Homework #6
Instructions. This is due on November 20
at the beginning of class. It is worth 20 points. Late papers lose 3 points per
day or partial day late. You must work by yourself, but feel free to consult
with me. Explain what you did and why you did it. Your paper should start with
an introduction and finish with a conclusion. Start early, ask questions and do
well.
Introduction. In this assignment, you will solve an
important problem in genetics. In general, calculus gives you the tools to
solve complex problems in a variety of different subjects. This problem
investigates whether a particular family line (e.g., you and your descendants)
will continue forever or become “extinct.” Assume that p0
gives the probability that a family member has 0 children, p1
gives the probability that a family member has 1 child, p2
gives the probability that a family member has 2 children, and so on. Assume
that no one has more than 4 children.
If x is the probability that the line goes extinct, an equation
for x is
p0 + p1x
+ p2x2 + p3x3
+ p4x4 = x
Define the function F(x) to be the left-hand side of this
equation. That is, define
F(x) = p0 + p1x + p2x2 + p3x3 + p4x4 . The probability of extinction is the smallest solution of the equation F(x) = x with x between 0 and 1.
Problems. Using the above information, explain why F(1) = p0 + p1 + p2
+ p3 + p4 = 1. (Hint: think about
probabilities.)
For x > 0, show
that F’(x) > 0. This says that F is an increasing
function.
For x > 0, show
that F’’(x) > 0. This says that the graph of F is
concave up.
Let p0 =
0.4. Sketch two possible graphs of F(x) for 0 < x <
1. Note that both graphs must satisfy F(0) = 0.4 and F(1) = 1,
and both should be increasing and concave up. In one graph, have the graph of F(x)
intersect the line y = x for some x < 1 (this x
is the probability of extinction). In the other graph, have the graph of F(x)
only intersect y = x at x = 1 (in this case, the
probability of extinction is 1).
Use these graphs (nothing
else) to explain why extinction is certain if F’(1) < 1 but that
there is a positive probability of survival if F’(1) > 1.
Make up your own (nonzero)
choices for the probabilities (p0, p1, p2,
p3, p4) so that there is a chance of
survival. Use the calculator to estimate the probability of survival in this
case.