Math 23
B. Dodson
Week 7 Homework:
14.6 directional derivatives
14.7 max/min
14.8 Lagrange multipliers
15.1 approximating sums for double integrals
Problem 14.6.8:
(a) Find the gradient of
f
(
x, y
) =
y
ln
x
.
(b) Evaluate the gradient at
P
(1
,

3)
.
(c) Find the rate of change of
f
at
P
in
the direction of the unit vector
±u
=
<

4
5
,
3
5
> .
Solution:
The gradient is
<
y
x
,
ln
x >,
which at
P
(1
,

3) is
<

3
1
,
ln 1
>
=
<

3
,
0
> .
So the rate of change is
D
±u
f
(1
,

3)
=
→
grad
·
±u
=
<

3
,
0
><

4
5
,
3
5
>
=
12
5
.
We say that
f
is increasing at a rate of 2.4 in the
direction of
±u.
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Problem 14.6.8d:
Find the maximum rate of change of
f
at
P
and the direction in which it occurs.
Solution.
We had the gradient
<
y
x
,
ln
x >,
which at
P
(1
,

3) is
<

3
1
,
ln 1
>
=
<

3
,
0
> .
so the max rate of change is the length of the
gradient, 3, which occurs in the direction of

±
i
=
<

1
,
0
>,
the direction of the gradient.
Week 7 Homework:
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 Spring '06
 YUKICH
 Calculus, Derivative, Integrals, lagrange multipliers

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