yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Worked example: forming a slope field | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

In drawing the slope field for the differential equation, the derivative of y with respect to x is equal to y minus 2x. I would place short line segments at select points on the xy-plane.

At the point (-1, 1), I would draw a short segment of slope blank. Like always, pause this video and see if you can fill out these three blanks.

When you're drawing the short segments to construct this slope field, you figure out their slope based on the differential equation. So, you're saying when x is equal to -1 and y is equal to 1, what is the derivative of y with respect to x? That's what this differential equation tells us.

So, for this first case, the derivative of y with respect to x is going to be equal to y, which is 1, minus 2 times x. x is -1, so this is going to be negative 2, but you're subtracting it, so it's going to be plus 2. Therefore, the derivative of y with respect to x at this point is going to be 3.

I would draw a short segment or a short segment of slope 3. We keep going. At the point (0, 2), let's see. When x is 0 and y is 2, the derivative of y with respect to x is going to be equal to y, which is 2, minus 2 times 0. Well, that's just going to be 2.

Now, last but not least, for this third point, the derivative of y with respect to x is going to be equal to y, which is 3, minus 2 times x. x here is 2. So, 2 times 2 is 4, and 3 minus 4 is equal to negative 1.

And that's all that problem asks us to do. Now, if we actually had to do it, it would look something like—I'll try to draw it real fast.

So, let's see. Let me make sure I go to make sure I have space for all of these points here. So, that's my coordinate axes, and I want to get the point (0, 2). That's (0, 2). Actually, I want to go all the way to (2, 3). So, let me get some space here. So, 1, 2, 3; and then 1, 2, 3.

Then we have to go to (-1, 1). We might go right over here. For this first one, this exercise isn't asking us to do it, but I'm just making it very clear how we would construct the slope field.

So, the point (-1, 1)—a short segment of slope 3. Slope 3 would look something like that. Then at the point (0, 2)—a slope of 2. (0, 2), the slope is going to be 2, which looks something like that.

Then at the point (2, 3)—at (2, 3), a short segment of slope negative 1. So, (2, 3), a segment of slope negative 1, would look something like that.

You would keep doing this at more and more points. If you had a computer to do it, that's what the computer would do, and you would draw these short line segments to indicate what the derivative is at those points. You get a sense of, I guess you say, the solution space for that differential equation.

More Articles

View All
6 Stocks Super Investors are Buying!
Listen closely because I’m about to let you in on one of the biggest secrets when it comes to investing. If you want to know what stocks you should be buying, pay attention to what the greatest investors are purchasing for their own portfolios. Investors …
Cool Things on YouTube and More! DONG #19
Hey, Vsauce. Michael here. Do you remember that Italian researcher I met in an airplane a couple of months ago? Well, I had to learn more, and so we’re gonna meet tomorrow here in Rome, Italy. In the meantime, here’s some cool things on the internet that …
Measuring area with tiled square units
What we’re going to do in this video is look at two rectangles that have the exact same area, and we’re going to measure each of them with a different square unit. So, this top unit right over here, this is a square foot. That means its height is one foo…
Smart Fish | Wicked Tuna
Come on, bite me! There he is, down! Run on, die! Yo, come on! Definitely a tuna. This fish is exactly what we need: a nice inshore bite, and it’s got some weight. If we can get a tail rope on this fish, it could be a $5,000 paycheck for us. Got a big fi…
How To Make Traditional Greek Salad | Chef Wonderful
[Music] [Applause] [Music] [Applause] [Music] [Applause] [Music] [Applause] [Music] Chef: “I want to play in the garden, and I’m really excited today. Do you know why? We’re going to be cooking one of the most primal and the important dishes mankind has …
Ideal sources | Circuit analysis | Electrical engineering | Khan Academy
There’s two kinds of ideal sources we’re going to talk about. One is an ideal voltage source, and the other is an ideal current source. An ideal voltage source, the symbol looks like a circle; like that, we put a voltage indication right inside there. Tha…