Comparing exponential and linear function

Company A is offering ten thousand dollars for the first month and will increase the amount each month by five thousand dollars. Company B is offering five hundred dollars for the first month and will double their payment each month. For which monthly payment will Company B's payment first exceed Company A's payment?

So pause this video and try to work that out.

All right, let's work this out together. So, let me set up a little bit of a table. The first column is month. The second column is how much Company A is going to pay, and then the third column—let's think about how much Company B is going to pay.

Well, they tell us a few things. They say Company A is offering ten thousand dollars for the first month. So, in month one, Company A is offering ten thousand dollars. We'll assume—well, I'll just write the dollars there.

And then Company B is offering five hundred dollars for the first month, five hundred dollars for the first month. But then they tell us Company A is offering—well, we'll increase the amount each month by 5,000. So, month two will be 5,000 more. We'll get to fifteen thousand.

Month three, we'll get to— we'll get to twenty thousand dollars. Month four, we'll get to twenty-five thousand dollars. Month five, I think you get the point. We'll get to thirty thousand dollars. Month six, we'll get to thirty-five thousand dollars.

Month seven, we'll get to forty thousand dollars. Let me scroll down a little bit. Month eight, I'll stop there. Month eight, we will get to forty-five thousand dollars.

Now, let me extend these lines a little bit. Now let's think about what's going to happen with Company B. Company B is offering five hundred for the first month but will double their payment each month. So, the second month is going to be double that—so that's going to be one thousand dollars.

Then we're gonna double that again—two thousand dollars. We're gonna double that again—four thousand dollars. Double that again—eight thousand dollars. Then we double that again—sixteen thousand dollars.

Double that again—thirty-two thousand dollars. Double that—oh, if I skipped one, I went from four thousand to sixteen thousand—four thousand, eight thousand dollars. Then we double it again—sixteen thousand dollars again. Thirty-two— I sound like my two-year-old again.

All right—thirty-two thousand. Then we get to sixty-four thousand. At that point, something interesting happens. Actually, good that I went to the eighth month because every month before the eighth month, Company A's payment was higher.

Until that eighth month, in that eighth month, Company B is going to pay more. So, first we can just answer their question for which monthly payment will Company B's payment first exceed Company A's payment? Well, that is month eight.

Month eight. And there's a broader lesson going on here. You might recognize that the rate at which Company A's payment is increasing is linear. Every month it increases by the same amount—so plus five thousand, plus five thousand.

It increases by five thousand, the same amount. Company B is increasing exponentially. It's increasing by the same factor every time. So, we're multiplying by the same value every time—we're multiplying by two.

We're multiplying by two, multiplying by two. And so there's actually a very interesting thing here that you can make the general statement that an exponential function will want—something that is exponentially increasing will eventually always surpass something that is linearly increasing.

And it doesn't matter what the initial situation is. It also doesn't even matter that rate of exponential increase. It will eventually always pass up something that's increasing linearly.

You could think about that visually if you like. If I were to draw a visual function, a linear function—this is the x-axis, this is the y-axis. A linear function—well, it's going to be described by a line. So, it could look something like this—a linear function is always going to be a line of some slope.

An exponential function, even though it might start a little bit slower, it's eventually—it's eventually going to pass up the linear function. This is going to be the case even if the linear function has a pretty high slope or pretty high starting point.

If it's something like that, and even if the exponential function is starting pretty slow, it will eventually—and even if it's compounding or growing relatively slow—but exponentially, if it's growing two percent or three percent, it still will eventually pass up the linear function, which is pretty cool.