6.8 Exponential Growth and Decay

One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications.

Exponential Growth Model

Many systems exhibit exponential growth. These systems follow a model of the form $y = y_{0} e^{k t},$ where $y_{0}$ represents the initial state of the system and $k$ is a positive constant, called the growth constant. Notice that in an exponential growth model, we have

y^{'} = k y_{0} e^{k t} = k y .

That is, the rate of growth is proportional to the current function value. This is a key feature of exponential growth. Exponential Growth and Decay involves derivatives and is called a differential equation. We learn more about differential equations in Introduction to Differential Equations.

Population growth is a common example of exponential growth. Consider a population of bacteria, for instance. It seems plausible that the rate of population growth would be proportional to the size of the population. After all, the more bacteria there are to reproduce, the faster the population grows. Figure 6.79 and Table 6.1 represent the growth of a population of bacteria with an initial population of $200$ bacteria and a growth constant of $0.02.$ Notice that after only $2$ hours $(120$ minutes), the population is $10$ times its original size!

This figure is a graph. It is the exponential curve for y=200e^0.02t. It is in the first quadrant and an increasing function. It begins on the y-axis. — Figure 6.79An example of exponential growth for bacteria.

Table 6.1Exponential Growth of a Bacterial Population

Time (min)	Population Size (no. of bacteria)
$10$	$244$
$20$	$298$
$30$	$364$
$40$	$445$

Note that we are using a continuous function to model what is inherently discrete behavior. At any given time, the real-world population contains a whole number of bacteria, although the model takes on noninteger values. When using exponential growth models, we must always be careful to interpret the function values in the context of the phenomenon we are modeling.

Let’s now turn our attention to a financial application: compound interest. Interest that is not compounded is called simple interest. Simple interest is paid once, at the end of the specified time period (usually $1$ year). So, if we put $$ 1000$ in a savings account earning $2 %$ simple interest per year, then at the end of the year we have

1000 (1 + 0.02) = $ 1020.

Compound interest is paid multiple times per year, depending on the compounding period. Therefore, if the bank compounds the interest every $6$ months, it credits half of the year’s interest to the account after $6$ months. During the second half of the year, the account earns interest not only on the initial $$ 1000,$ but also on the interest earned during the first half of the year. Mathematically speaking, at the end of the year, we have

1000 (1 + \frac{0.02}{2})^{2} = $ 1020.10.

Similarly, if the interest is compounded every $4$ months, we have

1000 (1 + \frac{0.02}{3})^{3} = $ 1020.13,

and if the interest is compounded daily $(365$ times per year), we have $$ 1020.20.$ If we extend this concept, so that the interest is compounded continuously, after $t$ years we have

1000 n \to \infty lim (1 + \frac{0.02}{n})^{n t} .

Now let’s manipulate this expression so that we have an exponential growth function. Recall that the number $e$ can be expressed as a limit:

e = m \to \infty lim (1 + \frac{1}{m})^{m} .

Based on this, we want the expression inside the parentheses to have the form $(1 + 1 / m) .$ Let $n = 0.02 m .$ Note that as $n \to \infty,$ $m \to \infty$ as well. Then we get

1000 n \to \infty lim (1 + \frac{0.02}{n})^{n t} = 1000 m \to \infty lim (1 + \frac{0.02}{0.02 m})^{0.02 m t} = 1000 [m \to \infty lim (1 + \frac{1}{m})^{m}]^{0.02 t} .

We recognize the limit inside the brackets as the number $e .$ So, the balance in our bank account after $t$ years is given by $1000 e^{0.02 t} .$ Generalizing this concept, we see that if a bank account with an initial balance of $$ P$ earns interest at a rate of $r %,$ compounded continuously, then the balance of the account after $t$ years is

Balance = P e^{r t} .

If a quantity grows exponentially, the time it takes for the quantity to double remains constant. In other words, it takes the same amount of time for a population of bacteria to grow from $100$ to $200$ bacteria as it does to grow from $10, 000$ to $20, 000$ bacteria. This time is called the doubling time. To calculate the doubling time, we want to know when the quantity reaches twice its original size. So we have

2 y_{0} 2 ln 2 t = = = = y_{0} e^{k t} e^{k t} k t \frac{ln 2}{k} .

Exponential Decay Model

Exponential functions can also be used to model populations that shrink (from disease, for example), or chemical compounds that break down over time. We say that such systems exhibit exponential decay, rather than exponential growth. The model is nearly the same, except there is a negative sign in the exponent. Thus, for some positive constant $k,$ we have $y = y_{0} e^{- k t} .$

As with exponential growth, there is a differential equation associated with exponential decay. We have

y^{'} = - k y_{0} e^{- k t} = - k y .

The following figure shows a graph of a representative exponential decay function.

This figure is a graph in the first quadrant. It is a decreasing exponential curve. It begins on the y-axis at 2000 and decreases towards the t-axis. — Figure 6.80An example of exponential decay.

Let’s look at a physical application of exponential decay. Newton’s law of cooling says that an object cools at a rate proportional to the difference between the temperature of the object and the temperature of the surroundings. In other words, if $T$ represents the temperature of the object and $T_{a}$ represents the ambient temperature in a room, then

T^{'} = - k (T - T_{a}) .

Note that this is not quite the right model for exponential decay. We want the derivative to be proportional to the function, and this expression has the additional $T_{a}$ term. Fortunately, we can make a change of variables that resolves this issue. Let $y (t) = T (t) - T_{a} .$ Then $y^{'} (t) = T^{'} (t) - 0 = T^{'} (t),$ and our equation becomes

y^{'} = - k y .

From our previous work, we know this relationship between y and its derivative leads to exponential decay. Thus,

y = y_{0} e^{- k t},

and we see that

T - T_{a} T = = (T_{0} - T_{a}) e^{- k t} (T_{0} - T_{a}) e^{- k t} + T_{a}

where $T_{0}$ represents the initial temperature. Let’s apply this formula in the following example.

Just as systems exhibiting exponential growth have a constant doubling time, systems exhibiting exponential decay have a constant half-life. To calculate the half-life, we want to know when the quantity reaches half its original size. Therefore, we have

\frac{y _{0}}{2} \frac{1}{2} - ln 2 t = = = = y_{0} e^{- k t} e^{- k t} - k t \frac{ln 2}{k} .

Note: This is the same expression we came up with for doubling time.

Key Concepts

Exponential growth and exponential decay are two of the most common applications of exponential functions.
Systems that exhibit exponential growth follow a model of the form $y = y_{0} e^{k t} .$
In exponential growth, the rate of growth is proportional to the quantity present. In other words, $y^{'} = k y .$