WEBVTT 00:00:05.701 --> 00:00:09.818 Today we're going to start talking about exponential functions. 00:00:09.818 --> 00:00:12.958 So, in order to introduce this, I have a short video. 00:00:14.195 --> 00:00:16.769 ♪ (music) ♪ 00:00:16.769 --> 00:00:21.739 - This is an old story, but it reminds us of the surprises we can get 00:00:21.739 --> 00:00:27.717 when even a small number like two is multiplied by itself many times. 00:00:27.717 --> 00:00:33.210 King Shahram of India was so pleased when his Grand Vizier Sissa Ben Dahir 00:00:33.210 --> 00:00:39.150 presented him with the game of chess that he asked Ben Dahir to name his own reward. 00:00:39.150 --> 00:00:43.979 The request was so modest that the happy king immediately complied. 00:00:44.515 --> 00:00:49.561 What the grand vizier had asked was this, that one grain of wheat be placed on 00:00:49.561 --> 00:00:53.576 the first square of the chess board, two grains on the second square, 00:00:53.576 --> 00:00:59.189 four on the third, eight on the fourth, 16 on the fifth square, and so on. 00:00:59.189 --> 00:01:05.009 Doubling the amount of wheat on each succeeding square until all 64 squares 00:01:05.009 --> 00:01:06.422 were accounted for. 00:01:06.422 --> 00:01:09.558 When the king's steward had gotten to the 17th square, 00:01:09.558 --> 00:01:11.842 the table was well filled. 00:01:11.842 --> 00:01:18.017 By the 26th square, the chamber held considerable wheat and a nervous king 00:01:18.017 --> 00:01:20.739 ordered the steward to speed up the count. 00:01:20.739 --> 00:01:26.047 When 42 squares were accounted for, the palace itself was swamped. 00:01:26.047 --> 00:01:30.417 Now fit to be tied, King Shahram learns from the court mathematician 00:01:30.417 --> 00:01:35.629 that had the process continued, the wheat required would have covered 00:01:35.629 --> 00:01:39.236 all India to a depth of over 50 feet. 00:01:41.095 --> 00:01:46.622 Incidentally, laying this many grains of wheat end to end also does something 00:01:46.622 --> 00:01:48.218 rather spectacular. 00:01:48.218 --> 00:01:53.021 They would stretch from the Earth, beyond the sun, past the orbits 00:01:53.021 --> 00:01:58.505 of the planets, far out across the galaxy, to the star Alpha Centauri, 00:01:58.505 --> 00:02:00.456 four light-years away. 00:02:00.456 --> 00:02:05.436 They would then stretch back to Earth, back to Alpha Centauri, 00:02:05.436 --> 00:02:07.851 and back to the Earth again. 00:02:07.851 --> 00:02:11.851 ♪ (music) ♪ 00:02:13.946 --> 00:02:17.228 - So, what was going on in that video? 00:02:17.228 --> 00:02:22.211 You can see that, rather than going up by the same amount for each square 00:02:22.211 --> 00:02:27.761 that they advanced on the chess board, that they went up by a doubled amount. 00:02:28.229 --> 00:02:31.902 So, that's a different type of function than we've been looking at. 00:02:31.902 --> 00:02:36.042 Rather than going up by one and then another one and another one, 00:02:36.042 --> 00:02:39.424 they went up by one and then they doubled and doubled again 00:02:39.424 --> 00:02:40.662 and doubled again. 00:02:40.662 --> 00:02:44.321 So, we cannot represent that with the type of linear function that 00:02:44.321 --> 00:02:46.090 we've been looking at before. 00:02:46.840 --> 00:02:51.469 In a linear function, such as this one, if I went up by one square, 00:02:51.469 --> 00:02:54.568 I would increase here by 1.5. 00:02:54.568 --> 00:02:59.889 So, if you think about how this plays out in a table, for each one unit increase 00:02:59.889 --> 00:03:03.986 in t, I am increasing by the same amount. 00:03:04.636 --> 00:03:09.676 Now, if I were to look at the percentage increase, therefore, I would actually 00:03:09.676 --> 00:03:14.978 indicate that I'm changing the percent that I'm going up by each time. 00:03:14.978 --> 00:03:24.527 So, going up by 1.5 from 2 is different than going up by 1.5 from 6.5. 