DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2513

FINAL EXAMINATION
DECEMBER 1995 TIME: 3 HOURS

MARKS
(6)1.If and y = 2st, evaluate .
(6)2. Let . Find
(a)the rate of change of f at in the direction of ;
(b)the direction in which f changed most rapidly at P;
(c)the maximum rate of change of f at P.

(5)3.Find the equation of the tangent line to the curve of intersection of and at .
(5)4.Find and classify the extreme points of the function .
(5)5.Use Lagrange multipliers to find the maximum value of f = xyz subject to the constraint .
(6)6.Change the order of integration and evaluate the integral .
(5)7.Find the volume of the solid under the cone and above the ring
.
(5)8.Use triple integrals to find the volume of the tetrahedron bounded by the planes
(6)9.Use cylindrical coordinates to find the volume of the solid bounded by
(6)10.(a)Let and . Evaluate, if possible, BA and .
(b)Let M be a non-singular matrix such that

Find the value(s) of .
(c)Let X be a 3 x 3 matrix such that . Find the value of .
(5)11.Solve the linear system:
(5)12.Use the inverse method to solve the system:
(5)13.Show that 2 is an eigenvalue of
Find the associated eigenvector.