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Let me start by basically listing the main things
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we've learned over the past three weeks or so.
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And I'll add a few compliments of information about that,
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because there's a few small details
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that I didn't quite clarify and that I should probably
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make a bit clearer, especially what
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happened at the very end of yesterday's class.
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So here's a list of things that should
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be on your review sheet, for the exam.
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So the first thing we learned about the main topic of this unit
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is about functions of several variables.
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So we've learned how to think of functions of two
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of revivals in terms of plotting them, in particular,
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well, not only the graph, but also the contour plot
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and how to read a contour plot.
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And we've learned how to study variations of these functions
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using partial derivatives.
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So remember, we've defined the partial of f
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with respect to some variables, say x,
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to be the rate of change with respect to x
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when we hold all the other variables constant.
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So if you have a function of x and y,
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this symbol means you differentiate with respect to x,
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trading y as a constant.
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And we've learned how to package partial derivatives
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into a vector, the gradient vector.
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So for example, if we have a function of variables,
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that's just the vector with components
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of the partial derivatives.
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And we've seen how to use the gradient vector
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of the partial derivatives to derive various things,
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such as approximation formulas.
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So the change in f when we change x, y, and z slightly
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is approximately equal to, well, there's several terms.
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And I can rewrite this in vector form
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as the gradient dot product, the amount by which the position
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vector has changed.
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So basically, what causes f to change
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is that I'm changing x, y, and z by small amounts,
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and how sensitive f is to each variable
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is precisely what the partial derivatives measure.
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And in particular, we can use, so this approximation
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is called the tangent plane approximation,
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because it tells us, in fact, it amounts
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to identifying the graph of a function with its tangent plane.
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It means that we assume that the function
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depends more or less linearly on x, y, z.
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And if we set these things equal, what we get is actually,
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we are replacing the function by its linear approximation.
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We're replacing the graph by its tangent plane.
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Except, of course, we haven't seen the graph of a function
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of three variables, because that would live in four
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dimensional space.
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So when we think of the graph, that's
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really with a function of two variables.
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That also tells us how to find tangent planes
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to level surfaces.
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We call that the tangent plane to a surface
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given by the equation f of x, y, z equal c.
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At a given point, can be found by looking first
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for its normal vector.
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And we know that the normal vector is actually,
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well, one normal vector is given by the gradient of a function,
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because we know that the gradient is actually pointing
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perpendicularly to the level sets towards higher values
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of a function.
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It gives us the direction of fastest
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increase of a function.
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OK, any questions about these topics?
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OK, so let me add actually a cultural note
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to what we've seen so far about partial derivatives
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and how to use them, which is maybe something
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I should have mentioned a couple of weeks ago.
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So why do we like partial derivatives?
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Well, one obvious reason is we can do all these things.
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But another reason is that really, you
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need partial derivatives to do physics
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and to understand much of the world that's around you,
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because a lot of things actually
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are governed by what's called partial differential equations.
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OK, so that's, if you want, a cultural remark about what
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this is good for.
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So a partial differential equation
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is an equation that involves the partial derivatives
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of a function.
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So you have some function that's unknown that depends
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on a bunch of variables, and a partial differential equation
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is some relation between its partial derivatives.
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So let me see.
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These are equations involving the partial derivatives
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of an unknown function.
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So let me give you an example to see how that works.
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So for example, the heat equation
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is one example of a partial differential equation.
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It's the equation, well, let me write for you
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for space version of it.
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And so the equation, partial f, partial t
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equals some constant times the sum of the second
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partials with respect to x, y, and z.
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So this is an equation where we are trying
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to solve for a function f that depends, actually,
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some four variables, x, y, z, and t.
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And what should you think of?
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What should you have in mind?
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Well, this equation governs temperature.
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So if you think that f of x, y, z, t
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will be the temperature at a point in space, at position x, y, z,
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and at time t, then this tells you how temperature changes
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over time.
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It tells you that at any given point,
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the rate of change of temperature over time
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is given by this complicated expression
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in the partial derivatives in terms of the space coordinates
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x, y, z.
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So if you know, for example, the initial distribution
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of temperature in this room, and if you assume
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that nothing is generating heat or taking heat away,
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so if you don't have any air conditioning or heating going on,
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then it will tell you how the temperature will change
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over time and eventually stabilize to some final value.
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Yes?
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Why do we take the partial derivative twice?
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Why do we take the partial derivative twice?
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Well, that's a question, I'd say.
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It's a question for a physics person.
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But actually, I'm going to, no.
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In a few weeks, we'll actually see a derivation of where
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this equation comes from and try to justify it.
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But really, that's something you will see in a physics class.
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The reason for that is basically physics
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of how heat is transported between particles in a fluid,
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or actually in any medium.
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This constant k actually is called the heat conductivity.
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It tells you how well the heat flows through the material
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that you're looking at.
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So anyway, I'm giving it to you just to show you
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an example of a real life problem, where, in fact,
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you have to solve one of these things.
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So now, how to solve partial differential equations
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is not a topic for this class.
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It's not even a topic for 18 or 3, which
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is called differential equations without partial, which
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means that actually you will learn tools
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to study and solve these equations
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between only one variable involved.
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And you'll see it's already quite hard.
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And if you want more, then later on,
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we have many final classes about partial differential equations.
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But one thing at a time.
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So just I wanted to point out to you
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that very often, functions that you see in real life
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satisfy many nice relations between their partial derivatives.
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So that was, in case you were wondering why
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on the syllabus for today, it said,
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partial differential equations now
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we've officially covered the topic.
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That's basically all we need to know about it.
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OK?
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But we'll get back to that a bit later.
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You'll see.
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OK.
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If there's no further questions, let me continue and go back
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to my list of topics.
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Oh, sorry.
