EXTRA CREDIT: You are on the honor code not
to read a solution before submitting a solution.
- Week One: Assume A wins p% of their games, and B wins q%. Which of the
following four formulas models well the probability that A beats B: (p+pq)/(p+q+2pq),
(p+pq)/(p+q-2pq), (p-pq)/(p+q+2pq), (p-pq)/(p+q-2pq)?
Click here for the solution. Click here
for an excel spreadsheet which you can use to get a feel of the problem.
- Week Two: the solution to the problem on why there is no closest point is
that it is not sufficient to just check critical points; one must also check
the endpoints. We have the constraint that y must be non-negative. Plugging in
y=0 does give the closest point. For the question on elementary comparing eπ
and πe,
click here. The problem of dividing N into summands to maximize the
product is more involved;
click here for a sketch of the proof.
- Week Two: See the extra credit problems
mentioned in the HW due Monday, February 14.
- Week Three:
Prove that the product of the slopes of two perpendicular lines in the plane
that are not parallel to the coordinate axes is -1. What is the generalization
of this to lines in three-dimensional space? What is the analogue of the
product of the slopes of the line equaling -1?
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solution and comments on the problem.
-
Week Three:
Prove or disprove: If P, Q and R are any three vectors, then (P x Q) x R = P x
(Q x R). Click here for
the solution and comments on the problem.
-
Week Four:
Section 2.2 of Marsden and Tromba: #21 (finding a function which is always b/w 0 and 1 and is 1
at a prescribed point and 0 at another prescribed pont):
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solution and comments on the problem.
-
Week Five:
(1)
Let f: R4 --> R3 and g: R2 --> R4
be two continuous functions, and let U be an open subset of R2.
Must f(g(U)) be an open subset of R3? Note h(U) is defined as the
set of all outputs of h where the input ranges over all points in U. (2)
Section 2.5, #24.
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solution and comments on the problem.