Here's a riddle: I'm offering you a job. It's a nasty one, like mucking out chicken coops (big ones, hundreds of chickens – and "mucking out" means shoveling ammonia-smelly poop) 12 hours a day, for 30 days — no lunch breaks. At the end of the first day, I'll pay you one cent. One cent. At the end of the second, I'll double that and give you two cents, and so on for 30 days.
Think it through. On the seventh day, I'll pay you 64 cents, and you will have done 84 total hours of wretched labor for a grand total of $1.27. Congratulations on your great decision and awesome survival instinct. Ready to quit yet?
Skip ahead to day 14. At the end of the day, I'll pay you $81.92. Now that's more like it. And your cumulative total is now up to $163.83 ... better, and remember that tomorrow your wage will double to $163.84 for one day's work. Would you take the job?
In the table are the daily payments and cumulative earnings for just the last 10 days of your employment in the coops.
On the last day alone, I'd have to pay you over 5 million dollars! All-told, you'd earn more than $10.7 million! Take the job!
That's exponential growth, and you'll see it in many different phenomena, like financial growth (compound interest) and population growth. Its cousin, exponential decay is used to model things like radioactive decay, the decrease in drug concentration in the bloodstream over time and the depreciation (loss of value) of property over time. And there are many other examples.
Knowing how to manipulate and solve exponential functions and equations is a very important part of your mathematics toolkit.
The most important feature of an exponential function is that the independent variable is in the exponent of some base, usually an agreed-upon (but constant) base, like 2, 10 or the special number "e".
The graphs of exponential functions are characterized by a period of relatively slow growth followed by much more rapid growth, overtaking polynomial functions of even the highest degree.
In an exponential function, the variable is in the exponent. The base is constant.
In this illustration, the functions f(x) = x2 (black) and f(x) = 2x (orange) are plotted together. Although these curves are very similar over a small range (they are equal at x = 2 and x = 4), they are very different outside of it. Moving from left to right, exponential growth always outpaces polynomial growth, if x gets large enough.
Notice the difference in the two functions when x < 0. Recall from the laws of exponents that a-1 means to take the reciprocal of a. So as x grows in the negative direction, the function asymptotically approaches zero: ½, ⅓, ¼, ...
The value of an exponential functions is never less than zero unless the function is specifically multiplied by a negative number - a vertical scaling parameter. Remember that a negative exponent does not make the number negative: a-1 = 1/a.
Exponential graphs all have a common shape, and sketching them is pretty easy. Check out the graph of f(x) = 2x on the right.
Without any transformations (covered below), every exponential function f(x) = bx, where b is a constant base (2 in this case), passes through y = 1, because anything raised to the zero power is 1: b0 = 1.
On the right we have the rapid exponential growth. On the left, as the exponent approaches -∞, the function approaches zero. Remember that a negative exponent like b-1 doesn't mean a negative number; it means 1/b, so as x → ∞, the value of the function approaches zero, thus we have asymptotic behavior on the left.
All of the usual transformations of functions apply to exponential functions (below). One of the most important parts of an exponential function is the base. While any exponential phenomenon can be modeled using any base at all (by adjusting the rest of the transformation parameters to fit the data), we tend to stick to a small set of agreed-upon bases, like 2, 10 and e. There will be much more to say about bases, especially e, later.
Remember that all of the transformations work in the same way on any function. Things are always more the same in algebra than different.
Consider the growth of money deposited in a bank account earning some annual interest rate, r%. (We use the letter r rather than a number so that our result will be good for any number.) One of the first challenges in solving problems with exponential functions is learning the new language. "Annual interest" means that every year, what's in the bank account is multiplied by the interest rate, r, and that amount is added to what's already there (the principal).
For every year after the initial deposit, this means that you're calculating this year's interest based not only on the initial amount deposited, but on last year's interest as well. That's called compound growth or compound interest.
Try to work your way through the table below. The algebra is pretty straightforward - just finding finding common factors between terms.
