Math 113 – Calculus III Exam 2 Practice Problems Spring 2003
1. Suppose ~u is a unit vector, and ~v and ~w are two more vectors that are not necessarily
unit vectors. Simplify the following expression as much as possible:
((~v ·~u)~u)·(~v ×~w)−(~w ×~v)·(~v −(~u ·~v)~u).
2. Let P= (1,1,1), Q= (1,−3,0) and R= (2,2,2).
(a) Find the equation of the plane that contains the points P,Q, and R.
(b) Find the area of the triangle formed by the three points.
(c) Find the distance from the plane found in (a) to the point (3,4,5).
3. We say two planes are perpendicular if their normal vectors are perpendicular. Given
the following two planes (which are not perpendicular):
x+ 2y+ 4z= 1,−x+y−2z= 5,
find the equation of a plane that is perpendicular to both of these planes, and that
contains the point (3,2,1).
4. Let
g(x, y, z) = e−(x+y)2+z2(x+y).
(a) What is the instantaneous rate of change of gat the point (2,−2,1) in the direction
of the origin?
(b) Suppose that a piece of fruit is sitting on a table in a room, and at each point
(x, y, z) in the space within the room, g(x, y, z ) gives the strength of the odor of
the fruit. Furthermore, suppose that a certain bug always flies in the direction in
which the fruit odor increases fastest. Suppose also that the bug always flies with
aspeed of 2 feet/second.
What is the velocity vector of the bug when it is at the position (2,−2,1)?
5. The path of a particle in space is given by the functions x(t) = 2t,y(t) = cos(t), and
z(t) = sin(t). Suppose the temperature in this space is given by a function H(x, y, z).
(a) Find dH
dt , the rate of change of the temperature at the particle’s position. (Since
the actual function H(x, y, z ) is not given, your answer will be in terms of deriva-
tives of H.)
(b) Suppose we know that at all points, ∂ H
∂x >0, ∂H
∂y <0 and ∂H
∂z >0. At t= 0, is dH
dt
positive, zero, or negative?
6. Let
f(x, y) = x3−xy + cos(π(x+y)).
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