Question 1

Solve the following initial boundary value problem using the method of separation of variables

∂²u/∂x² = ∂u/∂t + βu , 0 < x < 1, t > 0 ,

u(0,t) = u(1,t) = 0, t > 0 ,

                1            0 < x ≤ 1/ 2

u(x, 0) =

                0            1/2 < x < 1

where β is a constant.

Question 2

(a) Find the points on the cone x² = y² + z² that are closest to the point (0,4,2)

(b) Consider

f(x, y) = x²e^y.

(i) Find the rate of change of the function f(x,y) at the point P(1, 0) in the direction of the vector u = (1,1) .

(ii) In which direction does the function f(x, y) increase most rapidly at the point P(1,0)?

(iii) What is the maximum rate of change of f(x,y) at the point P(1,0)?

(c) Let f (x, y) = xy(3 - x - y).

Find all critical points of f(x,y) and determine their nature (local max or local min or saddle point).

Question 3

(a) Consider ₀∫²_y/2∫¹ y cos(x³ -1) dxdy .

Sketch the region of integration and then evaluate the integral.

(b) Evaluate z (y² + z² )dxdydz by firstly converting the integral to an equivalent integral in cylindrical coordinates.

Question 4

Evaluate

∫∫∫ y cos(x + z)dV
E

where E is the region bounded by the parabolic cylinder y = √x and the planes

y = 0, z = 0 and x + z = Π/2.

Question 5

(a) Evaluate the line integral ∫ F • dr
                                        C

where F = e^x-1 i + xyj and C is the path given by the vector function

r(t) = t²i + t³j , 0 ≤ t ≤ 1.

(b) Evaluate the line integral with respect to arc length where C is given by ∫ zds
                                                                                                           C

x = t cos t, y = t sin t, z = t,

0 ≤ t ≤ √7.

(c) Use Green's theorem to evaluate the following line integral along the given positively oriented curve

∫(e^x + xy)dx + (e^y + 2x² )dy
C

where C is the boundary of the region enclosed by the parabolas y = x² and y = x.