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I should have written down that this equation
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is solved by temperature at the point x, y, z, at time t.
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OK.
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And there are actually many of our interesting partial differential
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equations.
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You will maybe sometimes learn about the wave equation
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that governs how waves propagate in space
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about the diffusion equation, which tells you how, you know,
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when you have maybe a mixture of two fluids,
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how they somehow mix over time.
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And so on.
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There's basically two, every problem you
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might want to consider.
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But there's a partial differential equation to solve.
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OK.
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Anyway, sorry.
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Back to my list of topics.
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So one important application we've seen of partial derivatives
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is to try to optimize things, try
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to solve minimum, maximum problems.
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So remember that we've introduced an open of critical points
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of a function.
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So a critical point is when all the partial derivatives are 0.
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And then there are various kinds of critical points.
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There are maxima.
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There's minimal.
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But there's also saddle points.
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And we've seen a method using second derivatives
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to decide which kind of critical point we have.
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So I should say that's for a function of two variables
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to try to decide whether a given critical point
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is a minimum, a maximum, or a saddle point.
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And we've also seen that actually that's not enough
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to find the minimum of a maximum of a function
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because the minimum of a maximum could occur
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on the boundary.
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So just to give you a small reminder,
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when you have a function of one variable,
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if you're trying to find the minimum and the maximum
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of say a function with graph looks like this,
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well, you're going to tell me quite obviously
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that the maximum is this point up here.
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And that's a point where the first derivative is 0,
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that's a critical point.
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And we use the second derivative to see
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that this critical point is a local maximum.
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But then when we are looking for the minimum of a function,
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well, it's not at a critical point.
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It's actually here at the boundary of the range of values
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that we are going to consider.
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So here, the minimum is at the boundary.
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And the maximum is at a critical point.
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So similarly, when you have a function of several variables,
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say of two variables, for example,
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then the minimum and the maximum will
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be achieved either at a critical point.
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And then we can use these methods to find where they are,
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or somewhere on the boundary of the set of values
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that are allowed.
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So it could be that we actually achieve a minimum
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by making x and y as small as possible,
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maybe letting them go to 0 if they had to be positive,
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or maybe by making them go to infinity.
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So we have to keep our minds open and look
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at various possibilities.
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So we're going to do a problem like that.
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We're going to go over a practice problem
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from the practice test to clarify this.
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So another important cultural application
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of minimum maximum problems in two variables
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that we've seen in class is the least squares
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method to find the best fit line or the best fit anything,
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really, to find when you have a set of data points,
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what's the best linear approximation for this data point?
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And here I have some good news for you.
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While you should definitely know what this is about,
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it will not be on the test.
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APPLAUSE
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But doesn't mean that you should forget everything
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we've seen about it.
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OK?
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Now, what's next on my list of topics?
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So we've seen differentials.
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So remember, the differential of f by definition
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would be this kind of quantity.
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So at first, it looks just like a new way
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to package partial derivatives together
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into some new kind of object.
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Now, what is this good for?
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Well, it's a good way to remember approximation formulas.
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It's a good way to also study how variations in x, y, z
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relate to variations in f.
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In particular, we can divide this by variations,
265
00:16:29.440 --> 00:16:32.600
of, actually, by dx, or by dy, or by dz,
266
00:16:32.600 --> 00:16:35.560
in any situation that we want to find,
267
00:16:35.560 --> 00:16:38.600
or by d of some other variable, to get chain rules.
268
00:16:44.160 --> 00:16:47.280
So the chain rule says, for example,
269
00:16:47.280 --> 00:16:50.600
well, there's many different situations.
270
00:16:50.600 --> 00:16:58.200
But for example, if x, y, and z depend on some other variables,
271
00:16:58.200 --> 00:17:05.960
say, other variables, maybe even u and v,
272
00:17:05.960 --> 00:17:09.720
then that means that f becomes a function of u and v.
273
00:17:09.720 --> 00:17:14.640
And then we can ask ourselves how sensitive is f
274
00:17:14.640 --> 00:17:16.640
to the value of u?
275
00:17:16.640 --> 00:17:22.640
And well, we can answer that.
276
00:17:22.640 --> 00:17:26.640
So the chain rule is something like this.
277
00:17:26.640 --> 00:17:29.440
And let me explain to you again where this comes from.
278
00:17:37.120 --> 00:17:41.920
So basically, what this quantity means
279
00:17:41.920 --> 00:17:45.000
is if we change u and keep v constant,
280
00:17:45.000 --> 00:17:47.080
what happens to the value of f?
281
00:17:47.080 --> 00:17:50.040
Well, why would the value of f change in the first place
282
00:17:50.040 --> 00:17:53.200
when f is just a function of x, y, z and not directly of u?
283
00:17:53.200 --> 00:17:57.440
Well, it changes because x, y, and z depend on u.
284
00:17:57.440 --> 00:18:00.000
So first we have to figure out how quickly x, y, and z
285
00:18:00.000 --> 00:18:02.360
change when we change u.
286
00:18:02.360 --> 00:18:05.600
Well, how quickly we do that is precisely partial x
287
00:18:05.600 --> 00:18:08.680
partial u, partial y partial u, partial z partial u.
288
00:18:08.680 --> 00:18:12.640
These are the rates of change of x, y, z when we change u.
289
00:18:12.640 --> 00:18:16.920
And now when we change x, y, and z, that causes f to change.
290
00:18:16.920 --> 00:18:18.760
How much does f change?
291
00:18:18.760 --> 00:18:22.400
Well, partial f, partial x tells us how quickly f changes
292
00:18:22.400 --> 00:18:24.160
if I just change x.
293
00:18:24.160 --> 00:18:28.400
So I get this, that's the change in f
294
00:18:28.400 --> 00:18:33.040
caused just by the fact that x changes when u changes.
