Cell Division

How quickly does a chick embryo grow? Remember that the way any embryo develops is by repeated cell division. It starts off as a single cell which spilts into two. Each of those two cells then splits into two, and so on. We have illustrated this schematically with a square box representing the initial egg cell

Move your cursor into the box to `fertilize the egg' and start the cell divison procedure. The box divides into 2 regions, then 4, then 8, etc. The number of regions grows in the same way that the number of cells in a real embryo does. At the left, we graph the number of cells against the number of times cell division has occurred. Notice how quickly the number of cells grows!

Exponential Functions

We can very efficiently describe the way the number of cells grows if we use some mathematical notation. Let's denote the number of times division has occured by the letter t, and let's denote the total number of cells by N(t). Thus N(1) denotes the number of cells after 1 division; so N(1)=2. Similarly, N(2)=2x2=4, N(3)=2x2x2=8, etc. A more compact way to write 2x2x2 is as 2³; 2x2x2x2 is 2⁴ and so on. With this convenient notation, we can write:

Following this pattern, we can write down a formula for N(t), namely:

In mathematical language, N(t) is an exponential function of t.

In the graph next to the cell division simulation, we have plotted N(t) for t=1,2,3,.... We can join up the dots to get a smooth curve, from which we can read off values for 2^t corresponding to any value for t, whole number or not. (Later we will see a more precise way to define 2^t for any value of t.)

In the exponential function N(t)=2^t, the number 2 is called the base of the exponential. We can change this base and get other functions of t, all of which are called exponentials. For example, using base 3, we get the exponential function 3^t.

If Java worked here, you'd see something like this. The graphs of exponential functions are shown in this demonstration. You can change the base of the exponential (denoted by b) by using the slider at the top.

Notice that the graphs extend into the region where t is negative. This doesn't make much sense if t represents the number of times that a chick egg has divided. Mathematically, however, we can make sense of it as follows. For the exponential with base b, i.e. for the function b^t, we use the rule that:

This rule tells us how to compute b^t for negative values of t. With this rule it is still true that b^t+1 = b b^t.

If you experiment with a few different choices for the base, you will notice that the graphs of all the exponential functions have a few things in common:

1. The graphs all pass through the same point when t=0.

2. Going off to one direction (left or right), the value of the function (the height of the curve) grows very rapidly, but going off to the other direction, the value decays slowly towards zero, getting ever closer to the horizontal axis but never actually reaching it.

   Notice that for some choices of base b, the value increases when we move to the left and decays to the right, while for other choices of b, the value increases to the right and decays to the left. See if you can figure out what determines which alternative applies to a given exponential function.

3. There is a third, more subtle, property shared by all the exponential functions. To understand what this is, we need to look at the straight lines tangent to the graphs. In particular, we need to look at the directions of these tangent lines, as measured by their slopes.

If Java worked here, you'd see something like this. Select a point on the graph by clicking on it. The demonstration shows the tangent line to the graph at the selected point. The three numbers shown at the bottom of the demonstration are:

• the value of the exponential, N = b^t, at that point,
• the slope (denoted m) of the tangent line at that point, and
• the ratio m/b^t

Notice that along a given curve, while the first two numbers change, the third one remains the same no matter which point you select! This is the third special property of exponentials:

The ratio (the third number displayed in the applet) is called the constant of proportionality.

The Special Base

If you explore this property of exponentials in more detail, you will notice that this constant of proportionality changes depending on the base of the exponential. For instance, for the exponential to base 2 the constant is .693147, while for the exponential to base 3 the constant is 1.0986.

This prompts the following question:

• Is there some base for which the constant of proportionality is exactly 1?

The answer to this question is YES! This base, for which the exponential function has slope equal to its value, is traditionally denoted by the letter e. This number e has many wonderful properties. You can explore the graph of the exponential function e^t, and verify that it has slope equal to value by clicking the e button in the demonstration above.