DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2513

FINAL EXAMINATION
DECEMBER 1996 TIME: 3 HOURS

MARKS
(14)1.(a)Show that the vectors and are orthogonal.
(b)Find two unit vectors orthogonal to both
(c)For what values of c is the angle between the vectors and equal to ?
(d)Find the vector projection of onto .
(e)Find the area of triangle PQR, where
(8)2.Given , find
(a)the parametric and symmetric equations of the line through A and parallel to ;
(b)the equation of the plane containing A and normal to .
(8)3.(a)Find the equation of the plane that passes through the point and contains the line .
(b)Find the point at which the line intersects the plane 2x + y - z + 5 = 0.
(10)4.(a)If , show .
(b)Use differentials to approximate the value of the function at .
(c)Given , find at .
(14)5.Given , find
(a)grad f at ;
(b)The directional derivative of f at the point in the direction of ;
(c)the maximum value of the directional derivative of f at and the direction in which the maximum value occurs;
(d)find the equation of the normal to xy + yz + zx = 1 at the point .
(11)6.(a)Find the nature of the critical points of the function
(b)Use the method of Lagrange multipliers to find three positive numbers, whose sum is 100 and whose product is maximum.
(12)7.(a)Find the volume bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1.
(b)Sketch the region of integration and evaluate the integral
by reversing the order of integration.
(c)Evaluate the triple integral
where E lies under the plane z = x + 2y and above the region in the xy-plane bounded by the curves and x = 1.
(10)8.(a)Evaluate , where
using cylindrical co-ordinates.
(b)Use spherical co-ordinates to evaluate
where E is bounded below by the cone and above by the sphere .
(13)9.(a)Evaluate
where C is the arc of the parabola from to .
(b)(i)Show that is exact; find a function f such that
(ii)Use part (i) to evaluate
where C consists of the line from to followed by the parabola .
(c)Use Green's Theorem to evaluate
where C is the triangle with vertices and .