0:00:05.701,0:00:09.818 Today we're going to start talking [br]about exponential functions. 0:00:09.818,0:00:12.958 So, in order to introduce this, [br]I have a short video. 0:00:14.195,0:00:16.769 ♪ (music) ♪ 0:00:16.769,0:00:21.739 - This is an old story, but it reminds[br]us of the surprises we can get 0:00:21.739,0:00:27.717 when even a small number like two[br]is multiplied by itself many times. 0:00:27.717,0:00:33.210 King Shahram of India was so pleased when [br]his Grand Vizier Sissa Ben Dahir 0:00:33.210,0:00:39.150 presented him with the game of chess that[br]he asked Ben Dahir to name his own reward. 0:00:39.150,0:00:43.979 The request was so modest that [br]the happy king immediately complied. 0:00:44.515,0:00:49.561 What the grand vizier had asked was this,[br]that one grain of wheat be placed on 0:00:49.561,0:00:53.576 the first square of the chess board,[br]two grains on the second square, 0:00:53.576,0:00:59.189 four on the third, eight on the fourth, [br]16 on the fifth square, and so on. 0:00:59.189,0:01:05.009 Doubling the amount of wheat on each[br]succeeding square until all 64 squares 0:01:05.009,0:01:06.422 were accounted for. 0:01:06.422,0:01:09.558 When the king's steward had gotten to[br]the 17th square, 0:01:09.558,0:01:11.842 the table was well filled. 0:01:11.842,0:01:18.017 By the 26th square, the chamber held[br]considerable wheat and a nervous king 0:01:18.017,0:01:20.739 ordered the steward to speed up the count. 0:01:20.739,0:01:26.047 When 42 squares were accounted for,[br]the palace itself was swamped. 0:01:26.047,0:01:30.417 Now fit to be tied, King Shahram [br]learns from the court mathematician 0:01:30.417,0:01:35.629 that had the process continued,[br]the wheat required would have covered 0:01:35.629,0:01:39.236 all India to a depth of over 50 feet. 0:01:41.095,0:01:46.622 Incidentally, laying this many grains of[br]wheat end to end also does something 0:01:46.622,0:01:48.218 rather spectacular. 0:01:48.218,0:01:53.021 They would stretch from the Earth,[br]beyond the sun, past the orbits 0:01:53.021,0:01:58.505 of the planets, far out across the galaxy,[br]to the star Alpha Centauri, 0:01:58.505,0:02:00.456 four light-years away. 0:02:00.456,0:02:05.436 They would then stretch back to Earth,[br]back to Alpha Centauri, 0:02:05.436,0:02:07.851 and back to the Earth again. 0:02:07.851,0:02:11.851 ♪ (music) ♪ 0:02:13.946,0:02:17.228 - So, what was going on in that video? 0:02:17.228,0:02:22.211 You can see that, rather than going up[br]by the same amount for each square 0:02:22.211,0:02:27.761 that they advanced on the chess board,[br]that they went up by a doubled amount. 0:02:28.229,0:02:31.902 So, that's a different type of function[br]than we've been looking at. 0:02:31.902,0:02:36.042 Rather than going up by one and then [br]another one and another one, 0:02:36.042,0:02:39.424 they went up by one and then [br]they doubled and doubled again 0:02:39.424,0:02:40.662 and doubled again. 0:02:40.662,0:02:44.321 So, we cannot represent that with[br]the type of linear function that 0:02:44.321,0:02:46.090 we've been looking at before. 0:02:46.840,0:02:51.469 In a linear function, such as this one, [br]if I went up by one square, 0:02:51.469,0:02:54.568 I would increase here by 1.5. 0:02:54.568,0:02:59.889 So, if you think about how this plays out[br]in a table, for each one unit increase 0:02:59.889,0:03:03.986 in t, I am increasing by the same amount. 0:03:04.636,0:03:09.676 Now, if I were to look at the percentage [br]increase, therefore, I would actually 0:03:09.676,0:03:14.978 indicate that I'm changing the percent[br]that I'm going up by each time. 0:03:14.978,0:03:24.527 So, going up by 1.5 from 2 is different[br]than going up by 1.5 from 6.5. 