WEBVTT
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Okay, um, I'll get started. Um, so, just a couple of preliminary things.
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The office hours on Thursday and Friday will be via zoom.
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And I'm still trying to figure out whether those will continue to be on soon. Or else, start to do them in person.
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I do expect to return to in person lecture.
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Next week, and I hope for the rest of the semester.
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Today was just an unfortunate unexpected situation.
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But there's really nothing else keeping me from teaching in person. So do you plan on doing that.
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Um, let's see.
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Any, any, buddy. Want to ask me anything right now.
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Okay. Um, so
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today I'm going to.
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Today we want to start talking about functions.
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Um, and so then so this is where we start doing multi variable cactus for real.
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So, um,
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so more than one beer.
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And if you're a call, last time I talked about a function as being a box.
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And I'm going to call the box. So, hey, is the name of the function.
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And I'm purposely not using the letter F to emphasize function can be named anything.
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And I'm in.
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In, calc one and count to use functions where the input was a single number.
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We're going to study in this class functions where the inputs are consists of more than one beer.
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More than one number. And so if you're PNQR inputs
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to function a.
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Then there's an output on. Now the output, there can be any number of outputs.
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And we will see that, I mean you've already seen that with the vector value function, where the output is a whole vector which means it's a two or three numbers here that the big, we're going to start off where the output is just a single note.
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So this this diagram represents a function that takes to input numbers and and generates as an output a single number.
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Okay. And, you know, when we write something like this.
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You know, so this means, sir, are equal to the output
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of a with TQSMM and this is something I just want to point out to whenever you write down an equation, that's equivalent to a full English sentence.
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Okay. And, and in general I do want you to write in full sentences. And so that means usually
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equations.
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Okay. Um, and then.
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Let's just do a quick example.
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Um, I can write a us to, to be a squared plus t minus t square
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time.
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And if I have a function like that and a two, three, is the obvious two square plus two times three minus three squared, which is four plus six minus nine.
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Um, another thing you might write is a test plus tt
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con.
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I'm now, um, How do I compute that. So, the point is here, the s&t in this formula, the last one I wrote down. The s&p are not the same as some Ts in the original definition.
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So, I'm so in order to figure this out.
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The trick is first change the names of the variables in the original formula from it.
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So, so this thing is going to be the same thing as this, but just for the names change so it's going to be square plus pq minus cues.
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So that's so the two main things are just the same thing written twice.
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Just where the names changed.
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And so now I can write a plus tt.
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And then so now I can just sit best place to to be PQTT.
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And so I see I get asked us to square was asked once two times t minus t square.
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Okay. Um, so it's going to be important to remember
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how to deal with formulas, like this, where if you divide the original function using letters. S and T and then later, you want to evaluate the function in terms of s&t but the s&p are not the same as empty as before, you got to know how to deal with.
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Okay, so, um, so that was my just
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something to take note.
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Um, now. Um, if we have a function with two inputs, and one output. So, we can draw a graph
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to.
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I'm so I'm.
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So for example, let's look at the function f of x y equals the square root of one minus x squared minus once
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I'm in here the domain.
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I'm discouraged it's only going to be defined as x squared plus y squared.
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This lesson and both on one.
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And what we can do is write down z equals f of x, on.
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So this.
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To find the surface
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and.
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x y z space.
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Right. And I'm the see what it is. I'm here that means that z equals one minus x squared plus y squared.
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And if I do a little algebra, see that this equation.
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But there's another condition is it see it has to be positive because the skirt is always positive.
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So, um, and but x&y can be any
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domain in any anything in the unit test. So, what you have
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is, if you think of a unit circle
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in the xy plane.
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And then the graph is going to be a surface. So, I'm.
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So, this region here in the xy plane is the domain.
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Okay.
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And then the surface on the graph is going to be the sphere. So this is the upper half
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this.
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So, so, then so this is. So again, any point x y, in the next five plane.
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Then you go vertically.
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And it where it's a surface is where z is equal to f of x one.
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So, it's just a two dimensional analog of a graph that you learned in.
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So a graph of a function of one variable is a curve and the xy plane, where for each x, y is the output of the function. So here, it's a surface where for each x and y.
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z coordinate is the output of the function.
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And this is
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simple example.
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But.
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Another example
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is we can draw on a circular
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Blackboard.
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So this would be z equals x squared plus one.
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And we saw this already, but let's just so,
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so Ashley may not call it z because I want to say we're looking at a function.
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And so, the graph.
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So the graph
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is this.
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Yeah. And we, we saw what that was.
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Looks like
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this, where if you slice it horizontally.
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So, um, so a horizontal
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face is a circle.
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and
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can and you can see it's a radius
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squared.
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And the reason you know to make a circle of radius.
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Radius z is because the equation is a square plus y squared equals z, z has to be positive. So, this.
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And so, the radius of the circle is our.
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Okay, and and Vertical, vertical face
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is.
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So for example, if y equals zero so you looking at the XZ plane.
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Then we get z equals x.
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And so that would be perfectly
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fine.
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Another example
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is the graph of a linear function.
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So.
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So for example, I could have a function like this.
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So this is, is z equals six minus three x minus two y, or three X to Y z equals six. So we know that this is
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a point.
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So graph of a linear function is always a plane.
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Okay. Um, and let's suppose.
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Well, okay. So what is this playing.
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Um. Now, it's actually very hard to draw planes in general, three screens. So, I'm always there. Whenever I give a lecture, I want to give an example of a graph of a linear function or equation of a plane.
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I am always going to choose my examples very carefully so I can draw.
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And now why is this one easy to draw this easier than other points. Well, the idea is that if you look at where this claim process, each axis. It's not a simple place.
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So for example, where does it cross the axon.
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Well, the x axis is where y amp z or zero. So you get three x equals six.
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So what's
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here. If you said y and z equals zero, then you see me with
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x equals two.
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So we know the plane process there.
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Then we just do it again, and on the y axis is when x and 00. And so we can see that.
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Playing processor wipers, three.
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And then finally ZFX and wires zero process, sex, and so therefore the playing.
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There's going to be the plane that contains this trying.
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so just point.
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But then, so, because this plane crosses the axes, the positive half of each access.
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It's easy to draw because the process to first talked and in a nice triangle.
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And so, almost every example I draw will be something like this, and to draw the plane you just check where the plane crosses, he chances. And if they all cross positive on the positive half of the axis, then you can always draw a triangle.
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Okay.
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So that's one way to. So, um, so just backing up for a sec. So, um, well let me say there are three ways to describe a function.
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What's fun for you means.
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So one is a formula, which you're familiar with. So for example, f of x y equals three x y squared.
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To is a graph with, which we've just seen.
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And the third one I want to discuss our current course. And they're also called level curves.
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And I'm the discussion is very simple. These are just a horizontal traces
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different lights.
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Lights just means different values.
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Okay.
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So, um, so, so let's let's.
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Um, and the idea, so.
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So look at the picture is something like this.
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a graph of a surface, it's supposed to look like that.
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Then.
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So normally, um, so actually let me say, I'm usually equally spaced
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values.
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So, so what you would do here is you would.
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Slice horizontally.
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Um, so if you imagine that this is like a piece of cake or something, then you're making slices of equal thickness. Every slice as the same thickness, and you're looking at the curves that form the edge of the slice.
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Every, every place you have a slice. So,
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so the idea now is to put all these slices, these curves. So you can see that the red lines show you where the horizontal plane intersects the surface.
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And so now what you can do is you draw the xy plane.
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And then you get.
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So you get something like this.
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So, if you take the slice right at the tip, you get a single point. And otherwise, you get this. So for example, if I for this one was iz equals three.
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z equals two.
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z equals one.
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And then this one could be z equals zero.
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So then the contract box, you could put label them accordingly.
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So the data z equals zero equals one equals to C equals two.
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And there's going to be number one.
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Anyway.
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Okay.
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Um, and so this is called
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for con format.
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And I'm.
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And this is a very useful way to describe a function.
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And in fact it's useful enough that it's used quite commonly, it's called a topographical map.
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And it's used by hikers.
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When you hike in
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on a trail somewhere you know wilderness or apart. So, why is that it's because
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it shows you where the hills are, and the valleys are so, so for example, you might have a topographical map that looks like this. And
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I'm, and I'm.
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It shows you, and then it's labeled with, you know, five equals.
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100 equals 200.
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And then here this dot might be labeled equals 400.
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And let's see this one is equals 200.
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And then this could be a equals 100.
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And then this could be a equals zero.
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So, on the right so this, so that would tell you that this one is the top of the hill.
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Because the height is the biggest and.
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And then here. This will be the bottom
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of a bowl possible.
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So, it tells you
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how you know which way is up which way is down. Notice and if your trail follows one of these contours, then the heights not changing.
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So you know that's going to be a relatively easy walk, you're just going to be. So if you're walking, say about on on a path that's along with contour.
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You won't be going up or down. On the other hand, if you go perpendicularly to the contour, you're going to be either going up or down. Basically at the fastest rate possible.
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So that's going perpendicular so the contour is basically going up or down as quickly as possible. And then that's a good thing.
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If you're moving along the Silk Route to the contour then you're going up or down, but at a slower pace.
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Right so so that's why it's very useful when you hope you have an idea of what how hard the height is going to be what to expect.
