AP Calculus AB Slope Fields

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to actually close out the course of AP Calculus formally.0005

This is going to be our last lesson.0010

We are going to be talking about slope fields.0013

For the last few lessons, we have been talking about differential equations, what they are.0015

We have taken a look at separation of variables.0019

We have taken a look at one particular type of differential equation, to get a feel for it.0022

Talk about qualitative solutions.0026

Now we are going to round it out with this thing called a slope field.0030

Let us jump right on in.0033

It is not always possible to solve a differential equation explicitly.0039

For certain types of differential equations like the ones that we dealt with, the separable ones,0044

yes, we can separate them, we can integrate and we can solve explicitly for a family of solutions.0049

Also, if we have an initial value, we can also find a particular solution.0055

But the truth is, most of the time in the real world, it is not possible to solve it explicitly.0059

But there are other techniques we can use and slope field is going to be one of those techniques.0064

Let us go ahead and work in blue.0073

It is not always possible to solve the de explicitly.0078

That is to find a nice formula that relates x and y, relating x and y.0100

y is some function of x.0119

It is not always possible.0121

In this lesson, we introduce a geometric graphical technique0123

that still allows us to extract information about the solution for a given differential equation.0149

Slope fields, let us look at a particular initial value problem.0185

Let us look at the following ivp.0199

Remember, an initial value problem consists of a differential equation and it also consists of an initial value.0207

For some value of x, we know that it is this y.0215

In this particular case, y(0) = 3.0221

We know that this differential equation, whatever solution we come up with,0224

if we happen to be able to come up with one, when x is 0, the y value is going to be 3.0229

It is going to pass through that point.0234

Let me write this as dy dx which is, y’ = x + y and y(0) = 3.0238

This equation is not separable.0251

There is really not much we can do with it.0253

This equation, at least at this level, this equation is not separable.0256

We cannot solve, in other words, we cannot integrate for an explicit solution.0269

What this differential equation is saying is the following.0289

Again, it is always nice, when you are faced with any kind of equation, any kind of formula,0292

to ask yourself what is actually going on.0296

Let me go ahead and move to the next page and rewrite the equation up here.0302

I have got dy dx is equal to x + y.0306

This de says the following.0311

At a given point xy, the slope of the solution curve y(x) is x + y.0321

Now we know that when we solve a particular differential equation, we are going get some function of x.0351

It is going to be something like, let us say y = sin(x), whatever it is.0355

This is just an example.0361

It is going to be some function of x.0362

y expressed as a function of x is just a curve.0365

We know that, we have been dealing with them for years and years.0367

There you have it, it is a curve.0375

What this is telling me dy dx, we know that the derivative is a slope.0377

It is telling me that the derivative of whatever function it is that I'm trying to find, y(x),0383

its derivative which is its slope, at a particular point xy is actually equal to the x value + the y value.0391

A differential equation actually gives me a way of finding the slope of the curve at that point.0399

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.0407

We are going to make little marks representing that slope.0411

And then, we are going to follow those little marks and actually draw out a solution curve graphically, instead of solving for it explicitly.0414

That is what is going on.0421

Let us reiterate, if we can.0424

If we could graph y(x), the solution in the xy plane,0427

its slope at various points xy is given by x + y.0442

I can point 5,4, the slope of the curve, as it passes through 5,4 is going to be 5 + 4.0460

It is going to be 9.0469

We know that the curve is actually headed pretty steeply up, that is all this is saying.0471

That is all a differential equation really is.0476

Now what we want to do is, let us see if we can recover the graph.0479

If we can recover the graph of y(x) pictorially, instead of explicitly, analytically.0485

Let us see if we can recover the graphs by plotting the slopes at various points.0498

Let me draw a little xy coordinate system.0517

Here is our differential equation, let me rewrite it down here.0525

I’m going to rewrite it in red.0527

We have y’ = x + y.0529

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 0,0534

which means that at that point, my slope is horizontal.0544

I’m going to put a little dash representing this slope.0549

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.0552

Now I know that when the curve passes through 0,0, actually it is going to just touch it.0560

