[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:05.70,0:00:09.82,Default,,0000,0000,0000,,Today we're going to start talking \Nabout exponential functions.
Dialogue: 0,0:00:09.82,0:00:12.96,Default,,0000,0000,0000,,So, in order to introduce this, \NI have a short video.
Dialogue: 0,0:00:14.20,0:00:16.77,Default,,0000,0000,0000,,♪ (music) ♪
Dialogue: 0,0:00:16.77,0:00:21.74,Default,,0000,0000,0000,,- This is an old story, but it reminds\Nus of the surprises we can get
Dialogue: 0,0:00:21.74,0:00:27.72,Default,,0000,0000,0000,,when even a small number like two\Nis multiplied by itself many times.
Dialogue: 0,0:00:27.72,0:00:33.21,Default,,0000,0000,0000,,King Shahram of India was so pleased when \Nhis Grand Vizier Sissa Ben Dahir
Dialogue: 0,0:00:33.21,0:00:39.15,Default,,0000,0000,0000,,presented him with the game of chess that\Nhe asked Ben Dahir to name his own reward.
Dialogue: 0,0:00:39.15,0:00:43.98,Default,,0000,0000,0000,,The request was so modest that \Nthe happy king immediately complied.
Dialogue: 0,0:00:44.52,0:00:49.56,Default,,0000,0000,0000,,What the grand vizier had asked was this,\Nthat one grain of wheat be placed on
Dialogue: 0,0:00:49.56,0:00:53.58,Default,,0000,0000,0000,,the first square of the chess board,\Ntwo grains on the second square,
Dialogue: 0,0:00:53.58,0:00:59.19,Default,,0000,0000,0000,,four on the third, eight on the fourth, \N16 on the fifth square, and so on.
Dialogue: 0,0:00:59.19,0:01:05.01,Default,,0000,0000,0000,,Doubling the amount of wheat on each\Nsucceeding square until all 64 squares
Dialogue: 0,0:01:05.01,0:01:06.42,Default,,0000,0000,0000,,were accounted for.
Dialogue: 0,0:01:06.42,0:01:09.56,Default,,0000,0000,0000,,When the king's steward had gotten to\Nthe 17th square,
Dialogue: 0,0:01:09.56,0:01:11.84,Default,,0000,0000,0000,,the table was well filled.
Dialogue: 0,0:01:11.84,0:01:18.02,Default,,0000,0000,0000,,By the 26th square, the chamber held\Nconsiderable wheat and a nervous king
Dialogue: 0,0:01:18.02,0:01:20.74,Default,,0000,0000,0000,,ordered the steward to speed up the count.
Dialogue: 0,0:01:20.74,0:01:26.05,Default,,0000,0000,0000,,When 42 squares were accounted for,\Nthe palace itself was swamped.
Dialogue: 0,0:01:26.05,0:01:30.42,Default,,0000,0000,0000,,Now fit to be tied, King Shahram \Nlearns from the court mathematician
Dialogue: 0,0:01:30.42,0:01:35.63,Default,,0000,0000,0000,,that had the process continued,\Nthe wheat required would have covered
Dialogue: 0,0:01:35.63,0:01:39.24,Default,,0000,0000,0000,,all India to a depth of over 50 feet.
Dialogue: 0,0:01:41.10,0:01:46.62,Default,,0000,0000,0000,,Incidentally, laying this many grains of\Nwheat end to end also does something
Dialogue: 0,0:01:46.62,0:01:48.22,Default,,0000,0000,0000,,rather spectacular.
Dialogue: 0,0:01:48.22,0:01:53.02,Default,,0000,0000,0000,,They would stretch from the Earth,\Nbeyond the sun, past the orbits
Dialogue: 0,0:01:53.02,0:01:58.50,Default,,0000,0000,0000,,of the planets, far out across the galaxy,\Nto the star Alpha Centauri,
Dialogue: 0,0:01:58.50,0:02:00.46,Default,,0000,0000,0000,,four light-years away.
