MA 227, Fall 2001
TEST 2
Make sure to show all your work and underline the final results of each problem. Write your
name on this sheet and use it as a cover page when you turn in your work. Do not write
your results or computations on this paper. Good luck!
1. Find the linearization of f(x, y) = ln(x2โ3y2) at (2,1) and use it to approximate
f(1.9,2.1).
2. If f(x) = ex2/2and x= sin(rs) find
โ2f
โrโ s
in terms or rand s.
3. If f(x, y) = x2โy2/3, compute the gradient vector โf(2,3) and use it to find the tangent
line to the level curve f(x, y) = 1 at the point (2,3). Sketch the level curve, the tangent line
and the gradient vector.
4. Find the point on the surface y+x2+ 2z2= 1 where the tangent plane is perpendicular
to the vector h1,1/2,โ1i. Then give the equation of the tangent plane at that point.
5. Find the critical points of the function
f(x, y) = x2+y2+x2y+ 4
and decide for each critical point whether it is a local maximum, a local minimum or a saddle
point.
6. Find the absolute maximum and minimum values of
f(x, y) = 4xy2โx2y2โxy3
on the closed triangle in the xy-plane with vertices (0,0), (0,6), and (6,0).
7. Sketch the region of integration for the iterated integral
Z3
0
dy Z9
y2
dx y cos(x2).
Then evaluate the integral by first reversing the order of integration.
8. Find the volume of the solid under the paraboloid z= 3x2+y2and above the region
bounded by y=xand x=y2โy.
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