0:03:25.194,0:03:30.894 So, if you can see that I've done these [br]calculations already, going from 2 to 3.5 0:03:30.894,0:03:37.730 is an increase of the amount of 1.5,[br]but it's actually a 75% increase. 0:03:37.730,0:03:42.592 But by the time I get up to these values, [br]going from eight to 9.5, 0:03:42.592,0:03:45.442 that's only a 19% increase. 0:03:45.442,0:03:49.959 So, when I have a linear function, what[br]you'll see is that you have a constant 0:03:49.959,0:03:56.296 amount of increase but you do not have[br]a constant percentage of increase. 0:03:56.803,0:04:00.673 Now, what's going on in an exponential[br]function and what was happening in 0:04:00.673,0:04:05.065 that video, is that for every one unit [br]increase in t, 0:04:05.065,0:04:07.514 we will see a different constant. 0:04:08.165,0:04:13.629 So, in this case, if I use my exponential[br]equation and I go up by one every time, 0:04:13.629,0:04:18.094 you can see that I go up [br]by, first, just one. 0:04:18.094,0:04:26.794 Then I go up by 1.5 and then 2.25 [br]and then 3.36 and then 5.06. 0:04:27.277,0:04:32.327 So, I'm actually not increasing by[br]the same amount each time. 0:04:32.327,0:04:36.092 That's because rather than adding [br]a value each time, 0:04:36.092,0:04:38.823 I'm multiplying an additional time. 0:04:38.823,0:04:45.101 So, here I'm multiplying by 1.5 [br]an additional time each time I increase. 0:04:45.101,0:04:49.613 Well, because I'm multiplying by [br]a constant amount, what that actually 0:04:49.613,0:04:55.007 does end up doing is increasing by [br]a constant percentage. 0:04:55.007,0:04:59.930 So, going from 2 to 3 increases by 50%. 0:04:59.930,0:05:08.387 But also going from 10.13 to 15.19,[br]I'm increasingly by 50%. 0:05:08.988,0:05:12.927 So, in an exponential function, [br]I increase by a constant percentage, 0:05:12.927,0:05:14.527 but differing amounts. 0:05:15.356,0:05:18.465 Let's look at this with what [br]we call a decay function. 0:05:18.465,0:05:22.351 So, this is also an exponential function,[br]but one that's decreasing. 0:05:22.351,0:05:26.905 So, in this case, in my decay function,[br]now you can see that I'm multiplying by 0:05:26.905,0:05:28.958 a decimal repeatedly. 0:05:28.958,0:05:32.019 Well, if I repeatedly multiply by [br]a decimal, I should be getting 0:05:32.019,0:05:33.902 smaller and smaller values. 0:05:33.902,0:05:40.914 So, here, I decreased by one but then[br]I decreased by half, by .25, by .125, 0:05:40.914,0:05:42.298 and so on. 0:05:42.298,0:05:47.368 What that means is is I'm basically [br]slowing down my growth over time, 0:05:47.368,0:05:51.799 but I'm decreasing still by [br]a constant percentage. 0:05:51.799,0:05:55.520 So, here, I decreased by 50% each time. 0:05:56.222,0:06:00.153 Now, if I just had one that I was [br]multiplying by every time, 0:06:00.153,0:06:03.865 then I would actually have no change in [br]percentage because I would just 0:06:03.865,0:06:07.992 have a constant amount because multiplying[br]by one is the same every time. 0:06:09.158,0:06:11.464 So, what is an exponential function? 0:06:11.464,0:06:15.743 An exponential function can be one that is[br]recognized by a constant 0:06:15.743,0:06:18.768 percentage rate of change. 0:06:18.768,0:06:23.147 Remember, this is as opposed to our linear[br]functions which had a constant 0:06:23.147,0:06:25.230 amount of change. 0:06:25.230,0:06:30.827 So, rather than changing by one each time[br]or by five each time, 0:06:30.827,0:06:35.988 maybe I'm changing by 5% each time.[br]And those are different. 0:06:37.122,0:06:41.904 An exponential growth model is one that[br]has a positive percent rate of change, 0:06:41.904,0:06:46.594 while a decay model is one that has[br]a negative percent rate of change. 0:06:46.594,0:06:51.243 So, a exponential growth model will be[br]increasing over time, 0:06:51.243,0:06:54.400 and a decay model will be [br]decreasing over time. 0:06:55.596,0:06:57.970 So, here's the general [br]form of our equation. 