Math 200, Final Exam
December 2008
No calculators or notes of any kind are allowed. Time: 2.5 hours.
1. [11] Asurfaceisgivenby
z=x2−2xy +y2.
(a) Find the equation of the tangent plane to the surface at x=a, y =2a.
(b) For what value of ais the tangent plane parallel to the plane x−y+z=1?
2. [12] The pressure in a solid is given by
P(s, r)=sr(4s2−r2−2)
where sis the specificheatandris the density. We expect to measure (s, r)to
be approximately (2,2) and would like to have the most accurate value for P.
There are two different ways to measure sand r. Method 1 has an error in sof
±0.01 andanerrorinrof ±0.1, while method 2 has an error of ±0.02 for both
sand r.
Should we use method 1 or method 2? Explain your reasoning carefully.
3. [11] u(x, y)is defined as
u(x, y)=eyF(xe−y2)
for an arbitrary function F(z).
(a) If F(z)=ln(z),find ∂u
∂x and ∂u
∂y .
(b) For an arbitrary F(z)show that u(x, y)satisfies
2xy ∂u
∂x +∂u
∂y =u.
4. [12] The air temperature T(x, y, z )at a location (x, y, z)is given by:
T(x, y, z)=1+x2+yz.
(a) A bird passes through (2,1,3) travelling towards (4,3,4) with speed 2.
At what rate does the air temperature it experiences change at this instant?
(b) If instead the bird maintains constant altitude (z=3)as it passes
through (2,1,3) while also keeping at a fixed air temperature, T=8,what are
its two possible directions of travel?
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