00:03:25.194 --> 00:03:30.894 So, if you can see that I've done these calculations already, going from 2 to 3.5 00:03:30.894 --> 00:03:37.730 is an increase of the amount of 1.5, but it's actually a 75% increase. 00:03:37.730 --> 00:03:42.592 But by the time I get up to these values, going from eight to 9.5, 00:03:42.592 --> 00:03:45.442 that's only a 19% increase. 00:03:45.442 --> 00:03:49.959 So, when I have a linear function, what you'll see is that you have a constant 00:03:49.959 --> 00:03:56.296 amount of increase but you do not have a constant percentage of increase. 00:03:56.803 --> 00:04:00.673 Now, what's going on in an exponential function and what was happening in 00:04:00.673 --> 00:04:05.065 that video, is that for every one unit increase in t, 00:04:05.065 --> 00:04:07.514 we will see a different constant. 00:04:08.165 --> 00:04:13.629 So, in this case, if I use my exponential equation and I go up by one every time, 00:04:13.629 --> 00:04:18.094 you can see that I go up by, first, just one. 00:04:18.094 --> 00:04:26.794 Then I go up by 1.5 and then 2.25 and then 3.36 and then 5.06. 00:04:27.277 --> 00:04:32.327 So, I'm actually not increasing by the same amount each time. 00:04:32.327 --> 00:04:36.092 That's because rather than adding a value each time, 00:04:36.092 --> 00:04:38.823 I'm multiplying an additional time. 00:04:38.823 --> 00:04:45.101 So, here I'm multiplying by 1.5 an additional time each time I increase. 00:04:45.101 --> 00:04:49.613 Well, because I'm multiplying by a constant amount, what that actually 00:04:49.613 --> 00:04:55.007 does end up doing is increasing by a constant percentage. 00:04:55.007 --> 00:04:59.930 So, going from 2 to 3 increases by 50%. 00:04:59.930 --> 00:05:08.387 But also going from 10.13 to 15.19, I'm increasingly by 50%. 00:05:08.988 --> 00:05:12.927 So, in an exponential function, I increase by a constant percentage, 00:05:12.927 --> 00:05:14.527 but differing amounts. 00:05:15.356 --> 00:05:18.465 Let's look at this with what we call a decay function. 00:05:18.465 --> 00:05:22.351 So, this is also an exponential function, but one that's decreasing. 00:05:22.351 --> 00:05:26.905 So, in this case, in my decay function, now you can see that I'm multiplying by 00:05:26.905 --> 00:05:28.958 a decimal repeatedly. 00:05:28.958 --> 00:05:32.019 Well, if I repeatedly multiply by a decimal, I should be getting 00:05:32.019 --> 00:05:33.902 smaller and smaller values. 00:05:33.902 --> 00:05:40.914 So, here, I decreased by one but then I decreased by half, by .25, by .125, 00:05:40.914 --> 00:05:42.298 and so on. 00:05:42.298 --> 00:05:47.368 What that means is is I'm basically slowing down my growth over time, 00:05:47.368 --> 00:05:51.799 but I'm decreasing still by a constant percentage. 00:05:51.799 --> 00:05:55.520 So, here, I decreased by 50% each time. 00:05:56.222 --> 00:06:00.153 Now, if I just had one that I was multiplying by every time, 00:06:00.153 --> 00:06:03.865 then I would actually have no change in percentage because I would just 00:06:03.865 --> 00:06:07.992 have a constant amount because multiplying by one is the same every time. 00:06:09.158 --> 00:06:11.464 So, what is an exponential function? 00:06:11.464 --> 00:06:15.743 An exponential function can be one that is recognized by a constant 00:06:15.743 --> 00:06:18.768 percentage rate of change. 00:06:18.768 --> 00:06:23.147 Remember, this is as opposed to our linear functions which had a constant 00:06:23.147 --> 00:06:25.230 amount of change. 00:06:25.230 --> 00:06:30.827 So, rather than changing by one each time or by five each time, 00:06:30.827 --> 00:06:35.988 maybe I'm changing by 5% each time. And those are different. 00:06:37.122 --> 00:06:41.904 An exponential growth model is one that has a positive percent rate of change, 00:06:41.904 --> 00:06:46.594 while a decay model is one that has a negative percent rate of change. 00:06:46.594 --> 00:06:51.243 So, a exponential growth model will be increasing over time, 00:06:51.243 --> 00:06:54.400 and a decay model will be decreasing over time. 00:06:55.596 --> 00:06:57.970 So, here's the general form of our equation. 