If an initial amount, Ao is deposited in a bank account earning r% interest, and not touched afterward, interest is earned each year. But after the first year interest is also earned on previous interest – that's compound interest
|0||Ao (initial investment)||Ao(1 + r)0|
|1||Ao + Aor = Ao(1 + r)||Ao(1 + r)1|
|2||Ao(1 + r) + r·Ao(1 + r) = Ao(1 + r)(1 + r) = Ao(1 + r)2||Ao(1 + r)2|
|3||Ao(1 + r)2 + r·Ao(1 + r)2 = Ao(1 + r)3||Ao(1 + r)3|
|. . .|
|n years||Ao(1 + r)n (the pattern continues)||Ao(1 + r)n|
For a system that grows at a rate, r, added periodically over t periods, the amount A(t) at any time t is
$100 is deposited in a savings account that pays 5% interest per year. If no additional money is added, how much money will be in the account at the end of the 10th year?
The phrase "annual interest" is a clue that this is a simple growth problem. Later we'll encounter compound growth – more on that later. The best approach is to start with the simple growth formula, and identify the things we know:
Now it's just a matter of completing the calculation to find A(10), the amount in the account after t = 10 years.
That's it. Just remember to convert the percent interest to decimal form: 5% = 5/100ths, or 0.05.
If an initial investment of $1000 earned $185 in interest after 8 years, what was the average rate of interest over that time?
Take the same approach for this problem; write the simple growth equation and identify what we know. The one trick here is that if we earn $185 in interest on a $1000 investment, the total amount in the account after 8 years is A(8) = $1185, the sum of the initial investment and the interest.
Now the variable for which we must solve is inside a binomial raised to the 8th power:
To isolate r, we have to get rid of the 8th power, and we do that by raising both sides of the equation to the ⅛ power (remember the laws of exponents).
So an interest (growth) rate of 2.14% per year would do the trick.
At the right temperature (37˚C, the temperature of your gut) and with enough nutrients, E. coli populations will double every 20 minutes. Starting with an initial population of one bacterium, with time, t, in units of 20 min., How many bacteria will be present in 8 hours?
This problem has a couple of tricks you'll need to know. First, we have the word "double", which has a definite meaning in the simple growth equation, and second, we'll have to use "doubling" to calculate the growth rate before we can calculate the final population of E. coli. Finally, we're going to let 1 unit of time equal 20 minutes in this problem. It's not necessary at all, but it does simplify the arithmetic.
Start with the simple growth equation (I've used P for "population.")
Now we can solve for the rate. Not surprisingly, using this time scale, it's 100% over 20 minutes – the population doubles.
Now we can use r to construct our growth formula:
E. coli bacteria magnified 15,000 times
Keeping with our bank-account example, what if, instead of adding the interest to the balance at the end of each year, we added it in chunks periodically throughout the year. In banking, that's called compounding the interest. The idea will extend far beyond banking, so it's worth studying.
For example, let's say we have a bank account that pays 4% annual interest, and we do what banks call "compounding quarterly." That is, we split the 4% up into four equal pieces (1%) and pay that 1% in interest every three months. This begs two questions: (1) is there an easy way to do it, and (2) is it to our advantage?
To develop a formula for compounding, we can start with the simple growth formula and just modify it:
For a system that grows at a rate, r, compounded n times in each of t periods, the amount A(t) at any time t is
Now, of what value is this to a bank customer (and of what value is compounding in the mathematics of exponential growth)? Take a look at the figure below to see how.
Let's say we deposit $1000 into an account that pays 4% interest per year, and we wait 10 years. According to the simple interest formula, we'd end up with about $1480. The graph shows that there is about a $9 advantage in switching to compounding quarterly (4 times a year).
Notice that as we increase the frequency of compounding, the amount of money gained in compounding more often (the "marginal gain") is measurable, but less. The dashed line is an asymptote.
That asymptote represents the limit as compounding becomes continuous, and that is the essence of the next section.
Note that the horizontal axis of this graph has no real scale; the data are just spread evenly across it to make it easy to read.
If you take a look at the compound interest graph above, you'll notice that there is a limit to the amount of money that can be made by increasing the number of compounding steps. The word "steps" here is the key.
With compound growth, we wait a while, then add interest, then wait a while more and add interest, and so on. It's a stair-step process. But what if we took steps so frequently that the process was more like a ramp than a stairway?
That would be continuous growth, and the problem we'd be trying to solve is this one:
It's a tricky process to go through, but the answer to this question turns out to be very elegant. This equation, in the limit where n → ∞, converts to
where e is a very special number, the base of all continuously-growing exponential functions, e = 2.7182818 ... It's a transcendental number, like π, and you'll probably use it a lot, so get used to it.
