MATH 2513
FINAL EXAMINATION April 1997 | TIME: 3 HOURS |
ANSWER ALL QUESTIONS WITHOUT THE USE OF A CALCULATOR. SHOW ALL WORK AS CREDIT WILL BE GIVEN TO PRESENTATION OF SOLUTIONS. NO CREDIT WILL BE GIVEN FOR ILLEGIBLE WORK. |
MARKS | |||
(12) | 1. | Given , find: | |
(a) | and ; | ||
(b) | the vector projection of on ; | ||
(c) | the volume of the parallelipiped for which are concurrent edges; | ||
(d) | the cosine of the angle between and ; | ||
(e) | a vector in the plane of and that is orthogonal to . | ||
(12) | 2. | (a) | Determine the symmetric equations of the line through the two points, A(1,-1,2) and B(3,-2,1). |
(b) | Determine the scalar equation of the plane through the three points and R(2,-1,-1). | ||
Find the point of intersection of the line, with the plane 3x + y - 2z + 5 = 0. | |||
(d) | Identify and neatly sketch the surface given by 4x^{2} - y^{2} - z^{2} + 8x + 2z + 7 = 0. | ||
Find for the function, z = (x^{2} + y^{2})e^{xy}. | |||
Given where and ; find by means of the chain rule. | |||
(c) | Let z be given implicitly as a function of x and y by the relationship, x^{2} + 2y^{2} + z^{3} - 6xy + 2yz + 6 = 0. Determine the derivative, . | ||
(13) | 4. | (a) | Compute the directional derivative of the function |
f(x,y,z) = x^{2} + 2y^{2} - z^{2} + 4xyz | |||
in the direction of at the point (-1,2,1). | |||
(b) | Determine the equation of the tangent plane to the surface | ||
z = x^{2} - 4xy + 2y^{2} | |||
at the point (1,1,-1). | |||
(c) | Use differentials to find an approximate value for the number . | ||
(11) | 5. | Find all critical points of the function | |
f(x,y) = y^{3} + 3x^{2}y - 3y^{2} - 3x^{2} + 2 | |||
and determine the nature of each critical point. | |||
(12) | 6. | (i) | Draw a neat sketch of the region, R, in the first quadrant of the x-y plane, that is bounded below by the line y = x and above by the parabola, y = -x^{2} + 4x. Hence evaluate the double integral, |
(ii) | Change the order of integration in the double integral, | ||
(10) | 7. | Let R denote the region in the space of the 3-dimensional coordinates (x,y,z) that is bounded on its sides by the planes and x + y + z = 2, and bounded from below by the plane z = 1. Calculate, the triple integral, | |
(17) | 8. | (i) | Use double integration in polar coordinates to determine the area of the region in the x-y plane that lies outside the circle x^{2} + y^{2} = 1 and inside the circle x^{2} + y^{2} - 2y = 0. |
(ii) | Use triple integration in spherical coordinates to determine the volume of the region in the space of the x,y,z-coordinate system, which is bounded below by the cone x^{2} + y^{2} = z^{2} and above by the sphere x^{2} + y^{2} + z^{2} = 1. | ||
(100) |