10.1: Unlimited Growth and Doubling

Learning Objectives

1. Explain the link between population doubling and integer powers of the base 2.
2. Given information about the doubling time of a population and its initial size, determine the size of that population at some later time.
3. Appreciate the connection between \(2^{n}\) for integer values of \(n\) and \(2^{x}\) for a real number \(x\).

Mastered Material Check

1. If a population has size \(P\), what do we mean by a doubled poulation size??
2. How large would the population be it it doubled twice?

The Andromeda Strain

The Andromeda Strain scenario (described by Crichton in the opening quotation) motivates our investigation of population doubling and uncontrolled growth. Consider \(2^{n}\) where \(n=1,2 \ldots\) is an integer. We will study values of this discrete function as the "variable", \(n\) in the exponent changes. We list some values and display a graph of \(2^{n}\) versus \(n\) in Figure 10.1. Notice that an initially "gentle" growth becomes extremely steep in just a few steps, as shown in the accompanying graph.

Note: properties of \(2^{n}\) (and related expressions) are reviewed in Appendix B.1 where common manipulations are illustrated. We assume the reader is familiar with this material.

Figure 10.1: Powers of 2 including both negative and positive integers: here we show \(2^{n}\) for \(-4<n<10\).

The function \(2^{n}\) first grows slowly, but then grows faster and faster as \(n\) increases. As a side remark, the fact that \(2^{10} \approx 1000=10^{3}\), will prove useful for simple approximations. With this preparation, we can now check the accuracy of Crichton’s statement about bacterial growth.

A screencast summary of population doubling and the Andromeda Strain. Edu.Cr.

Example 10.1

(Growth of E. coli) Use the following facts to check the assertion made by Crichton’s statement at the beginning of this chapter.

• Mass of 1 E. coli cell: 1 picogram \(=10^{-12} \mathrm{gm}=10^{-15} \mathrm{~kg}\).
• Mass of Planet Earth : \(6 \cdot 10^{24} \mathrm{~kg}\).

Solution

Based on the above two facts, we surmise that the size of an E. coli colony (number of cells, \(m\) ) that together form a mass equal to Planet Earth would be

\[m=\frac{6 \cdot 10^{24} \mathrm{~kg}}{10^{-15} \mathrm{~kg}}=6 \cdot 10^{39} . \nonumber \]

Each hour corresponds to 3 twenty-minute generations. In a period of 24 hours, there are \(24 \times 3=72\) generations, each doubling the colony size. After 1 day of uncontrolled growth, the number of cells would be \(2^{72}\). We can find a decimal approximation using the observation that \(2^{10} \approx 10^{3}\) :

\[2^{72}=2^{2} \cdot 2^{70}=4 \cdot\left(2^{10}\right)^{7} \approx 4 \cdot\left(10^{3}\right)^{7}=4 \cdot 10^{21} . \nonumber \]

Using a scientific calculator, the value is found to be \(4.7 \cdot 10^{21}\), so the approximation is relatively good.

Apparently, the estimate made by Crichton is not quite accurate. However it can be shown that it takes less than 2 days to produce a number far in excess of the "size of Planet Earth". The exact number of generations is left as an exercise for the reader and is discussed in Example 10.12.

Mastered Material Check

3. Compare the function \(f(n)=2^{n}\) and \(g(n)=n^{2}\) for \(n=1,2, \ldots, 5\). How do these differ?

Mastered Material Check

4. Why would the approximation \(2^{10} \approx 10^{3}\) be helpful?

Mastered Material Check

5. How many cells of E. coli would there be after 20 minutes? 1 hour? 2 hours?

Mastered Material Check

6. Verify that it takes less than 2 days to produce a number far in excess of the size of Planet Earth.

The function \(2^{x}\) and its "relatives"

We would like to generalize the function \(2^{n}\) to a continuous function, so that the tools of calculus - such as derivatives - can be used. To this end, we start with values that can be calculated based on previous mathematical experience, and then "fill in gaps".

Figure 10.2: (a) Values of the function \(2^{x}\) for discrete value of \(x\). We can compute many values (e.g. for \(x=0, \pm 1, \pm 2\), by simple arithmetical operations, and for \(x=\pm 1 / 2, \pm 3 / 2\) by evaluating square roots). (b) The function \(2^{x}\) is connected smoothly to form a continuous curve.

From previous familiarity with power functions such as \(y=x^{2}\) (not to be confused with \(2^{x}\) ), we know the value of

\[2^{1 / 2}=\sqrt{2} \approx 1.41421 \ldots \nonumber \]

We can use this value to compute

\[2^{3 / 2}=(\sqrt{2})^{3}, \quad 2^{5 / 2}=(\sqrt{2})^{5}, \nonumber \]

and all other fractional exponents that are multiples of \(1 / 2\). We can add these to the graph of our previous powers of 2 to fill in additional points. This is shown on Figure 10.2(a).

Similarly, we could also calculate exponents that are multiples of \(1 / 4\) since

\[2^{1 / 4}=\sqrt{\sqrt{2}} \nonumber \]

is a value that we can obtain. Adding these values leads to an even finer set of points. By continuing in the same way, we "fill in" the graph of the emerging function. Connecting the dots smoothly allows us to define a value for any real \(x\), of a new continuous function,

\[y=f(x)=2^{x} . \nonumber \]

Here \(x\) is no longer restricted to an integer, as shown by the smooth curve in Figure 10.2(b).

Example 10.2

(Generalization to other bases) Plot "relatives" of \(2^{x}\) that have other bases, such as \(y=3^{x}, y=4^{x}\) and \(y=10^{x}\) and comment about the function \(y=a^{x}\) where \(a>0\) is a constant (called the base).

Solution

We first form the discrete function \(a^{n}\) for integer values of \(n\), simply by multiplying \(a\) by itself \(n\) times. This is analogous to Figure 10.1. So long as \(a\) is positive, we can "fill in" values of \(a^{x}\) when \(x\) is rational (in the same way as we did for \(2^{x}\) ), and we can smoothly connect the points to lead to the continuous function \(a^{x}\) for any real \(x\). Given some positive constant \(a\), we define the new function \(f(x)=a^{x}\) as the exponential function with base \(a\). Shown in Figure \(10.3\) are the functions \(y=2^{x}, y=3^{x}, y=4^{x}\) and \(y=10^{x}\).

Figure 10.3: The function \(y=f(x)=a^{x}\) is shown here for a variety of bases, \(a=2,3,4\), and 10.

Mastered Material Check

7. Given \(2^{1 / 2} \approx 1.41421\), find \(2^{3 / 2}\) and \(2^{5 / 2}\) without using fractional powers.
8. What method might you use to determine a decimal approximation of \(2^{1 / 4}\) without computing fractional powers?
9. Why do we need to assume that \(a>0\) for the exponential function \(y=a^{x}\) ?