We'll use this new function for continuous growth to model many types of exponential growth situations, especially populations. When populations become large enough, and births and deaths occur more-or-less simultaneously, the growth is approximately continuous, and the P(t) = Poert is a good model.
If you want to know more about where e comes from, go here:
For a system that grows continuously at a rate, r, the amount A(t) at any time t is
Let's say you own a pizza restaurant and you buy a new oven. Your business is now worth more and will therefore be taxed accordingly. But the government (you and me) recognizes that the value of the oven will decrease over time, and provides a way to compensate you for that. It's called depreciation. Such an oven might depreciate by 5% per year. Calculate the taxable value of an oven originally purchased for $10,000 after 10 years with a 5% depreciation rate.
This is really just a simple growth problem, except that the growth is negative, so instead of the function A(t) = Ao(1 + r)t, we use the depreciation form:
We start by writing the depreciation equation and what we know:
Then it's just a matter of finishing the calculation:
After ten years the pizza oven is worth a little more than half of its original value.
Compare the amount of money in a bank account after ten years at an interest rate of 4% per year, when that interest is compounded annually and quarterly. Assume that $10,000 is initially deposited and that no more deposits are made.
The first part of the problem is a simple growth problem, so we'll write that function and what label what we know:
Now it's easy to solve:
The second part is a compound-growth problem with n = 4 (quarterly):
And solving that gives us:
The compound interest option gives us more money at the end of ten years. Cool.
Compare the interest earned after ten years in a savings account started with $1000.00 (no additional deposits) if the 5% interest is compounded quarterly (4 times/year), weekly (52 times/year), daily and continuously.
Continuously – here let's pause to write our function and plug in values more carefully:
and the result is:
Notice that the final amounts increase, but by less and less as we add more compounding steps, and that the difference between daily compounding and continuous growth is less than a cent, but the continuous case is still just a little greater – check it out for yourself.
A population of seabirds has grown reliably at a rate of 4% per year. If the population today is 16,000 birds, what was it 5 years ago?
This is a continuous growth problem because it says so – watch for that! Notice also that we never actually know anything about the actual bird population. We just have changes, but that will be fine.
We begin by writing the continuous-growth function and sketching in things we know
Now let's rearrange the function before plugging in numbers (always a good idea):
Now plug in what we know
and the result is
Continuous growth models like this are almost always used to model populations.
In problems that involve exponential decay, it is conventional to use the half-life measure. Here's how it works. Think about a pile of Iodine-131 (131I). This radioactive isotope decays spontaneously to 131Xe by emission of a beta particle followed by some gamma radiation. It's not possible to tell when a single atom of 131I will decay, but we know that on average for a large enough quantity, one half of the initial amount will have decayed to 131Xe in 8.02 days, the half-life of 131I. Every 8.02 days, half of what remained will have decayed and become 131Xe.
It's not important that you know the chemistry or physics of these problems right now, just that at any given time, what's left will be cut in half (turn to something else) in about 8 days.
Look at the graph to see how it works. It's pretty clear that this is a upside-down exponential graph.
To derive a formula for finding the amount of a radioactive substance still left after some number of half lives, we start with the formula for simple growth:
Now if 1/2 of the sample is removed every time unit, then r = -0.5, so we have
Then we can introduce the half life, which we'll call k, to put the exponent in units of number of half lifes:
It's that exponent that's usually a little tricky. Think of it this way: If the half life is 8 days and we wait for 16 days, so t/k = 16/8 = 2 half lifes. We divide by k to convert the exponent to half lifes.
For exponential decay expressed in half-lifes, the function is:
If the half life of a substance is 100 days, how much of the initial sample will be present after one year?
Notice that we haven't been given an initial amount (Ao) here. That's OK because the answer, expressed as a fraction (percent) of the starting amount, would be the same for any starting amount. For convenience, let's say we started with 1 (gram, Kilogram, whatever). Plugging in k = 100 days, t = 365 days and Ao = 1, we have:
Now it's just a matter of plugging in and solving for A(365).
So after one year, only 7.9% of the isotope will be present. The rest will have decayed to other stuff.
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