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Unformatted text preview: Be sure this eXamination has 2 pages. The University of British Columbia
Final Examinations - December 2000
Mathematics 200 Instructors: V. Jungié (§101), J. Fournier (§102), X. Wang (§103), D. Pe—
terson (§104), K. Liu (§106 and 107), R. Bradean (§90A). Closed book examination Time: 2.5 hours 2: 150 minutes. Special Instructions: No books or notes allowed, except for one sheet of
paper. No other aids allowed; in particular, no calculators. Write your
answers in the answer booklet(s). If you use more than one booklet, put
your name and the number of booklets used on each booklet. Show enough
of your work to justify your answers. 1. (15 points) (a) Find a vector perpendicular at the point (1, 1, 3) to the
surface with equation 3:2 + 22 = 10. - ~ (b) Find a vector tangent at the same point to the curve of intersection
of the surface in part (a) with the surface 3/2 + 22 = 10. (c) Find parametric equations for the line tangent to that curve at
that point. ' ' 2. (10 points) Use differentials to estimate the volume of metal in a closed metal can with diameter 8 cm and height 12 cm if the metal is 0.04 cm
thick. 3. (10 points) We say that u is inversely proportional to v if there is a
constant It so that u = 19/12. Suppose the temperature T in a metal
ball is inversely proportional to the distance from the centre of the ball,
which we take to be the origin. The temperature at the point (1, 2,
is 120°. ' (a) Find the constant of proportionality. (b) Find the rate of change of T at (1,2, 2) in the direction towards
thepoint (2,1,3). (G) Show that at most points in the ball, the direction of greatest
increase of temperature is towards the origin. - 1 - Continued on Page 2 Mathematics 200 — All Sections December 2000 4. (15 points) Find and classify all critical points of f(a:,y) = 3:3 — 3173/2 — 3x2 — 3342. 5. (15 points) A closed rectangular box having a volume of 4 cubic metres
is to be built with material that costs $8 per square metre for the sides
but $12 per square metre for the top and bottom. Find the least
expensive dimensions for the box. 6. (15 points) The iterated integral 1 x/E
I = / sin(y3 — 3y) dy] da:
0 —x/5 is equal to ff sin(y3 — 3y) dA for a suitable region R in the my-plane.
R (a) Sketch the region R. Write the integral I with the orders of integration reversed, and
with suitable limits of integration. (c) Find I. 7. (15 points) Consider the top half of a ball of radius 2 centred at the
origin. Suppose that the ball has variable density equal to 9z units of
mass per unit volume. (a) Set 'up a triple integral giving the mass of this half—ball.
(b) Find out What fraction of that mass lies inside the cone 2 = «2:2 + y2. ﬂ 8. (5 points) Let in = f (ac, y, t) with a: and y depending on t. Suppose that
p at some point (cc, 3/) and at some time t, the partial derivatives fx, fy d
and ft are equal to 2, —3 and 5 respectively, while if = 1 and 5% = 2.
Find and explain the difference between id? and ft. The End ...
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