Douglass Houghton Workshop, Section 2, Tue 11/29/11
Worksheet Out Among the Stars
1.
Shortest Network.
We found last week that some
V
newtorks can be improved to
a
Y
, and some can’t. In particular, our big result was that:
If
A
,
B
, and
C
form an isosceles triangle, then the
V
network is
not minimizing if the vertex angle is less than
120
◦
.
Now we need to generalize to all possible placements of cities.
(a) Prove that the network to the right is NOT minimizing.
You don’t need to find the optimal network, just prove
that this one can be improved. Hint: consider the portion
of the network that is inside the circle.
(b) What allowed that trick to work?
Phrase your answer
like this:
“Any network which contains
can be
improved.”
(c) Put it all together, and explain where the soap puts the
roundabout.
2. Write the following sums in sigma (
∑
) notation.
(a) 1 + 2 + 3 + 4 +
· · ·
10
(b) 1 + 2 + 3 + 4 +
· · ·
+
n
(c) 3 + 5 + 7 + 9 +
· · ·
+ 21
(d) 4 + 9 + 16 + 25 +
· · ·
+ 100
(e) 2
.
3 + 2
.
8 + 3
.
3 + 3
.
8 + 4
.
3 + 4
.
8 +
· · ·
+ 10
.
3
(f)
f
(
a
1
) +
f
(
a
2
) +
f
(
a
3
) +
· · ·
+
f
(
a
n
)
3. (From the Winter, 2011 Math 115 Final) The table below gives the expected growth
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 Winter '08
 Conger
 Order theory, Solid angle

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