295
00:18:33.040 --> 00:18:35.120
But then why also changes?
296
00:18:35.120 --> 00:18:36.440
Why changes at this rate?
297
00:18:36.440 --> 00:18:39.240
And that causes f to change at that rate.
298
00:18:39.240 --> 00:18:40.760
And z changes as well.
299
00:18:40.760 --> 00:18:43.160
And that causes f to change at that rate.
300
00:18:43.160 --> 00:18:45.560
And v effects add up together.
301
00:18:45.560 --> 00:18:47.320
Does that make sense?
302
00:18:47.320 --> 00:18:47.840
OK.
303
00:18:57.080 --> 00:18:59.680
And so in particular, we can use the chain rule
304
00:18:59.680 --> 00:19:01.360
to do changes of variables.
305
00:19:01.360 --> 00:19:04.720
If we have, say, a function defined in terms of polar
306
00:19:04.720 --> 00:19:06.560
coordinates r and theta.
307
00:19:06.560 --> 00:19:09.080
And we'd like to switch it to rectangular coordinates x
308
00:19:09.080 --> 00:19:11.880
and y, then we can use chain rules
309
00:19:11.880 --> 00:19:13.400
to relate the partial derivatives.
310
00:19:19.360 --> 00:19:22.760
And finally, last but not least, we've
311
00:19:22.760 --> 00:19:26.480
seen how to deal with non-independent variables.
312
00:19:31.600 --> 00:19:37.400
So when our variables say x, y, z are related by some equation.
313
00:19:37.400 --> 00:19:39.560
So one way we can deal with this is
314
00:19:39.560 --> 00:19:43.560
to solve for one of the variables and go back to independent
315
00:19:43.560 --> 00:19:44.600
variables.
316
00:19:44.600 --> 00:19:46.080
But we can't always do that.
317
00:19:46.080 --> 00:19:49.400
Of course, on the exam, you can be sure that I will make sure
318
00:19:49.400 --> 00:19:52.400
that you cannot solve for the variable you want to remove.
319
00:19:52.400 --> 00:19:54.800
That would be too easy.
320
00:19:54.800 --> 00:19:58.840
So then when we have to keep all of them, but handle,
321
00:19:58.840 --> 00:20:05.080
take into account this relation, we've seen two useful methods.
322
00:20:05.080 --> 00:20:12.360
One of them is to find the minimum of a maximum of a function
323
00:20:12.360 --> 00:20:14.760
when the variables are not independent.
324
00:20:14.760 --> 00:20:17.200
And that's the method of Lagrange multipliers.
325
00:20:17.200 --> 00:20:40.000
So remember to find the minimum or maximum of a function f
326
00:20:40.000 --> 00:20:49.080
subject to the constraint g equals constant.
327
00:20:49.080 --> 00:20:55.440
Well, we've write down equations that say that the gradient of f
328
00:20:55.440 --> 00:20:59.720
is actually proportional to the gradient of g.
329
00:20:59.720 --> 00:21:06.440
So there's a new variable here, lambda, the multiplier.
330
00:21:06.440 --> 00:21:11.840
And so for example, if f g, well, I guess here I had
331
00:21:11.840 --> 00:21:12.920
functions of three variables.
332
00:21:12.920 --> 00:21:17.720
So this becomes three equations, f sub x equals lambda g sub x,
333
00:21:17.720 --> 00:21:24.880
f sub y equals lambda g sub y, and f sub z equals lambda g sub z.
334
00:21:24.880 --> 00:21:27.800
And when we plug in the formulas for f and g,
335
00:21:27.800 --> 00:21:30.360
well, we're left with three equations involving the four
336
00:21:30.360 --> 00:21:33.160
variables, x, y, z, and lambda.
337
00:21:33.160 --> 00:21:36.800
What's wrong? Well, we don't have actually four independent
338
00:21:36.800 --> 00:21:37.280
variables.
339
00:21:37.280 --> 00:21:44.160
We also have this relation g equals whatever
340
00:21:44.160 --> 00:21:48.280
the constraint was relating x, y, and z together.
341
00:21:48.280 --> 00:21:50.320
So then we can try to solve this.
342
00:21:50.320 --> 00:21:52.720
And depending on situations, it's sometimes
343
00:21:52.720 --> 00:21:57.640
easy and sometimes it's very hard or even impossible.
344
00:21:57.640 --> 00:22:02.080
So in particular, on the test, I haven't decided yet.
345
00:22:02.080 --> 00:22:04.400
But it could well be that the problem about Lagrange
346
00:22:04.400 --> 00:22:07.360
Multiplayer will just ask you to write the equations,
347
00:22:07.360 --> 00:22:08.600
not to solve them.
348
00:22:08.600 --> 00:22:14.400
So well, I don't know yet.
349
00:22:14.400 --> 00:22:15.680
I'm not promising anything.
350
00:22:15.680 --> 00:22:20.040
But just before you start solving,
351
00:22:20.040 --> 00:22:22.880
check whether the problem asks you to solve them on it.
352
00:22:22.880 --> 00:22:34.440
If it doesn't, then probably you wouldn't.
353
00:22:52.880 --> 00:22:58.600
Whoops.
354
00:22:58.600 --> 00:23:00.800
OK.
355
00:23:00.800 --> 00:23:06.600
So another topic that we saw just yesterday
356
00:23:06.600 --> 00:23:11.480
is constraint partial derivatives.
357
00:23:11.480 --> 00:23:13.720
And I guess I have to re-explain a little bit,
358
00:23:13.720 --> 00:23:18.800
because my guess is that things were not
359
00:23:18.800 --> 00:23:22.320
extremely clear at the end of class yesterday.
360
00:23:22.320 --> 00:23:25.520
So now we're in the same situation.
