[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.24,0:00:04.42,Default,,0000,0000,0000,,- [Voiceover] Part c, let y\Nequals f of x be the particular
Dialogue: 0,0:00:04.42,0:00:07.04,Default,,0000,0000,0000,,solution to the differential\Nequation with the
Dialogue: 0,0:00:07.04,0:00:11.49,Default,,0000,0000,0000,,initial condition f of\Ntwo is equal to three.
Dialogue: 0,0:00:11.49,0:00:15.71,Default,,0000,0000,0000,,Does f have a relative\Nminimum, a relative maximum,
Dialogue: 0,0:00:15.71,0:00:18.77,Default,,0000,0000,0000,,or neither at x equals two?
Dialogue: 0,0:00:18.77,0:00:20.89,Default,,0000,0000,0000,,Justify your answer.
Dialogue: 0,0:00:20.89,0:00:22.20,Default,,0000,0000,0000,,Well, to think about\Nwhether we have a realtive
Dialogue: 0,0:00:22.20,0:00:24.26,Default,,0000,0000,0000,,minimum or relative maximum, we can say,
Dialogue: 0,0:00:24.26,0:00:26.71,Default,,0000,0000,0000,,well, what's the derivative at that point?
Dialogue: 0,0:00:26.71,0:00:28.63,Default,,0000,0000,0000,,If it's zero, then it's a good candidate
Dialogue: 0,0:00:28.63,0:00:30.06,Default,,0000,0000,0000,,that we're dealing with a,
Dialogue: 0,0:00:30.06,0:00:32.55,Default,,0000,0000,0000,,that it could be a relative\Nminimum or maximum,
Dialogue: 0,0:00:32.55,0:00:34.75,Default,,0000,0000,0000,,if it's not zero, then it's neither.
Dialogue: 0,0:00:34.75,0:00:36.11,Default,,0000,0000,0000,,And then if it is zero,\Nif we wanna figure out
Dialogue: 0,0:00:36.11,0:00:37.70,Default,,0000,0000,0000,,relative minimum or relative maximum,
Dialogue: 0,0:00:37.70,0:00:40.84,Default,,0000,0000,0000,,we can evaluate the sign\Nof the second derivative.
Dialogue: 0,0:00:40.84,0:00:42.36,Default,,0000,0000,0000,,So let's just think about this.
Dialogue: 0,0:00:42.36,0:00:45.68,Default,,0000,0000,0000,,So, we want to evaluate f prime,
Dialogue: 0,0:00:45.68,0:00:49.31,Default,,0000,0000,0000,,we wanna figure out what f prime of,
Dialogue: 0,0:00:49.31,0:00:53.14,Default,,0000,0000,0000,,f prime of two is equal to.
Dialogue: 0,0:00:53.14,0:00:56.71,Default,,0000,0000,0000,,So, we know that f prime of x,
Dialogue: 0,0:00:56.71,0:01:01.58,Default,,0000,0000,0000,,f prime of x, which is\Nthe same thing as dy dx,
Dialogue: 0,0:01:01.58,0:01:05.00,Default,,0000,0000,0000,,is equal to two times x minus y.
Dialogue: 0,0:01:05.00,0:01:06.98,Default,,0000,0000,0000,,We saw that in the last problem.
Dialogue: 0,0:01:06.98,0:01:10.03,Default,,0000,0000,0000,,And so f prime of two,\NI'll write it this way,
Dialogue: 0,0:01:10.03,0:01:14.18,Default,,0000,0000,0000,,f prime of two is going to be equal to
Dialogue: 0,0:01:14.18,0:01:16.06,Default,,0000,0000,0000,,two times two,
Dialogue: 0,0:01:16.06,0:01:19.12,Default,,0000,0000,0000,,two times two minus
Dialogue: 0,0:01:19.12,0:01:21.76,Default,,0000,0000,0000,,whatever y is when x is equal to two.
Dialogue: 0,0:01:21.76,0:01:24.67,Default,,0000,0000,0000,,Well do we know what y\Nis when x equals to two?
Dialogue: 0,0:01:24.67,0:01:26.87,Default,,0000,0000,0000,,Sure, they tell us right over here.
Dialogue: 0,0:01:26.87,0:01:29.06,Default,,0000,0000,0000,,Y is equal to f of x,
Dialogue: 0,0:01:29.06,0:01:31.03,Default,,0000,0000,0000,,so when x is equal to two,
Dialogue: 0,0:01:31.03,0:01:32.66,Default,,0000,0000,0000,,when x is equal to two,
Dialogue: 0,0:01:32.66,0:01:35.20,Default,,0000,0000,0000,,y is equal to three,
Dialogue: 0,0:01:35.20,0:01:37.94,Default,,0000,0000,0000,,so two times two minus three.
