$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

This section discusses the two main modeling uses of exponentials; exponential growth, and exponential decay.

You can watch a video on this section!
_

#### Exponentials in models

Exponential functions have a very specific purpose in models. Exponentials are used when growth/decay rate of something changes depending on how much of it you have. For example, a population of animals will increase faster the more animals you have. After all, if you have more animals, they can produce more baby animals, which thus increases the population faster. Similarly, the more debt you have, the more interest it accrues, so debt increases “exponentially”.

Often people use the term “increases exponentially” to mean that something grows fast. However, things can grow fast and not be exponential, and things can grow slowly but grow exponentially. Take the situation $10x^2$ (which is a polynomial, and thus not exponential) and $1.1^x$ (which is an exponential). Let’s compare the values of these two things for $x=1$ to $x=6$.

 $x$-values $10x^2$ $1.1^x$ $1$ $10$ $1.1$ $2$ $40$ $1.21$ $3$ $90$ $1.331$ $4$ $160$ $1.4641$ $5$ $250$ $1.61051$ $6$ $360$ $1.771561$

As we can see, over the span of $x=1$ to $x=6$, the polynomial expression has gone from $10$ to $360$ whereas the exponential expression only went from $1.1$ to just over $1.77$. But we can also see that, as $x$ got bigger (and thus $1.1^x$ got bigger), increasing $x$ by 1 kept increasing the exponential value by more than it increased before. The jump from $x=1$ to $x=2$ was $0.11$, but the jump from $x=5$ to $x=6$ was an increase of more than $0.16$... granted not exactly amazing, but that’s a relative increase of over 50%!

Despite what it looks like above however, eventually exponential growth will always outpace polynomial growth... but it can take a very long time to start doing so. Nonetheless it is important that growth speed isn’t what makes something “grow exponentially”, rather it is the fact that the growth speed depends explicitly on how much of whatever is growing is present.

Lets see a few more examples that are, perhaps, more relevant to you.

Buried in the above example is an explanation of the compound interest formula. As with almost everything in this course, this formula seems mysterious at first, but is actually fairly straight forward once you know what you are looking at. The compound interest formula is as follows: If you have a principle value of $P$, an annual interest rate of $r$ (as a decimal) and it is compounded $N$ times a year, then the new principle value $P_n$ after $t$ years is: I realize that probably looks like another alphabet soup equation, but let’s go through it carefully.

##### Exponential Growth

Compound interest is an example of exponential growth. First we will work through a general example, and then we will look at specific application.

As we have seen above, we can build an exceptionally generic “exponential growth/decay equation.” Specifically, given a growth/decay multiplier $r$ and initial population/value $P$, then after a number of iterations $N$ the population is: In the above equation, the growth/decay multiplier $r$ is often the hardest part to understand. One should thing of $r$ as being the multiple of the population that changes (either increase or decrease) each iteration. Let’s consider several examples.

The formulas that you have probably seen before can be explained in this same general context as well. Consider the classic “compound interest formula” from before; the $P_N = P\left (1 + \frac {r}{N}\right )^{t \cdot N}$. If we actually interpret what these values are, we can see that $P$ is the initial value/population in both formulas, and in both formulas it has starts with $1 +$ in the parentheses. The first difference comes from the $r$ in the general formula versus $\frac {r}{N}$ in the compound-interest formula.

In the general formula the $r$ stands for the growth/decay multiple; how much is added/subtracted each iteration. In the compound interest case, this is the annual percentage rate, divided by the number of times a year interest is compounded... which in the compound interest formula is exactly $\frac {r}{N}$. So, in fact, these two values are equal.

The other place the two formulas are different are the exponents. In the general case it is just $N$, for the number of iterations that are applied. In the compound interest case, this is going to be the number of years we are spanning, times the number of times a year interest is applied... which again is exactly the $t \cdot N$ from the compound interest formula.

So one can see that these two formulas are exactly the same. Which may lead one to wonder why you should learn a “new” formula if you already remember the compound interest formula. To answer that question though, let’s have a look at half-life next.

The half-life formula that is typically taught is $A_0\left (\frac {1}{2}\right )^{\frac {t}{t_0}}$, or the even more obscure (and pointlessly convoluted) $Pe^{kt}$. In the second form $k$ is a “catch-all magic constant” that “fixes” everything to make it work, but it is incredibly difficult to see how or why the equation is working... which forces students to just memorize the formula and makes it incredibly difficult to use. The first example is at least a bit more enlightening, but fairly unwieldy. We will compare the “General Growth/Decay” equation to the first one to see what is happening.

In the general form $P$ is multiplied against everything and is the initial value, in the “classic half-life formula” the same initial value is $A_0$. In the general formula the quantity inside the parentheses would be $(1 + r)$ for the growth/decay multiplier $r$. As we saw in one of the examples above, the decay multiplier for half-life is $-0.5$, which means the base of the exponent in the general equation will simplify to $\frac {1}{2}$, which is the same base as in the classic formula. Finally, the number of iterations that occur ($N$ in the general formula) is equal to the years that elapsed divided by the half-life in years. In the classic formula that is represented by the elapsed time $t$ and the half-life time $t_0$. Again these things are a bit obscure (although not so bad as the $Pe^{kt}$ formula), but again we see that the general formula and the classic half-life formula result in the same formula in the half-life case.

If we continue on with each of the applications, we would see that the general equation (when one correctly determines $r$) will always be equivalent to the various different dedicated “special formulas” that are normally given for each specific case. Thus one could memorize all the various special-case formulas; compound interest, half-life, population increases, etc. Or one could learn the general formula... which even if you forget, can be figured out again by doing several iterations of the growth multiplier and re-discovering the pattern, which is how we originally deduced it back in the “General Exponential Growth Example” earlier in this topic.