361
00:23:25.520 --> 00:23:32.040
We have a function, let's say, f of x, y, z,
362
00:23:32.040 --> 00:23:35.800
where the variables x, y, and z are not independent,
363
00:23:35.800 --> 00:23:39.240
but are constrained by some relation of this form.
364
00:23:39.240 --> 00:23:42.040
So some quantity involving x, y, and z
365
00:23:42.040 --> 00:23:45.000
is equal to maybe 0 or some other constant.
366
00:23:45.000 --> 00:23:47.200
And then what we want to know is what
367
00:23:47.200 --> 00:23:50.240
is the rate of change of f with respect
368
00:23:50.240 --> 00:23:55.920
to one of the variables, say x, y, or z,
369
00:23:55.920 --> 00:23:59.200
when I keep the others constant.
370
00:23:59.200 --> 00:24:02.400
Well, I can't keep all the others constant,
371
00:24:02.400 --> 00:24:08.400
because that wouldn't be compatible with this condition.
372
00:24:08.400 --> 00:24:11.360
I mean, that would be the usual, also
373
00:24:11.360 --> 00:24:16.520
called formal partial derivative of f, ignoring the constraint.
374
00:24:16.520 --> 00:24:21.520
To take this into account means that if we variable well
375
00:24:21.520 --> 00:24:24.520
keeping another one fixed, then the third one,
376
00:24:24.520 --> 00:24:27.640
since it depends on them, must also change somehow.
377
00:24:27.640 --> 00:24:30.280
And we must take that into account.
378
00:24:30.280 --> 00:24:38.200
So let's say, for example, we want to find some going
379
00:24:38.200 --> 00:24:39.840
to do a different example from yesterday,
380
00:24:39.840 --> 00:24:42.000
so that if you really didn't like that one,
381
00:24:42.000 --> 00:24:43.080
you don't have to see it again.
382
00:24:43.080 --> 00:24:49.680
So let's say that we want to find the partial derivative
383
00:24:49.680 --> 00:24:53.800
of f with respect to z keeping y constant.
384
00:24:53.800 --> 00:24:56.120
So what does that mean?
385
00:24:56.120 --> 00:25:04.360
That means y is constant, z varies.
386
00:25:04.360 --> 00:25:09.520
And x somehow is mysteriously a function of y and z
387
00:25:09.520 --> 00:25:11.640
through this equation.
388
00:25:11.640 --> 00:25:14.000
And then, of course, because it depends on y,
389
00:25:14.000 --> 00:25:16.240
that means x will vary.
390
00:25:16.240 --> 00:25:20.040
Sorry, x depends on y and z and z varies.
391
00:25:20.040 --> 00:25:22.120
So now we're asking ourselves, what's
392
00:25:22.120 --> 00:25:42.560
the rate of change of f with respect to z in this situation?
393
00:25:42.560 --> 00:25:44.080
And so we have two methods to do that.
394
00:25:47.040 --> 00:25:49.200
So let me start with one with the frontials,
395
00:25:49.200 --> 00:25:53.600
which hopefully you kind of understood yesterday.
396
00:25:53.600 --> 00:25:56.880
But if not, here's the second chance.
397
00:26:03.440 --> 00:26:05.760
So using the frontials means that we'll
398
00:26:05.760 --> 00:26:11.520
try to express df in terms of dz in this particular situation.
399
00:26:11.520 --> 00:26:14.600
So what do we know about df in general?
400
00:26:14.600 --> 00:26:22.520
Well, we know that df is epsilon x dx plus epsilon y dy plus
401
00:26:22.520 --> 00:26:25.320
partial f partial z dz.
402
00:26:25.320 --> 00:26:27.360
That's the general statement.
403
00:26:27.360 --> 00:26:28.840
But of course, we're in a special case.
404
00:26:28.840 --> 00:26:32.880
We're in a special case where first y is constant.
405
00:26:32.880 --> 00:26:45.400
Y is constant means that we can set dy to be 0.
406
00:26:45.400 --> 00:26:49.880
So this goes away and becomes 0.
407
00:26:49.880 --> 00:26:52.920
The second thing is, actually, we don't care about x.
408
00:26:52.920 --> 00:26:57.560
We'd like to get rid of x because it's this dependent variable.
409
00:26:57.560 --> 00:27:02.600
What we really want to do is express df only in terms of dz.
410
00:27:02.600 --> 00:27:14.680
So what we need is to relate dx with dz.
411
00:27:14.680 --> 00:27:18.240
Well, to do that, we need to look at how the variables are
412
00:27:18.240 --> 00:27:19.520
related.
413
00:27:19.520 --> 00:27:22.440
So we need to look at the constraint g.
414
00:27:22.440 --> 00:27:24.160
Well, how do we do that?
415
00:27:24.160 --> 00:27:27.000
We look at the differential of g.
416
00:27:27.000 --> 00:27:35.120
So dg is g sub x dx plus g sub y dy plus g sub z dz.
417
00:27:35.120 --> 00:27:39.880
And that's 0 because we are setting g to always stay constant.
418
00:27:39.880 --> 00:27:42.080
g doesn't change.
419
00:27:42.080 --> 00:27:45.920
If g doesn't change, then we have a relation between dx,
420
00:27:45.920 --> 00:27:48.080
dy and dz.
421
00:27:48.080 --> 00:27:50.560
Well, in fact, we said we are going to look only at the case
422
00:27:50.560 --> 00:27:51.760
where y is constant.
423
00:27:51.760 --> 00:27:54.320
So y doesn't change.
424
00:27:54.320 --> 00:27:56.600
This becomes 0.
425
00:27:56.600 --> 00:27:59.560
Well, now we have our relation between dx and dz.