Dialogue: 0,0:02:00.46,0:02:05.44,Default,,0000,0000,0000,,They would then stretch back to Earth,\Nback to Alpha Centauri,
Dialogue: 0,0:02:05.44,0:02:07.85,Default,,0000,0000,0000,,and back to the Earth again.
Dialogue: 0,0:02:07.85,0:02:11.85,Default,,0000,0000,0000,,♪ (music) ♪
Dialogue: 0,0:02:13.95,0:02:17.23,Default,,0000,0000,0000,,- So, what was going on in that video?
Dialogue: 0,0:02:17.23,0:02:22.21,Default,,0000,0000,0000,,You can see that, rather than going up\Nby the same amount for each square
Dialogue: 0,0:02:22.21,0:02:27.76,Default,,0000,0000,0000,,that they advanced on the chess board,\Nthat they went up by a doubled amount.
Dialogue: 0,0:02:28.23,0:02:31.90,Default,,0000,0000,0000,,So, that's a different type of function\Nthan we've been looking at.
Dialogue: 0,0:02:31.90,0:02:36.04,Default,,0000,0000,0000,,Rather than going up by one and then \Nanother one and another one,
Dialogue: 0,0:02:36.04,0:02:39.42,Default,,0000,0000,0000,,they went up by one and then \Nthey doubled and doubled again
Dialogue: 0,0:02:39.42,0:02:40.66,Default,,0000,0000,0000,,and doubled again.
Dialogue: 0,0:02:40.66,0:02:44.32,Default,,0000,0000,0000,,So, we cannot represent that with\Nthe type of linear function that
Dialogue: 0,0:02:44.32,0:02:46.09,Default,,0000,0000,0000,,we've been looking at before.
Dialogue: 0,0:02:46.84,0:02:51.47,Default,,0000,0000,0000,,In a linear function, such as this one, \Nif I went up by one square,
Dialogue: 0,0:02:51.47,0:02:54.57,Default,,0000,0000,0000,,I would increase here by 1.5.
Dialogue: 0,0:02:54.57,0:02:59.89,Default,,0000,0000,0000,,So, if you think about how this plays out\Nin a table, for each one unit increase
Dialogue: 0,0:02:59.89,0:03:03.99,Default,,0000,0000,0000,,in t, I am increasing by the same amount.
Dialogue: 0,0:03:04.64,0:03:09.68,Default,,0000,0000,0000,,Now, if I were to look at the percentage \Nincrease, therefore, I would actually
Dialogue: 0,0:03:09.68,0:03:14.98,Default,,0000,0000,0000,,indicate that I'm changing the percent\Nthat I'm going up by each time.
Dialogue: 0,0:03:14.98,0:03:24.53,Default,,0000,0000,0000,,So, going up by 1.5 from 2 is different\Nthan going up by 1.5 from 6.5.
Dialogue: 0,0:03:25.19,0:03:30.89,Default,,0000,0000,0000,,So, if you can see that I've done these \Ncalculations already, going from 2 to 3.5
Dialogue: 0,0:03:30.89,0:03:37.73,Default,,0000,0000,0000,,is an increase of the amount of 1.5,\Nbut it's actually a 75% increase.
Dialogue: 0,0:03:37.73,0:03:42.59,Default,,0000,0000,0000,,But by the time I get up to these values, \Ngoing from eight to 9.5,
Dialogue: 0,0:03:42.59,0:03:45.44,Default,,0000,0000,0000,,that's only a 19% increase.
Dialogue: 0,0:03:45.44,0:03:49.96,Default,,0000,0000,0000,,So, when I have a linear function, what\Nyou'll see is that you have a constant
Dialogue: 0,0:03:49.96,0:03:56.30,Default,,0000,0000,0000,,amount of increase but you do not have\Na constant percentage of increase.
