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\noindent {\bf Problem statement} a) Suppose $\displaystyle{f(x) =
{{e^{10 x}} \over {1 + e^{10x}}}}$. Graph this function when $-5\le x
\le 5$, and find the notable features of this graph, including any
local extrema, points of inflection, and asymptotes. Sketch a
plausible graph of $\displaystyle{{{e^{10,000 x}} \over {1 + e^{10,000
x}}}}$

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\noindent b) Suppose $\displaystyle{g(x) = {{x^{10}} \over {1 +
x^{10}}}}$. Graph this function for $-5\le x\le 5$, and find the
notable features of this graph, including any local extrema, points of
inflection, and asymptotes. Sketch a plausible graph of
$\displaystyle{{{x^{10,000}} \over {1 + x^{10,000}}}}$

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\noindent {\bf Note} Such functions may serve as appropriate models
for biophysical phenomena where rate constants in reactions are very
different from everyday time scales. The curves sketched in a) are
called {\it logistic curves}.






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