426
00:27:59.560 --> 00:28:02.320
We know how x depends on z.
427
00:28:02.320 --> 00:28:04.800
And when we know how x depends on z,
428
00:28:04.800 --> 00:28:10.400
we can plug that into here and get how f depends on z.
429
00:28:10.400 --> 00:28:10.920
Let's do that.
430
00:28:20.920 --> 00:28:21.420
OK.
431
00:28:21.420 --> 00:28:32.940
So again, saying that g cannot change and keeping y constant
432
00:28:32.940 --> 00:28:37.420
tells us gx dx plus gz dz is 0.
433
00:28:37.420 --> 00:28:41.820
And we'd like to solve for dx in terms of dz.
434
00:28:41.820 --> 00:28:53.020
That tells us dx should be minus gz dz divided by gx.
435
00:28:53.020 --> 00:28:56.820
So if you want, this is the rate of change of x with respect
436
00:28:56.820 --> 00:29:00.300
to z when we keep y constant.
437
00:29:00.300 --> 00:29:04.540
In our terminology, this is partial z.
438
00:29:04.540 --> 00:29:06.260
No.
439
00:29:06.260 --> 00:29:12.700
It's partial x over partial z with y-held constant.
440
00:29:12.700 --> 00:29:16.380
OK.
441
00:29:16.380 --> 00:29:19.260
This is the rate of change of x with respect to z.
442
00:29:19.260 --> 00:29:23.500
So now when we know that, we are going to plug that into
443
00:29:23.500 --> 00:29:24.500
this equation.
444
00:29:24.500 --> 00:29:37.260
And that will tell us that df is f sub x times dx.
445
00:29:37.260 --> 00:29:38.340
Well, what is dx?
446
00:29:38.340 --> 00:29:46.740
dx is now minus gz over gx dz plus fz dz.
447
00:29:46.740 --> 00:29:56.620
So that will be minus fx gz over gx plus fz times dz.
448
00:29:56.620 --> 00:30:01.660
And so this coefficient here is the rate of change of f
449
00:30:01.660 --> 00:30:06.420
with respect to z in the situation we are considering.
450
00:30:06.420 --> 00:30:12.020
So this quantity is what we call partial f over partial z
451
00:30:12.020 --> 00:30:14.860
with y-held constant.
452
00:30:14.860 --> 00:30:17.660
That's what we wanted to find.
453
00:30:17.660 --> 00:30:20.660
OK.
454
00:30:20.660 --> 00:30:26.020
So now let's see another way to do the same calculation.
455
00:30:26.020 --> 00:30:55.060
And then you can choose which one you prefer.
456
00:30:55.060 --> 00:31:00.180
So the other method is using the chain rule.
457
00:31:06.380 --> 00:31:06.700
OK.
458
00:31:06.700 --> 00:31:14.300
So let's use the chain rule to understand how f depends on z
459
00:31:14.300 --> 00:31:16.980
when y is held constant.
460
00:31:16.980 --> 00:31:18.860
So let me first fight the chain rule brutally.
461
00:31:18.860 --> 00:31:21.340
And then we'll try to analyze what's going on.
462
00:31:21.340 --> 00:31:28.060
So you can just use the version that I have affair as a template
463
00:31:28.060 --> 00:31:31.220
to see what's going on.
464
00:31:31.220 --> 00:31:37.260
But I'm going to explain it more carefully again.
465
00:31:49.940 --> 00:31:50.260
OK.
466
00:31:50.260 --> 00:31:56.500
So that's the most mechanical and no.
467
00:31:56.500 --> 00:31:58.860
Mindless way of fighting down the chain rule.
468
00:31:58.860 --> 00:32:01.780
I'm just saying here, let's say that I'm varying z
469
00:32:01.780 --> 00:32:02.940
keeping y constant.
470
00:32:02.940 --> 00:32:04.460
And I want to know how f changes.
471
00:32:04.460 --> 00:32:09.300
Well, f might change because x might change, y might change,
472
00:32:09.300 --> 00:32:11.260
and z might change.
473
00:32:11.260 --> 00:32:13.900
Now how quickly does x change?
474
00:32:13.900 --> 00:32:16.340
Well, the rate of change of x in the situation
475
00:32:16.340 --> 00:32:20.500
is partial x, partial z with y held constant.
476
00:32:20.500 --> 00:32:27.300
So if I change x at this rate, then f will change at that rate.
477
00:32:27.300 --> 00:32:29.500
Now, y might change.
478
00:32:29.500 --> 00:32:32.140
So the rate of change of y would be the rate of change
479
00:32:32.140 --> 00:32:35.460
of y respect to z holding y constant.
480
00:32:35.460 --> 00:32:36.580
Wait a second.
481
00:32:36.580 --> 00:32:39.180
If y is held constant, then y doesn't change.
482
00:32:39.180 --> 00:32:40.540
So actually, this guy is zero.
483
00:32:40.540 --> 00:32:43.260
And you didn't really have to write that term.
484
00:32:43.260 --> 00:32:46.740
But I wrote it just to be systematic, OK?
485
00:32:46.740 --> 00:32:50.580
If y had been somehow able to change at a certain rate,
486
00:32:50.580 --> 00:32:54.020
then that would have caused f to change at that rate.
487
00:32:54.020 --> 00:32:58.420
Of course, if y is held constant, then nothing happens here.
488
00:32:58.420 --> 00:33:02.860
And finally, while z is changing at a certain rate,
489
00:33:02.860 --> 00:33:08.460
this rate is this one, and that causes f to change at that rate.
490
00:33:08.460 --> 00:33:10.340
And then we add the effects together.
491
00:33:10.340 --> 00:33:13.220
OK, see, it's nothing but the good old chain rule.
