**
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 is the angle between the vectors and equal to ?
c | ||||

(d) | Find the vector projection of onto . | ||||

(e) | Find the area of triangle , where
PQR | ||||

(8) | 2. | Given , find | |||

(a) | the parametric and
symmetric equations of the line through and parallel to
;
A | ||||

(b) | the equation of the plane containing and normal to
.
A | ||||

(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 at ;
f | ||||

(b) | The directional derivative of at the point in the
direction of ;
f | ||||

(c) | the maximum value of the directional derivative of at
and the direction in which the maximum value occurs;
f | ||||

(d) | find the equation of the normal to at the
point .
xy + yz + zx = 1 | ||||

(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 and z = 0.
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 lies under the plane E and above the region in
the z = x + 2y-plane bounded by the curves and xy.
x = 1 | |||||

(10) | 8. | (a) | Evaluate ,
where | ||

using cylindrical co-ordinates. | |||||

(b) | Use spherical co-ordinates to evaluate | ||||

where is bounded below by the cone
and above by the sphere .
E | |||||

(13) | 9. | (a) | Evaluate | ||

where is the arc of the parabola from to
.
C | |||||

(b) | (i) | Show that is exact; find a
function such that
f | |||

(ii) | Use part (i) to evaluate | ||||

where consists of the line from to followed by the
parabola .
C | |||||

(c) | Use Green's Theorem to evaluate | ||||

where is the triangle with vertices and .
C |