WEBVTT mathematics/ap-calculus-ab/hovasapian
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Hello, welcome back to www.educator.com, and welcome back to AP Calculus.
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Today, we are going to actually close out the course of AP Calculus formally.
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This is going to be our last lesson.
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We are going to be talking about slope fields.
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For the last few lessons, we have been talking about differential equations, what they are.
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We have taken a look at separation of variables.
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We have taken a look at one particular type of differential equation, to get a feel for it.
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Talk about qualitative solutions.
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Now we are going to round it out with this thing called a slope field.
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Let us jump right on in.
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It is not always possible to solve a differential equation explicitly.
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For certain types of differential equations like the ones that we dealt with, the separable ones,
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yes, we can separate them, we can integrate and we can solve explicitly for a family of solutions.
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Also, if we have an initial value, we can also find a particular solution.
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But the truth is, most of the time in the real world, it is not possible to solve it explicitly.
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But there are other techniques we can use and slope field is going to be one of those techniques.
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Let us go ahead and work in blue.
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It is not always possible to solve the de explicitly.
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That is to find a nice formula that relates x and y, relating x and y.
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y is some function of x.
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It is not always possible.
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In this lesson, we introduce a geometric graphical technique
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that still allows us to extract information about the solution for a given differential equation.
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Slope fields, let us look at a particular initial value problem.
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Let us look at the following ivp.
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Remember, an initial value problem consists of a differential equation and it also consists of an initial value.
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For some value of x, we know that it is this y.
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In this particular case, y(0) = 3.
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We know that this differential equation, whatever solution we come up with,
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if we happen to be able to come up with one, when x is 0, the y value is going to be 3.
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It is going to pass through that point.
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Let me write this as dy dx which is, y’ = x + y and y(0) = 3.
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This equation is not separable.
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There is really not much we can do with it.
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This equation, at least at this level, this equation is not separable.
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We cannot solve, in other words, we cannot integrate for an explicit solution.
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What this differential equation is saying is the following.
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Again, it is always nice, when you are faced with any kind of equation, any kind of formula,
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to ask yourself what is actually going on.
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Let me go ahead and move to the next page and rewrite the equation up here.
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I have got dy dx is equal to x + y.
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This de says the following.
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At a given point xy, the slope of the solution curve y(x) is x + y.
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Now we know that when we solve a particular differential equation, we are going get some function of x.
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It is going to be something like, let us say y = sin(x), whatever it is.
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This is just an example.
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It is going to be some function of x.
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y expressed as a function of x is just a curve.
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We know that, we have been dealing with them for years and years.
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There you have it, it is a curve.
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What this is telling me dy dx, we know that the derivative is a slope.
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It is telling me that the derivative of whatever function it is that I'm trying to find, y(x),
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its derivative which is its slope, at a particular point xy is actually equal to the x value + the y value.
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A differential equation actually gives me a way of finding the slope of the curve at that point.
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What I'm going to do, what we are going to do in slope field is we are going to plot out a bunch of points.
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We are going to make little marks representing that slope.
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And then, we are going to follow those little marks and actually draw out a solution curve graphically, instead of solving for it explicitly.
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That is what is going on.
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Let us reiterate, if we can.
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If we could graph y(x), the solution in the xy plane,
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its slope at various points xy is given by x + y.
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I can point 5,4, the slope of the curve, as it passes through 5,4 is going to be 5 + 4.
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It is going to be 9.
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We know that the curve is actually headed pretty steeply up, that is all this is saying.
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That is all a differential equation really is.
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Now what we want to do is, let us see if we can recover the graph.
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If we can recover the graph of y(x) pictorially, instead of explicitly, analytically.
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Let us see if we can recover the graphs by plotting the slopes at various points.
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Let me draw a little xy coordinate system.
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Here is our differential equation, let me rewrite it down here.
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I’m going to rewrite it in red.
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We have y’ = x + y.
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If I take the point 0,0, the point 0,0, if I put y’ at 0,0 is equal to 0 + 0 which is 0,
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which means that at that point, my slope is horizontal.
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I’m going to put a little dash representing this slope.
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As if I'm literally drawing the tangent line at the curve, except I’m not drawing an entire tangent line, I’m just drawing a piece of the tangent line.
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Now I know that when the curve passes through 0,0, actually it is going to just touch it.