492
00:33:13.220 --> 00:33:15.700
Just I've put this extra subscript to tell us
493
00:33:15.700 --> 00:33:19.740
what is held constant and what isn't.
494
00:33:19.740 --> 00:33:21.780
OK?
495
00:33:21.780 --> 00:33:23.660
Now, of course, we can simplify it a bit more
496
00:33:23.660 --> 00:33:28.180
because here, how quickly does z change, if I'm changing z?
497
00:33:28.180 --> 00:33:30.180
Well, the rate of change of z with respect to itself
498
00:33:30.180 --> 00:33:31.060
is just 1.
499
00:33:31.060 --> 00:33:34.660
OK?
500
00:33:34.660 --> 00:33:37.140
OK?
501
00:33:37.140 --> 00:33:40.900
So in fact, the really mysterious part of this
502
00:33:40.900 --> 00:33:43.780
is the one here, which is the rate of change of x
503
00:33:43.780 --> 00:33:45.340
with respect to z.
504
00:33:45.340 --> 00:33:47.660
And to find that, we have to understand the constraint.
505
00:33:50.940 --> 00:33:53.900
So how can we find the rate of change of x with respect to z?
506
00:33:53.900 --> 00:33:56.140
Well, we could use different roles, like we did here,
507
00:33:56.140 --> 00:34:01.540
but we can also keep using the chain rule.
508
00:34:14.340 --> 00:34:18.500
So how can I do that?
509
00:34:18.500 --> 00:34:23.020
Well, I can just look at how g would change with respect
510
00:34:23.020 --> 00:34:24.660
to z when y is held constant.
511
00:34:24.660 --> 00:34:26.700
So I just do the same calculation
512
00:34:26.700 --> 00:34:28.060
but with g instead of f.
513
00:34:33.460 --> 00:34:35.820
But before I do it, let's ask ourselves first,
514
00:34:35.820 --> 00:34:38.740
what is this equal to?
515
00:34:38.740 --> 00:34:43.820
Well, if g is held constant, then when we vary z,
516
00:34:43.820 --> 00:34:45.500
keeping y constant and changing x,
517
00:34:45.500 --> 00:34:47.100
well, g still doesn't change.
518
00:34:47.100 --> 00:34:48.060
It's held constant.
519
00:34:48.060 --> 00:34:56.300
So in fact, that should be 0.
520
00:34:56.300 --> 00:35:00.100
OK, but if we just say that, we're not going to get to that.
521
00:35:00.100 --> 00:35:05.060
So let's see how we can compute that using the chain rule.
522
00:35:05.060 --> 00:35:07.460
Well, the chain rule tells us g changes
523
00:35:07.460 --> 00:35:11.100
because x, y, and z change.
524
00:35:11.100 --> 00:35:12.780
How does it change because of x?
525
00:35:12.780 --> 00:35:18.260
Well, partial g partial x times the rate of change of x.
526
00:35:18.260 --> 00:35:19.900
How does it change because of y?
527
00:35:19.900 --> 00:35:23.580
Well, partial g partial y times the rate of change of y.
528
00:35:23.580 --> 00:35:25.060
But of course, if you're smarter than me,
529
00:35:25.060 --> 00:35:26.900
then you don't need to actually write this one
530
00:35:26.900 --> 00:35:28.180
because y is held constant.
531
00:35:30.940 --> 00:35:33.300
And then there's the rate of change
532
00:35:33.300 --> 00:35:34.540
because z changes.
533
00:35:34.540 --> 00:35:41.540
And how quickly z changes here, of course, is 1.
534
00:35:41.540 --> 00:35:44.700
OK?
535
00:35:44.700 --> 00:35:46.700
So out of this, you get that.
536
00:35:50.700 --> 00:35:52.940
Well, I'm tired of writing partial g partial x.
537
00:35:52.940 --> 00:35:56.340
Let me just write g sub x times partial x
538
00:35:56.340 --> 00:36:05.220
partial z y constant plus g sub z.
539
00:36:05.220 --> 00:36:10.780
And now we found how x depends on z.
540
00:36:10.780 --> 00:36:14.100
Partial x partial z with y held constant
541
00:36:14.100 --> 00:36:19.020
is negative g sub z over g sub x.
542
00:36:19.020 --> 00:36:27.180
Now we plug this into that and we get our answer.
543
00:36:27.180 --> 00:36:30.580
Because it goes all the way up here.
544
00:36:30.580 --> 00:36:31.860
OK?
545
00:36:31.860 --> 00:36:33.060
And then we get the answer.
546
00:36:33.060 --> 00:36:35.580
I'm not going to, well, I guess I can write it again.
547
00:36:35.580 --> 00:36:53.540
So that was partial f partial x times this guy minus g sub z
548
00:36:53.540 --> 00:36:58.820
over g sub x plus partial f over partial z.
549
00:36:58.820 --> 00:37:02.500
And you can observe that this is exactly the same formula
550
00:37:02.500 --> 00:37:05.140
that we had over here.
551
00:37:05.140 --> 00:37:08.100
In fact, let's compare these two methods side by side.
552
00:37:08.100 --> 00:37:10.020
I claimed we did exactly the same thing,
553
00:37:10.020 --> 00:37:12.820
just with different notations.
554
00:37:12.820 --> 00:37:15.500
See, if you take the differential of f
555
00:37:15.500 --> 00:37:18.380
and you divide it by dz in this situation
556
00:37:18.380 --> 00:37:20.380
where y is held constant and so on,
557
00:37:20.380 --> 00:37:23.420
you get exactly this chain rule up there.
558
00:37:23.420 --> 00:37:30.260
But chain rule up there is this guy, df, divided by dz
559
00:37:30.260 --> 00:37:33.380
with y held constant.
560
00:37:33.380 --> 00:37:37.900
And the term involving dy was replaced by zero on both sides
561
00:37:37.900 --> 00:37:42.540
because we knew actually that y is held constant.