Dialogue: 0,0:03:56.80,0:04:00.67,Default,,0000,0000,0000,,Now, what's going on in an exponential\Nfunction and what was happening in
Dialogue: 0,0:04:00.67,0:04:05.06,Default,,0000,0000,0000,,that video, is that for every one unit \Nincrease in t,
Dialogue: 0,0:04:05.06,0:04:07.51,Default,,0000,0000,0000,,we will see a different constant.
Dialogue: 0,0:04:08.16,0:04:13.63,Default,,0000,0000,0000,,So, in this case, if I use my exponential\Nequation and I go up by one every time,
Dialogue: 0,0:04:13.63,0:04:18.09,Default,,0000,0000,0000,,you can see that I go up \Nby, first, just one.
Dialogue: 0,0:04:18.09,0:04:26.79,Default,,0000,0000,0000,,Then I go up by 1.5 and then 2.25 \Nand then 3.36 and then 5.06.
Dialogue: 0,0:04:27.28,0:04:32.33,Default,,0000,0000,0000,,So, I'm actually not increasing by\Nthe same amount each time.
Dialogue: 0,0:04:32.33,0:04:36.09,Default,,0000,0000,0000,,That's because rather than adding \Na value each time,
Dialogue: 0,0:04:36.09,0:04:38.82,Default,,0000,0000,0000,,I'm multiplying an additional time.
Dialogue: 0,0:04:38.82,0:04:45.10,Default,,0000,0000,0000,,So, here I'm multiplying by 1.5 \Nan additional time each time I increase.
Dialogue: 0,0:04:45.10,0:04:49.61,Default,,0000,0000,0000,,Well, because I'm multiplying by \Na constant amount, what that actually
Dialogue: 0,0:04:49.61,0:04:55.01,Default,,0000,0000,0000,,does end up doing is increasing by \Na constant percentage.
Dialogue: 0,0:04:55.01,0:04:59.93,Default,,0000,0000,0000,,So, going from 2 to 3 increases by 50%.
Dialogue: 0,0:04:59.93,0:05:08.39,Default,,0000,0000,0000,,But also going from 10.13 to 15.19,\NI'm increasingly by 50%.
Dialogue: 0,0:05:08.99,0:05:12.93,Default,,0000,0000,0000,,So, in an exponential function, \NI increase by a constant percentage,
Dialogue: 0,0:05:12.93,0:05:14.53,Default,,0000,0000,0000,,but differing amounts.
Dialogue: 0,0:05:15.36,0:05:18.46,Default,,0000,0000,0000,,Let's look at this with what \Nwe call a decay function.
Dialogue: 0,0:05:18.46,0:05:22.35,Default,,0000,0000,0000,,So, this is also an exponential function,\Nbut one that's decreasing.
Dialogue: 0,0:05:22.35,0:05:26.90,Default,,0000,0000,0000,,So, in this case, in my decay function,\Nnow you can see that I'm multiplying by
Dialogue: 0,0:05:26.90,0:05:28.96,Default,,0000,0000,0000,,a decimal repeatedly.
Dialogue: 0,0:05:28.96,0:05:32.02,Default,,0000,0000,0000,,Well, if I repeatedly multiply by \Na decimal, I should be getting
Dialogue: 0,0:05:32.02,0:05:33.90,Default,,0000,0000,0000,,smaller and smaller values.
Dialogue: 0,0:05:33.90,0:05:40.91,Default,,0000,0000,0000,,So, here, I decreased by one but then\NI decreased by half, by .25, by .125,
Dialogue: 0,0:05:40.91,0:05:42.30,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:05:42.30,0:05:47.37,Default,,0000,0000,0000,,What that means is is I'm basically \Nslowing down my growth over time,
Dialogue: 0,0:05:47.37,0:05:51.80,Default,,0000,0000,0000,,but I'm decreasing still by \Na constant percentage.
Dialogue: 0,0:05:51.80,0:05:55.52,Default,,0000,0000,0000,,So, here, I decreased by 50% each time.