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It is going to have a slope of 0 at that point.
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Let us try the point 1,0.
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Let us make this 1, 2, 3.
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Let us go -1, -2, -3.
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Let us go 1, 2, 3, up, -1, -2, and -3.
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At the point 1,0, the derivative is 1 + 0, it is 1.
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It looks like this.
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At the point 2,0, the slope is going to be 2.
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It is going to be a little steeper like that.
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Let us go over here to 0,1 is going to be 0 + 1.
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At the point 0,1, the slope of the curve is going to be 1.
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0,2 is going to be 2, it is going to be a lot steeper.
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How about -1,0, at the point -1,0, the slope is -1 + 0 is -1.
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It looks like this.
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For 0 and -1, -1.
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For -2,0, it is -2.
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0 and -2 is -2.
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Let us go 1,1, at the point 1,1, the slope is 1 + 1, it is 2.
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I do this for all the different points of the xy plane.
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When I do that, I get this field of slopes.
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I'm going to get a bunch of lines everywhere, of different slopes, whatever they happen to be, based on the differential equation.
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What this does, let us look at what this actually looks like.
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I used a piece of software, just an online slope field generator.
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This is what it actually looks like, when I plot the fields.
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This is what it looks like.
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At all these different points in the xy plane, in this particular case, we went from x value -5 to 5, y value -5 to 5.
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At all the different points, we put little dashes.
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In this particular case, the particular piece of software that I use, instead of short dashes, it actually uses arrows.
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I personally prefer the arrows, it is totally up to you.
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There are other generators online that just do the dashes like we did before.
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I will show you one of those, in just a minute.
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What is happening here is this, this is a pictorial representation of the slope field of the differential equation.
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Our initial value problem was the following.
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The initial value problem was y’ = x + y, that is the slope field that we see here.
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We said y(0) is equal to, I cannot remember if it was 1 or 3.
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It actually does not really matter.
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In this particular case, we are going to go to 0,1.
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0, it looks like 1 is somewhere around right there.
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We know that the curve passes through that point.
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What does the curve look like?
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Follow the trajectories of the slope fields.
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It is going to look something like this.
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I know it pass through here, I know it goes up like that.
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Here, it looks like it comes down and goes up, there you go.
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Just by following the arrows nearby, here, this arrow is pointing this way.
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We will go that way.
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This arrow is pointing that way.
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We just follow the arrows and it gives us a general trajectory of the particular curve.
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I was not able to solve for this explicitly, I do not know what this curve is, in terms of y as a function of x.
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But I do have an idea of what it looks like.
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In fact, I have an idea what the entire family looks like.
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The arrows or the dashes give you various trajectories for the general solution, in other words, the family of curves.
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What you had here, depending on what you want to follow, just follow along.
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Those are the family of curves.
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For a particular initial value problem, you pick the point that it passes through and
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you draw a particular curve that looks like it satisfies at following the arrows.
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That is all that is happening here.
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Let us go here, we have an extra one.
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Again, what you have is, just follow the arrows, that is all you are doing.
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Just follow the arrows, that will give you a nice family of curves.
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You just have to take a couple of minutes to see which way the arrows are going but nothing particularly strange.
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You know in this particular case, all of these curves tend to be coming near, they tend to stabilize right about there.
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They all seem to be coming from there, and then, diverging and going in whenever direction that they need to be going in.
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We said that, in this particular case, the particular piece of software that I use online,
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the slope generator that I use uses arrows.
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I personally like the arrows, but most of them actually just have dashes and they look like this.
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This is the same slope field except without the arrows, just the dashes.
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Notice it looks exactly the same.
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You just have to follow the slopes to draw out a particular curve.
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The arrows are just reminders that in general, we proceed from left to right.
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In general, we go from negative x to positive x.
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We are going in this direction.
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Some curve is going to start from the left and it is going to come and it is going to head out to the right.
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It is totally a personal choice.
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Again, I personally prefer the arrows, you might prefer something else.
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I want to go ahead and give you the web site for the slope field generator that I use,
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the one with the arrows and it is as follows.
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Www.math.missouri.edu/~bartonae/dfield.html.
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It actually looks as follows, when you pull it up.
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This is actually what it looks like.
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It is actually right up there.
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Basically what this does is, it allows you to pick,
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You enter your differential equation there.