Dialogue: 0,0:01:37.94,0:01:40.65,Default,,0000,0000,0000,,And so this is going to be\Nequal to four minus three
Dialogue: 0,0:01:40.65,0:01:42.13,Default,,0000,0000,0000,,is equal to one.
Dialogue: 0,0:01:42.13,0:01:47.13,Default,,0000,0000,0000,,And so since the derivative\Nat two is not zero,
Dialogue: 0,0:01:47.19,0:01:50.08,Default,,0000,0000,0000,,this is not going to be a\Nminimum, a relative minimum
Dialogue: 0,0:01:50.08,0:01:53.11,Default,,0000,0000,0000,,or a relative maximum, so you could say
Dialogue: 0,0:01:53.11,0:01:58.11,Default,,0000,0000,0000,,since, since f prime of two,
Dialogue: 0,0:01:59.24,0:02:02.51,Default,,0000,0000,0000,,f prime of two does not equal zero,
Dialogue: 0,0:02:02.51,0:02:07.41,Default,,0000,0000,0000,,this, we have a, f has,\Nlet me write it this way
Dialogue: 0,0:02:07.41,0:02:12.41,Default,,0000,0000,0000,,f has neither, neither minimum,
Dialogue: 0,0:02:14.11,0:02:16.27,Default,,0000,0000,0000,,or relative minimum I guess I could say,
Dialogue: 0,0:02:16.27,0:02:19.54,Default,,0000,0000,0000,,relative min or
Dialogue: 0,0:02:19.54,0:02:23.04,Default,,0000,0000,0000,,relative max
Dialogue: 0,0:02:23.04,0:02:26.32,Default,,0000,0000,0000,,at x equals two.
Dialogue: 0,0:02:26.32,0:02:29.06,Default,,0000,0000,0000,,Alright, let's do the next one.
Dialogue: 0,0:02:29.06,0:02:32.94,Default,,0000,0000,0000,,Find the values of the constants m and b,
Dialogue: 0,0:02:32.94,0:02:37.49,Default,,0000,0000,0000,,for which y equals m\Nx plus b is a solution
Dialogue: 0,0:02:37.49,0:02:39.58,Default,,0000,0000,0000,,to the differential equation.
Dialogue: 0,0:02:39.58,0:02:41.90,Default,,0000,0000,0000,,Alright, this one is interesting.
Dialogue: 0,0:02:41.90,0:02:44.08,Default,,0000,0000,0000,,So let's, actually let's\Njust write down everything we
Dialogue: 0,0:02:44.08,0:02:47.26,Default,,0000,0000,0000,,know before we even think\Nthat y equals m x plus b
Dialogue: 0,0:02:47.26,0:02:49.90,Default,,0000,0000,0000,,could be a solution to\Nthe differential equation.
Dialogue: 0,0:02:49.90,0:02:54.51,Default,,0000,0000,0000,,So, we know that, we know that dy
Dialogue: 0,0:02:54.51,0:02:58.94,Default,,0000,0000,0000,,over dx is equal to two x minus y,
Dialogue: 0,0:02:58.94,0:03:00.27,Default,,0000,0000,0000,,they told us that.
Dialogue: 0,0:03:00.27,0:03:02.28,Default,,0000,0000,0000,,We also know that the second derivative,
Dialogue: 0,0:03:02.28,0:03:07.28,Default,,0000,0000,0000,,the second derivative of y,\Nwith respect to x, is equal to
Dialogue: 0,0:03:07.52,0:03:12.52,Default,,0000,0000,0000,,two minus dy dx. We\Nfigured that out in part b
Dialogue: 0,0:03:12.82,0:03:14.33,Default,,0000,0000,0000,,of this problem.
Dialogue: 0,0:03:14.33,0:03:17.27,Default,,0000,0000,0000,,And then, we could also express this
Dialogue: 0,0:03:17.27,0:03:18.91,Default,,0000,0000,0000,,we saw that we could also write that
Dialogue: 0,0:03:18.91,0:03:22.60,Default,,0000,0000,0000,,as two minus two x plus\Ny, if you just substitute,
Dialogue: 0,0:03:22.60,0:03:25.38,Default,,0000,0000,0000,,if you substitute this in for that.
Dialogue: 0,0:03:25.38,0:03:27.83,Default,,0000,0000,0000,,So it's two minus two x plus y.
