1.[N] Consider the logistic map with s = 4, x_{n+1} =
4*x_{n}*(1 - x_{n}). This system is deterministic -- if we know
x_{n} exactly, then we can find x_{n+1} exactly. Starting with
some x_{0} between 0 and 1 (and not including 1/2), use a calculator to
generate x_{1}, ..., x_{10}, and record your answers to the
fourth decimal place. Now change x_{0} in the third decimal place,
generate a new sequence x_{1}, ..., x_{10}, and record your
answers to the fourth decimal place. Comparing the last term of the new sequence
with the last term of the old sequence, comment on the accuracy with which the
initial value must be known to make an accurate prediction of x_{10}.

2.[N] For the logistic map with s = 4, start with x_{0} = 0.0001 and
use a calculator to generate x_{1}, ..., x_{10}. For the tent map
with s = 2, start with x_{0} = 0.0001 and use your calculator to generate
x_{1}, ..., x_{10}. Compare these answers. What can you conclude
by comparing these sequences?

3.[C] For s = 0.999, both the tent map and the logistic map have attracting
fixed points at x = 0, while for s > 1, neither do. For a starting value of
x_{0} = 0.5, compare the number of iterations needed to converge for
these two maps using TENT/HISTOGRAM and LOGISTIC/HISTOGRAM. Why do think there
is such a difference?

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