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Discussion — Tuesday, September 28th
Subject:
Taylor series, the second derivative test, and changing coordinates.
1. Consider
f
(
x
,
y
)
=
5
(
sin
2
(
x
)
+
y
2
)

3
e

(
x
+
1)
y
+
3
e

y
+
1.
(a) Show that (0,0) is a critical point for
f
.
(b) Calculate each of
f
xx
,
f
xy
,
f
yy
at (0,0) and use this to write out the 2
nd
order Taylor
approximation for
f
at (0,0).
(c) To make sure the next two problems go smoothly, check your answer to (b) with the
instructor.
2. Let
g
(
x
,
y
) be the approximation you obtained for
f
(
x
,
y
) near (0,0) in 1(b).
(a) It’s not clear from the formula whether
g
, and hence
f
, has a min, max, or a saddle
at (0,0). Test along several lines until you are convinced you’ve determined which
type it is.
(b) Check that you’re right in (a) using the 2
nd
derivative test. The next problem will
help explain why this test works.
3. Consider alternate coordinates on
R
2
where (
u
,
v
) corresponds to
u
(1,1)
+
v
(

1,1).
(a) Sketch the
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