0:06:57.970,0:07:01.587 You're going to see I use t in my general[br]form and that's because, essentially, 0:07:01.587,0:07:05.577 we'll always be looking at models[br]with time as our independent variable. 0:07:06.162,0:07:09.588 Here we have some new parameters [br]we're looking at, as well. 0:07:09.588,0:07:12.918 Because we're talking about time, [br]I'm going to call a, 0:07:12.918,0:07:17.660 which is my y-intercept, my initial value.[br]And that's because with time, 0:07:17.660,0:07:22.602 when time is zero, I get my y-intercept,[br]so I get a. 0:07:22.602,0:07:25.202 A is therefore my initial value. 0:07:25.775,0:07:30.243 The growth factor is, what am I [br]multiplying by each time? 0:07:30.243,0:07:33.314 And when I know what I'm multiplying [br]by each time, 0:07:33.314,0:07:35.612 I can get to what is more important[br]to me, 0:07:35.612,0:07:39.674 which is my growth rate, or what is[br]the percentage that I'm increasing 0:07:39.674,0:07:41.905 or decreasing by each time. 0:07:41.905,0:07:46.770 So, my growth rate is just related to that[br]growth factor and can be calculated 0:07:46.770,0:07:47.958 from it. 0:07:47.958,0:07:50.197 So, if I have my growth factor, 0:07:50.197,0:07:54.473 I can just subtract one and I'll get[br]my growth rate. 0:07:55.704,0:07:58.498 Now, this r is not the same as [br]our correlation r, 0:07:58.498,0:08:00.086 so don't let that fool you. 0:08:01.552,0:08:02.930 So, here's our definitions. 0:08:02.930,0:08:07.646 Our growth factor, or b, is the amount [br]that we multiply by each y-value 0:08:07.646,0:08:09.396 to get the new y-value. 0:08:10.483,0:08:15.640 Our growth rate is the percentage increase[br]or decrease that we get 0:08:15.640,0:08:18.760 for each one unit increase in x. 0:08:18.760,0:08:22.567 And these are the two things that[br]we can get from our b. 0:08:22.567,0:08:25.968 The one that we're going to be interested [br]in interpreting primarily is 0:08:25.968,0:08:29.480 our growth rate when we do [br]our parameter estimates. 0:08:30.865,0:08:33.440 So, here are three examples of[br]exponential models. 0:08:33.440,0:08:37.407 I have a growth, a stagnant, [br]and a decay model. 0:08:37.407,0:08:42.516 One thing to note is that none of[br]these have negative values possible. 0:08:42.516,0:08:47.591 In an exponential function, we can [br]actually never cross that axis. 0:08:47.591,0:08:50.028 So, we can never have a negative value. 0:08:50.028,0:08:54.047 Notice that these are all the same [br]concavity, so they are all trending 0:08:54.047,0:08:58.197 that same direction because we cannot[br]cross this axis. 0:08:58.197,0:09:02.248 And, in fact, in a decay model, [br]we will continuously get closer 0:09:02.248,0:09:06.839 and closer and closer to zero, [br]but we will never actually get to zero. 0:09:08.864,0:09:11.357 So, let's work with some general [br]functions. 0:09:11.357,0:09:15.575 I want to know what the growth factor[br]would be if I said that water usage is 0:09:15.575,0:09:18.301 increasing by 5% per year. 0:09:18.301,0:09:20.398 So, what am I giving you here? 0:09:20.398,0:09:24.619 Well, I'm giving you the growth rate,[br]that percentage of change. 0:09:24.619,0:09:29.951 And if I want to convert this to b,[br]remember that b equals 1 plus r. 0:09:29.951,0:09:32.289 And that is r written as a decimal. 0:09:32.289,0:09:36.353 So, if I write my r as a decimal, [br]I would have .05, 0:09:36.353,0:09:41.760 which tells me that my b here, [br]my growth factor, would be 1.05. 0:09:42.505,0:09:43.871 Now, you try. 0:10:00.058,0:10:02.929 So, in this case, I've given you[br]a decay model. 0:10:03.457,0:10:08.920 If I have decay, where I know that it's[br]shrinking by 78%, 0:10:08.920,0:10:12.092 that actually means that [br]I have a negative r value. 