Week 2 -- Math Readings and Assignments: NCLC 120

There are two topics this week:

(1) Mathematical Models for Growth
      (linear and exponential growth + assignments)

(2) The Mathematics Model of Mendelian Genetics
      (an introduction to probability)
      

Notice that both of these topics use the phrase: "Mathematical Model."   Recall, this notion was discussed in Math Stuff for Week1.  In that section we said that a  mathematical model is a mathematical description (often a formula) which attempts to summarize a particular scientific idea.  See if you can see how this notion applies to our two topics for this week

TOPIC ONE: Models for Growth

Often, things change as time passes.  Perhaps, as time passes, the number of bacteria present in a dish changes with time. Or, maybe, the number of hairs on one's head is a function of time.  Or, even, the number of elephants in South Africa.  In this section, we'll discuss formulas (models) that can be used to describe such change.

We will denote the amount of what we are keeping track of  by the letter A, and we will denote by A(t) the amount present at time t.  We will in each case have to tell what units of time we are using: seconds, year, centuries, etc.  We'll need also to state the units of the amount.  

So, for example, let's say we are measuring the number of mosquitoes in Fairfax County.   Assume are units of measurement for the amount of mosquitoes is the kilogram.  If t=0 means January 1, 1998, and if the unit of time specified is the day, then, naturally, A(0) is the weight in kilograms of mosquitoes present on January 1, 1998 in Fairfax County, and, similarly, A(33) is the weight in kilograms of those mosquitoes on February 2, 1998.  Clearly, A(t) does indeed vary with time, being rather small when t is, say under 100, and the getting larger as t increases (as we would then be approaching the summer.)

Often, one wants a formula for A(t), and sometimes one wants a picture.  Having both.is even better.

Here is a fast example of table (not as good as a formula)  and picture:  Let's keep thinking about this mosquito example.

Pretend we've somehow been able to measure A(t) at various time t.  Perhaps we have the following table:

There is no apparent formula (that's ok), but we can certainly draw a picture (graph) of the data:

Linear Growth:

This is actually the simplest case.  We say that A(t) grows linearly if the graph of A(t) as a function of t is a straight line.  

It is a theorem that the ONLY time this happens is when the formula for A(t) looks like:

A(t) = mt + b, where m and b are
           two constants ( fixed numbers).

So, for, example, if we knew that, in our mosquito case, that the formula for A(t) was:
A(t) = 6t +10, then the graph of A(t) as a function of t would be a straight line.   In fact, if you make a table and then graph this data, you would get the following straight line:  [Try that yourself.] [Really] [Actually, if you want to learn how to read mathematics, you need to verify things on your own.  So, I REALLY suggest that if this is not crystal clear to you, that you do indeed make a table of A(t) as a function of t and then draw the graph yourself.]

If you want to look at the spreadsheet made these graphs, click on: mosquitoes.xls

For LINEAR GROWTH, you may recall that in the expression A(t) = mt + b, both of the constants have a geometric  interpretation: The the number m is called the slope [it tells how much the line is angled up or down] and the number b is called the y-intercept, and tells where the line crosses the vertical axis (also called the the y-axis). In our example of A(t) = 6t + 10, the slope is thus 6 and the intercept is 10.  Can you see that the graph crossed the axis at the point (0,10)?  NOTE: You may recall that for linear growth, when m>0, the line tilts up to the right and when m<0, it tilts down to the left.]

Exponential Growth

It is this sort of growth that readily occurs in biological examples, and you'll see that it is the most "dangerous" for it results in HUGE values of A(t) at time passes.

By definition, we say that A(t) grows exponentially as a function of t if the formula for A(t) is of the following format:

A(t) = C·b^t where C and b are constants, with b>0.

Just to make sure you understand the above formula, to find, say A(7), first raise b to the power 7 and then multiply that result by the constant C.  The number b is called the base of the growth.

So, for example, consider A(t) given by A(t) = 5(8^t).  

Then, as above, A(7) = 5(8⁷)
                                 = 5[2,097,152]
                                 = 10,485,760

Notice how HUGE the value of A(7) is.    This should give you a taste of why exponential functions grow so fast.  

Notice that when we defined the notion of exponential growth, we used a characterization of the formula for the expression for A(t), rather than a geometric notion, as we did for linear growth (where we used the geometric definition of line).   The reason is this that there is no standard name for the type of curve that an exponential graph looks like,  Sometimes it is sort of like a the letter J and some times it is sort of like the letter L (i.e., rising steeply or falling sharply).

Ok, let's see what the shape is of exponential growth.  We'll pick that same A(t) = 5(8^t).   One of the issues is what range of t to plot this over.   If we make t as large as only 7, the value of A(t) is over 10 million.  Let's do that anyway.  We'll graph this from t = 0 to t = 7.  The graph will be in that same spreadsheet mosquitoes.xls.

