DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2513

FINAL EXAMINATION
April 1997
TIME: 3 HOURS

ANSWER ALL QUESTIONS WITHOUT THE USE OF A CALCULATOR. SHOW ALL WORK AS CREDIT WILL BE GIVEN TO PRESENTATION OF SOLUTIONS. NO CREDIT WILL BE GIVEN FOR ILLEGIBLE WORK.

 MARKS (12) 1. Given , find: (a) and ; (b) the vector projection of on ; (c) the volume of the parallelipiped for which are concurrent edges; (d) the cosine of the angle between and ; (e) a vector in the plane of and that is orthogonal to . (12) 2. (a) Determine the symmetric equations of the line through the two points, A(1,-1,2) and B(3,-2,1). (b) Determine the scalar equation of the plane through the three points and R(2,-1,-1). (c) Find the point of intersection of the line, with the plane 3x + y - 2z + 5 = 0. (d) Identify and neatly sketch the surface given by 4x2 - y2 - z2 + 8x + 2z + 7 = 0. (13) 3. (a) Find for the function, z = (x2 + y2)exy. (b) Given where and ; find by means of the chain rule. (c) Let z be given implicitly as a function of x and y by the relationship, x2 + 2y2 + z3 - 6xy + 2yz + 6 = 0. Determine the derivative, . (13) 4. (a) Compute the directional derivative of the function f(x,y,z) = x2 + 2y2 - z2 + 4xyz in the direction of at the point (-1,2,1). (b) Determine the equation of the tangent plane to the surface z = x2 - 4xy + 2y2 at the point (1,1,-1). (c) Use differentials to find an approximate value for the number . (11) 5. Find all critical points of the function f(x,y) = y3 + 3x2y - 3y2 - 3x2 + 2 and determine the nature of each critical point. (12) 6. (i) Draw a neat sketch of the region, R, in the first quadrant of the x-y plane, that is bounded below by the line y = x and above by the parabola, y = -x2 + 4x. Hence evaluate the double integral, . R (ii) Change the order of integration in the double integral, (10) 7. Let R denote the region in the space of the 3-dimensional coordinates (x,y,z) that is bounded on its sides by the planes and x + y + z = 2, and bounded from below by the plane z = 1. Calculate, the triple integral, R (17) 8. (i) Use double integration in polar coordinates to determine the area of the region in the x-y plane that lies outside the circle x2 + y2 = 1 and inside the circle x2 + y2 - 2y = 0. (ii) Use triple integration in spherical coordinates to determine the volume of the region in the space of the x,y,z-coordinate system, which is bounded below by the cone x2 + y2 = z2 and above by the sphere x2 + y2 + z2 = 1. (100)