0:10:12.092,0:10:15.807 So, here I would end up with a b of 0.22. 0:10:16.333,0:10:21.116 When I have a decay model, [br]my r can be negative. 0:10:21.116,0:10:26.136 So, I will expect to see a negative rate,[br]because I am changing 0:10:26.136,0:10:28.503 in a downward trajectory. 0:10:28.503,0:10:33.730 Now, remember I can never actually have [br]a negative value in my chart, 0:10:33.730,0:10:39.683 so that means that b cannot be negative, [br]even though r can. 0:10:39.683,0:10:41.844 Well, why can't b be negative? 0:10:41.844,0:10:46.563 Well, think about if I had a chart here[br]and I had a negative b value. 0:10:46.563,0:10:49.695 Well, when I add b to the first,[br]I'd have a negative result, 0:10:49.695,0:10:52.220 but then when I square b,[br]I get a positive, 0:10:52.220,0:10:54.303 and when I cubed it, I'd have a negative, 0:10:54.303,0:10:57.149 and then when I had to the fourth, [br]I'd be up here. 0:10:57.149,0:10:59.877 So, I'd end up with some kind of[br]function like that. 0:10:59.877,0:11:05.157 Well, that is not an exponential model.[br]So, I cannot have a negative b. 0:11:05.157,0:11:10.326 When I have a negative r, [br]for a decay model, what that will do is 0:11:10.326,0:11:15.273 give me a b value that is between[br]zero and one. 0:11:16.024,0:11:22.546 So, I will have a b value that is [br]less than one but more than zero 0:11:22.546,0:11:24.315 when I have a decay model. 0:11:24.315,0:11:28.789 If I have a b value over one,[br]then I have a growth model. 0:11:31.917,0:11:35.160 So, here is another example of [br]an exponential function. 0:11:35.160,0:11:37.580 And I've gone ahead and drawn it out. 0:11:37.580,0:11:42.955 You can see here that from my equation,[br]I have 34.3 as my initial value. 0:11:42.955,0:11:46.808 So, if I were to sketch this or put this[br]into technology and get it, 0:11:46.808,0:11:50.104 you would see that that's[br]my initial value, my starting point. 0:11:50.821,0:11:55.209 My growth factor for this problem[br]would be 0.63, 0:11:55.209,0:11:59.214 indicating I have a decay model because[br]that b is less than one. 0:12:00.181,0:12:05.567 If I calculate my growth rate, then,[br]I can have my b, 0.63, 0:12:05.567,0:12:07.559 as equal to 1 plus r. 0:12:07.559,0:12:12.527 And when I subtract one, [br]I get a decay of negative 37%. 0:12:13.724,0:12:17.502 So, now you try it.[br]Identify the growth rate in this equation. 0:12:35.253,0:12:41.961 In this equation, you can see that[br]our growth rate is very small, at 0.2%. 0:12:41.961,0:12:44.807 What else do you notice about this chart? 0:12:44.807,0:12:47.790 Well, you should notice that[br]it kind of looks like a line, 0:12:47.790,0:12:51.584 and not like that exponential curve that[br]we're so used to seeing. 0:12:51.584,0:12:56.246 When you have a very small growth rate,[br]one that is very close to zero, 0:12:56.246,0:13:00.264 you will actually see a model that[br]doesn't have a very striking curve to it. 0:13:00.264,0:13:02.029 It'll almost like linear. 0:13:02.029,0:13:06.103 Now, if I zoomed this out to [br]a thousand points, you'd probably begin 0:13:06.103,0:13:08.071 to be able to determine the curve. 0:13:08.071,0:13:12.800 But for very small r values, [br]we'll have a pretty flat curve. 0:13:12.800,0:13:17.298 The larger your r value is,[br]the more significant your curve is, 0:13:17.298,0:13:19.167 either going up or down. 0:13:19.167,0:13:22.379 So, if you were comparing two models[br]drawn on the same chart, 0:13:22.379,0:13:28.429 you could see that the more curving one[br]should have a higher absolute value of r, 0:13:28.429,0:13:31.432 because it's changing at a faster rate. 0:13:31.432,0:13:34.465 Alright. As always, [br]bring us your questions to class.