Partial Derivatives Review Worksheet
Name:
Math 2263
June 25, 2010
1. Consider the parallelpiped which has sides (1, 3, −2), (3, 3, 6), and
(4, −2, −1). What is its volume?
2. Mathematical physicists have determined that, in order for a function u(x, t) to describe the flow of heat through a metal bar along
2
= k 2 ∂∂xu2 .
the x-axis, the function must satisfy the heat equation: ∂u
∂t
You are an experimental physicist collecting data about the heat flow,
2
and you find that the function u(x, t) = e−k t sin(x) models the flow
of heat through the bar. Is this consistent with the theory?
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3. Let f (x, y) = x2 + 12 y 2 − 2x. Find a point on the graph z = f (x, y)
where the tangent plane is horizontal.
4. Let f (x, y) = x/y+y/x. Using a linear approximation about the point
(1/2, 1/4), estimate the value of f (.48, .3).
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5. An itsy bitsy spider is at the point x = 2, y = −1, z = 7 in a parabolic
water spout with height z = x2 + 3y 2 . In order to get to the top of
the spout as fast as possible (before the rain comes down to wash the
spider out), in which direction should the spider set out? Should it
follow a straight line path from then on?
6. Find and classify the critical points of f (x, y) = 3x2 + 2xy + 2x + y 2 +
y + 4.
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7. Find the following limits, or prove that they do not exist.
(a) lim(x,y)→(0,0)
y2
x4 +3y 2
(b) lim(x,y)→(1,2)
x2 y
2x+y
(c) lim(x,y)→(0,0)
x3 sin2 y
2x2 +y 2
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