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So, let's do some examples. So first, let's look at the contour of a linear.
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So here, f of x y is going to be AXPYYC.
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So, so the contour.
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Let's say. So, H equals f of x y is going to be a mx plus b yc because he or a mx plus b equals c minus x minus.
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And you can see that right away that this means that the contours
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are parallel lines.
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Yea big never change that they're the same for every contour. So you'll get parallel lines, and, and.
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And you can see that as he changes
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on the line.
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Just shifts
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shifts upward
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by distance age.
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So what this means that if.
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So,
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for evenly spaced values.
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Well, Hu get evenly spaced
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on parallel ones.
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So, the contour of
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linear function is always going to look something like this.
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If you choose evenly spaced. So this could be z equals minus one. C equals zero. z equals one equals two. So, so, for evenly spaced
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values of z.
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back.
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So that's one example.
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Okay. Um. The next example let's, let's do a rambling.
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And so let's do.
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After Effects one.
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Okay.
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Um,
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I see a question here and I'll just say I've no idea when
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somebody else says the answer.
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Please feel free. So, um, this is nine minus x squared minus y squared. And let's just restrict our attention to the desk
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where this is where zS positive
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one.
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Yeah.
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Okay, so.
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So, the graph.
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You know what the graph looks like.
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It just looks like a
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parabola.
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Um, and so, You also know that if you slice it
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would just get a circle.
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So, um, so the contours
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are all circles.
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Right now we want to get more information about it, because there are lots of surfaces, where the contours are also also.
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So let's, let's do this
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more carefully.
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And actually I guess I do both.
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So let's try. Um, so let's first do z equals three.
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So then this is
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x squared.
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tables three equals nine.
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So this means that nine minus x squared minus y squared equals nine, which means that x squared plus y squared equals zero.
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So you get a single point.
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So, let me on x one point.
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So, the first contour is just this one. So that's the equals nine.
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Ok.
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Now let's do z equals six.
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z equals six.
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Means nine minus x squared minus. So let me remind you that we're doing the contours of nine minus x squared minus one.
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So that means nine minus x squared minus y squared equals six.
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So that means, x squared plus y squared equals three, which is the square,
00:31:50.000 --> 00:31:53.000
which is 1.7.
00:31:53.000 --> 00:31:59.000
So, um, so if I mark off
00:31:59.000 --> 00:32:06.000
to 123123.
00:32:06.000 --> 00:32:11.000
That's roughly, you
00:32:11.000 --> 00:32:20.000
This is supposed to be a circle. And so this is z equals six.
00:32:20.000 --> 00:32:21.000
Right.
00:32:21.000 --> 00:32:26.000
Now let's do z equals three.
00:32:26.000 --> 00:32:38.000
And that's mine, minus x squared minus y squared equals three x x squared plus y squared equals six.
00:32:38.000 --> 00:32:42.000
Which is us
00:32:42.000 --> 00:32:44.000
six square.
00:32:44.000 --> 00:32:52.000
What's the square root of six square square to six.
00:32:52.000 --> 00:32:56.000
That's going to be
00:32:56.000 --> 00:33:03.000
roughly 2.5 Square.
00:33:03.000 --> 00:33:12.000
So that's going to be here.
00:33:12.000 --> 00:33:14.000
One.
00:33:14.000 --> 00:33:19.000
Let's do the equals zero.
00:33:19.000 --> 00:33:22.000
m
00:33:22.000 --> 00:33:29.000
equals zero is nine minus x squared minus y squared equals zero.
00:33:29.000 --> 00:33:42.000
That's x squared plus y squared equals nine, which is three square. So that's going to be a circle of radius.
00:33:42.000 --> 00:33:50.000
Right, so this is the zero z equals three. z equals six.
00:33:50.000 --> 00:33:53.000
Okay. I'm.
00:33:53.000 --> 00:34:01.000
Now, I'm actually, um,
00:34:01.000 --> 00:34:04.000
let's do one more.
00:34:04.000 --> 00:34:16.000
But is he equals minus three, then I get x squared plus y squared equals 12, which is
00:34:16.000 --> 00:34:20.000
12 square, which is too.
00:34:20.000 --> 00:34:27.000
So, um, so if I extend
00:34:27.000 --> 00:34:32.000
this out, it's gonna be like.
00:34:32.000 --> 00:34:37.000
Okay, so that's.
00:34:37.000 --> 00:34:44.000
Mom. Now, let me. Okay, so that's what we're map of a parabola it looks.
00:34:44.000 --> 00:34:49.000
Let me let me draw the parabola it again.
00:34:49.000 --> 00:34:53.000
So the prevalent looks.
00:34:53.000 --> 00:34:56.000
This.
00:34:56.000 --> 00:35:07.000
Notice that I'm as you go out from the origin or away from the z axis, the current, the surface gets deep.
00:35:07.000 --> 00:35:14.000
So, so your surface is deeper.
00:35:14.000 --> 00:35:18.000
And over here, it's shallow.
00:35:18.000 --> 00:35:25.000
Okay, so no other words if this were inhale, when you start at the top. It starts off.
00:35:25.000 --> 00:35:34.000
Going down slowly. And then as you go further down you can see that for the same amount of horizontal distance, you're going down faster.
00:35:34.000 --> 00:35:48.000
Okay, now how you can see that for the content. So notice what happens with the conference, the contour start off farther apart from each other. So for example just going from nine to six.
00:35:48.000 --> 00:35:58.000
You have to go further. So, near the top to go down, say three yards.
00:35:58.000 --> 00:36:05.000
And let's say three meters, you have to go horizontal distance of 1.7.
00:36:05.000 --> 00:36:17.000
On the other hand, when you're all the way out here to go down three meters. You only have to go walk horizontally point four.
00:36:17.000 --> 00:36:34.000
Okay, so, um, so in other words, the further apart the contours are the longer you have to walk horizontally to go down the vertical distance. The closer the contours are the shorter distance you need to go to go down to certain amount.
00:36:34.000 --> 00:36:46.000
So, so, So the conclusion is closely
00:36:46.000 --> 00:36:51.000
spaced on tours.
00:36:51.000 --> 00:36:55.000
And let me emphasize, or,
00:36:55.000 --> 00:36:58.000
or evenly.
00:36:58.000 --> 00:37:07.000
Space values, see.
00:37:07.000 --> 00:37:16.000
So, implies the sea surface
00:37:16.000 --> 00:37:23.000
and widely space
00:37:23.000 --> 00:37:31.000
on tours.
00:37:31.000 --> 00:37:35.000
named a shower
00:37:35.000 --> 00:37:38.000
surface.
00:37:38.000 --> 00:37:42.000
this is easy to forget.
00:37:42.000 --> 00:37:53.000
But, again, you want to think about how far you have to walk to go down a certain amount or up those certain.
00:37:53.000 --> 00:38:15.000
And the idea is that if you have to walk further to go down a certain distance than the surface is going down very gently. On the other hand if you can go down the same distance vertically by going shorter distance horizontally, then it's the.
00:38:15.000 --> 00:38:22.000
And so in fact you can see that with a picture like this.
00:38:22.000 --> 00:38:24.000
So,
00:38:24.000 --> 00:38:32.000
so the picture of a time format shows you the horizontal distance is not the vertical.
00:38:32.000 --> 00:38:41.000
So, so it's a little confusing, but this is something you want to try to remember.
00:38:41.000 --> 00:38:58.000
So, linear function evenly spaced on tours, so you're always going down the aisle or uphill at the same rate on, and then if the contour spacing changes, then it's telling you sometimes you're going to be going up don't quickly.
00:38:58.000 --> 00:39:05.000
And sometimes that's more slow.
00:39:05.000 --> 00:39:09.000
And
00:39:09.000 --> 00:39:14.000
I'm.
00:39:14.000 --> 00:39:18.000
Now one thing you can do with contours.
00:39:18.000 --> 00:39:40.000
This you can estimate the value.
00:39:40.000 --> 00:39:48.000
And this is where easy, there's not much tables back. So let's slide.
00:39:48.000 --> 00:39:53.000
And I have mark it off.
00:39:53.000 --> 00:40:17.000
12449 suppose I have confidence. So maybe this this
00:40:17.000 --> 00:40:24.000
inspires let's see, 01.
00:40:24.000 --> 00:40:28.000
The two.
00:40:28.000 --> 00:40:38.000
So let's see plus minus one, minus.
00:40:38.000 --> 00:40:49.000
Sarah. Let's make this.
00:40:49.000 --> 00:41:17.000
Okay. So, suppose I pick a point value for x and y. So, let's suppose I asked what f of two comma three.
00:41:17.000 --> 00:41:29.000
So at this time for now. And you're asked to estimate the value of the function. When access three and wise to, well you just find that corresponding place.
00:41:29.000 --> 00:41:32.000
So it's right here.
00:41:32.000 --> 00:41:39.000
You see that it's between the contours leak was minus two, and z equals minus one.
00:41:39.000 --> 00:41:50.000
So, and maybe allies slightly closer to minus two, one. So this lies. This is between
00:41:50.000 --> 00:41:56.000
the equals minus one. z equals minus two.
00:41:56.000 --> 00:42:05.000
And it's a little closer
00:42:05.000 --> 00:42:09.000
to z equals.