Dialogue: 0,0:03:27.83,0:03:29.28,Default,,0000,0000,0000,,So let me write it that way.
Dialogue: 0,0:03:29.28,0:03:33.60,Default,,0000,0000,0000,,This is also equal to\Ntwo minus two x plus y.
Dialogue: 0,0:03:33.60,0:03:35.78,Default,,0000,0000,0000,,So that's everything that we know
Dialogue: 0,0:03:35.78,0:03:39.08,Default,,0000,0000,0000,,before we even thought that\Nmaybe there's a solution
Dialogue: 0,0:03:39.08,0:03:42.19,Default,,0000,0000,0000,,of y equals m x plus b.
Dialogue: 0,0:03:42.19,0:03:44.51,Default,,0000,0000,0000,,So now let's start with\Ny equals m x plus b.
Dialogue: 0,0:03:44.51,0:03:48.92,Default,,0000,0000,0000,,So if y is equal to m x\Nplus b, y equals m x plus b,
Dialogue: 0,0:03:48.92,0:03:50.55,Default,,0000,0000,0000,,so this is the equation of a line,
Dialogue: 0,0:03:50.55,0:03:54.100,Default,,0000,0000,0000,,then dy dx is going to be equal to,
Dialogue: 0,0:03:54.100,0:03:57.36,Default,,0000,0000,0000,,well the derivative of this\Nwith respect to x is just m,
Dialogue: 0,0:03:57.36,0:04:00.01,Default,,0000,0000,0000,,the derivative of this with\Nrespect to x, this is constant
Dialogue: 0,0:04:00.01,0:04:01.58,Default,,0000,0000,0000,,this is not going to\Nchange with respect to x,
Dialogue: 0,0:04:01.58,0:04:02.84,Default,,0000,0000,0000,,it's just zero.
Dialogue: 0,0:04:02.84,0:04:03.65,Default,,0000,0000,0000,,And that makes sense,
Dialogue: 0,0:04:03.65,0:04:06.86,Default,,0000,0000,0000,,the rate of change of y with\Nrespect to x is the slope,
Dialogue: 0,0:04:06.86,0:04:09.46,Default,,0000,0000,0000,,is the slope of our line.
Dialogue: 0,0:04:09.46,0:04:12.16,Default,,0000,0000,0000,,So, can we use, and this\Nis really all that we know,
Dialogue: 0,0:04:12.16,0:04:14.23,Default,,0000,0000,0000,,we could keep, actually\Nwe could go even further,
Dialogue: 0,0:04:14.23,0:04:16.07,Default,,0000,0000,0000,,we could take the second derivative here,
Dialogue: 0,0:04:16.07,0:04:19.18,Default,,0000,0000,0000,,the second derivative\Nof y with respect to x.
Dialogue: 0,0:04:19.18,0:04:21.01,Default,,0000,0000,0000,,Well, that's going to be zero.
Dialogue: 0,0:04:21.01,0:04:23.55,Default,,0000,0000,0000,,The second derivative of\Na, of a linear function,
Dialogue: 0,0:04:23.55,0:04:25.58,Default,,0000,0000,0000,,well it's going to be\Nzero, you see that here.
Dialogue: 0,0:04:25.58,0:04:28.59,Default,,0000,0000,0000,,So this is, this is all of\Nthe information that we have.
Dialogue: 0,0:04:28.59,0:04:31.35,Default,,0000,0000,0000,,We get this from the previous\Nparts of the problem,
Dialogue: 0,0:04:31.35,0:04:33.89,Default,,0000,0000,0000,,and we get this just taking the\Nfirst and second derivatives
Dialogue: 0,0:04:33.89,0:04:37.32,Default,,0000,0000,0000,,of, of y equals m x plus b.
Dialogue: 0,0:04:37.32,0:04:41.23,Default,,0000,0000,0000,,So given this, can we figure out,
Dialogue: 0,0:04:41.23,0:04:45.39,Default,,0000,0000,0000,,can we figure out what m and b are?
Dialogue: 0,0:04:45.39,0:04:50.20,Default,,0000,0000,0000,,Alright, so we could,\Nif we said m is equal to
Dialogue: 0,0:04:50.20,0:04:55.20,Default,,0000,0000,0000,,two x minus y, that doesn't seem right.
Dialogue: 0,0:04:55.55,0:04:57.95,Default,,0000,0000,0000,,This one is a tricky one.