Dialogue: 0,0:05:56.22,0:06:00.15,Default,,0000,0000,0000,,Now, if I just had one that I was \Nmultiplying by every time,
Dialogue: 0,0:06:00.15,0:06:03.86,Default,,0000,0000,0000,,then I would actually have no change in \Npercentage because I would just
Dialogue: 0,0:06:03.86,0:06:07.99,Default,,0000,0000,0000,,have a constant amount because multiplying\Nby one is the same every time.
Dialogue: 0,0:06:09.16,0:06:11.46,Default,,0000,0000,0000,,So, what is an exponential function?
Dialogue: 0,0:06:11.46,0:06:15.74,Default,,0000,0000,0000,,An exponential function can be one that is\Nrecognized by a constant
Dialogue: 0,0:06:15.74,0:06:18.77,Default,,0000,0000,0000,,percentage rate of change.
Dialogue: 0,0:06:18.77,0:06:23.15,Default,,0000,0000,0000,,Remember, this is as opposed to our linear\Nfunctions which had a constant
Dialogue: 0,0:06:23.15,0:06:25.23,Default,,0000,0000,0000,,amount of change.
Dialogue: 0,0:06:25.23,0:06:30.83,Default,,0000,0000,0000,,So, rather than changing by one each time\Nor by five each time,
Dialogue: 0,0:06:30.83,0:06:35.99,Default,,0000,0000,0000,,maybe I'm changing by 5% each time.\NAnd those are different.
Dialogue: 0,0:06:37.12,0:06:41.90,Default,,0000,0000,0000,,An exponential growth model is one that\Nhas a positive percent rate of change,
Dialogue: 0,0:06:41.90,0:06:46.59,Default,,0000,0000,0000,,while a decay model is one that has\Na negative percent rate of change.
Dialogue: 0,0:06:46.59,0:06:51.24,Default,,0000,0000,0000,,So, a exponential growth model will be\Nincreasing over time,
Dialogue: 0,0:06:51.24,0:06:54.40,Default,,0000,0000,0000,,and a decay model will be \Ndecreasing over time.
Dialogue: 0,0:06:55.60,0:06:57.97,Default,,0000,0000,0000,,So, here's the general \Nform of our equation.
Dialogue: 0,0:06:57.97,0:07:01.59,Default,,0000,0000,0000,,You're going to see I use t in my general\Nform and that's because, essentially,
Dialogue: 0,0:07:01.59,0:07:05.58,Default,,0000,0000,0000,,we'll always be looking at models\Nwith time as our independent variable.
Dialogue: 0,0:07:06.16,0:07:09.59,Default,,0000,0000,0000,,Here we have some new parameters \Nwe're looking at, as well.
Dialogue: 0,0:07:09.59,0:07:12.92,Default,,0000,0000,0000,,Because we're talking about time, \NI'm going to call a,
Dialogue: 0,0:07:12.92,0:07:17.66,Default,,0000,0000,0000,,which is my y-intercept, my initial value.\NAnd that's because with time,
Dialogue: 0,0:07:17.66,0:07:22.60,Default,,0000,0000,0000,,when time is zero, I get my y-intercept,\Nso I get a.
Dialogue: 0,0:07:22.60,0:07:25.20,Default,,0000,0000,0000,,A is therefore my initial value.
Dialogue: 0,0:07:25.78,0:07:30.24,Default,,0000,0000,0000,,The growth factor is, what am I \Nmultiplying by each time?
Dialogue: 0,0:07:30.24,0:07:33.31,Default,,0000,0000,0000,,And when I know what I'm multiplying \Nby each time,
Dialogue: 0,0:07:33.31,0:07:35.61,Default,,0000,0000,0000,,I can get to what is more important\Nto me,
Dialogue: 0,0:07:35.61,0:07:39.67,Default,,0000,0000,0000,,which is my growth rate, or what is\Nthe percentage that I'm increasing
Dialogue: 0,0:07:39.67,0:07:41.90,Default,,0000,0000,0000,,or decreasing by each time.
Dialogue: 0,0:07:41.90,0:07:46.77,Default,,0000,0000,0000,,So, my growth rate is just related to that\Ngrowth factor and can be calculated
Dialogue: 0,0:07:46.77,0:07:47.96,Default,,0000,0000,0000,,from it.