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In this particular slope field generator, you have to be very explicit.
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If I had something like y’ = let us say x³ + y, I would not actually enter this up here as x³ + y.
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I have to be very explicitly, I would actually enter this as x × x × x.
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I have to use a little times symbol, the * symbol, + y.
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Whenever you have trigonometric functions like sin(x),
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you have to actually put the parentheses sin(x), and that explains right here.
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That is the only thing again.
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Now the differential equation is going to be altogether strange and difficult.
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You enter the differential equation there.
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You pick your x min, your x max.
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It allows you to pick your y min and y max.
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It also allows you to pick the length of your arrows.
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It is nice to play around with different lengths of arrows.
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In this particular case, I have a length of 15, whatever the unit happens to be.
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The length of the arrow, of course, it also allows you to choose the number of arrows per horizontal and vertical distance.
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In this case, it is 25, that means there are 25 arrows here and 25 arrows vertically.
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Playing with these numbers will give you different pictures, different degrees of density.
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I think it is a great idea to play with that because sometimes more arrows will make the solution more clear.
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Less arrows will make it more clear, longer errors will make it more clear.
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Shorter arrows might make it more clear.
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The one thing that is interesting about this particular slope field generator is they actually,
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if you check this, it allows you to choose variable length arrows.
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What that does is, as the slope gets steeper, notice in this case, here we have a slope of let us say 0,
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over here we have a slope of let us say 2, here we have a slope of maybe 8, 6 or 7 or 8.
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The arrows, the length of the arrow is proportional to the slope.
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Here is going to be a very short arrow.
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Here is going to be a longer arrow.
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Here it is going to be a huge arrow.
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If you want to take a look at what that looks like, again, you can, just play with it.
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That is about it.
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Some of your homework assignments, you may have to do this by hand.
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But in general, just go ahead and use a slope field generator.
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It is a wonderful tool, that is what these tools are there for.
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We do not want to just use technology for the sake of using technology.
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But in this particular case, it is very appropriate.
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Otherwise, we are just going to be there forever, writing these things out.
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Let us take a look at, basically that is it.
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Essentially, all you have to do is plot the slope field and follow the slopes.
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Just follow the slopes or the arrows, whichever you like, through the field for the various solutions.
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It gives you a geometric solution.
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You have the curve, you do not have the analytic function but you have the curve.
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For all practical purposes, for most applied purposes, really, all you need is the curve.
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Again, the truth is, in real life, those of you who would actually go on to science and engineering work,
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in your professional work, you are going to do things numerically.
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It is going to be pretty rare that you actually end up solving a differential equation explicitly
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and being able to use it to predict the behavior, during other times or other places in space.
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You are going to use a slope field or you are going to use some numerical procedure.
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Follow the slopes and arrows through the field for the various solutions.
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y as a function of x, that would be nice if I actually printed properly.
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For an initial value problem, if you are actually given an initial value in addition to the differential equation,
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you plot the y(x) that passes through the given initial value.
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The given initial value will be y of some x0 is equal to y0.
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Let us go ahead and do a quick example.
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Draw or generate the slope field for the differential equation y’ = x cos y.
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Graph at least three particular solutions for this differential equation.
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There is nothing to actually do, as far as writing it out by hand for us.
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I just enter this into the slope field generator and what I got was the following.
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That is the slope field generator.
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I set my x, max and min to -10 to 10.
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I did -10 to 10 on y, I picked an arrow length of 15 and I picked 30 arrows per horizontal and vertical distance.
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I decided not to go with the variable length arrows.
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This is what I got.
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Essentially, what is happening is this.
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Draw out some solutions.
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It looks like this, all of a sudden jumps up from here and jumps down from here.
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Chances are there is some equilibrium solution.
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Just follow the arrows.
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There, that is a particular solution.
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That is it, that is all you are doing.
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It is all a differential equation is.
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A differential equation is just a slope field.
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It tells you that at a given point xy in the plane, the slope of the curve is that number, whatever it is.
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In this particular case, it was x cos y, y’ = x cos y, that is all.
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Thank you so much for joining us here at www.educator.com.
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Once again, this was the final lesson of the AP calculus, formally.
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After this, we are going to start working completely through some practice AP calculus exams.
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Thank you so much for joining us, we will see you next time, bye.