1. Find the directional derivative of each of the functions at the given point in the direction

of the vector v.

a) f(x, y) = 1 + 2x

√y, (3, 4), v = h4, −3i.

b) g(x, y, z) = x arctan( y

z

), (1, 2, −2), r = h1, 1, −1i

2. Find the maximum rate of change of f = sin(xy) at (1, 0) and the direction in which it

occurs.

3. Find all points at which the direction of fastest change of the function

f(x, y) = x

2 + y

2 − 2x − 4y is h1, 1i

4. The temperature T in a metal ball is inversely proportional to the distance from the

center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2)

is 120◦

.

a) Find the rate of change of of T at (1, 2, 2) in the direction toward the point (2, 1, 3).

b) Show that at any point in the ball the direction of greatest increase in temperature

is given by a vector that points toward the origin.

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