Question

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Answer 1

Answer:

a)

Explanation:

(a) At the point x = 1, y = 1, z = 1, find the unit vector that points in the

direction for which f is increasing at the fastest rate.

Solution: The function f increases at the fastest rate in the direction of

∇f(1, 1, 1)

|∇f(1, 1, 1)|

=

(1, 1, 2)

√

6

.

(b) For x = (1, 1, 1), find a vector v for which d

dtf(x+tv) = 17 at t = 0. (There

are many possible answers. Just find one.)

Solution: If we let v = (a, b, c), then

17 = d

dt(f(x + tv)|t=0 = ∇f(1, 1, 1) · v = (1, 1, 2) · (a, b, c) = a