00:42:09.000 --> 00:42:19.000
So, so we'll just say that three comma two is roughly 1.7.
00:42:19.000 --> 00:42:28.000
Okay, so it's a eyeball approximation, but for many purposes, that can be about.
00:42:28.000 --> 00:42:35.000
So, This is a useful, but very simple.
00:42:35.000 --> 00:42:43.000
Minus one points that is correct.
00:42:43.000 --> 00:43:01.000
Okay, so, um, so that's a basic introduction to font or maps for a function of two variables.
00:43:01.000 --> 00:43:08.000
Now we want to turn two functions of three variables.
00:43:08.000 --> 00:43:18.000
So, function.
00:43:18.000 --> 00:43:32.000
So for example, XYZEXY was wise for blue screens.
00:43:32.000 --> 00:43:33.000
So something like this.
00:43:33.000 --> 00:43:39.000
So you have three inputs and a single.
00:43:39.000 --> 00:43:49.000
I'm now so I'm so description.
00:43:49.000 --> 00:43:58.000
So as we just saw here you can use a formula.
00:43:58.000 --> 00:44:04.000
Well, what about a graph.
00:44:04.000 --> 00:44:19.000
So, the graph of a function with two inputs, is a surface in three dimensional space graph of a function of three valuable is in four dimensional space.
00:44:19.000 --> 00:44:23.000
I don't know about you but I have no idea how to drive.
00:44:23.000 --> 00:44:26.000
So this doesn't work.
00:44:26.000 --> 00:44:30.000
So we can't describe
00:44:30.000 --> 00:44:32.000
function or three variables as a graph.
00:44:32.000 --> 00:44:37.000
However, we can do.
00:44:37.000 --> 00:44:41.000
Why is that a contour.
00:44:41.000 --> 00:44:43.000
So for example.
00:44:43.000 --> 00:44:48.000
So example
00:44:48.000 --> 00:44:53.000
was a, f of x y z
00:44:53.000 --> 00:45:11.000
is x squared plus y square the square, then the contour with technicals one corresponds to this,
00:45:11.000 --> 00:45:17.000
which we know this is a sphere
00:45:17.000 --> 00:45:19.000
radius.
00:45:19.000 --> 00:45:20.000
One.
00:45:20.000 --> 00:45:26.000
So, in general,
00:45:26.000 --> 00:45:29.000
the contour
00:45:29.000 --> 00:45:42.000
of a function x y z contours are surfaces
00:45:42.000 --> 00:45:45.000
In Greece.
00:45:45.000 --> 00:45:59.000
So just the contour of a function of two variables with a curve and two dimensional space function or three variables, the contour is our services.
00:45:59.000 --> 00:46:01.000
Freedom.
00:46:01.000 --> 00:46:02.000
Okay.
00:46:02.000 --> 00:46:07.000
And so you can draw those.
00:46:07.000 --> 00:46:09.000
And then again there use.
00:46:09.000 --> 00:46:13.000
Okay, so, um, so.
00:46:13.000 --> 00:46:24.000
So we can draw pictures in pencil.
00:46:24.000 --> 00:46:38.000
But as I said earlier, it can be hard to draw pictures and I'm so,
00:46:38.000 --> 00:46:48.000
So let's do some examples. Simple.
00:46:48.000 --> 00:46:55.000
Okay so linear from.
00:46:55.000 --> 00:47:07.000
Now, f of x y z Mikey x plus y su su fee.
00:47:07.000 --> 00:47:11.000
So, for
00:47:11.000 --> 00:47:23.000
me, that was the one that says he was gay. Because some pants and age by and.
00:47:23.000 --> 00:47:33.000
And so then, this is a linear equation,
00:47:33.000 --> 00:47:38.000
size, this is a constant.
00:47:38.000 --> 00:47:49.000
So, Um, and so what we see is that every con for
00:47:49.000 --> 00:47:52.000
the same
00:47:52.000 --> 00:47:58.000
normal direction.
00:47:58.000 --> 00:48:02.000
Namely, ABC
00:48:02.000 --> 00:48:05.000
coefficients have a version.
00:48:05.000 --> 00:48:18.000
Um, and so, and the height of the plane. So they're parallel planes
00:48:18.000 --> 00:48:23.000
and
00:48:23.000 --> 00:48:29.000
evenly spaced
00:48:29.000 --> 00:48:33.000
while he is the height
00:48:33.000 --> 00:48:35.000
niche.
00:48:35.000 --> 00:48:48.000
I'm using the word height but, you know, it's a four dimensional high. So, you can't use the, we can't think of a literal
00:48:48.000 --> 00:48:50.000
are.
00:48:50.000 --> 00:48:59.000
Well, actually you can win sorry we're looking at points evenly spaced.
00:48:59.000 --> 00:49:02.000
Give.
00:49:02.000 --> 00:49:11.000
Yeah, but they just not give evenly spaced
00:49:11.000 --> 00:49:15.000
out.
00:49:15.000 --> 00:49:32.000
So that's exactly the analog of what happened to for contour contours of a linear function up to everything up just went up on one.
00:49:32.000 --> 00:49:33.000
Okay.
00:49:33.000 --> 00:49:39.000
So that's something is pretty straightforward.
00:49:39.000 --> 00:49:40.000
Um.
00:49:40.000 --> 00:49:47.000
Now let's do a simple example of a
00:49:47.000 --> 00:49:50.000
kind, contours.
00:49:50.000 --> 00:49:53.000
So,
00:49:53.000 --> 00:50:03.000
suppose we have f of x, x y z
00:50:03.000 --> 00:50:19.000
equals x squared plus y squared plus four is.
00:50:19.000 --> 00:50:28.000
So contour bx squared plus y squared plus for z.
00:50:28.000 --> 00:50:33.000
Because, ah,
00:50:33.000 --> 00:50:39.000
so this is some kind of surface, and three dimensional space.
00:50:39.000 --> 00:50:49.000
And if we solve for, here we can see that we can solve for
00:50:49.000 --> 00:51:02.000
z rather usually.
00:51:02.000 --> 00:51:04.000
And again.
00:51:04.000 --> 00:51:09.000
So for example, H equals zero.
00:51:09.000 --> 00:51:17.000
z equals minus or for expert plus y squared
00:51:17.000 --> 00:51:20.000
h equals one
00:51:20.000 --> 00:51:29.000
equals one for one,
00:51:29.000 --> 00:51:41.000
which equals two.
00:51:41.000 --> 00:51:47.000
Okay. Nobody said all the formulas here are the same except you're adding a constant.
00:51:47.000 --> 00:51:54.000
And your, and if we choose, given the space values for age, we get evenly space values of the constant.
00:51:54.000 --> 00:52:03.000
So what we can. We're just evenly spaced
00:52:03.000 --> 00:52:05.000
copies.
00:52:05.000 --> 00:52:24.000
And this is the formula for an upside down around.
00:52:24.000 --> 00:52:31.000
So, I'm so might have one one,
00:52:31.000 --> 00:52:40.000
then inside of it is another one.
00:52:40.000 --> 00:52:54.000
Then side one side. So you just haven't given a space problem it's just, they're all the same parabola, you're just moving the same ground
00:52:54.000 --> 00:53:07.000
right so it's just so it's the same surface shifted.
00:53:07.000 --> 00:53:13.000
This is not always true, isn't just true for this particular exam.
00:53:13.000 --> 00:53:15.000
Sample exam.
00:53:15.000 --> 00:53:31.000
In general, if the formula for a function of three variables is a little bit too complicated. then the surfaces can get
00:53:31.000 --> 00:53:34.000
a little bit.
00:53:34.000 --> 00:53:40.000
Okay.
00:53:40.000 --> 00:53:45.000
Okay. Um, that's, That's.
00:53:45.000 --> 00:53:55.000
So, I just want to make one more comment. Okay so, um, you may have noticed that when I tried to draw contours.
00:53:55.000 --> 00:53:59.000
So let's say that one.
00:53:59.000 --> 00:54:03.000
See, zero.
00:54:03.000 --> 00:54:08.000
z equals
00:54:08.000 --> 00:54:16.000
minus one
00:54:16.000 --> 00:54:23.000
equals minus one
00:54:23.000 --> 00:54:26.000
minus two.
00:54:26.000 --> 00:54:33.000
What do you notice that I'm always careful to do it.
00:54:33.000 --> 00:54:37.000
I'm always making sure that cars don't promise.
00:54:37.000 --> 00:54:43.000
Okay. So the question is, why can't they was I have two daughters.
00:54:43.000 --> 00:54:55.000
And this is equals one in z equals two, What goes wrong, why can't this happen.
00:54:55.000 --> 00:55:12.000
I think this cannot happen because cannot be a cliff structure in the 3d curve, like, at X and the Y, just to the value.
00:55:12.000 --> 00:55:13.000
It is, yeah.
00:55:13.000 --> 00:55:15.000
So, yeah.
00:55:15.000 --> 00:55:20.000
So let's keep in mind so let me.
00:55:20.000 --> 00:55:24.000
Well, so we're doing this time for.
00:55:24.000 --> 00:55:30.000
So this is a convert to
00:55:30.000 --> 00:55:33.000
function of two variables.
