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Math 121 Calculus for Biology
Spring Semester, 2007
Lab Help
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06-Mar-06
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San Diego State University
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Laboratory Help Page for Lab 6
The first question in this lab
examines the Improved Malthusian Nonautonomus
growth model.
The second lab problem studies an allometric modeling
for the heitgh and weight of girls.
Question 1:
Malthusian Growth and Nonautonomous Growth Models (F4).
This problem asks you to repeat what we have done in the
notes with the U. S. census data to census data from another country.
You will
be producing similar graphs (see the graph titled "Growth Rate for U.
S." and the following graph called "Discrete Growth Models for U.
S."), which should give you a guideline. Be sure to answer all of the
questions carefully.
Use the power of Excel to find your
growth rates. Suppose that once again
you have entered the dates in Column A (starting in A2) and the
population
values in Column B (starting in B2). I would suggest that you copy all
but the
last date again into another column, say Column G. (I like to have
different
computations away from the main table of data. Note that A2 = G2 are
the same
dates.) In Column H, you start in H2 and enter "=B3/B2-1" to let
Excel compute the growth rate associated with the first date. You
simply fill
down from here to obtain all the growth rates for your country. At the
bottom
of the calculations in Column H you can use the Excel AVERAGE command
to
compute the average growth rate for the country. (I find it easiest to
simply
put this average value in Column I and fill down, so that I can
highlight
columns G, H, and I to produce the graph like the one in the notes.)
Use
trendline on the growth rate data to find the best straight line. In order to get good results, it is very important
that after
you use trendline to find the best straight line through the growth
data, you
double click (or right click) on the equation (be careful here), then
when the
Format menu pops up, you can choose "number" of the 3 folders. Under
number, you select "number" again, which should give you a menu to
select the way you want your numbers displayed. You should use the
scientific
notation option. Its this equation that will determine the
behavior of
your nonautonomous model, so you need 6 significant figures for an
accurate
model.
Now you return to the Columns A and
B in your data set. In Column C, you
repeat the procedure you did in Question 1 above using the average
value just
computed for your growth rate r.
In
Column D, you will produce the nonautonomous model using the equation
you just
found from trendline. Say trendline found the equation k(t) = 3.148 -
0.002347
t, then after the first entry in D2 (which is the same as B2 and C2),
in D3,
you enter "=(1 + 3.148 - 0.002347*A2)*D2." You fill down from here to
complete the nonautonomous model. You may want to use my discrete
template with
the nonautonomous model as a guide, making the appropriate changes.
From there
you simply need to create tables and graphs of the models and answer
the
questions posed in this Lab question.
Question 2:
Weight and Height of Girls (I2).
The first part of this problem gives you more experience
with allometric modeling using heights and weights of girls. Part c is related
to the previous problem giving you more intuition on rate of gain of weight,
which is again a derivative. The rate of gain of weight is similar to the computations
we did for the lecture notes.