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Dialogue: 0,0:00:06.62,0:00:08.82,Default,,0000,0000,0000,,Hi! Welcome back to recitation.
Dialogue: 0,0:00:08.82,0:00:14.79,Default,,0000,0000,0000,,In lecture, you've been talking about implicitly defined functions and implicit differentiation.
Dialogue: 0,0:00:14.79,0:00:20.03,Default,,0000,0000,0000,,One of the reasons that these are important, or that implicit differentiation
Dialogue: 0,0:00:20.03,0:00:23.40,Default,,0000,0000,0000,,is important, is that sometimes you have a function defined implicitly
Dialogue: 0,0:00:23.40,0:00:27.98,Default,,0000,0000,0000,,and you can't solve for it. You don't have any algebraic method for computing
Dialogue: 0,0:00:27.98,0:00:31.13,Default,,0000,0000,0000,,the function values as a formula.
Dialogue: 0,0:00:31.13,0:00:35.78,Default,,0000,0000,0000,,For example, this function that I've written on the board
Dialogue: 0,0:00:35.78,0:00:39.86,Default,,0000,0000,0000,,that I've called w of x is defined implicitly by the equation that
Dialogue: 0,0:00:39.86,0:00:46.04,Default,,0000,0000,0000,,w of x plus one, quantity times e to the w of x, is equal to x for all x.
Dialogue: 0,0:00:46.04,0:00:51.39,Default,,0000,0000,0000,,So this function, some of its values you can guess.
Dialogue: 0,0:00:51.39,0:00:56.71,Default,,0000,0000,0000,,Like at x equals zero, the function value is going to be negative one.
Dialogue: 0,0:00:56.71,0:01:01.02,Default,,0000,0000,0000,,The reason is that this can't ever be zero, so the only way to get this side to be zero
Dialogue: 0,0:01:01.02,0:01:03.87,Default,,0000,0000,0000,,is if w is negative one, if this term is zero.
Dialogue: 0,0:01:03.87,0:01:08.48,Default,,0000,0000,0000,,So, some of its values are easy to compute, but some of its values aren't.
Dialogue: 0,0:01:08.48,0:01:12.76,Default,,0000,0000,0000,,So for example, if I asked you what w of three halves is,
Dialogue: 0,0:01:12.76,0:01:17.88,Default,,0000,0000,0000,,it's very hard, there's no algebraic way you can manipulate this equation
Dialogue: 0,0:01:17.88,0:01:21.76,Default,,0000,0000,0000,,that will let you figure that out. So in that situation you might still care
Dialogue: 0,0:01:21.76,0:01:26.37,Default,,0000,0000,0000,,about what the function value is. So what can you do?
Dialogue: 0,0:01:26.37,0:01:28.74,Default,,0000,0000,0000,,Well, you can try and find a numerical approximation.
Dialogue: 0,0:01:28.74,0:01:32.65,Default,,0000,0000,0000,,So in this problem I'd like you to try and estimate the value w of three halves
Dialogue: 0,0:01:32.65,0:01:47.63,Default,,0000,0000,0000,,by using a linear approximation of the function w of x in order to compute this value.
Dialogue: 0,0:01:47.63,0:01:58.60,Default,,0000,0000,0000,,As a hint, I've given you that w of one is zero. Right? If you put in
Dialogue: 0,0:01:58.60,0:02:08.80,Default,,0000,0000,0000,,x equals one and w of one equals zero on the left hand side, you do indeed get one, as you should.
Dialogue: 0,0:02:08.80,0:02:19.06,Default,,0000,0000,0000,,So, good! So that will give you a hint about where you could base your linear approximation.
Dialogue: 0,0:02:19.06,0:02:22.95,Default,,0000,0000,0000,,Why don't you pause the video, and take a few minutes to work this out.
Dialogue: 0,0:02:22.95,0:02:32.04,Default,,0000,0000,0000,,Come back and we can work it out together.