00:55:33.000 --> 00:55:53.000
And let's remind ourselves that if I pick a point here, then what, this is a minus three, and this is to. Then, what is this, telling me, that's telling me that minus three, two is equal to two.
00:55:53.000 --> 00:55:57.000
So every point on a contour.
00:55:57.000 --> 00:56:05.000
Below, it's telling me what the value of the function is at that point. But now, if you go to this point.
00:56:05.000 --> 00:56:16.000
Let's say that to then, since it's on the con for z equals one.
00:56:16.000 --> 00:56:21.000
Then this is a 2.0 is one.
00:56:21.000 --> 00:56:29.000
But the fact that it's on the contours equals two things enough to comma zero is too.
00:56:29.000 --> 00:56:40.000
Well function is not a lot of two separate belts. So this is impossible.
00:56:40.000 --> 00:56:44.000
It's been so the contour
00:56:44.000 --> 00:56:46.000
map.
00:56:46.000 --> 00:56:53.000
This is impossible.
00:56:53.000 --> 00:56:58.000
Now on the other hand, it is possible.
00:56:58.000 --> 00:57:12.000
So, let's do another example is f of x y equals x squared minus false.
00:57:12.000 --> 00:57:24.000
So, Um, so let's do this three conference.
00:57:24.000 --> 00:57:30.000
I'm using all hyperbole
00:57:30.000 --> 00:57:34.000
that we saw that.
00:57:34.000 --> 00:57:40.000
And, um,
00:57:40.000 --> 00:57:43.000
so
00:57:43.000 --> 00:57:46.000
let's see. So,
00:57:46.000 --> 00:57:52.000
what we see is that on
00:57:52.000 --> 00:57:55.000
this one here.
00:57:55.000 --> 00:58:03.000
Now let's recall that that gives you two straight lines.
00:58:03.000 --> 00:58:07.000
So, this is going to get worse.
00:58:07.000 --> 00:58:21.000
So this is ass Nichols, then the other two problems.
00:58:21.000 --> 00:58:24.000
First,
00:58:24.000 --> 00:58:38.000
and, um, let's see. So, the x equals y and y equals zero is a solution to that equation.
00:58:38.000 --> 00:58:46.000
Let me just finish this. So that means, this, this PR is
00:58:46.000 --> 00:58:50.000
equals one.
00:58:50.000 --> 00:59:04.000
And also, minus one word so this one is also f equals one
00:59:04.000 --> 00:59:12.000
is all different colors. And so then this hyperbole is that because minus one.
00:59:12.000 --> 00:59:17.000
And so this is f equals.
00:59:17.000 --> 00:59:30.000
Okay, so, uh, so yeah these contours. I'm notice a couple of things Ican Ican floor for one value of that consists of two different herbs. That's all right.
00:59:30.000 --> 00:59:39.000
And a concert can consist of two curves that cross the catches the function has to have the same value on both.
00:59:39.000 --> 00:59:46.000
So, curves are a lot of cross, if the value of the function is the same on both groups.
00:59:46.000 --> 00:59:54.000
And as I said, and if you slice a saddle surface. You can see this working.
00:59:54.000 --> 00:59:58.000
Why the contours.
00:59:58.000 --> 01:00:03.000
Okay, let's take a break until five after.
01:00:03.000 --> 01:00:33.000
And then we'll respond.
01:03:38.000 --> 01:03:40.000
Hello professor.
01:03:40.000 --> 01:03:44.000
Yes.
01:03:44.000 --> 01:03:46.000
Oh question about come for.
01:03:46.000 --> 01:03:48.000
Yes.
01:03:48.000 --> 01:04:08.000
So can I also interpret the idea of like the to the conference cannot cross as like, so for the definitional function. The forever value of x, there's only one unique why the corresponding to it so that's also the reason why the to come competition with
01:04:08.000 --> 01:04:11.000
each other.
01:04:11.000 --> 01:04:26.000
Um, yeah, I'm not really worried because I'm really worried because I'm going.
01:04:26.000 --> 01:04:35.000
So, I guess my examples were little too easy but if you have a function like this.
01:04:35.000 --> 01:04:38.000
Professor, we cannot see your iPad.
01:04:38.000 --> 01:04:51.000
Okay, yeah, your iPhone and iPad, connect to Wi Fi network.
01:04:51.000 --> 01:04:54.000
Okay, so if you look at this.
01:04:54.000 --> 01:04:59.000
I'm a contour let's say y squared plus x equals one.
01:04:59.000 --> 01:05:02.000
Here
01:05:02.000 --> 01:05:06.000
you have. So, here is a graph looks like.
01:05:06.000 --> 01:05:19.000
What does it look like when x equals one, you get a sideways parabola.
01:05:19.000 --> 01:05:27.000
Um, so here for one value of x i. So let's say here are two wise.
01:05:27.000 --> 01:05:44.000
So, so, contours can be almost any curve and and and so here the contours wall look like problems, but sideways.
01:05:44.000 --> 01:05:59.000
Okay, so. So here it's a, you know, y squared, plus x equals
01:05:59.000 --> 01:06:01.000
y
01:06:01.000 --> 01:06:07.000
security, plus x equals zero,
01:06:07.000 --> 01:06:16.000
y squared plus x
01:06:16.000 --> 01:06:35.000
Okay, so, um, no it really is just the fact that, so it really is just the fact that you can't have, You know, you just can't have a function.
01:06:35.000 --> 01:06:40.000
You can't have this.
01:06:40.000 --> 01:06:46.000
Okay, just just not allowed.
01:06:46.000 --> 01:07:00.000
So professor, I change the statement to one t equal See, there's only one specifics xy solution. No, no, no.
01:07:00.000 --> 01:07:07.000
I mean, that's why you got a whole curve
01:07:07.000 --> 01:07:09.000
surface.
01:07:09.000 --> 01:07:17.000
So if you have f of x, y, z, let's just take a simple one.
01:07:17.000 --> 01:07:24.000
Okay, so you said z equals one, you get x plus.
01:07:24.000 --> 01:07:27.000
Let me see.
01:07:27.000 --> 01:07:30.000
You're talking about
01:07:30.000 --> 01:07:33.000
a function of three variables or function of to her.
01:07:33.000 --> 01:07:37.000
Folks, with three variable actually.
01:07:37.000 --> 01:07:47.000
Okay, So, um, well if I do this
01:07:47.000 --> 01:07:51.000
equals one, then I get x plus y.
01:07:51.000 --> 01:07:54.000
z equals one.
01:07:54.000 --> 01:08:00.000
And so, so there are many solutions.
01:08:00.000 --> 01:08:05.000
And if I said z equal to zero.
01:08:05.000 --> 01:08:14.000
I get x plus y equals one. So there are many solutions.
01:08:14.000 --> 01:08:22.000
It's, it's not about, it's, it's not about uniqueness or solution.
01:08:22.000 --> 01:08:28.000
It's about the fact that a function is not allowed to have two different values.
01:08:28.000 --> 01:08:31.000
So we're not, it's not about an equation.
01:08:31.000 --> 01:08:38.000
Well I mean it's sort of. It's just that, you know, again,
01:08:38.000 --> 01:08:41.000
You know, it going back to the box thing.
01:08:41.000 --> 01:08:49.000
The thing about a function is that if you feed in two inputs and suppose you get four hour.
01:08:49.000 --> 01:08:51.000
You walk away.
01:08:51.000 --> 01:08:59.000
So have some lunch, come back and you hear the same.
01:08:59.000 --> 01:09:19.000
This, the same inputs function as the property that you will still get the same amount. So this cannot change. So you're not allowed to have it give a four in the morning and give you a five in the afternoon.
01:09:19.000 --> 01:09:23.000
Okay, so this is
01:09:23.000 --> 01:09:27.000
not allowed.
01:09:27.000 --> 01:09:29.000
Okay, so this is sunny.
01:09:29.000 --> 01:09:37.000
Okay. So, in particular, if you write down any formula, no matter what the formula you write down.
01:09:37.000 --> 01:09:47.000
You get this property automatically.
01:09:47.000 --> 01:09:49.000
Okay, so here.
01:09:49.000 --> 01:09:53.000
If I write down any formula, and if I write down.
01:09:53.000 --> 01:09:56.000
123.
01:09:56.000 --> 01:10:00.000
There's no way possible to get two different answers.
01:10:00.000 --> 01:10:07.000
If you you know a formula, you put in the same values for the variables, you always get the same value for the formula.
01:10:07.000 --> 01:10:28.000
Okay, so any function for which there is a formula has this property. But actually, any function because we to meet with demand that property for functions that if you choose this, the, there's only one possible output for the same.
01:10:28.000 --> 01:10:29.000
Okay.
01:10:29.000 --> 01:10:32.000
You can have f of one.
01:10:32.000 --> 01:10:39.000
So, you can't have f of one equals three and four one equals five.
01:10:39.000 --> 01:10:48.000
This is just this violates the definition of a function.
01:10:48.000 --> 01:11:03.000
a professor like it based on your slides you said the outages can be one or more numbers. So what does that mean, what does that mean. Okay, so that's a good question, and a good comparison.
01:11:03.000 --> 01:11:18.000
So what that means is that you have, let's say you have three inputs, what's called an x y z, and you have two simultaneous outputs.
