Math 210 Midterm II March 29, 1993

Name SS number

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1) (10pt) A point has spherical coordinates (1, \frac{π}{6} , \frac{π}{6} ). What are the cylindric and rectangular coordinates for the point?

2) (10pt) Find the following limit (if it exists):

lim_{(x,y)→(0,0)} \frac{x^{2}y}{x^{4} + y^{2}} .

3) (10pt) Let z = e^{x^{2}} sin(x − y). Find \frac{∂z}{∂x}.

4) (10pt) Let z = xe^{y} + yx, x = st^{2} and y = t + s. Find \frac{∂z}{∂t}.

5) (10pt) Find the equation of the plane tangent to the surface z = \sqrt{xy − x} at the point (1, 2, 1).

6) (10pt) The heat at a point (x, y) in the plane is given by h(x, y) = x^{2}ye^{−x^{2}−y^{2}}. A particle is at the point (1, 2). In what direction is the heat incresasing the most rapidly? What is the rate of increase in that direction?

7) (15pt) Find the local maximums, local minimums and saddle points of f(x, y) = x^{2}y − 2x^{2} + 2yx − 3y.

8) (10pt) Maximize f(x, y, z) = x + 3y + 5z subject to the constraint x^{2} + y^{2} + z^{2} = 1.

9) (15pt) Find the absolute maximum and minimum for the function f(x, y) = x^{2} − yx on the rectangle with corners (1,2), (1,-3), (-1,-3), (-1,2).