### 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.