MAT 205 SPRING 2000 MIDTERM II

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SSN:

THERE ARE FOUR PROBLEMS PLUS ONE EXTRA CREDIT PROBLEM SHOW YOUR WORK!!!

1. Suppose

z = 2x^{2}y + xe^{y}
x(u, v) = v sin u
y(u, v) = uv

Compute \frac{∂z}{∂u} and \frac{∂z}{∂v}.

2. A box has dimensions l, w, and h which vary in the following way:

l(t) = t^{2} + 2
h(t) = e^{2t} + 1
w(t) = 4t^{4} + 2t^{2} + 2

Let V = V (l, w, h) = l × w × h be the volume function of the box. Find the rate of change with respect to time of the volume of the box at t = 0 seconds.

3. The region D in ℝ^{2} shown on the blackboard is bounded by x = 1, y = e^{x}, and y = 1 − x^{2}. Compute

∫ ∫_{D} xdA 4. Let z = f(x, y) = xy^{3} − x^{2}y. a. Find the gradient of f at the point (x, y) = (3, 2).

b. Find the equation of the tangent plane to the graph of the surface at the point (3, 2, 6).

c. Let u be the unit vector pointing in the direction given by θ = −\frac{π}{3}. Find the derivative of f in the direction u at the point (x, y) = (3, 2).

Extra Credit: Let f(x, y) be a differentiable function on ℝ^{2}. Two unit vectors u and v are given by

u = \frac{\sqrt{2}}{2}i + \frac{\sqrt{2}}{2}j

v = \frac{\sqrt{2}}{2}i − \frac{\sqrt{2}}{2}j

Assume that D_{u}f(0, 0) = \frac{3\sqrt{2}}{2} and D_{v}f(0, 0) = \frac{\sqrt{2}}{2}. Find ∇f(0, 0).