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Notebook[{
Cell[CellGroupData[{
Cell["Calculus: The Language of Change", "Title"],
Cell[TextData[{
StyleBox["Exercise Set 2.3\n",
FontVariations->{"Underline"->True}],
"The Continuous Variable Model"
}], "Subtitle"],
Cell["by Kevra Tanke", "Subsubtitle"],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Exercise 1 ", "Subtitle"]], "Subtitle"],
Cell["\<\
What is the meaning of fractional values of S, I, and R in \
Excercise 2.1.1 of Section 2.1? Are the fractional variables s, i, and r \
meaningful in a small population?\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution ", "Subsection"],
Cell[TextData[{
"Looking back at 2.1.1 we find that \n\n\tS[0] = 10000 = number of \
suseptable individuals\n \tI[0] = 1000 = number of infectious individuals\n\t\
R[0] = 19000 = number of removed individuals\n\tn = 30000 = total population\n\
\t\n\ts = ",
Cell[BoxData[
\(TraditionalForm\`S\/n\)]],
"= ",
Cell[BoxData[
\(TraditionalForm\`10000\/30000\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`1\/3\)]],
"\n\t\n\ti = ",
Cell[BoxData[
\(TraditionalForm\`I\/n\)]],
"= ",
Cell[BoxData[
\(TraditionalForm\`1000\/30000\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`1\/30\)]],
"\n\t\n\tr = ",
Cell[BoxData[
\(TraditionalForm\`R\/n\)]],
"= ",
Cell[BoxData[
\(TraditionalForm\`19000\/30000\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`19\/30\)]],
"\n\t\n"
}], "Text"],
Cell["\<\
Fractional values of S, I and R represent less than a complete person and we \
should therefore round off the values to the nearest \"whole person.\"
s, r, and i are the fractions of the population that are susceptable, \
removed, and infectious, respectively. Representing a fraction of the total \
population, each of these ratios is then a smaller population. However, if \
the total population is too small, say less than 30, the ratios may not be \
accurate because you cannot have a fraction of a person removed. Another \
thing to consider is whether the population is so small that everyone comes \
into contact with everyone everyday. That would also affect the ratios. As \
in most experiments, larger samples generally yield more accurate findings. \
\
\>", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Exercise 2 ", "Subtitle"]], "Subtitle"],
Cell["\<\
The total population is part of a model with S, I, and R, but not \
with s, i, and r. Explain how we might use the history of a rubella epidemic \
with one size population to make predictions about a new and different size \
one.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution ", "Subsection"],
Cell[TextData[{
"To make predictions with a new size population use the fractional values \
to find the ratios of the population that are infected, susceptable, and \
removed. Use these ratios with the new population.\n\nFor example: to find \
the new number of suseptable people, ",
Cell[BoxData[
\(TraditionalForm\`S\_n\)]],
", for a new population of 20000:\n\t\t \n\t\t",
Cell[BoxData[
\(TraditionalForm\`10000\/30000\)]],
"= ",
Cell[BoxData[
\(TraditionalForm\`S\_n\/20000\)]],
" \[Implies] ",
Cell[BoxData[
\(TraditionalForm\`\(10000\ *\ 20000\)\/30000\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`S\_n\)]],
" = 6667"
}], "Text"],
Cell["", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Exercise 3", "Subtitle"]], "Subtitle"],
Cell["Computer Program.", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[{
StyleBox["Exercise 4 : ", "Subtitle"],
"The Geometry of Euler's Method "
}], "Subtitle"],
Cell[TextData[{
"Let x and y be real variables. You are given that when x=0, then y=1, \n\
\n\ty[0] = 1 and that y changes with respect to x by the rule ",
Cell[BoxData[
\(TraditionalForm\`dx\/dy\)]],
" = -y\n\t\nSketch the graph of y = y[x] by graphically using Euler's \
Method."
}], "Text"],
Cell[CellGroupData[{
Cell["Solution ", "Subsection"],
Cell[CellGroupData[{
Cell["Sketch the graph for dx = 0.5.", "Subsubsection"],
Cell[BoxData[{
\(Clear[f, G, x, y]; \nf[x_, y_] := \(-y\); \n
G[{x_, y_}] := {x + dx, y + f[x, y]\ dx}; \nx0 = 0; \ny0 = 1; \n
finalTime = 1; \ndx = .5; \nsteps = Floor[finalTime\/dx]; \n
Print["\