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Cell["\<\
The Mathematics of Modeling
Part 1: Time-Stepping Methods\
\>", "Subtitle",
TextAlignment->Center],
Cell["\<\
\tIn many ways, science is the art of predicting the future. By \
observing how something changes over the course of time, we learn about the \
conditions that affect those changes and learn to expect, and thus predict, \
similar changes for similar things given similar conditions. The more \
information we have, the more likely it is that we can make a prediction that \
is correct.\
\>", "Text",
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Cell[TextData[{
StyleBox["\tSuppose I ask you to predict the weather for tomorrow. If \
you've been working hard in your windowless office, and only vaguely know \
that it is early March, you might predict 'sunny, with some wind'. If you \
had recently been outside and noticed that today's conditions were actually \
cloudy and windy, you would be inclined to predict much the same for \
tomorrow. If you were able to access an internet weather station that showed \
a major stormfront moving in your direction, you might intead predict 'very \
windy, with rain and thunderstorms'. The conditions you know at the start of \
your prediction are called ",
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FontSize->14,
FontSlant->"Italic"],
StyleBox[".",
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Cell[TextData[StyleBox["\tThe accuracy of a prediction will also depend on \
the length of time that you are trying to predict over. Continuing with our \
weather example - you would probably be able to make a very good prediction \
of the weather five minutes from now, even with only limited personal \
observations. Predicting twenty-four hours from now is much less likely to be \
accurate, and predicting a month from now becomes little more than a guess, \
even if you have the complete array of technology to determine your initial \
conditions.",
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StyleBox["Time Stepping\n", "Section"],
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StyleBox[" try to take advantage of the idea that changes in a system can \
be predicted pretty well, if the prediction time is small. Presume for \
example that we are tracking a hurricane in the Atlantic, and that we expect \
landfall in twenty-four hours, but would like to predict where the hurricane \
will be at that point. We start by dividing our prediction time into many \
smaller pieces - small enough so that we think we can make a good prediction, \
but not too small! - the more pieces we have the more work we have to do, and \
it doesn't help if it takes more than 5 minutes to figure out what happens \
(happened!) in those five minutes! A good sized piece for this example might \
be an hour.",
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although with the calculus assumption that this change is infinitesimally \
small and that all we care about is the relationship between the two changes \
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dealing with gravity, we might say that the accelerations are constant - for \
every time step there might be no acceleration in the x direction and \
acceleration equal to gravity in the negative (downwards) y direction.\n\t\t\t\
\t",
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\tThen we estimate initial values for velocity (in meters per \
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\>", "Text",
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