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Section0.2Exponential Functions

Objectives

  • How can exponential functions be used to model growth and decay of populations, investments, radioactive isotopes, and many other physical phenomena?

  • How can we build exponential functions from data?

The exponential function is a powerful tool in the mathematician's arsenal for modeling growth and decay phenomena. The common applications of the exponential funciton range from population modeling, to tracking drug levels in the blood stream, to using carbon dating to estimate the age of an artifact. The common mathematical fact about all of these situations is that the growth (or decay) rate is a constant multiple. For example, if we are measuring exponential population growth then the ratio of two successive populations must be constant. Linear functions have a similar behavior, except that in linear functions the difference (not the ratio) between two successive values is constant (the slope).

Preview Activity0.2.1

Suppose that the populations of two towns are both growing over time. The town of Exponentia is growing at a rate of 2% per year, and the town of Lineola is growing at a rate of 100 people per year. In 2014, both of the towns have 2,000 people.

  1. Complete the table for the population of each of these towns over the next several years.

    2014 2015 2016 2017 2018 2019 2020 2021 2022
    Exponentia 2000
    Lineola 2000

  2. Write a linear function for the population of Lineola. Interpret the slope in the context of this problem.

  3. The ratio of successive populations for Exponentia should be equal. For example, dividing the population in 2015 by that of 2014 should give the same ratio as when the population from 2016 is divided by the population of 2015. Find this ratio. How is this ratio related to the 2% growth rate?

  4. Based on your data from part (a) and your ratio in part (c), write a function for the population of Exponentia.

  5. When will the population of Exponentia exceed that of Lineola?

Subsection0.2.1Exponential Functions

Consider the example where the population of a bacteria colony is doubling every week. If in the first week there are 100 bacteria, then there are 200 bacteria by the end of the second week, 400 by the end of the third and so on. In Table 1 we can see a simple way to model this type of growth:

\begin{gather*} P(t) = 100 \cdot 2^t \text{ (\(t=\) number of weeks) } \end{gather*}
Week Bacteria
0 \(100\)
1 \(100 \cdot 2=200\)
2 \(200 \cdot 2 = 100 \cdot 2^2=400\)
3 \(400 \cdot 2 = 100 \cdot 2^3=800\)
\( \vdots \) \( \vdots \)
Table0.2.1Bacteria population doubling

The time, \(t\), in the previous equation is measured in weeks. It is easy to see that the ratio of the populations for each successive week is constant at \(P(t+1)/P(t) = 2\text{.}\) This is indicative of exponential growth. Of course, this population growth could have been modeled using time measured in days instead. The population still doubles every week so for this new model the value at \(t=7\) should be double the value at \(t=0\text{.}\) The next equation shows this new model with only a slight modification adjusting for the new time measurement.

\begin{gather*} P(t) = 100 \cdot 2^{t/7} \text{ (\(t=\) number of days) } \end{gather*}

This type of modeling and thought process can be used to describe most exponential growth and decay situations. One general formula for an exponential function is

\begin{gather*} f(x) = A \cdot r^{kx} \end{gather*}

where \(A\) is some given initial value, \(r\) is the common ratio, and \(k\) is a constant given by the frequency in which the common ratio is applied. In the previous population doubling example, \(A=100\text{,}\) \(r=2\text{,}\) and \(k=1/7\text{.}\)

Exponential Growth and Decay

A few simple guidelines should make it clear when an exponential function in the form \(f(x) = A \cdot r^{kx} \) is modeling growth or decay.

  • If \(r > 1\) then the function exhibits exponential growth.

  • If \(0 \lt r \lt 1\) then the function exhibits exponential decay.

Growth and Decay Rates

  • If a population is growing by \(p\%\) per unit time, then \(r = 1+p/100\text{.}\)

  • If a population is decreasing by \(p\%\) per unit time, then \(r = 1-p/100\text{.}\)

Activity0.2.2

Consider the exponential functions plotted in the figure below.

  1. Which of the functions have common ratio \(r > 1\)?

  2. Which of the functions have common ratio \(0\lt r\lt 1\)?

  3. Rank each of the functions in order from largest to smallest \(r\) value.

Hint
Solution
Example 0.2.3

One application to exponential decay is to calculate the intensity of radiation from radioactive isotopes. Most isotopes emit particles and decay into stable forms. We measure the rate of decay from the particles by the isotope's half-life, which is how long it takes half of the isotope to decay. The half-life for Sodium-25 (\(Na^{25}\)) is almost exactly one minute. Write a function that models that amount of \(Na^{25}\) over time if you start with exactly 36 grams.

Solution
Activity0.2.3

A sample of \(Ni^{56}\) has a half-life of 6.4 days. Assume that there are 30 grams present initially.

  1. Write a function describing the number of grams of \(Ni^{56}\) present as a function of time. Check your function based on the fact that in 6.4 days there should be 50% remaining.

  2. What percent of the substance is present after 1 day?

  3. What percent of the substance is present after 10 days?

Hint
Solution
Activity0.2.4

Uncontrolled geometric growth of the bacterium Escherichia coli (E. Coli) is the theme of the following quote taken from the best-selling author Michael Crichton's science fiction thriller, The Andromeda Strain:

The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.