01:11:18.000 --> 01:11:22.000
So it's really a pair of functions.
01:11:22.000 --> 01:11:23.000
So here.
01:11:23.000 --> 01:11:41.000
So the input
01:11:41.000 --> 01:11:45.000
Okay, then. so let's, let's do an example.
01:11:45.000 --> 01:11:50.000
Suppose I have three books, all on XYZ.
01:11:50.000 --> 01:11:58.000
And then I can have two outputs, one of them will be x plus one plus one and the other one be x times point.
01:11:58.000 --> 01:12:06.000
So that's an example of a function that takes three inputs and has to our words.
01:12:06.000 --> 01:12:09.000
Okay, so that's.
01:12:09.000 --> 01:12:13.000
So, so that's different from.
01:12:13.000 --> 01:12:30.000
So, the lecture up on now has been focused only on functions with a single output, and what it's saying is that if the output as a function is a single number, then you're not allowed to have the same inputs and two possible output.
01:12:30.000 --> 01:12:42.000
But you are allowed to have a box that takes him three numbers, and then spits out two at the same time.
01:12:42.000 --> 01:12:46.000
Okay so that then really it's two different functions.
01:12:46.000 --> 01:13:01.000
So, you know, this one you could think of as being inside secretly there are two separate boxes.
01:13:01.000 --> 01:13:05.000
Then this one goes down.
01:13:05.000 --> 01:13:15.000
Okay, so, so that's what I mean, and we'll be seeing this. You've already seen that with vector field of vector value function.
01:13:15.000 --> 01:13:21.000
So, if you recall, the velocity vector.
01:13:21.000 --> 01:13:27.000
Really spits out three different numbers.
01:13:27.000 --> 01:13:29.000
And so we're.
01:13:29.000 --> 01:13:36.000
And then we, we put this together.
01:13:36.000 --> 01:13:49.000
So, velocity function is really a function that takes in one input and spits out three numbers. Yeah.
01:13:49.000 --> 01:13:52.000
Yeah.
01:13:52.000 --> 01:13:57.000
Okay.
01:13:57.000 --> 01:13:59.000
This.
01:13:59.000 --> 01:14:19.000
You know I'm going through this more quickly if I haven't said it already, I do want to remind you that it's actually very difficult to learn, and anything and to really understand things from a lecture, especially at the speed that I have to go to go
01:14:19.000 --> 01:14:20.000
at.
01:14:20.000 --> 01:14:27.000
Um, and so what you're learning and lecture is what I say is you learning what you have to learn.
01:14:27.000 --> 01:14:43.000
Um, you know I do provide notes, and the recording so that you can review it after the lecture and back reviewing, as well as struggling with the homework.
01:14:43.000 --> 01:14:48.000
You're not supposed to know how to do the homework, just from listening the lecture.
01:14:48.000 --> 01:14:58.000
You're you you figure out how to do the homework, because you struggle with it, trying to figure out what the heck, the lecture had to do with the homework.
01:14:58.000 --> 01:15:03.000
And then eventually, if you figure it out.
01:15:03.000 --> 01:15:11.000
And of course, if you want to struggle less, you ask your classmates.
01:15:11.000 --> 01:15:27.000
For help you can come to office hours for help. And, you know any way you can. To help yourself figure things out. But, um, but, you know, you do have to, there's a lot going on in this course.
01:15:27.000 --> 01:15:39.000
And so even yeah so what I've done so far does take at least a little time, at least to me, to really understand.
01:15:39.000 --> 01:15:44.000
Any more questions about this.
01:15:44.000 --> 01:15:47.000
Professor. Yes.
01:15:47.000 --> 01:15:58.000
So, Can we also kind of, like, Zero for a while.
01:15:58.000 --> 01:16:02.000
You're talking about like, f of x.
01:16:02.000 --> 01:16:07.000
Yeah, so, contrary.
01:16:07.000 --> 01:16:10.000
Yeah, yeah, no, that's a good question.
01:16:10.000 --> 01:16:13.000
So you can have a.
01:16:13.000 --> 01:16:15.000
Let's do x square.
01:16:15.000 --> 01:16:22.000
OK, so now you just have a line.
01:16:22.000 --> 01:16:28.000
And so then you have contours. Let's do with you.
01:16:28.000 --> 01:16:38.000
Let's do minus one, zero and one. So, so this is x squared equals minus one.
01:16:38.000 --> 01:16:44.000
There are no solutions. So, no console.
01:16:44.000 --> 01:16:55.000
Because zero is x squared equals zero. So that's just the origin.
01:16:55.000 --> 01:16:58.000
equals zero.
01:16:58.000 --> 01:17:15.000
And then, equals one, is x squared equals one, which means x equals one, or minus. So the contour, that is this.
01:17:15.000 --> 01:17:18.000
Okay, so you're right there points.
01:17:18.000 --> 01:17:24.000
And on the equation you may get no points, 1.2 points three points.
01:17:24.000 --> 01:17:27.000
Okay.
01:17:27.000 --> 01:17:30.000
So, two points.
01:17:30.000 --> 01:17:41.000
Like I say, and then two points.
01:17:41.000 --> 01:17:48.000
No, no, this is a function right, it's, it's what's not allowed.
01:17:48.000 --> 01:17:50.000
Like so.
01:17:50.000 --> 01:17:56.000
Right, like for ya there.
01:17:56.000 --> 01:18:07.000
with this one. So, if I take this point, and I claim f equals three and one and that's no good.
01:18:07.000 --> 01:18:16.000
So, so this. Yeah, so this is the thing that's now
01:18:16.000 --> 01:18:25.000
not allowed to label the same point with two different models.
01:18:25.000 --> 01:18:29.000
But you are allowed to have more than one point with the same function.
01:18:29.000 --> 01:18:35.000
And then that. So this general principle works in Ireland, as well.
01:18:35.000 --> 01:18:42.000
So you are allowed to have two curves, two different curves at the same value of the function.
01:18:42.000 --> 01:18:53.000
But you're not allowed to have one curve with two different balance.
01:18:53.000 --> 01:18:59.000
Is it just like
01:18:59.000 --> 01:19:00.000
our one.
01:19:00.000 --> 01:19:07.000
Yeah, I don't know why you're echoing me.
01:19:07.000 --> 01:19:13.000
So, uh, so everything is more or less the same when you have a function of reverb.
01:19:13.000 --> 01:19:20.000
So, you are allowed to have two surfaces that intersect.
01:19:20.000 --> 01:19:24.000
So,
01:19:24.000 --> 01:19:40.000
you are, as long as the function as the same value on both surface, and they can intersect along occur, they could then there's like the one point. So another possible example, this.
01:19:40.000 --> 01:19:54.000
So if you have two surfaces that intersect so this is ethical three in this surface also has to be true.
01:19:54.000 --> 01:20:04.000
So I need to have two services touch the two contour. Two contours, touch, they have to have the same value of the function.
01:20:04.000 --> 01:20:08.000
On the other hand, you are allowed to have.
01:20:08.000 --> 01:20:17.000
But you're also allowed to have to surface that don't touch and have the same value for different.
01:20:17.000 --> 01:20:19.000
Okay.
01:20:19.000 --> 01:20:28.000
So it's all just a higher dimensional versions of the properties for points on the line.
01:20:28.000 --> 01:20:31.000
So,
01:20:31.000 --> 01:20:37.000
yeah.
01:20:37.000 --> 01:20:41.000
Any more questions.
01:20:41.000 --> 01:20:43.000
Okay.
01:20:43.000 --> 01:20:48.000
Ah, that was easy part of the lecture.
01:20:48.000 --> 01:20:52.000
Things are much harder. Ah.
01:20:52.000 --> 01:21:01.000
So, uh, so what I want to talk about, I'm going to switch topics, almost completely.
01:21:01.000 --> 01:21:04.000
I'm going to talk about margin there.
01:21:04.000 --> 01:21:15.000
And the concept of aluminum theory you learn this in calculus one by, it's,
01:21:15.000 --> 01:21:17.000
it's hard stuff though.
01:21:17.000 --> 01:21:20.000
I'm very easy.
01:21:20.000 --> 01:21:29.000
So, um, so I want to describe this in terms of measure.
01:21:29.000 --> 01:21:32.000
And I'm going to draw the real line.
01:21:32.000 --> 01:21:40.000
Okay, but don't want to be able to think of it more gym. So suppose or something we're trying to measure.
01:21:40.000 --> 01:21:46.000
And let's suppose the exact value of that measurement is x zero.
01:21:46.000 --> 01:21:56.000
So it could be the length of a piece of work that you you're trying to measure. Now there is no way to measure the length of a piece of work with that.
01:21:56.000 --> 01:22:01.000
So, so, so if.
01:22:01.000 --> 01:22:15.000
Let's call x one is a real measurement.
01:22:15.000 --> 01:22:26.000
That means it's not going to be equal to zero, close to, then, um, so let me draw a picture, suppose I have next one.
01:22:26.000 --> 01:22:30.000
So many.
01:22:30.000 --> 01:22:34.000
Put it there, with.
01:22:34.000 --> 01:22:35.000
Mom, let's put it here.
01:22:35.000 --> 01:22:46.000
Let's put it here. So, I'm the error of that measurement is simply the distance
01:22:46.000 --> 01:22:49.000
between x one and zero.
