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Practice Problems on Gradients and Directional Derivatives
1) Consider the function
f
(
x,y
) = ln(
p
x
2
+
y
2
). Find its gradient at the
point (
x,y
) = (1
,

1). At this point, what is the directional derivative of
f
in
the direction (3
/
5
,
4
/
5)?
2) Pikabo Street is skiing on a Colorado mountain. Colorado is a large square
state with a surface of many high bumps. Surveyers have mapped out the state
into “sections” which are 1 mile by 1 mile squares. The “origin” for each section
is its Southwest corner. Each point within a section is assigned a number (
x,y
)
where
x
feet is the number of feet that this point lies East from the origin and
y
is the number of feet that it lies North of the origin is denoted by (
x,y
).
The altitude of the earth’s surface at the point (
x,y
) is given by the formula
z
=
f
(
x,y
). In the region of interest to us,
f
is a continuously diﬀerentiable
function.
Suppose that
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 Fall '09
 Bergstrom

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