|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.
|(12)||1.||Given , find:|
|(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
y - 2z + 5 = 0.
||Identify and neatly sketch the surface given by
4x2 - y2 -
z2 + 8x + 2z + 7 = 0.
for the function,
z = (x2 + y2)exy.
by means of the chain
||Let z be given implicitly as a function of x and y by the
x2 + 2y2 + z3 - 6xy + 2yz + 6 = 0. Determine the
||Compute the directional derivative
of the function
f(x,y,z) = x2 + 2y2 - z2 + 4xyz
in the direction of
||Determine the equation of the tangent plane to the surface
z = x2 - 4xy + 2y2
at the point (1,1,-1).
||Use differentials to find an approximate value for the number
||Find all critical points of the function
f(x,y) = y3 + 3x2y - 3y2 - 3x2 + 2
and determine the nature of each critical point.
||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 = -x2 + 4x. Hence evaluate the
||Change the order of integration in the double integral,
||Let R denote the region in the space
of the 3-dimensional coordinates (x,y,z)
that is bounded on its sides by the
x + y + z = 2, and bounded from below by the
plane z = 1. Calculate, the triple integral,
||Use double integration in polar coordinates to
determine the area of the region in the x-y plane that lies outside
x2 + y2 = 1 and inside the circle
x2 + y2 - 2y = 0.
||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
x2 + y2 = z2 and
above by the sphere
x2 + y2 + z2 = 1.