01:22:49.000 --> 01:22:54.000
And that's the absolute value of x one
01:22:54.000 --> 01:22:57.000
point. Mom, so.
01:22:57.000 --> 01:23:01.000
So then the error
01:23:01.000 --> 01:23:04.000
is
01:23:04.000 --> 01:23:09.000
this.
01:23:09.000 --> 01:23:14.000
So then the next thing I want to talk about is a margin of error.
01:23:14.000 --> 01:23:20.000
So, I'm suppose
01:23:20.000 --> 01:23:23.000
equals the margin.
01:23:23.000 --> 01:23:25.000
So what is margin of error.
01:23:25.000 --> 01:23:39.000
Well, when you're trying to measure a piece of wood, let's say you want to one meter long piece of wood, while you know you're never going to get it exactly one year, but you know you want it within somewhere.
01:23:39.000 --> 01:23:59.000
So let's say you want it within a 10th of me. So then, if, you know, in science classes, you, you know, seeing things like this.
01:23:59.000 --> 01:24:01.000
One plus or minus point.
01:24:01.000 --> 01:24:06.000
So here, point one indicates the margin of error.
01:24:06.000 --> 01:24:24.000
Okay. And so, so then we say that x zero is within the margin of error.
01:24:24.000 --> 01:24:26.000
Yes, the error.
01:24:26.000 --> 01:24:30.000
Sorry. This is this one.
01:24:30.000 --> 01:24:34.000
Yes, there is simply less than.
01:24:34.000 --> 01:24:34.000
And then my says,
01:24:34.000 --> 01:24:42.000
And my sister. Okay.
01:24:42.000 --> 01:24:58.000
So now I want to discuss the concept of a limit. From this point of view. So, I'm. So, Suppose that x one
01:24:58.000 --> 01:25:05.000
x two, etc. is infinite
01:25:05.000 --> 01:25:10.000
sequence of.
01:25:10.000 --> 01:25:17.000
When I say values, which means either numbers.
01:25:17.000 --> 01:25:20.000
points,
01:25:20.000 --> 01:25:25.000
or backers.
01:25:25.000 --> 01:25:29.000
Then, um, so I want to be able to say for.
01:25:29.000 --> 01:25:50.000
And when I want to discuss this the concept of what happens divide, take the limit of these measures so you think of this infinite sequence is a bunch of measurements, and what you're trying to ask is do these measurements get closer and closer to the
01:25:50.000 --> 01:26:04.000
actual end. So we're going to. So what we want to understand is what is this statement saying, Okay, um, and let me note here that.
01:26:04.000 --> 01:26:16.000
Now when I write x minus y, this has three different means. This is the absolute value
01:26:16.000 --> 01:26:22.000
of x and y are numbers.
01:26:22.000 --> 01:26:33.000
And its distance from x to y
01:26:33.000 --> 01:26:41.000
yF right so what what points or the.
01:26:41.000 --> 01:26:49.000
So in other words, it's that notation is exactly what it means from.
01:26:49.000 --> 01:26:56.000
Okay. Um, so, um, so what is the limit me.
01:26:56.000 --> 01:26:57.000
Yeah.
01:26:57.000 --> 01:27:04.000
So, um, so yeah, so we knew.
01:27:04.000 --> 01:27:05.000
Okay.
01:27:05.000 --> 01:27:13.000
As a measurement.
01:27:13.000 --> 01:27:14.000
And I'm.
01:27:14.000 --> 01:27:23.000
Suppose we choose the margin of error.
01:27:23.000 --> 01:27:26.000
So a number M.
01:27:26.000 --> 01:27:35.000
And the idea is we want to look at measurements that are within that margin. Okay, so I'm.
01:27:35.000 --> 01:27:50.000
Okay, so. So what do we want. We want all measurements. So, all is very important.
01:27:50.000 --> 01:28:03.000
Be honest certain point
01:28:03.000 --> 01:28:09.000
to be within
01:28:09.000 --> 01:28:17.000
on margin.
01:28:17.000 --> 01:28:19.000
And I'm.
01:28:19.000 --> 01:28:31.000
So, then, then, well, and we want more than that. We want this
01:28:31.000 --> 01:28:35.000
to be true
01:28:35.000 --> 01:28:39.000
for any
01:28:39.000 --> 01:28:41.000
mark.
01:28:41.000 --> 01:28:48.000
So, no matter how small.
01:28:48.000 --> 01:28:54.000
So, the picture
01:28:54.000 --> 01:28:59.000
looks roughly like this.
01:28:59.000 --> 01:29:13.000
So you have this limit now, and you haven't measured it so maybe next one is here. It's two is your experience here, it's for us here. It's five is here.
01:29:13.000 --> 01:29:15.000
It's six is here.
01:29:15.000 --> 01:29:17.000
It's seven is here.
01:29:17.000 --> 01:29:33.000
Okay. And so then you choose a margin of error, which means you draw a circle of radius around, l.
01:29:33.000 --> 01:29:43.000
And you want to say that after some point in the sequence, every value in the sequences within this little circle.
01:29:43.000 --> 01:29:52.000
Okay. And so, no matter how small or circle I draw somewhere down the road.
01:29:52.000 --> 01:29:55.000
Every element, every measurement is within that.
01:29:55.000 --> 01:29:57.000
So, it could be.
01:29:57.000 --> 01:30:08.000
If I draw the circle really small the first million values of the first 1 million measurements are not inside that circle. But after that, they all are in.
01:30:08.000 --> 01:30:11.000
And then if I draw the smaller circle.
01:30:11.000 --> 01:30:21.000
They'd be the first billion. Don't online side that even smaller circle, but everything after that.
01:30:21.000 --> 01:30:29.000
So, so, the important thing is that you got to get the steps right. First you want to choose the margin of error.
01:30:29.000 --> 01:30:36.000
And then you ask to all do all the elements of the sequence passage one point.
01:30:36.000 --> 01:30:38.000
Online within the margin.
01:30:38.000 --> 01:30:49.000
So here's a formal mathematical speak well.
01:30:49.000 --> 01:30:52.000
So, I'm.
01:30:52.000 --> 01:31:05.000
So, given any margin of error.
01:31:05.000 --> 01:31:19.000
There is a point in the sequence the sequence is labeled by numbers. So there is a number of him, which will change depending on what the margin of error is.
01:31:19.000 --> 01:31:31.000
It's an integer, such that every man, every measurement asked that point.
01:31:31.000 --> 01:31:33.000
This within the margin.
01:31:33.000 --> 01:31:39.000
So for every
01:31:39.000 --> 01:31:42.000
a greater than.
01:31:42.000 --> 01:31:52.000
So, if you. It's like if you tossed the first million measurements and everything past it is within the margin of error.
01:31:52.000 --> 01:32:03.000
And it's just that so that's just like a million, but it's going to get bigger, the smaller there is the bigger Capital One.
01:32:03.000 --> 01:32:12.000
Okay, so this is, this is the precise meaning of the limit of takers infinity of x equals.
01:32:12.000 --> 01:32:18.000
And notice here, XL can be points, they can be vectors, they can be.
01:32:18.000 --> 01:32:22.000
It's all the same thing.
01:32:22.000 --> 01:32:27.000
The only thing that changes is what the two vertical bars.
01:32:27.000 --> 01:32:34.000
It's either absolute value or the length of effect.
01:32:34.000 --> 01:32:42.000
So, okay, so that's the first thing I want to say.
01:32:42.000 --> 01:32:54.000
The second thing now is I want to talk about the limit of a front
01:32:54.000 --> 01:33:00.000
end again let's start with.
01:33:00.000 --> 01:33:04.000
Well function of one.
01:33:04.000 --> 01:33:07.000
and I'm
01:33:07.000 --> 01:33:11.000
so,
01:33:11.000 --> 01:33:15.000
so this is one of the pictures version of a graph like this.
01:33:15.000 --> 01:33:20.000
And I'm
01:33:20.000 --> 01:33:24.000
so I'm
01:33:24.000 --> 01:33:27.000
was I well one.
01:33:27.000 --> 01:33:35.000
So, um, so we want her to suppose as the ground wipers.
01:33:35.000 --> 01:33:47.000
Then we want to understand what it means to say limit as x goes to zero, f of x equals.
01:33:47.000 --> 01:33:56.000
Okay. And again. So the idea is now the way on when you think of this is that there to margins of error.
01:33:56.000 --> 01:34:03.000
There's the margin of error.
01:34:03.000 --> 01:34:08.000
For the output
01:34:08.000 --> 01:34:11.000
of
01:34:11.000 --> 01:34:13.000
the margin.
01:34:13.000 --> 01:34:19.000
There are the input
01:34:19.000 --> 01:34:23.000
to it.
01:34:23.000 --> 01:34:27.000
And so the idea
01:34:27.000 --> 01:34:39.000
is, again, we want to choose a margin of error.
01:34:39.000 --> 01:34:43.000
For the output.
01:34:43.000 --> 01:34:47.000
So we went we have this function.
01:34:47.000 --> 01:35:00.000
And we want to make sure the output is accurate with a certain air, a minute, you know, we want there to be some smaller some margin.
01:35:00.000 --> 01:35:06.000
And I'm, and I'm going to call that epsilon.
