Drawing a contour diagram

Question

Assignment:

Question 1. Represent the following surfaces in the form z = f(x; y):                                  

(a) the lower half of the sphere of radius 2 centred at the point (3,1,0),

(b) the plane crossing axis x at x = 1, axis y at y = 2 and axis z at z = 3,

(c) the surface obtained by moving the hyperboloid z = 1/(x² + y²) the distance 3 vertically, in the direction opposite to z.

(d) Draw a contour diagram (either by hand or using a computer) for item (a).

(e) Represent each of the above functions as a level surface of some function of x; y; z.

Question 2. Find partial derivatives f_x, f_y, f_xx, f_yy, f_xy of the functions                                

(a) f = cos(xy) ;

(b) f = ln(x² + y) :

Question 3. A bar may lose of receive heat through its ends. The temperature T of the bar is the function of the distance x from one end, and time t:

T (x; t) = T₀e ^at cos x ;  -Π/2 < x < Π/2

where T₀ and a are constants and a > 0. All the quantities are non-dimensional.

(a) What can you say about the temperature after long time, that is at large t?

(b) Find T_x(x; t) at the points x = - Π /2 and x = Π /2. What do these partial derivatives tell us about the temperature behaviour?

(c) Find T_t(x; t). What does this partial derivative tell us about the tem-perature behaviour?

Note that T₀ and a are not specified, therefore use them as symbols in your solutions.

Provide complete and step by step solution for the question and show calculations and use formulas.