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# a) Find the equation of the tangent plane to the surface at the point P(3, –1). [10 marks]

Question 1:                                                                                                                     [15 marks]
A surface is defined by the following equation:
.
a)    Find the equation of the tangent plane to the surface at the point P(3, –1).        [10 marks]
b)    Find the total differential of function z (x, y) at that point.                                    [5 marks]

Question 2:                                                                                                                     [15 marks]
Find all critical points of the function   and determine their character, that is whether there is a local maximum, local minimum, saddle point or none of these at each critical point.

Question 3:                                                                                                                     [20 marks]
A mass of a thin plate M is given by the double integral in the conditional units:

a)    Determine the function describing the surface mass density.                               [3 marks]
b)    Present a sketch of the plate and shade it (by hand or using technology).            [7 marks]
c)    Change the order of integration and calculate the plate mass M.                        [10 marks]

Question 4:                                                                                                                     [20 marks]
Find the work done by the force field

acting on a massive body to move it from point M to point N along the path given by the equation
;
where the parameter t varies from 0 to 1.
Present your answer in exact form and then calculate up to four significant figures.
Question 5:                                                                                                                     [25 marks]
Calculate the double integral in polar coordinates

over the domain D bounded in the first quadrant by the following lines:
-    x = 0;
-    y = x;
-    x2 + y2 = 1/4.
Sketch the domain of integration. Present your answer in exact form and then calculate it up to two significant figures.

Question 6:                                                                                                                     [15 marks]
Find the total mass M of the wire, if the mass line density distribution is:
,
and the vector equation of the wire is r(t) = 4 cos t i + 4 sin t j + 3(t – 1) k, where 3/2  t  2 and the dimension is [|r|] = m. Present your answer in the dimensional exact form and then evaluate it approximately up to five significant figures.

Question 7:                                                                                                                     [25 marks]
i.    Determine which of the vector fields given below is conservative:

ii.    Show the points A(3, ) and B(4, –3 ) on the x, y-plane.
iii.    Calculate the work done (in dimensionless units) by the conservative field between these two points. Present your answer in exact form and then calculate up to four significant figures.

Question 8:                                                                                                                     [15 marks]
For the given vector field   calculate
(a)    the divergence at the point P(15, 2, –3) and                                                          [5 marks]
(b)    the curl at the same point P(15, 2, –3).                                                                [10 marks]

Determine, whether the vector field is solenoidal or conservative.

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