Dialogue: 0,0:02:32.04,0:02:36.99,Default,,0000,0000,0000,,All right, welcome back! So, hopefully, you've had a chance to work on this question a little bit.
Dialogue: 0,0:02:36.99,0:02:42.91,Default,,0000,0000,0000,,In order to do this linear approximation that we want, we need to know a base point
Dialogue: 0,0:02:42.91,0:02:46.49,Default,,0000,0000,0000,,and we need to know the derivative of the function at that base point.
Dialogue: 0,0:02:46.49,0:02:51.14,Default,,0000,0000,0000,,Those are the two pieces of data you need in order to construct a linear approximation.
Dialogue: 0,0:02:51.14,0:02:56.29,Default,,0000,0000,0000,,We have a good candidate for a base point here, which is the point one, zero.
Dialogue: 0,0:02:56.29,0:02:59.100,Default,,0000,0000,0000,,So this curve, whatever it looks like, it passes through the point one, zero.
Dialogue: 0,0:02:59.100,0:03:03.17,Default,,0000,0000,0000,,And that's the point we're going to use for our approximation.
Dialogue: 0,0:03:03.17,0:03:15.68,Default,,0000,0000,0000,,So we're going to use the linear approximation w of x is approximately equal to
Dialogue: 0,0:03:15.68,0:03:33.23,Default,,0000,0000,0000,,w prime of one, times x minus one, plus w of one when x is approximately equal to one.
Dialogue: 0,0:03:33.23,0:03:38.19,Default,,0000,0000,0000,,So this is the linear approximation we're going to use. And we have that w of one here is zero.
Dialogue: 0,0:03:38.19,0:03:48.11,Default,,0000,0000,0000,,So this is equal to w prime of one, times x minus one.
Dialogue: 0,0:03:48.11,0:03:51.26,Default,,0000,0000,0000,,(So the w one of zero just goes away.)
Dialogue: 0,0:03:51.26,0:03:58.17,Default,,0000,0000,0000,,In order to estimate w of x, in particular w of three halves, what we need to know is
Dialogue: 0,0:03:58.17,0:04:03.43,Default,,0000,0000,0000,,we need to know the derivative of w. And to get the derivative of w
Dialogue: 0,0:04:03.43,0:04:07.51,Default,,0000,0000,0000,,we need to use, well, we have only one piece of information about w,
Dialogue: 0,0:04:07.51,0:04:16.00,Default,,0000,0000,0000,,that it's defined by this implicit equation. So in order to get the derivative of w
Dialogue: 0,0:04:16.00,0:04:19.18,Default,,0000,0000,0000,,we have to use implicit differentiation. OK?
Dialogue: 0,0:04:19.18,0:04:25.54,Default,,0000,0000,0000,,So let's do that. So if we implicitly differentiate this equation --
Dialogue: 0,0:04:25.54,0:04:28.18,Default,,0000,0000,0000,,so let's start with -- the right hand side's going to be really easy --
Dialogue: 0,0:04:28.18,0:04:31.56,Default,,0000,0000,0000,,we're going to differentiate with respect to x. The right hand side's going to be one.
Dialogue: 0,0:04:31.56,0:04:34.69,Default,,0000,0000,0000,,On the left hand side, it's going to be a little more complicated.
Dialogue: 0,0:04:34.69,0:04:39.38,Default,,0000,0000,0000,,We have a product, and then this piece, we're going to have a chain rule situation.
Dialogue: 0,0:04:39.38,0:04:53.69,Default,,0000,0000,0000,,We have e to the w of x. So, [implicit diff 'n], we're going to take an implicit derivative.
Dialogue: 0,0:04:53.69,0:04:58.84,Default,,0000,0000,0000,,On the left, so, the product rule first. We take the derivative of the first part.
Dialogue: 0,0:04:58.84,0:05:05.69,Default,,0000,0000,0000,,So that's just w prime of x, times the second part. That's e to the w of x.
