DEPARTMENT OF MATHEMATICS & STATISTICS MATH 2513

FINAL EXAMINATION
December 1997
TIME: 3 HOURS

CALCULATORS ARE NOT ALLOWED

 MARKS (10) 1. (a) Calculate the angle between the vectors (b) Given , find the vector projection of onto . (c) Find the area of a triangle with vertices at . (d) Find the work done by the force , as its point of application moves from the point A = (2,1,3) to B = (4,-1,5). (20) 2. (a) Find the equation of the plane passing through the point A = (6,5,-2) and parallel to the plane x + y - z + 1 = 0. (b) Find the (i) parametric and (ii) symmetric equations of the line through the point A = (1,-2,3) and perpendicular to the plane 2x - y + 3z = 5. (c) Find the point at which the line intersects the plane 4x + 5y - 2z = 18. (d) Find the equation of the plane that passes through the point (1,6,-4) and contains the line . (e) Find the point of intersection of the lines L1 and L2, where (7) 3. (a) If z = f(x,y), where , (i) find and ; (ii) show that . (b) If , show (9) 4. (a) Find the equation of the tangent plane and the equations of the normal line to the surface z = x2 + xy - y2 at the point where x = 2 and y = -1. (b) Use differentials to find the approximate value of . (9) 5. (a) Given the function f(x,y,z) = xy + yz2 + xz3 and the point P = (2,0,3), find (i) the directional derivative of the function at P in the direction of the vector ; (ii) the maximum rate of change at P and the direction in which it occurs. (b) Given , use the chain rule to find at . (10) 6. (a) Find the local maximum and minimum values and saddle points of f(x,y) = 2x3 - 24xy + 16y2. (b) Use Lagrange multipliers to find the point on the plane 2x + 3y - z = 1, which is closest to the origin. (18) 7. (a) Sketch the region of integration and evaluate by reversing the order of integration. (b) Find the volume of the solid bounded by the planes . (c) Find the volume of the solid under the paraboloid z = x2 + y2 and above the region y = x2 and x = y2. (17) 8. (a) Sketch the solid whose volume is given by the integral and evaluate the integral. (b) Use cylindrical co-ordinates to find the volume of the region E bounded by the paraboloids z = x2 + y2 and z = 36 - 3x2 - 3y2. (c) Use spherical co-ordinates to evaluate E where E lies between the spheres and and above the cone . (100)