Fall 2007
ARE211
Problem Set #12
First CompStat Problem Set
Due date: Dec 04
(1) In problem 3 on your last problem set, you found the maximum and minimum distance from
the origin to the ellipse
x
2
1
+
x
1
x
2
+
x
2
2
= 3. Generalize this problem to “minimize/maximize
the distance from the origin to the ellipse
x
2
1
+
x
1
x
2
+
αx
2
2
= 3” and use the Envelope theorem
(starting from
α
= 1) to estimate the maximum and minimum distance from the origin to
the following
ellipse,
x
2
1
+
x
1
x
2
+ 0
.
9
x
2
2
= 3
.
(2)
a) Prove that the expression
α
2

αx
3
+
x
5
= 17 defines
x
implicitly as a function of
α
.
in a neighborhood of (¯
α,
¯
x
) = (5
,
2)
b) Estimate the
x
value which corresponds to
α
= 4
.
8 using a first order approximation.
(3) Consider the equation
α
3
1
+ 3
α
2
2
+ 4
α
1
x
2

3
x
2
α
2
= 1. Does this equation define
x
as an
implicit function of
α
1
, α
2
a) in a neighborhood of ( ¯
α
1
,
¯
α
2
) = (1
,
1)
b) in a neighborhood of ( ¯
α
1
,
¯
α
2
) = (1
,
0)
c) in a neighborhood of ( ¯
α
1
,
¯
α
2
) = (0
.
5
,
0)
If so, compute
δx
δα
1
and
δx
δα
2
at this point.
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 Fall '07
 Simon
 Supply And Demand, Minimum distance, implicit function, ellipse x2, CompStat Problem Set

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