01:35:06.000 --> 01:35:18.000
Okay, so the idea is that, um, and so we want to find a margin of error
01:35:18.000 --> 01:35:21.000
john paul Delta.
01:35:21.000 --> 01:35:27.000
for the input
01:35:27.000 --> 01:35:35.000
Such fan. Um, yeah.
01:35:35.000 --> 01:35:45.000
Um, so, if I have an input, then it's equal to x not within the margin of error, don't know.
01:35:45.000 --> 01:35:55.000
So, then, then the output will be within
01:35:55.000 --> 01:36:00.000
the margin of error for the.
01:36:00.000 --> 01:36:15.000
Okay. So you start with how close you want the output to be. And the question is can you find a margin of error for the input that will guarantee that the output is also within the margin of error.
01:36:15.000 --> 01:36:25.000
Right. And then the idea is that limit
01:36:25.000 --> 01:36:30.000
means
01:36:30.000 --> 01:36:33.000
no matter
01:36:33.000 --> 01:36:37.000
how small
01:36:37.000 --> 01:36:40.000
epsilon is.
01:36:40.000 --> 01:36:49.000
Well there is delta, such that
01:36:49.000 --> 01:36:55.000
if the input is within the margin of error.
01:36:55.000 --> 01:37:08.000
Then, you know.
01:37:08.000 --> 01:37:10.000
Bye.
01:37:10.000 --> 01:37:12.000
Bye. Um.
01:37:12.000 --> 01:37:29.000
Now this is kind of a hard concept to work with, um, you'll have to struggle with it more in an hour. If you're a math major, you learn how to work with this and analysis.
01:37:29.000 --> 01:37:32.000
Um.
01:37:32.000 --> 01:37:36.000
Let me also phrase it differently.
01:37:36.000 --> 01:37:39.000
Another way
01:37:39.000 --> 01:37:43.000
that will find more useful.
01:37:43.000 --> 01:37:46.000
So, The limit as.
01:37:46.000 --> 01:38:05.000
So, so now I'm, I want to do it for more than one. So, if I have a function for. So, so suppose I'm a function
01:38:05.000 --> 01:38:09.000
function of to hear.
01:38:09.000 --> 01:38:19.000
spouse I, um, I have xy going to next night.
01:38:19.000 --> 01:38:22.000
And then I want to say that the limit is L.
01:38:22.000 --> 01:38:26.000
So,
01:38:26.000 --> 01:38:27.000
this.
01:38:27.000 --> 01:38:35.000
So, this this is true
01:38:35.000 --> 01:38:39.000
for any sequence
01:38:39.000 --> 01:38:46.000
of influence
01:38:46.000 --> 01:38:53.000
search that.
01:38:53.000 --> 01:38:57.000
So I said there a limit
01:38:57.000 --> 01:39:02.000
is.
01:39:02.000 --> 01:39:05.000
Yes, excellent, why not.
01:39:05.000 --> 01:39:11.000
Um, so, this limit. I'm using the,
01:39:11.000 --> 01:39:16.000
the picture, I have here of a sequence of points, except now instead of x one, x 111 x to y to, etc.
01:39:16.000 --> 01:39:26.000
Except now instead of x one, x 111 x to y two, etc. But it's really the same picture.
01:39:26.000 --> 01:39:42.000
So if I have a sequence that's all getting in points where the points are all getting closer and closer to x. Now, why not, then I want to evaluate the output to
01:39:42.000 --> 01:39:48.000
convert to the limit.
01:39:48.000 --> 01:40:07.000
Okay, so, taking any sequence of points that converts the next not why not, the value of the function at those points for the sequence converges to the number, l, and that's what it means for all to be over.
01:40:07.000 --> 01:40:11.000
And
01:40:11.000 --> 01:40:23.000
this is all very abstract right now. And frankly, you don't really have to fully understand it but I just want you to have some idea of what's going on.
01:40:23.000 --> 01:40:31.000
So next concept is continuity.
01:40:31.000 --> 01:40:41.000
So, I'm will say that is continuous.
01:40:41.000 --> 01:40:51.000
If it never jumps in body.
01:40:51.000 --> 01:41:05.000
Mom, and simple pictures of this continuous functions. Might be a graph that looks like this. It's constantly equal to one.
01:41:05.000 --> 01:41:13.000
Then when exit zero, it suddenly jumps to two. So there's a break in the graph.
01:41:13.000 --> 01:41:26.000
You can introduce artificial discontinuities by just simply graph that looks like a parabola.
01:41:26.000 --> 01:41:35.000
But then at some point, when you make the function different.
01:41:35.000 --> 01:41:43.000
So this could be faster.
01:41:43.000 --> 01:41:57.000
f of x equals x squared plus one, if x is not equal to minus one and a half.
01:41:57.000 --> 01:42:02.000
So, you can just simply change the value of that.
01:42:02.000 --> 01:42:06.000
So these are examples of
01:42:06.000 --> 01:42:09.000
this continuous.
01:42:09.000 --> 01:42:16.000
Now, will we want to do is discusses or functional to her.
01:42:16.000 --> 01:42:21.000
So, So we'll say that.
01:42:21.000 --> 01:42:27.000
That's one is continuous.
01:42:27.000 --> 01:42:31.000
Not a point x naught, why not.
01:42:31.000 --> 01:42:35.000
if the limit as x y, why not why not.
01:42:35.000 --> 01:42:49.000
Want to excellent why not. Well, that's why. So is the value of the function.
01:42:49.000 --> 01:42:53.000
So, I'm.
01:42:53.000 --> 01:43:01.000
Um, so there's no. So, this is basically a fancy way to say there's no chance.
01:43:01.000 --> 01:43:04.000
Okay. And when visually.
01:43:04.000 --> 01:43:12.000
Um, was this continuous.
01:43:12.000 --> 01:43:17.000
If it's brown
01:43:17.000 --> 01:43:20.000
is a
01:43:20.000 --> 01:43:23.000
continuous surface.
01:43:23.000 --> 01:43:32.000
This is a big statement. So the no jobs or tears.
01:43:32.000 --> 01:43:40.000
So, an example of a background, it's just the same as
01:43:40.000 --> 01:43:42.000
for.
01:43:42.000 --> 01:43:50.000
So I can have a surface of Bravo right then change the value of point.
01:43:50.000 --> 01:43:55.000
Um, I didn't have a plane.
01:43:55.000 --> 01:44:02.000
Then, suddenly jumped the value of the plane.
01:44:02.000 --> 01:44:07.000
So, there are many different types of. Yeah.
01:44:07.000 --> 01:44:11.000
Just like Parker.
01:44:11.000 --> 01:44:17.000
And so if you can draw a graph, then, um,
01:44:17.000 --> 01:44:25.000
then you can see that, whether it's continuous.
01:44:25.000 --> 01:44:39.000
Not a mathematical proof. But, basically. So for this one. This function, as we saw the beginning the writer is just the upper half of his sphere.
01:44:39.000 --> 01:44:47.000
And that's a nice, smooth surface no jobs, no holes, tears. So that's fun.
01:44:47.000 --> 01:45:00.000
On the other hand, if you have f of x y equals one minus x squared minus y squared.
01:45:00.000 --> 01:45:07.000
If that's why is not equal to 00.
01:45:07.000 --> 01:45:16.000
x one equals zero, then graph now.
01:45:16.000 --> 01:45:21.000
There's a hole in it.
01:45:21.000 --> 01:45:27.000
And the value of the function what the origin is
01:45:27.000 --> 01:45:39.000
that, so that's it.
01:45:39.000 --> 01:45:46.000
Okay. Now, I want to. So that's the concept of continuity.
01:45:46.000 --> 01:45:53.000
You may be asked some questions about continuous functions are not continuous.
01:45:53.000 --> 01:45:57.000
And,
01:45:57.000 --> 01:46:09.000
well, actually when I'm back at this. So, um, So here's a general rule.
01:46:09.000 --> 01:46:17.000
What if work is defined
01:46:17.000 --> 01:46:25.000
using a single form.
01:46:25.000 --> 01:46:32.000
So what do I mean by single point. Let me something that does not look like this.
01:46:32.000 --> 01:46:35.000
This required to separate point.
01:46:35.000 --> 01:46:39.000
On the other hand, this one is a single point.
01:46:39.000 --> 01:46:43.000
So I don't have this brace and if and all that.
01:46:43.000 --> 01:46:46.000
Okay,
01:46:46.000 --> 01:46:50.000
and
01:46:50.000 --> 01:46:53.000
fix not, why not.
01:46:53.000 --> 01:47:00.000
Well, then, let me just say,
01:47:00.000 --> 01:47:02.000
then.
01:47:02.000 --> 01:47:09.000
If this continues,
01:47:09.000 --> 01:47:15.000
every point.
01:47:15.000 --> 01:47:20.000
When it's done.
01:47:20.000 --> 01:47:24.000
Now, this is a very important point. And it's them.
01:47:24.000 --> 01:47:34.000
So for example,
01:47:34.000 --> 01:47:39.000
So this is continuous.
01:47:39.000 --> 01:47:46.000
But it has to be where n squared plus y squared his last name.