562
00:37:42.540 --> 00:37:46.220
Now the real difficulty in both cases comes from dx.
563
00:37:46.220 --> 00:37:49.100
And what we do about dx is we use the constraint.
564
00:37:49.100 --> 00:37:53.020
So here we use it by writing dg equals zero.
565
00:37:53.020 --> 00:37:54.700
Here we write the chain rule for g.
566
00:37:54.700 --> 00:38:00.900
That's the same thing, just divided by dz with y held constant.
567
00:38:00.900 --> 00:38:05.020
So this formula or that formula are the same,
568
00:38:05.020 --> 00:38:08.580
just dividing by dz with y held constant.
569
00:38:08.580 --> 00:38:13.500
And then in both cases we use that to solve for dx.
570
00:38:13.500 --> 00:38:17.900
And then we plugged into the formula for df to express df dz,
571
00:38:17.900 --> 00:38:20.620
or partial f partial dz with y held constant.
572
00:38:24.260 --> 00:38:26.980
So the two methods are pretty much the same.
573
00:38:26.980 --> 00:38:30.500
Quick poll, we'll preface this one.
574
00:38:30.500 --> 00:38:33.700
We'll preface that one.
575
00:38:33.700 --> 00:38:35.420
Majority vote seems to be for differentials,
576
00:38:35.420 --> 00:38:36.940
but it doesn't mean that it's better.
577
00:38:36.940 --> 00:38:39.340
Both are fine.
578
00:38:39.340 --> 00:38:41.860
You can use whichever one you want.
579
00:38:41.860 --> 00:38:42.980
But you could give both a try.
580
00:38:46.660 --> 00:38:50.060
OK, any questions?
581
00:38:50.060 --> 00:38:50.700
Yes?
582
00:38:50.700 --> 00:38:53.700
Or maybe use a direct-direction.
583
00:38:53.700 --> 00:38:54.740
Sorry, can we use it?
584
00:38:54.740 --> 00:38:57.500
Like, in our direction?
585
00:38:57.500 --> 00:38:58.300
Yes.
586
00:38:58.300 --> 00:38:58.800
Thank you.
587
00:38:58.800 --> 00:39:01.260
I forgot to mention it.
588
00:39:01.260 --> 00:39:02.500
Where did that go?
589
00:39:02.500 --> 00:39:04.780
I think I erased that part.
590
00:39:04.780 --> 00:39:14.180
So yes, we need to know directional derivatives.
591
00:39:18.420 --> 00:39:20.580
The pretty much the only thing to remember about them
592
00:39:20.580 --> 00:39:25.300
is that the fDS in the direction of some unit vector u
593
00:39:25.300 --> 00:39:29.980
is just the gradient f dot product with u.
594
00:39:29.980 --> 00:39:35.420
So that's pretty much all we know about them.
595
00:39:35.420 --> 00:39:39.580
Any other topics that I forgot to list?
596
00:39:39.580 --> 00:39:41.540
No?
597
00:39:41.540 --> 00:39:42.540
Yes?
598
00:39:42.540 --> 00:39:44.900
The area is three borders at the time.
599
00:39:44.900 --> 00:39:46.260
Can I erase people at the time?
600
00:39:46.260 --> 00:39:47.740
No, I would need three hands to do that.
601
00:39:47.740 --> 00:39:57.220
OK, let's see.
602
00:39:57.220 --> 00:40:06.140
OK, I think what we should do for is look quickly
603
00:40:06.140 --> 00:40:07.660
at the practice test.
604
00:40:07.660 --> 00:40:10.140
I mean, given the time, you will mostly
605
00:40:10.140 --> 00:40:12.580
have to think about it yourselves.
606
00:40:12.580 --> 00:40:18.940
But so hopefully you have a copy of the practice exam.
607
00:40:18.940 --> 00:40:20.540
Oh, yes.
608
00:40:20.540 --> 00:40:23.540
OK, so let's look at the second.
609
00:40:23.540 --> 00:40:26.220
So the first problem is a simple problem.
610
00:40:26.220 --> 00:40:28.420
Find the gradient, find an approximation formula.
611
00:40:28.420 --> 00:40:30.540
Hopefully you know how to do that.
612
00:40:30.540 --> 00:40:32.860
The second problem is one about reading a contour plot.
613
00:40:32.860 --> 00:40:35.260
And so before I let you go for the weekend,
614
00:40:35.260 --> 00:40:37.380
I want to make sure that you actually know
615
00:40:37.380 --> 00:40:42.940
how to read a contour plot.
616
00:40:42.940 --> 00:40:52.340
So one thing I should first mention is this problem
617
00:40:52.340 --> 00:40:54.980
asks you to estimate partial derivatives by reading a contour
618
00:40:54.980 --> 00:40:55.260
plot.
619
00:40:55.260 --> 00:40:56.140
We haven't done that.
620
00:40:56.140 --> 00:40:57.540
So that won't be actually on the test.
621
00:40:57.540 --> 00:40:59.500
We'll be doing qualitative questions
622
00:40:59.500 --> 00:41:01.580
like what is the sign of a partial derivative?
623
00:41:01.580 --> 00:41:04.700
Is it 0, less than 0, more than 0?
624
00:41:04.700 --> 00:41:07.660
You don't need to bring a roller to estimate partial derivatives
625
00:41:07.660 --> 00:41:09.140
in the way that this problem asks you to.
626
00:41:27.620 --> 00:41:30.140
OK.
627
00:41:30.140 --> 00:41:31.140
I guess.
628
00:41:31.140 --> 00:41:38.420
So let's look at problem 2b.