It is going to have a slope of 0 at that point.0567

Let us try the point 1,0.0571

Let us make this 1, 2, 3.0573

Let us go -1, -2, -3.0577

Let us go 1, 2, 3, up, -1, -2, and -3.0580

At the point 1,0, the derivative is 1 + 0, it is 1.0585

It looks like this.0591

At the point 2,0, the slope is going to be 2.0594

It is going to be a little steeper like that.0598

Let us go over here to 0,1 is going to be 0 + 1.0600

At the point 0,1, the slope of the curve is going to be 1.0605

0,2 is going to be 2, it is going to be a lot steeper.0610

How about -1,0, at the point -1,0, the slope is -1 + 0 is -1.0615

It looks like this.0621

For 0 and -1, -1.0625

For -2,0, it is -2.0629

0 and -2 is -2.0633

Let us go 1,1, at the point 1,1, the slope is 1 + 1, it is 2.0635

I do this for all the different points of the xy plane.0642

When I do that, I get this field of slopes.0647

I'm going to get a bunch of lines everywhere, of different slopes, whatever they happen to be, based on the differential equation.0651

What this does, let us look at what this actually looks like.0661

I used a piece of software, just an online slope field generator.0667

This is what it actually looks like, when I plot the fields.0671

This is what it looks like.0677

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 5.0678

At all the different points, we put little dashes.0689

In this particular case, the particular piece of software that I use, instead of short dashes, it actually uses arrows.0692

I personally prefer the arrows, it is totally up to you.0700

There are other generators online that just do the dashes like we did before.0702

I will show you one of those, in just a minute.0706

What is happening here is this, this is a pictorial representation of the slope field of the differential equation.0708

Our initial value problem was the following.0717

The initial value problem was y’ = x + y, that is the slope field that we see here.0722

We said y(0) is equal to, I cannot remember if it was 1 or 3.0729

It actually does not really matter.0741

In this particular case, we are going to go to 0,1.0742

0, it looks like 1 is somewhere around right there.0746

We know that the curve passes through that point.0751

What does the curve look like?0754

Follow the trajectories of the slope fields.0755

It is going to look something like this.0760

I know it pass through here, I know it goes up like that.0762

Here, it looks like it comes down and goes up, there you go.0767

Just by following the arrows nearby, here, this arrow is pointing this way.0774

We will go that way.0780

This arrow is pointing that way.0782

We just follow the arrows and it gives us a general trajectory of the particular curve.0783

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.0789

But I do have an idea of what it looks like.0797

In fact, I have an idea what the entire family looks like.0799

The arrows or the dashes give you various trajectories for the general solution, in other words, the family of curves.0807

What you had here, depending on what you want to follow, just follow along.0838

Those are the family of curves.0856

For a particular initial value problem, you pick the point that it passes through and0858

you draw a particular curve that looks like it satisfies at following the arrows.0863

That is all that is happening here.0867

Let us go here, we have an extra one.0874

Again, what you have is, just follow the arrows, that is all you are doing.0884

Just follow the arrows, that will give you a nice family of curves.0889

You just have to take a couple of minutes to see which way the arrows are going but nothing particularly strange.0901

You know in this particular case, all of these curves tend to be coming near, they tend to stabilize right about there.0908

They all seem to be coming from there, and then, diverging and going in whenever direction that they need to be going in.0915

We said that, in this particular case, the particular piece of software that I use online,0923

the slope generator that I use uses arrows.0929

I personally like the arrows, but most of them actually just have dashes and they look like this.0931

This is the same slope field except without the arrows, just the dashes.0938

Notice it looks exactly the same.0942

You just have to follow the slopes to draw out a particular curve.0944

The arrows are just reminders that in general, we proceed from left to right.0951

In general, we go from negative x to positive x.0982

We are going in this direction.0984

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.0986

It is totally a personal choice.0991

Again, I personally prefer the arrows, you might prefer something else.0995

I want to go ahead and give you the web site for the slope field generator that I use,1000

the one with the arrows and it is as follows.1004

Www.math.missouri.edu/~bartonae/dfield.html.1011

It actually looks as follows, when you pull it up.1043

This is actually what it looks like.1046

It is actually right up there.1051

Basically what this does is, it allows you to pick,1053

You enter your differential equation there.1057

In this particular slope field generator, you have to be very explicit.1059

If I had something like y’ = let us say x³ + y, I would not actually enter this up here as x³ + y.1064

