 108८

Suppose that a particle moves through 3-space so that its position vector at time
tisr(t)=tit?jt+tek.

Find the scalar tangential and normal components of acceleration at time t = 1.

Answer any 2 complete questions each having 7 marks

Use a triple integral to find the volume of the solid within the cylinder x? + y? = 9 and

between the planes z= land 2-5
x ~
Evaluate [റ്‌ dA where R is the region enclosed by x -y=0,x - + = 1, + + = 1,
# +}

ಜತ = 3