629
00:41:38.420 --> 00:41:41.660
The problem 2b is asking you to find a point at which h
630
00:41:41.660 --> 00:41:44.100
equals 2200.
631
00:41:44.100 --> 00:41:46.020
Partial h partial x equals 0.
632
00:41:46.020 --> 00:41:49.460
And partial h partial y is less than 0.
633
00:41:49.460 --> 00:41:52.940
So let's try and see what is going on here.
634
00:41:52.940 --> 00:41:56.380
So a point where f equals 2200, well,
635
00:41:56.380 --> 00:41:58.220
that should be probably unverloved curve.
636
00:41:58.220 --> 00:42:01.820
But say 2200.
637
00:42:01.820 --> 00:42:06.820
And here's the level 2200.
638
00:42:06.820 --> 00:42:09.060
OK.
639
00:42:09.060 --> 00:42:12.100
Now I want partial h partial x to be 0.
640
00:42:12.100 --> 00:42:15.820
That means if I change x, keeping y constant,
641
00:42:15.820 --> 00:42:18.620
the value of h doesn't change.
642
00:42:18.620 --> 00:42:26.700
So which points on the level curve satisfy that property?
643
00:42:26.700 --> 00:42:28.740
Yeah, it's the top and the bottom.
644
00:42:28.740 --> 00:42:29.780
OK.
645
00:42:29.780 --> 00:42:35.620
If you are here, for example, and you move in the x direction,
646
00:42:35.620 --> 00:42:37.980
well, you see as you get to where from the left,
647
00:42:37.980 --> 00:42:41.420
the height first increases and then decreases.
648
00:42:41.420 --> 00:42:43.660
It goes for a maximum at that point.
649
00:42:43.660 --> 00:42:45.860
So at that point, the partial derivative
650
00:42:45.860 --> 00:42:48.260
is 0 with respect to x.
651
00:42:48.260 --> 00:42:49.500
And same here.
652
00:42:49.500 --> 00:42:52.980
OK.
653
00:42:52.980 --> 00:42:57.460
Now we also told, and let's find partial h partial y
654
00:42:57.460 --> 00:42:58.860
less than 0.
655
00:42:58.860 --> 00:43:03.300
So that means if we go north, we should go down.
656
00:43:03.300 --> 00:43:05.860
Well, which one is it?
657
00:43:05.860 --> 00:43:07.500
Top or bottom?
658
00:43:07.500 --> 00:43:08.020
Top.
659
00:43:08.020 --> 00:43:09.100
Yes.
660
00:43:09.100 --> 00:43:15.220
Here, if you go north, then you go from 2200 down to 2100.
661
00:43:15.220 --> 00:43:21.260
So this is where the point is.
662
00:43:21.260 --> 00:43:23.060
So now the problem here was also asking you
663
00:43:23.060 --> 00:43:25.060
to estimate partial h partial y.
664
00:43:25.060 --> 00:43:27.980
And just if you're curious how you would do that,
665
00:43:27.980 --> 00:43:30.740
well, you would try to figure out how long it takes
666
00:43:30.740 --> 00:43:33.540
before you reach the next level curve.
667
00:43:33.540 --> 00:43:33.860
OK.
668
00:43:33.860 --> 00:43:40.940
And you see that so here, from here to here,
669
00:43:40.940 --> 00:43:46.740
so to go from q to this new point, say, q prime,
670
00:43:46.740 --> 00:43:49.940
the change in y, well, you'd have to read the scale, which
671
00:43:49.940 --> 00:43:54.300
was down here.
672
00:43:54.300 --> 00:43:55.820
OK.
673
00:43:55.820 --> 00:44:00.460
That would be about something like 300.
674
00:44:00.460 --> 00:44:04.020
What's the change in height when you go from q to q prime?
675
00:44:04.020 --> 00:44:06.940
Well, you go down from 2200 to 2100.
676
00:44:06.940 --> 00:44:10.660
That's about, well, that's actually
677
00:44:10.660 --> 00:44:13.820
minus 100, exactly.
678
00:44:13.820 --> 00:44:14.540
OK.
679
00:44:14.540 --> 00:44:21.060
And so delta h about delta y is about minus 1.3.
680
00:44:21.060 --> 00:44:25.580
Well, minus 100 over 300, which is, whoops, getting out
681
00:44:25.580 --> 00:44:30.700
of minus 1.3.
682
00:44:30.700 --> 00:44:38.580
And that's an approximation for the partial derivative.
683
00:44:38.580 --> 00:44:42.780
So that's how you would do it.
684
00:44:42.780 --> 00:44:43.900
Now, OK.
685
00:44:43.900 --> 00:44:47.940
Let me go back to other things.
686
00:44:47.940 --> 00:44:50.460
So if you look at this practice exam, basically,
687
00:44:50.460 --> 00:44:51.780
there's a bit of everything.
688
00:44:51.780 --> 00:44:53.980
And it's kind of fairly representative
689
00:44:53.980 --> 00:44:57.620
of what might happen on Tuesday.
690
00:44:57.620 --> 00:45:00.180
So there will be a mix of easy problems
691
00:45:00.180 --> 00:45:02.340
and of harder problems.
692
00:45:02.340 --> 00:45:04.060
Expect something about computing,
693
00:45:04.060 --> 00:45:06.340
gradients, approximations, space of change.
694
00:45:06.340 --> 00:45:09.860
Expect a problem about reading a control plot.
695
00:45:09.860 --> 00:45:14.140
Expect one about a min max problem.
696
00:45:14.140 --> 00:45:15.900
Something about Lagrange multipliers.
697
00:45:15.900 --> 00:45:17.700
Something about the chain rule.
698
00:45:17.700 --> 00:45:19.860
And something about constrained partial derivatives.
699
00:45:19.860 --> 00:45:45.900
And pretty much all of the topics are going to be there.