Dialogue: 0,0:04:57.95,0:05:00.81,Default,,0000,0000,0000,,Well, let's see, we know\Nthat the second derivative
Dialogue: 0,0:05:00.81,0:05:03.45,Default,,0000,0000,0000,,is going to be equal to zero.
Dialogue: 0,0:05:03.45,0:05:05.54,Default,,0000,0000,0000,,We know that this is\Ngoing to be equal to zero
Dialogue: 0,0:05:05.54,0:05:08.32,Default,,0000,0000,0000,,for this particular solution.
Dialogue: 0,0:05:08.32,0:05:12.15,Default,,0000,0000,0000,,And we know dy dx is equal to m.
Dialogue: 0,0:05:12.15,0:05:13.72,Default,,0000,0000,0000,,We know this is m.
Dialogue: 0,0:05:13.72,0:05:15.22,Default,,0000,0000,0000,,And so there you have it,\Nwe have enough information
Dialogue: 0,0:05:15.22,0:05:16.03,Default,,0000,0000,0000,,to solve for m.
Dialogue: 0,0:05:16.03,0:05:19.01,Default,,0000,0000,0000,,We know that zero is equal to two minus m.
Dialogue: 0,0:05:19.01,0:05:22.30,Default,,0000,0000,0000,,So zero is equal to two minus m.
Dialogue: 0,0:05:22.30,0:05:25.12,Default,,0000,0000,0000,,And so we can add m to\Nboth sides and we get
Dialogue: 0,0:05:25.12,0:05:30.12,Default,,0000,0000,0000,,m is equal to, m is equal to two.
Dialogue: 0,0:05:30.83,0:05:34.94,Default,,0000,0000,0000,,So, that by itself, was quite useful.
Dialogue: 0,0:05:34.94,0:05:38.16,Default,,0000,0000,0000,,And then, what we could say, let's see,
Dialogue: 0,0:05:38.16,0:05:40.93,Default,,0000,0000,0000,,can we solve this further?
Dialogue: 0,0:05:40.93,0:05:43.50,Default,,0000,0000,0000,,Well we know that this,\Nright over here, dy dx,
Dialogue: 0,0:05:43.50,0:05:46.28,Default,,0000,0000,0000,,this is m, this is m.
Dialogue: 0,0:05:46.28,0:05:48.69,Default,,0000,0000,0000,,And it's equal to two.
Dialogue: 0,0:05:48.69,0:05:52.53,Default,,0000,0000,0000,,So, we could say that two\Nis equal to 2 x minus y.
Dialogue: 0,0:05:52.53,0:05:56.46,Default,,0000,0000,0000,,Two is equal to 2 x minus y.
Dialogue: 0,0:05:56.46,0:05:58.32,Default,,0000,0000,0000,,And then let's see, if we solve for y,
Dialogue: 0,0:05:58.32,0:06:01.34,Default,,0000,0000,0000,,add y to both sides,\Nsubtract two from both sides,
Dialogue: 0,0:06:01.34,0:06:05.15,Default,,0000,0000,0000,,we get y is equal to two x minus two.
Dialogue: 0,0:06:05.15,0:06:07.33,Default,,0000,0000,0000,,And there we have our whole solution.
Dialogue: 0,0:06:07.33,0:06:09.97,Default,,0000,0000,0000,,And so you have your m, right over there.
Dialogue: 0,0:06:09.97,0:06:10.95,Default,,0000,0000,0000,,That is m.
Dialogue: 0,0:06:10.95,0:06:14.57,Default,,0000,0000,0000,,And then we also have our b.
Dialogue: 0,0:06:14.57,0:06:16.82,Default,,0000,0000,0000,,This one was a tricky one.
Dialogue: 0,0:06:16.82,0:06:19.72,Default,,0000,0000,0000,,Anytime that you, you have to, ya know,
Dialogue: 0,0:06:19.72,0:06:21.76,Default,,0000,0000,0000,,do something like this and it\Ndoesn't just jump out at you,
Dialogue: 0,0:06:21.76,0:06:24.00,Default,,0000,0000,0000,,and if it wasn't obvious, it\Ndidn't jump out at me at first,
Dialogue: 0,0:06:24.00,0:06:26.01,Default,,0000,0000,0000,,when I looked at this problem,\NI said, well let me just
Dialogue: 0,0:06:26.01,0:06:28.26,Default,,0000,0000,0000,,write down everything that they told us,
Dialogue: 0,0:06:28.26,0:06:29.65,Default,,0000,0000,0000,,so they wrote this before,
Dialogue: 0,0:06:29.65,0:06:32.34,Default,,0000,0000,0000,,and then we say okay this\Nis going to be a solution.