Dialogue: 0,0:05:05.69,0:05:15.40,Default,,0000,0000,0000,,Plus the first part. That's w of x plus one, times the derivative of the second part.
Dialogue: 0,0:05:15.40,0:05:24.32,Default,,0000,0000,0000,,So the second part is e to the w of x. So that gives me an e to the w of x, times w prime of x.
Dialogue: 0,0:05:24.32,0:05:29.14,Default,,0000,0000,0000,,That's the chain rule. So that's what happens when I differentiate the left hand side.
Dialogue: 0,0:05:29.14,0:05:32.91,Default,,0000,0000,0000,,And on the right hand side, I take the derivative of x and I get one.
Dialogue: 0,0:05:32.91,0:05:40.20,Default,,0000,0000,0000,,OK? Good. So now I've got this equation and I need to solve this equation for w prime.
Dialogue: 0,0:05:40.20,0:05:45.12,Default,,0000,0000,0000,,Because if you look up here, that's what I want. I want a particular value of w prime.
Dialogue: 0,0:05:45.12,0:05:50.83,Default,,0000,0000,0000,,And as always happens in implicit differentiation, from the point of view of this w prime,
Dialogue: 0,0:05:50.83,0:05:54.44,Default,,0000,0000,0000,,it's only involved in the equation in a very simple way.
Dialogue: 0,0:05:54.44,0:06:01.47,Default,,0000,0000,0000,,So there's it multiplied by functions of x and w of x. But not... it's just, you know,
Dialogue: 0,0:06:01.47,0:06:05.92,Default,,0000,0000,0000,,it's just multipled by something that doesn't involve w prime at all.
Dialogue: 0,0:06:05.92,0:06:08.89,Default,,0000,0000,0000,,And here it's multiplied by something that doesn't involve w prime at all.
Dialogue: 0,0:06:08.89,0:06:14.08,Default,,0000,0000,0000,,So you can just collect your w primes and divide through. It's just like solving a linear equation.
Dialogue: 0,0:06:14.08,0:06:29.39,Default,,0000,0000,0000,,So here if we collect our w primes, this is w prime of x, times, it looks like, w of x, plus two
Dialogue: 0,0:06:29.39,0:06:36.41,Default,,0000,0000,0000,,times e to the w of x. Did I get that right?
Dialogue: 0,0:06:36.41,0:06:41.22,Default,,0000,0000,0000,,Looks good. OK, so that's still equal to one.
Dialogue: 0,0:06:41.22,0:06:48.45,Default,,0000,0000,0000,,That means w prime of x is just, well not just, it's equal to
Dialogue: 0,0:06:48.45,0:06:57.84,Default,,0000,0000,0000,,one over w of x plus two, times e to the w of x.
Dialogue: 0,0:06:57.84,0:07:04.03,Default,,0000,0000,0000,,OK, so this is true for every x. But I don't need this equation for every x.
Dialogue: 0,0:07:04.03,0:07:07.78,Default,,0000,0000,0000,,I just need the particular value of w prime at one.
Dialogue: 0,0:07:07.78,0:07:12.19,Default,,0000,0000,0000,,So I'm going to take this equation then and I'm just going to put in
Dialogue: 0,0:07:12.19,0:07:16.61,Default,,0000,0000,0000,,x equals one. So I put in x equals one. I'll do it over here.
Dialogue: 0,0:07:16.61,0:07:21.75,Default,,0000,0000,0000,,So I get w prime of one. And everywhere I had an x, I put in a one.
Dialogue: 0,0:07:21.75,0:07:26.73,Default,,0000,0000,0000,,So actually, in this equation, the only place x appears is in the argument of w.
Dialogue: 0,0:07:26.73,0:07:33.96,Default,,0000,0000,0000,,So this is w of one, plus two, times e to the w of one.
Dialogue: 0,0:07:33.96,0:07:38.71,Default,,0000,0000,0000,,OK, so in order to get w prime of one, I need to know what w of one is.