00:06:57.970 --> 00:07:01.587 You're going to see I use t in my general form and that's because, essentially, 00:07:01.587 --> 00:07:05.577 we'll always be looking at models with time as our independent variable. 00:07:06.162 --> 00:07:09.588 Here we have some new parameters we're looking at, as well. 00:07:09.588 --> 00:07:12.918 Because we're talking about time, I'm going to call a, 00:07:12.918 --> 00:07:17.660 which is my y-intercept, my initial value. And that's because with time, 00:07:17.660 --> 00:07:22.602 when time is zero, I get my y-intercept, so I get a. 00:07:22.602 --> 00:07:25.202 A is therefore my initial value. 00:07:25.775 --> 00:07:30.243 The growth factor is, what am I multiplying by each time? 00:07:30.243 --> 00:07:33.314 And when I know what I'm multiplying by each time, 00:07:33.314 --> 00:07:35.612 I can get to what is more important to me, 00:07:35.612 --> 00:07:39.674 which is my growth rate, or what is the percentage that I'm increasing 00:07:39.674 --> 00:07:41.905 or decreasing by each time. 00:07:41.905 --> 00:07:46.770 So, my growth rate is just related to that growth factor and can be calculated 00:07:46.770 --> 00:07:47.958 from it. 00:07:47.958 --> 00:07:50.197 So, if I have my growth factor, 00:07:50.197 --> 00:07:54.473 I can just subtract one and I'll get my growth rate. 00:07:55.704 --> 00:07:58.498 Now, this r is not the same as our correlation r, 00:07:58.498 --> 00:08:00.086 so don't let that fool you. 00:08:01.552 --> 00:08:02.930 So, here's our definitions. 00:08:02.930 --> 00:08:07.646 Our growth factor, or b, is the amount that we multiply by each y-value 00:08:07.646 --> 00:08:09.396 to get the new y-value. 00:08:10.483 --> 00:08:15.640 Our growth rate is the percentage increase or decrease that we get 00:08:15.640 --> 00:08:18.760 for each one unit increase in x. 00:08:18.760 --> 00:08:22.567 And these are the two things that we can get from our b. 00:08:22.567 --> 00:08:25.968 The one that we're going to be interested in interpreting primarily is 00:08:25.968 --> 00:08:29.480 our growth rate when we do our parameter estimates. 00:08:30.865 --> 00:08:33.440 So, here are three examples of exponential models. 00:08:33.440 --> 00:08:37.407 I have a growth, a stagnant, and a decay model. 00:08:37.407 --> 00:08:42.516 One thing to note is that none of these have negative values possible. 00:08:42.516 --> 00:08:47.591 In an exponential function, we can actually never cross that axis. 00:08:47.591 --> 00:08:50.028 So, we can never have a negative value. 00:08:50.028 --> 00:08:54.047 Notice that these are all the same concavity, so they are all trending 00:08:54.047 --> 00:08:58.197 that same direction because we cannot cross this axis. 00:08:58.197 --> 00:09:02.248 And, in fact, in a decay model, we will continuously get closer 00:09:02.248 --> 00:09:06.839 and closer and closer to zero, but we will never actually get to zero. 00:09:08.864 --> 00:09:11.357 So, let's work with some general functions. 00:09:11.357 --> 00:09:15.575 I want to know what the growth factor would be if I said that water usage is 00:09:15.575 --> 00:09:18.301 increasing by 5% per year. 00:09:18.301 --> 00:09:20.398 So, what am I giving you here? 00:09:20.398 --> 00:09:24.619 Well, I'm giving you the growth rate, that percentage of change. 00:09:24.619 --> 00:09:29.951 And if I want to convert this to b, remember that b equals 1 plus r. 00:09:29.951 --> 00:09:32.289 And that is r written as a decimal. 00:09:32.289 --> 00:09:36.353 So, if I write my r as a decimal, I would have .05, 00:09:36.353 --> 00:09:41.760 which tells me that my b here, my growth factor, would be 1.05. 00:09:42.505 --> 00:09:43.871 Now, you try. 00:10:00.058 --> 00:10:02.929 So, in this case, I've given you a decay model. 00:10:03.457 --> 00:10:08.920 If I have decay, where I know that it's shrinking by 78%, 00:10:08.920 --> 00:10:12.092 that actually means that I have a negative r value. 