I have to be very explicitly, I would actually enter this as x × x × x.1081

I have to use a little times symbol, the * symbol, + y.1086

Whenever you have trigonometric functions like sin(x),1091

you have to actually put the parentheses sin(x), and that explains right here.1095

That is the only thing again.1100

Now the differential equation is going to be altogether strange and difficult.1101

You enter the differential equation there.1105

You pick your x min, your x max.1107

It allows you to pick your y min and y max.1110

It also allows you to pick the length of your arrows.1113

It is nice to play around with different lengths of arrows.1117

In this particular case, I have a length of 15, whatever the unit happens to be.1119

The length of the arrow, of course, it also allows you to choose the number of arrows per horizontal and vertical distance.1124

In this case, it is 25, that means there are 25 arrows here and 25 arrows vertically.1130

Playing with these numbers will give you different pictures, different degrees of density.1136

I think it is a great idea to play with that because sometimes more arrows will make the solution more clear.1142

Less arrows will make it more clear, longer errors will make it more clear.1148

Shorter arrows might make it more clear.1153

The one thing that is interesting about this particular slope field generator is they actually,1157

if you check this, it allows you to choose variable length arrows.1161

What that does is, as the slope gets steeper, notice in this case, here we have a slope of let us say 0,1167

over here we have a slope of let us say 2, here we have a slope of maybe 8, 6 or 7 or 8.1174

The arrows, the length of the arrow is proportional to the slope.1179

Here is going to be a very short arrow.1184

Here is going to be a longer arrow.1186

Here it is going to be a huge arrow.1188

If you want to take a look at what that looks like, again, you can, just play with it.1191

That is about it.1196

Some of your homework assignments, you may have to do this by hand.1199

But in general, just go ahead and use a slope field generator.1201

It is a wonderful tool, that is what these tools are there for.1204

We do not want to just use technology for the sake of using technology.1208

But in this particular case, it is very appropriate.1211

Otherwise, we are just going to be there forever, writing these things out.1214

Let us take a look at, basically that is it.1220

Essentially, all you have to do is plot the slope field and follow the slopes.1230

Just follow the slopes or the arrows, whichever you like, through the field for the various solutions.1247

It gives you a geometric solution.1267

You have the curve, you do not have the analytic function but you have the curve.1271

For all practical purposes, for most applied purposes, really, all you need is the curve.1276

Again, the truth is, in real life, those of you who would actually go on to science and engineering work,1281

in your professional work, you are going to do things numerically.1286

It is going to be pretty rare that you actually end up solving a differential equation explicitly1292

and being able to use it to predict the behavior, during other times or other places in space.1296

You are going to use a slope field or you are going to use some numerical procedure.1303

Follow the slopes and arrows through the field for the various solutions.1310

y as a function of x, that would be nice if I actually printed properly.1317

For an initial value problem, if you are actually given an initial value in addition to the differential equation,1324

you plot the y(x) that passes through the given initial value.1331

The given initial value will be y of some x0 is equal to y0.1346

Let us go ahead and do a quick example.1358

Draw or generate the slope field for the differential equation y’ = x cos y.1360

Graph at least three particular solutions for this differential equation.1366

There is nothing to actually do, as far as writing it out by hand for us.1370

I just enter this into the slope field generator and what I got was the following.1374

That is the slope field generator.1380

I set my x, max and min to -10 to 10.1381

I did -10 to 10 on y, I picked an arrow length of 15 and I picked 30 arrows per horizontal and vertical distance.1385

I decided not to go with the variable length arrows.1393

This is what I got.1397

Essentially, what is happening is this.1398

Draw out some solutions.1403

It looks like this, all of a sudden jumps up from here and jumps down from here.1405

Chances are there is some equilibrium solution.1410

Just follow the arrows.1414

There, that is a particular solution.1418

That is it, that is all you are doing.1431

It is all a differential equation is.1434

A differential equation is just a slope field.1435

It tells you that at a given point xy in the plane, the slope of the curve is that number, whatever it is.1438

In this particular case, it was x cos y, y’ = x cos y, that is all.1449

Thank you so much for joining us here at www.educator.com.1460

Once again, this was the final lesson of the AP calculus, formally.1464

After this, we are going to start working completely through some practice AP calculus exams.1468

Thank you so much for joining us, we will see you next time, bye.1475