Dialogue: 0,0:07:47.96,0:07:50.20,Default,,0000,0000,0000,,So, if I have my growth factor,
Dialogue: 0,0:07:50.20,0:07:54.47,Default,,0000,0000,0000,,I can just subtract one and I'll get\Nmy growth rate.
Dialogue: 0,0:07:55.70,0:07:58.50,Default,,0000,0000,0000,,Now, this r is not the same as \Nour correlation r,
Dialogue: 0,0:07:58.50,0:08:00.09,Default,,0000,0000,0000,,so don't let that fool you.
Dialogue: 0,0:08:01.55,0:08:02.93,Default,,0000,0000,0000,,So, here's our definitions.
Dialogue: 0,0:08:02.93,0:08:07.65,Default,,0000,0000,0000,,Our growth factor, or b, is the amount \Nthat we multiply by each y-value
Dialogue: 0,0:08:07.65,0:08:09.40,Default,,0000,0000,0000,,to get the new y-value.
Dialogue: 0,0:08:10.48,0:08:15.64,Default,,0000,0000,0000,,Our growth rate is the percentage increase\Nor decrease that we get
Dialogue: 0,0:08:15.64,0:08:18.76,Default,,0000,0000,0000,,for each one unit increase in x.
Dialogue: 0,0:08:18.76,0:08:22.57,Default,,0000,0000,0000,,And these are the two things that\Nwe can get from our b.
Dialogue: 0,0:08:22.57,0:08:25.97,Default,,0000,0000,0000,,The one that we're going to be interested \Nin interpreting primarily is
Dialogue: 0,0:08:25.97,0:08:29.48,Default,,0000,0000,0000,,our growth rate when we do \Nour parameter estimates.
Dialogue: 0,0:08:30.86,0:08:33.44,Default,,0000,0000,0000,,So, here are three examples of\Nexponential models.
Dialogue: 0,0:08:33.44,0:08:37.41,Default,,0000,0000,0000,,I have a growth, a stagnant, \Nand a decay model.
Dialogue: 0,0:08:37.41,0:08:42.52,Default,,0000,0000,0000,,One thing to note is that none of\Nthese have negative values possible.
Dialogue: 0,0:08:42.52,0:08:47.59,Default,,0000,0000,0000,,In an exponential function, we can \Nactually never cross that axis.
Dialogue: 0,0:08:47.59,0:08:50.03,Default,,0000,0000,0000,,So, we can never have a negative value.
Dialogue: 0,0:08:50.03,0:08:54.05,Default,,0000,0000,0000,,Notice that these are all the same \Nconcavity, so they are all trending
Dialogue: 0,0:08:54.05,0:08:58.20,Default,,0000,0000,0000,,that same direction because we cannot\Ncross this axis.
Dialogue: 0,0:08:58.20,0:09:02.25,Default,,0000,0000,0000,,And, in fact, in a decay model, \Nwe will continuously get closer
Dialogue: 0,0:09:02.25,0:09:06.84,Default,,0000,0000,0000,,and closer and closer to zero, \Nbut we will never actually get to zero.
Dialogue: 0,0:09:08.86,0:09:11.36,Default,,0000,0000,0000,,So, let's work with some general \Nfunctions.
Dialogue: 0,0:09:11.36,0:09:15.58,Default,,0000,0000,0000,,I want to know what the growth factor\Nwould be if I said that water usage is
Dialogue: 0,0:09:15.58,0:09:18.30,Default,,0000,0000,0000,,increasing by 5% per year.
Dialogue: 0,0:09:18.30,0:09:20.40,Default,,0000,0000,0000,,So, what am I giving you here?
Dialogue: 0,0:09:20.40,0:09:24.62,Default,,0000,0000,0000,,Well, I'm giving you the growth rate,\Nthat percentage of change.