00:10:12.092 --> 00:10:15.807 So, here I would end up with a b of 0.22. 00:10:16.333 --> 00:10:21.116 When I have a decay model, my r can be negative. 00:10:21.116 --> 00:10:26.136 So, I will expect to see a negative rate, because I am changing 00:10:26.136 --> 00:10:28.503 in a downward trajectory. 00:10:28.503 --> 00:10:33.730 Now, remember I can never actually have a negative value in my chart, 00:10:33.730 --> 00:10:39.683 so that means that b cannot be negative, even though r can. 00:10:39.683 --> 00:10:41.844 Well, why can't b be negative? 00:10:41.844 --> 00:10:46.563 Well, think about if I had a chart here and I had a negative b value. 00:10:46.563 --> 00:10:49.695 Well, when I add b to the first, I'd have a negative result, 00:10:49.695 --> 00:10:52.220 but then when I square b, I get a positive, 00:10:52.220 --> 00:10:54.303 and when I cubed it, I'd have a negative, 00:10:54.303 --> 00:10:57.149 and then when I had to the fourth, I'd be up here. 00:10:57.149 --> 00:10:59.877 So, I'd end up with some kind of function like that. 00:10:59.877 --> 00:11:05.157 Well, that is not an exponential model. So, I cannot have a negative b. 00:11:05.157 --> 00:11:10.326 When I have a negative r, for a decay model, what that will do is 00:11:10.326 --> 00:11:15.273 give me a b value that is between zero and one. 00:11:16.024 --> 00:11:22.546 So, I will have a b value that is less than one but more than zero 00:11:22.546 --> 00:11:24.315 when I have a decay model. 00:11:24.315 --> 00:11:28.789 If I have a b value over one, then I have a growth model. 00:11:31.917 --> 00:11:35.160 So, here is another example of an exponential function. 00:11:35.160 --> 00:11:37.580 And I've gone ahead and drawn it out. 00:11:37.580 --> 00:11:42.955 You can see here that from my equation, I have 34.3 as my initial value. 00:11:42.955 --> 00:11:46.808 So, if I were to sketch this or put this into technology and get it, 00:11:46.808 --> 00:11:50.104 you would see that that's my initial value, my starting point. 00:11:50.821 --> 00:11:55.209 My growth factor for this problem would be 0.63, 00:11:55.209 --> 00:11:59.214 indicating I have a decay model because that b is less than one. 00:12:00.181 --> 00:12:05.567 If I calculate my growth rate, then, I can have my b, 0.63, 00:12:05.567 --> 00:12:07.559 as equal to 1 plus r. 00:12:07.559 --> 00:12:12.527 And when I subtract one, I get a decay of negative 37%. 00:12:13.724 --> 00:12:17.502 So, now you try it. Identify the growth rate in this equation. 00:12:35.253 --> 00:12:41.961 In this equation, you can see that our growth rate is very small, at 0.2%. 00:12:41.961 --> 00:12:44.807 What else do you notice about this chart? 00:12:44.807 --> 00:12:47.790 Well, you should notice that it kind of looks like a line, 00:12:47.790 --> 00:12:51.584 and not like that exponential curve that we're so used to seeing. 00:12:51.584 --> 00:12:56.246 When you have a very small growth rate, one that is very close to zero, 00:12:56.246 --> 00:13:00.264 you will actually see a model that doesn't have a very striking curve to it. 00:13:00.264 --> 00:13:02.029 It'll almost like linear. 00:13:02.029 --> 00:13:06.103 Now, if I zoomed this out to a thousand points, you'd probably begin 00:13:06.103 --> 00:13:08.071 to be able to determine the curve. 00:13:08.071 --> 00:13:12.800 But for very small r values, we'll have a pretty flat curve. 00:13:12.800 --> 00:13:17.298 The larger your r value is, the more significant your curve is, 00:13:17.298 --> 00:13:19.167 either going up or down. 00:13:19.167 --> 00:13:22.379 So, if you were comparing two models drawn on the same chart, 00:13:22.379 --> 00:13:28.429 you could see that the more curving one should have a higher absolute value of r, 00:13:28.429 --> 00:13:31.432 because it's changing at a faster rate. 00:13:31.432 --> 00:13:34.465 Alright. As always, bring us your questions to class.