[Hand out time sheets]
Questions on chapter 10?
If you have questions on sections 11.1 or 11.2, please send
me email, so I can prepare explanations to present in class.
Section 11.3: Partial derivatives
Main ideas:
•
Definition of partial derivatives
•
Geometrical meaning of partial derivatives as slopes
of traces
•
Higher partial derivatives
•
How to estimate partial derivatives
•
How to compute partial derivatives
•
Clairaut’s Theorem
•
Partial differential equations
Discuss definitions 1, 2, 3, and 4 on page 609, and the
variant notations
f
1
(
x
,
y
) and
f
2
(
x
,
y
).
Given a function
f
(
x
,
y
), we can write its graph as
{(
x
,
y
,
z
):
x
,
y
,
z
in
R
,
z
=
f
(
x
,
y
)}
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or
{(
x
,
y
,
f
(
x
,
y
)):
x
,
y
in
R
}.
We can intersect this surface with the plane
y
=
b
, obtaining
the trace
{(
x
,
b
,
f
(
x
,
b
)):
x
in
R
}.
Write this trace as
{(
x
,
b
,
g
(
x
)):
x
in
R
}
where
g
(
x
) is defined as
f
(
x
,
b
). If we look at this trace as a
2dimensional curve in the plane
y
=
b
, we can look at the
tangent to the curve at the point
P
= (
a
,
b
,
c
) = (
a
,
b
,
g
(
a
)).
The slope of this tangent line is
g
′
(
a
), which was Stewart’s
definition of
f
1
(
a
,
b
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 Fall '11
 Staff
 Derivative, x,y, Clairaut

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