Find the directional derivative of f(x y) in the direction of angle
Answers
Answer:
Let’s start off by supposing that we wanted the rate of change of
f
at a particular point, say
(
x
0
,
y
0
)
. Let’s also suppose that both
x
and
y
are increasing and that, in this case,
x
is increasing twice as fast as
y
is increasing. So, as
y
increases one unit of measure
x
will increase two units of measure.
To help us see how we’re going to define this change let’s suppose that a particle is sitting at
(
x
0
,
y
0
)
and the particle will move in the direction given by the changing
x
and
y
. Therefore, the particle will move off in a direction of increasing
x
and
y
and the
x
coordinate of the point will increase twice as fast as the
y
coordinate. Now that we’re thinking of this changing
x
and
y
as a direction of movement we can get a way of defining the change. We know from Calculus II that vectors can be used to define a direction and so the particle, at this point, can be said to be moving in the direction,
→
v
=
⟨
2
,
1
⟩