Dialogue: 0,0:09:24.62,0:09:29.95,Default,,0000,0000,0000,,And if I want to convert this to b,\Nremember that b equals 1 plus r.
Dialogue: 0,0:09:29.95,0:09:32.29,Default,,0000,0000,0000,,And that is r written as a decimal.
Dialogue: 0,0:09:32.29,0:09:36.35,Default,,0000,0000,0000,,So, if I write my r as a decimal, \NI would have .05,
Dialogue: 0,0:09:36.35,0:09:41.76,Default,,0000,0000,0000,,which tells me that my b here, \Nmy growth factor, would be 1.05.
Dialogue: 0,0:09:42.50,0:09:43.87,Default,,0000,0000,0000,,Now, you try.
Dialogue: 0,0:10:00.06,0:10:02.93,Default,,0000,0000,0000,,So, in this case, I've given you\Na decay model.
Dialogue: 0,0:10:03.46,0:10:08.92,Default,,0000,0000,0000,,If I have decay, where I know that it's\Nshrinking by 78%,
Dialogue: 0,0:10:08.92,0:10:12.09,Default,,0000,0000,0000,,that actually means that \NI have a negative r value.
Dialogue: 0,0:10:12.09,0:10:15.81,Default,,0000,0000,0000,,So, here I would end up with a b of 0.22.
Dialogue: 0,0:10:16.33,0:10:21.12,Default,,0000,0000,0000,,When I have a decay model, \Nmy r can be negative.
Dialogue: 0,0:10:21.12,0:10:26.14,Default,,0000,0000,0000,,So, I will expect to see a negative rate,\Nbecause I am changing
Dialogue: 0,0:10:26.14,0:10:28.50,Default,,0000,0000,0000,,in a downward trajectory.
Dialogue: 0,0:10:28.50,0:10:33.73,Default,,0000,0000,0000,,Now, remember I can never actually have \Na negative value in my chart,
Dialogue: 0,0:10:33.73,0:10:39.68,Default,,0000,0000,0000,,so that means that b cannot be negative, \Neven though r can.
Dialogue: 0,0:10:39.68,0:10:41.84,Default,,0000,0000,0000,,Well, why can't b be negative?
Dialogue: 0,0:10:41.84,0:10:46.56,Default,,0000,0000,0000,,Well, think about if I had a chart here\Nand I had a negative b value.
Dialogue: 0,0:10:46.56,0:10:49.70,Default,,0000,0000,0000,,Well, when I add b to the first,\NI'd have a negative result,
Dialogue: 0,0:10:49.70,0:10:52.22,Default,,0000,0000,0000,,but then when I square b,\NI get a positive,
Dialogue: 0,0:10:52.22,0:10:54.30,Default,,0000,0000,0000,,and when I cubed it, I'd have a negative,
Dialogue: 0,0:10:54.30,0:10:57.15,Default,,0000,0000,0000,,and then when I had to the fourth, \NI'd be up here.
Dialogue: 0,0:10:57.15,0:10:59.88,Default,,0000,0000,0000,,So, I'd end up with some kind of\Nfunction like that.
Dialogue: 0,0:10:59.88,0:11:05.16,Default,,0000,0000,0000,,Well, that is not an exponential model.\NSo, I cannot have a negative b.
Dialogue: 0,0:11:05.16,0:11:10.33,Default,,0000,0000,0000,,When I have a negative r, \Nfor a decay model, what that will do is
Dialogue: 0,0:11:10.33,0:11:15.27,Default,,0000,0000,0000,,give me a b value that is between\Nzero and one.
Dialogue: 0,0:11:16.02,0:11:22.55,Default,,0000,0000,0000,,So, I will have a b value that is \Nless than one but more than zero
Dialogue: 0,0:11:22.55,0:11:24.32,Default,,0000,0000,0000,,when I have a decay model.
Dialogue: 0,0:11:24.32,0:11:28.79,Default,,0000,0000,0000,,If I have a b value over one,\Nthen I have a growth model.