  1. Write an equation for the number of E. coli cells present if a single cell of E. coli divides every 20 minutes.

  2. How many E. coli would there be at the end of 24 hours?

  3. The mass of an E. coli bacterium is \(1.7 \times 10^{-12}\) grams, while the mass of the Earth is \(6.0 \times 10^{27}\) grams. Is Michael Crichton's claim accurate? Approximate the number of hours we should have allowed for this statement to be correct?

Hint
Solution

Subsection0.2.2Investments

Interest bearing bank accounts and investments follow exponential growth and decay models. In the case of a savings accounts the interest is typically compounded several times per year. This means that the investor is getting interest on their interest every time the bank computes the interest.

If the money is gaining \(p\%\) interest compounded \(n\) times per year then the common ratio for the exponential function is \(1 + p/n\text{.}\) The exponent needs to reflect the fact that the interest occurs at monthly intervals. This means that the exponential function is

\begin{gather*} A(t) = A_0 \left( 1+\frac{p}{n} \right)^{nt} \text{ (\(t=\) number of years) } . \end{gather*}

In the previous equation, \(A_0\) is the initial investment, \(A(t)\) is the value of the investment over time, \(p\) is the interest rate, and \(n\) is the number of times the interest is compounded per year.

Example 0.2.5

If $100 are invested into a bank account earning 2% interest compounded 12 times per year, how much does the investor have at the end of 1 year? 5 years? at retirement age? How does this change is we compound quarterly or daily instead of monthly?

Solution

Subsection0.2.3Exponential Functions with Base \(e\)

Exponential functions are commonly written with a base of \(e \approx 2.718281828459045\dots\text{.}\) This may seem like an arbitrary and bizarre choice at first glance, but we will see that this famous number (called Euler's Number 1 ) plays a central role in Calculus.

Euler's number can be derived from the equation \begin{equation*} A(t) = A_0 \left( 1 + \frac{p}{n} \right)^{nt} \end{equation*} if we assume that a fictitious bank gives \(100\%\) interest compounded infinitely many times per year on a one dollar investment. Mathematically this is written as

\begin{gather*} e = 1 \cdot \left( 1 + \frac{1}{n} \right)^n \text{ as } n \to \infty. \end{gather*}
\(n\) \(1\) \(10\) \(100\) \(1000\) \(\cdots\) \(10^{10}\) \(\cdots\)
\((1+\frac{1}{n})^n\) \(2\) \(2.5935\) \(2.7048\) \(2.7169\) \(\cdots\) \(2.71828\) \(\cdots\)
Table0.2.8Approximations of Euler's number, \(e\text{,}\) with various values of \(n\)
Exponential Functions with Euler's Number

Any exponential function can be rewritten in terms of Euler's number in the form

\begin{gather*} f(x) = A_0 e^{kx}. \end{gather*}

In this equation, \(k\) is called the continuous rate.

  • If \(k>0\) then \(f(x) = A_0e^{kx}\) models exponential growth.

  • If \(k\lt 0\) then \(f(x) = A_0e^{kx}\) models exponential decay.

Example0.2.9

A population of a city is 5000 people and is doubling in size every 5 years. Use the equations \begin{equation*} P(t) = A_0 r^{nt} \end{equation*} and \begin{equation*} P(t) = A_0 e^{kt} \end{equation*} to write two different functions modeling this population; one with base 2 and one with base \(e\text{.}\)

Solution

Subsection0.2.4Summary

  • An exponential function can be written in the form \(f(x) = A r^{kx}\) or \(g(x) = A e^{kx}\text{.}\)

    • In \(f(x)\text{,}\) if \(k>0\) and \(r>1\) then \(f(x)\) models exponential growth.

    • In \(f(x)\text{,}\) if \(k>0\) and \(0\lt r\lt 1\) then \(f(x)\) models exponential decay.

    • In \(g(x)\text{,}\) if \(k>0\) then \(g(x)\) models exponential growth.

    • In \(g(x)\text{,}\) if \(k\lt 0\) then \(g(x)\) models exponential decay.

  • Exponential functions have a constant common ratio for successive time values.

Subsubsection0.2.5Exercises

1

Suppose that \(h(t) = A \cdot r^t\text{.}\) If \(h(3)=4\) and \(h(5)=40\text{,}\)

  1. find \(r\text{.}\)

  2. find \(A\text{.}\)

  3. Does this function model exponential growth or decay? How can you tell?

2

The half-life of \(Br^{77}\) is 57 hours.

  1. If the initial amount is \(150\) grams, find the amount remaining after 171 hours.

  2. Write an equation to predict the amount remaining after \(t\) hours.

  3. Estimate within one hour how long it will take the amount to decrease to 10 grams.

3

Consider the data in Table 10

  1. Which (if any) of the functions could be linear? Explain how you know that these functions are linear, and find formulas for these functions.

  2. Which (if any) of the functions could be exponential? Explain how you know that these functions are exponential, and find formulas for these functions.

\(x\) \(f(x)\) \(g(x)\) \(h(x)\)
\(-2\) \(12\) \(16\) \(37\)
\(-1\) \(17\) \(24\) \(34\)
\(0\) \(20\) \(36\) \(31\)
\(1\) \(21\) \(54\) \(28\)
\(2\) \(18\) \(81\) \(25\)
Table0.2.10Data tables for \(f(x)\text{,}\) \(g(x)\text{,}\) and \(h(x)\)