Dialogue: 0,0:07:38.71,0:07:43.89,Default,,0000,0000,0000,,But I had that. I had it -- it was right back here. There was the - that was my hint to you.
Dialogue: 0,0:07:43.89,0:07:50.71,Default,,0000,0000,0000,,This is why we are using this point as a base point. Which is, we know the value of w for this value of x.
Dialogue: 0,0:07:50.71,0:08:04.27,Default,,0000,0000,0000,,So we take that value. So w of one at zero, so this is just one over two.
Dialogue: 0,0:08:09.93,0:08:14.48,Default,,0000,0000,0000,,OK? So I take that back upstairs to this equation that I had here.
Dialogue: 0,0:08:14.48,0:08:21.18,Default,,0000,0000,0000,,And I have that w of x is approximately equal to w prime of one, times x minus one.
Dialogue: 0,0:08:21.18,0:08:31.04,Default,,0000,0000,0000,,So w of x is approximately equal to (w prime of one we saw is one half)
Dialogue: 0,0:08:31.04,0:08:40.67,Default,,0000,0000,0000,,times x minus one. And that approximation was good near our base point,
Dialogue: 0,0:08:40.67,0:08:46.52,Default,,0000,0000,0000,,so that's good when x is near one.
Dialogue: 0,0:08:46.52,0:08:51.37,Default,,0000,0000,0000,,All right. So this is the linear approximation. I asked for the linear approximation
Dialogue: 0,0:08:51.37,0:08:54.92,Default,,0000,0000,0000,,-- its value at the particular point x equals three halves.
Dialogue: 0,0:08:54.92,0:09:04.21,Default,,0000,0000,0000,,So w of three halves is approximately one half times (well, three halves
Dialogue: 0,0:09:04.21,0:09:12.80,Default,,0000,0000,0000,,minus one is also a half) so this is a quarter.
Dialogue: 0,0:09:12.80,0:09:16.11,Default,,0000,0000,0000,,OK. So this is our estimate for w of three halves.
Dialogue: 0,0:09:16.11,0:09:18.70,Default,,0000,0000,0000,,W of three halves is approximately one fourth.
Dialogue: 0,0:09:18.70,0:09:21.53,Default,,0000,0000,0000,,If you wanted a better estimate, you could try iterating this process,
Dialogue: 0,0:09:21.53,0:09:27.52,Default,,0000,0000,0000,,or choosing some base point even closer if you could figure out
Dialogue: 0,0:09:27.52,0:09:35.31,Default,,0000,0000,0000,,the value of w and x near this point that you're interested in -- three halves.
Dialogue: 0,0:09:35.31,0:09:40.83,Default,,0000,0000,0000,,Just to sum up what we did was we had this implicitly defined function w.
Dialogue: 0,0:09:40.83,0:09:44.64,Default,,0000,0000,0000,,We wanted to estimate its value at a point where we couldn't compute it explicitly.
Dialogue: 0,0:09:44.64,0:09:47.68,Default,,0000,0000,0000,,So what we did, was we did our normal linear approximation method.
Dialogue: 0,0:09:47.68,0:09:53.50,Default,,0000,0000,0000,,We wrote down our normal linear approximation formula.
Dialogue: 0,0:09:53.50,0:09:58.82,Default,,0000,0000,0000,,The only thing that was slightly unusual is that we had to use implicit differentiation.
Dialogue: 0,0:09:58.82,0:10:03.56,Default,,0000,0000,0000,,In order to compute the derivative that appears in the linear approximation, we implicitly differentiated.
Dialogue: 0,0:10:03.56,0:10:08.02,Default,,0000,0000,0000,,OK? So that happened just like normal. And then at the end, we plugged in the values
Dialogue: 0,0:10:08.02,0:10:14.97,Default,,0000,0000,0000,,that we were interested in to actually compute the particular value of that approximation.
Dialogue: 0,0:10:14.97,9:59:59.99,Default,,0000,0000,0000,,So I'll end there.