Dialogue: 0,0:11:31.92,0:11:35.16,Default,,0000,0000,0000,,So, here is another example of \Nan exponential function.
Dialogue: 0,0:11:35.16,0:11:37.58,Default,,0000,0000,0000,,And I've gone ahead and drawn it out.
Dialogue: 0,0:11:37.58,0:11:42.96,Default,,0000,0000,0000,,You can see here that from my equation,\NI have 34.3 as my initial value.
Dialogue: 0,0:11:42.96,0:11:46.81,Default,,0000,0000,0000,,So, if I were to sketch this or put this\Ninto technology and get it,
Dialogue: 0,0:11:46.81,0:11:50.10,Default,,0000,0000,0000,,you would see that that's\Nmy initial value, my starting point.
Dialogue: 0,0:11:50.82,0:11:55.21,Default,,0000,0000,0000,,My growth factor for this problem\Nwould be 0.63,
Dialogue: 0,0:11:55.21,0:11:59.21,Default,,0000,0000,0000,,indicating I have a decay model because\Nthat b is less than one.
Dialogue: 0,0:12:00.18,0:12:05.57,Default,,0000,0000,0000,,If I calculate my growth rate, then,\NI can have my b, 0.63,
Dialogue: 0,0:12:05.57,0:12:07.56,Default,,0000,0000,0000,,as equal to 1 plus r.
Dialogue: 0,0:12:07.56,0:12:12.53,Default,,0000,0000,0000,,And when I subtract one, \NI get a decay of negative 37%.
Dialogue: 0,0:12:13.72,0:12:17.50,Default,,0000,0000,0000,,So, now you try it.\NIdentify the growth rate in this equation.
Dialogue: 0,0:12:35.25,0:12:41.96,Default,,0000,0000,0000,,In this equation, you can see that\Nour growth rate is very small, at 0.2%.
Dialogue: 0,0:12:41.96,0:12:44.81,Default,,0000,0000,0000,,What else do you notice about this chart?
Dialogue: 0,0:12:44.81,0:12:47.79,Default,,0000,0000,0000,,Well, you should notice that\Nit kind of looks like a line,
Dialogue: 0,0:12:47.79,0:12:51.58,Default,,0000,0000,0000,,and not like that exponential curve that\Nwe're so used to seeing.
Dialogue: 0,0:12:51.58,0:12:56.25,Default,,0000,0000,0000,,When you have a very small growth rate,\None that is very close to zero,
Dialogue: 0,0:12:56.25,0:13:00.26,Default,,0000,0000,0000,,you will actually see a model that\Ndoesn't have a very striking curve to it.
Dialogue: 0,0:13:00.26,0:13:02.03,Default,,0000,0000,0000,,It'll almost like linear.
Dialogue: 0,0:13:02.03,0:13:06.10,Default,,0000,0000,0000,,Now, if I zoomed this out to \Na thousand points, you'd probably begin
Dialogue: 0,0:13:06.10,0:13:08.07,Default,,0000,0000,0000,,to be able to determine the curve.
Dialogue: 0,0:13:08.07,0:13:12.80,Default,,0000,0000,0000,,But for very small r values, \Nwe'll have a pretty flat curve.
Dialogue: 0,0:13:12.80,0:13:17.30,Default,,0000,0000,0000,,The larger your r value is,\Nthe more significant your curve is,
Dialogue: 0,0:13:17.30,0:13:19.17,Default,,0000,0000,0000,,either going up or down.
Dialogue: 0,0:13:19.17,0:13:22.38,Default,,0000,0000,0000,,So, if you were comparing two models\Ndrawn on the same chart,
Dialogue: 0,0:13:22.38,0:13:28.43,Default,,0000,0000,0000,,you could see that the more curving one\Nshould have a higher absolute value of r,
Dialogue: 0,0:13:28.43,0:13:31.43,Default,,0000,0000,0000,,because it's changing at a faster rate.
Dialogue: 0,0:13:31.43,0:13:34.46,Default,,0000,0000,0000,,Alright. As always, \Nbring us your questions to class.