First, here is the table whose values we are going to graph:

And, here is the graph of this table.

Notice that the graph is sort of J shaped.  Can you see why from about t=0 to t=4, the graph looks like it is simply a horizontal line of ZERO height?   [Hint, look at the scale of the vertical axis.]

What about the geometric interpretations of those constants?  Are there any things we can say that is similar to the linear case?

Well, it is a bit more difficult.  Several things are not so hard to show are true, and they are very important.  We'll even call the following result a theorem, for it could in fact be proven.

THEOREM: Consider the function of t given by A(t) = C·b^t,
    where C and b are constants, with b>0.

THEN:
   
    (1)  A(t) grows exponentially as a function of t.
    (2)  The constant C is the initial value; that is: A(0) = C
    (3)  Assume C is positive.  Then
        (a) if b > 1, the graph of A(t) versus t looks sort of like a J (as above)
        (b) if b = 1, the graph of A(t) versus t is a horizontal line at height C.
        (c) if b < 1, the graph of A(t) versus t is sort of like an L (see below)

Before we give an example of that graph when b<1, let's say a few words about the proof of these statements.  Statement (1) is true because it is just the definition of exponential growth. Statement (2) is easy to show, for, let's just stick 0 in for t in the expression A(t) = C(b^t)   and see what happens,  [We should get C.]   Well, b raised to the 0 power is 1, and C times 1 is just C.  So Statement (2) is true.

Now for Statement (3a).  We'll not prove this here, but just mention that this statement is consistent with the graph we did above of A(t)=5(8^t). An actual proof of Statement (3a) requires calculus.   [In fact, one central theme in first-year calculus is curve sketching.]

Statement (3b) is not complicated at all.  For, when b=1, b^t is just 1^t which equals 1, and C times 1 is just C.  So, in this case A(t) = C, which is, indeed a horizontal line of height C.

For statement (3c), we'll illustrate it with a graph, also to be found in mosquitoes.xls.  NOTE; when b<0, the growth is actually called decay.  You'll see why this is called exponential decay when you see the next graph. [Note, this illustration is NOT a proof of the general case, only an indication that it might be true.]

Note that the above theorem says that the role of b is sort of like the slope in the linear case, for both b and the slope tell if the graph is going up (increasing, growing) or going down (decreasing, decaying).  And, the role of C is, in fact, exactly like the y-intercept in the linear case, for both tell where the graph crossed (intercepts) the y-axis.

Let's graph A(t) = 5(0.06)^t.  Note b=0.06, which is, of course, less than 1.  We'll graph this from t=0 to t=7.

Here is the table:

And, here is the graph.  Now do you see why it is called exponential decay?

Let's quickly refine the definition we gave earlier:

DEFINITION: Assume A(t) = C·b^t where C and b are constants, with b>0.
    (1) If b>1, we say that A(t) grows exponentially.
    (2) If b<1, we say that A(t) decays exponentially.

ASSIGNMENTS for Topic One: Exponential Growth

(1) Consider the linear model: A(t)=5t+3

(a) Graph this line on a piece of graph paper.

(b) What are its slope and intercept.  Does it tilt the way it should?

(2) Assume A(t) = 6·2^t.  Pretend t is measured in seconds and A(t) is measured in grams.

(a) Show that when t=0, A(t)=6 and that when t=3, A(t) = 48.

(b) Graph this, either with Excel or on a piece of graph paper.  Do this over the range from t=0 to t=6.  [First create a table and then plot the resulting points.   Be careful about how you label each axis.  That is, carefully put numbers along the horizontal axis as well as the vertical axis.] [Actually, you might try to do this BOTH in Excel and with graph paper.]

(c) How long (how many seconds) does it take for the amount to double from 6 grams to 12 grams?  How long to go from 48 to 96 grams.  [These times will be the same.   This time is called the double time.]

(d) What is the weight in one year?

(e) [Challenge Question] How many seconds does it take to go from 6 grams to 7 grams?

(f) [Challenge Question] How long does it take to go from 7 grams to 14 grams.  Is that STILL the double time gotten in part (c)?

Challenge Assignments (assigned, perhaps, by your seminar leader)

(1)  It's a THEOREM that the graph of every exponential growth or decay function never falls below the horizontal axis.  Yet, the graph above of A(t) = 5(0.06)^t does just that!  Can you explain that apparent contradiction?

(2) If A is a linear function of t, say A=mt+b and if t is a linear function of u, say t=ru+s, then one can show that, consequently, A is a linear function of s.  In fact:
    A=mt+b=m[ru+s]+b=(mr)u+(ms+b), which is certainly a linear function of u.
The question, here, is: Is this same thing true for exponential functions?  That is, if A is an exponential function of t and t is an exponential function of u, does it follow that A is an exponential function of u?

TOPIC TWO: Models for Mendelian Genetics

Click HERE to get the page for Topic Two.