Dialogue: 0,0:06:32.34,0:06:35.70,Default,,0000,0000,0000,,And so let me see if I\Ncan somehow solve, so,
Dialogue: 0,0:06:35.70,0:06:37.06,Default,,0000,0000,0000,,let's see what I didn't use.
Dialogue: 0,0:06:37.06,0:06:39.87,Default,,0000,0000,0000,,I didn't use, I didn't use that.
Dialogue: 0,0:06:39.87,0:06:42.12,Default,,0000,0000,0000,,I did use this.
Dialogue: 0,0:06:42.12,0:06:44.57,Default,,0000,0000,0000,,I absolutely used that.
Dialogue: 0,0:06:44.57,0:06:47.67,Default,,0000,0000,0000,,I did use that, I did use\Nthat, and I did use that.
Dialogue: 0,0:06:47.67,0:06:49.67,Default,,0000,0000,0000,,So this was a little bit\Nof a fun little puzzle
Dialogue: 0,0:06:49.67,0:06:52.16,Default,,0000,0000,0000,,where I just wrote down all\Nthe information they gave us
Dialogue: 0,0:06:52.16,0:06:54.48,Default,,0000,0000,0000,,and I tried to figure out, based on that,
Dialogue: 0,0:06:54.48,0:06:59.26,Default,,0000,0000,0000,,whether I could figure out m and m and b.
Dialogue: 0,0:06:59.26,0:07:01.14,Default,,0000,0000,0000,,And this is pretty neat,\Nthat this a solution
Dialogue: 0,0:07:01.14,0:07:02.72,Default,,0000,0000,0000,,two x minus two.
Dialogue: 0,0:07:02.72,0:07:05.63,Default,,0000,0000,0000,,If we go to our slope\Nfield above, it's not,
Dialogue: 0,0:07:05.63,0:07:07.03,Default,,0000,0000,0000,,it wouldn't have jumped out at me.
Dialogue: 0,0:07:07.03,0:07:09.26,Default,,0000,0000,0000,,But if you think about, if you think about
Dialogue: 0,0:07:09.26,0:07:11.30,Default,,0000,0000,0000,,so two x minus two, it's\Ny intercept would be
Dialogue: 0,0:07:11.30,0:07:14.36,Default,,0000,0000,0000,,negative two like that, let me\Ndo this in a different color,
Dialogue: 0,0:07:14.36,0:07:17.72,Default,,0000,0000,0000,,and so the line would look something,
Dialogue: 0,0:07:17.72,0:07:20.40,Default,,0000,0000,0000,,would like something like this.
Dialogue: 0,0:07:20.40,0:07:22.73,Default,,0000,0000,0000,,The line would look something like this.
Dialogue: 0,0:07:22.73,0:07:25.55,Default,,0000,0000,0000,,And you can verify that\Nany one of these points,
Dialogue: 0,0:07:25.55,0:07:28.20,Default,,0000,0000,0000,,at any one of these points, the slope,
Dialogue: 0,0:07:28.20,0:07:33.20,Default,,0000,0000,0000,,the slope is equal, the\Nslope is equal to two.
Dialogue: 0,0:07:33.27,0:07:36.67,Default,,0000,0000,0000,,If we're at the point two comma two,
Dialogue: 0,0:07:36.67,0:07:39.17,Default,,0000,0000,0000,,well it's gonna be two\Ntimes two minus two is two.
Dialogue: 0,0:07:39.17,0:07:42.68,Default,,0000,0000,0000,,One comma zero, two times\None minus zero is two.
Dialogue: 0,0:07:42.68,0:07:46.27,Default,,0000,0000,0000,,Negative two comma, or sorry\Nzero comma negative two
Dialogue: 0,0:07:46.27,0:07:49.24,Default,,0000,0000,0000,,well, zero minus negative two, that's two.
Dialogue: 0,0:07:49.24,0:07:51.20,Default,,0000,0000,0000,,So you see this is\Npretty neat, the slope is
Dialogue: 0,0:07:51.20,0:07:53.57,Default,,0000,0000,0000,,changing all around it, but this is
Dialogue: 0,0:07:53.57,0:07:55.36,Default,,0000,0000,0000,,this is a linear solution
Dialogue: 0,0:07:55.36,0:07:57.40,Default,,0000,0000,0000,,to that original differential\Nequation, that was,
Dialogue: 0,0:07:57.40,0:07:59.73,Default,,0000,0000,0000,,that was pretty cool.