For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

### Slope Fields

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Slope Fields 0:35
- Slope Fields
- Graphing the Slope Fields, Part 1
- Graphing the Slope Fields, Part 2
- Graphing the Slope Fields, Part 3
- Steps to Solving Slope Field Problems 20:24
- Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y 22:38

### AP Calculus AB Online Prep Course

### Transcription: 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 technique*0123

*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 and*0858

*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 explicitly*1292

*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

0 answers

Post by Muhammad Ziad on April 12 at 10:29:51 PM

Hello professor,

I was wondering: in the explanation about slope fields, how do you know that the DE, dy/dx=x+y is not separable?

Thanks so much for your lectures, they are so helpful!

Best,

M. Ziad

1 answer

Last reply by: Professor Hovasapian

Fri Jun 3, 2016 6:05 PM

Post by Mohamed Talni on May 25, 2016

Hello Professor,

excellent explanation. Do you know when the course BC Calculus will be published on the website.

Kind regards,

M.Tal