MATH 210  FINAL EXAM  FALL 1993  page 1

NAME(Print):

SIGNATURE:  SSN:

CIRCLE LECTURER BELOW:

Atkin  9:00  121TH  59047
Starchenko  9:00  117TH  05711
Lin  10:00  204TH  59054
Masur  10:00  215TH  59063
Smith  11:00  204TH  59072
Hamilton  11:00  211TH  59020
Tartakoff  12:00  204TH  59089

There are 12 problems on the exam. Each problem is worth 15 points. SHOW ALL WORK. Please show your university photo ID, if requested to do so. If you do not have this ID with you, you must see your instructor immediately and obtain permission to take the exam.

1 2 3 4 5 6 7 8 9 10 11 12

TOTAL 1. Find the angle between the vectors: 3i + 4j − 2k and i − 3j + 4k. You

may express your answer in degrees correct to .0001 degree or in exact form.

2. Find parametric equations for the line through ( −1 , 1 , 3 ) and ( 2 , −1 , 4 ) and find the point where it intersects the xy-plane.

3. Let T be the triangle whose vertices are ( 1 , 2 , 3 ), ( 4 , −7 , 2 ) and ( 2 , 2 , 2 ).

a) Find the area of T.

b) Find an equation of the plane which contains T.

4. Find the length of the curve

r(t) =< cos 3t, sin 3t, 4t > 0 ≤ t ≤ π/2.

MATH 210  FINAL EXAM  FALL 1993  page 5

5. Suppose z is implicitly defined as a function of x and y by

z^{3} − 2z^{2}x + 3zy + 2xy = 72.

Find \frac{∂z}{∂x} and \frac{∂z}{∂y}.

6. Let w = xy^{2} + xz^{2} + yz and P = ( 2 , −1 , 1 ). Find the directional derivative of w in the direction of i − j + k.

7. Find the absolute maximum and absolute minimum of 3x − 2y + 2z + 4 on the ellipsoid x^{2} + 2y^{2} + 3z^{2} = 12. Also find all the points where they occur.

8. Let f(x, y) = xy − x^{2}y − xy^{2}. Find all critical points and classify (local maxima, local minima, saddle points).

9. Evaluate ∫_{0}^{3} ∫_{0}^{x^{2}−2x} xy dydx.

10. Evaluate ∫_{0}^{2} ∫_{0}^{√\overline{4−x^{2}}} ∫_{0}^{√\overline{4−x^{2}−y^{2}}} z dzdydx.

11. Find the centroid(center of mass) of the lamina

−π/2 ≤ x ≤ π/2, 0 ≤ y ≤ cos x,

whose density function ρ is constant.

12. Find the surface area of the portion of the paraboloid x^{2} + y^{2} = 9 − z that lies above the xy-plane.