01:47:46.000 --> 01:47:59.000
Otherwise this, if you're outside. It's on the fine. So it's not a functions undefined somewhere, it's not.
01:47:59.000 --> 01:48:06.000
Another example would be x plus one over x minus one.
01:48:06.000 --> 01:48:11.000
This is undefined.
01:48:11.000 --> 01:48:15.000
If x equals one.
01:48:15.000 --> 01:48:19.000
So the domain
01:48:19.000 --> 01:48:26.000
is the set of all small, where x is not equal.
01:48:26.000 --> 01:48:34.000
And it's a single formula. If this continues.
01:48:34.000 --> 01:48:39.000
Now, I'm
01:48:39.000 --> 01:48:44.000
a general rule, this is informal.
01:48:44.000 --> 01:48:45.000
Okay.
01:48:45.000 --> 01:48:48.000
What do I mean by that.
01:48:48.000 --> 01:48:51.000
When you're answering a test question.
01:48:51.000 --> 01:49:00.000
You're not really allowed to use it so I'm just telling you this so you know, at the very least, if you don't have an explanation.
01:49:00.000 --> 01:49:18.000
And there's a test question that says is that continuous at x, x, y equals one part two. And you see that one calm is to is in the domain in a single formula, then you know you'll get the right answer.
01:49:18.000 --> 01:49:20.000
Even if you don't provide these.
01:49:20.000 --> 01:49:26.000
Okay, so this is a something you should remember.
01:49:26.000 --> 01:49:39.000
So basically, point it's very hard to use formulas to define this continuous function.
01:49:39.000 --> 01:49:49.000
So actually this rule says. Awesome.
01:49:49.000 --> 01:49:54.000
Now, but sometimes. I'm a function.
01:49:54.000 --> 01:49:59.000
Okay so but a function.
01:49:59.000 --> 01:50:02.000
So let me write down anything.
01:50:02.000 --> 01:50:06.000
I'm so,
01:50:06.000 --> 01:50:08.000
So it's fun as I have.
01:50:08.000 --> 01:50:16.000
What's y equals c squared y
01:50:16.000 --> 01:50:19.000
squared.
01:50:19.000 --> 01:50:23.000
x squared plus one
01:50:23.000 --> 01:50:37.000
square.
01:50:37.000 --> 01:50:39.000
I notice that I can't.
01:50:39.000 --> 01:50:48.000
The first with the formula is undefined. When x and y are both zero, because the denominator is zero.
01:50:48.000 --> 01:50:54.000
But I've added a second line to say I want the function to be zero at that point.
01:50:54.000 --> 01:50:59.000
So, This might be continuous. it might not.
01:50:59.000 --> 01:51:08.000
Okay. Now, everywhere but at the origin, the function has a single formula, and the end is fine.
01:51:08.000 --> 01:51:16.000
So we know. So right away. We know that this continuous
01:51:16.000 --> 01:51:25.000
fact someone is not well we don't know is whether it's continuous.
01:51:25.000 --> 01:51:32.000
So, is continuous
01:51:32.000 --> 01:51:43.000
at 00, using the definition of limit as x goes to 000.
01:51:43.000 --> 01:51:46.000
Wow, is zero.
01:51:46.000 --> 01:51:50.000
That's the definition of continuous, we started.
01:51:50.000 --> 01:51:57.000
Okay, So, we can prove that, then we can say that.
01:51:57.000 --> 01:52:01.000
And this does turn out to be true in this case.
01:52:01.000 --> 01:52:03.000
Okay.
01:52:03.000 --> 01:52:15.000
And so now what I want to discuss. In the last few minutes of class and it's going to continue into the beginning of next class is how to compute something.
01:52:15.000 --> 01:52:26.000
So, now, the truth is, is that it's very hard to compute limits for formulas that have to work.
01:52:26.000 --> 01:52:33.000
And the truth is, is that you're only going to have to deal with some simple case.
01:52:33.000 --> 01:52:54.000
And so, um, so let me do some mixing. Basically, um, except for what I've said so far. Well, I'm going to give you the rules of thumb. So, to repeat myself.
01:52:54.000 --> 01:52:58.000
expert 911.
01:52:58.000 --> 01:53:03.000
y squared minus three.
01:53:03.000 --> 01:53:08.000
Okay, you're handed a limit on exit.
01:53:08.000 --> 01:53:10.000
What's the first thing you should.
01:53:10.000 --> 01:53:13.000
Okay, there's a simple formula here.
01:53:13.000 --> 01:53:23.000
You're in so the first thing you try is to find out whether 00 is in the domain at this point.
01:53:23.000 --> 01:53:24.000
Okay.
01:53:24.000 --> 01:53:31.000
And so what does that mean means you plugged 00 into your form.
01:53:31.000 --> 01:53:38.000
So first five plugging in
01:53:38.000 --> 01:53:45.000
pretty good yes zero minus zero plus one, zero minus three, you get one.
01:53:45.000 --> 01:53:57.000
It worked. That means it's the limit means that the limit is one.
01:53:57.000 --> 01:53:58.000
Okay.
01:53:58.000 --> 01:54:08.000
So, every exam will have at least one problem like that or you didn't have to breathe.
01:54:08.000 --> 01:54:12.000
Let's do another one.
01:54:12.000 --> 01:54:17.000
Next one is 00.
01:54:17.000 --> 01:54:32.000
x squared plus two wine, over x squared minus once.
01:54:32.000 --> 01:54:46.000
Okay, so we're asking is what happens to the value of this one though, is x points get really close to the origin. So you can think of as a sequence of points, so.
01:54:46.000 --> 01:54:51.000
So, what happens if we look at this formula for a sequence of points.
01:54:51.000 --> 01:54:59.000
And the second thing you look for is. So, here if I plug in. Again zero divided by user.
01:54:59.000 --> 01:55:03.000
So, doesn't tell me.
01:55:03.000 --> 01:55:13.000
The next thing you do when you have this is when is the denominator.
01:55:13.000 --> 01:55:18.000
Because that's where all the difficult.
01:55:18.000 --> 01:55:22.000
Here you see that it's x squared minus y squared equals zero.
01:55:22.000 --> 01:55:34.000
So, I'm so that means it's the denominator is zero along those things we're trying to take a limit at the origin.
01:55:34.000 --> 01:55:39.000
Now, what we see is that
01:55:39.000 --> 01:55:43.000
it's at the formula.
01:55:43.000 --> 01:55:47.000
Notice that the formula.
01:55:47.000 --> 01:55:49.000
Well was,
01:55:49.000 --> 01:55:52.000
you get a nonzero number.
01:55:52.000 --> 01:55:59.000
I'm along these two lines.
01:55:59.000 --> 01:56:07.000
So because you get ratio nonzero.
01:56:07.000 --> 01:56:09.000
Yeah.
01:56:09.000 --> 01:56:10.000
visor visor.
01:56:10.000 --> 01:56:18.000
So, and notice that no matter how close you get to the origin, where you're trying to take this moment.
01:56:18.000 --> 01:56:23.000
You always have points where the function is under.
01:56:23.000 --> 01:56:33.000
Okay. And so, um, so you have a sequence of points approaching your where the function is never defined.
01:56:33.000 --> 01:56:38.000
And so, um, so. So there is a.
01:56:38.000 --> 01:56:45.000
So there is a sequence
01:56:45.000 --> 01:56:50.000
is a sequence.
01:56:50.000 --> 01:56:59.000
Books kyk, going to the origin, where the formula.
01:56:59.000 --> 01:57:06.000
There's always on the phone.
01:57:06.000 --> 01:57:11.000
And so this means the limit does not exist.
01:57:11.000 --> 01:57:15.000
There is no.
01:57:15.000 --> 01:57:18.000
So, if you can find points.
01:57:18.000 --> 01:57:26.000
I'm going towards the origin, where the denominator is always zero, then the limit. There is no.
01:57:26.000 --> 01:57:30.000
That wasn't.
01:57:30.000 --> 01:57:34.000
So that's the
01:57:34.000 --> 01:57:35.000
sub.
01:57:35.000 --> 01:57:45.000
That's an easy way to detect a limit that does not exist. So these are the two things.
01:57:45.000 --> 01:57:50.000
Next time, things are going to discuss more complicated ways to deal with limits.
01:57:50.000 --> 01:58:04.000
But if you just know these two principles, if you can figure out, and understand and you figure out how to use these two principles, you'll probably get about half the limits in the test.
01:58:04.000 --> 01:58:21.000
Correct. actually more likely two thirds. So, these two are the simplest thing for luck on next time, we'll discuss how to deal with more complicated limits, and how to figure them.
01:58:21.000 --> 01:58:23.000
Okay.
01:58:23.000 --> 01:58:35.000
Okay, I'm going to stop here.
01:58:35.000 --> 01:58:41.000
Any
01:58:41.000 --> 01:58:54.000
hospital room to fight to prove, there's a limit, not. No, no, because rule does not work with functions.
01:58:54.000 --> 01:58:57.000
Now that's why it's so hard.
01:58:57.000 --> 01:59:02.000
No simple rules or procedures.
01:59:02.000 --> 01:59:06.000
You'll see that next time.
01:59:06.000 --> 01:59:14.000
Professor it should be negative one over three on the top.
01:59:14.000 --> 01:59:44.000
Yes. So, that's right.