A horizontal line would be tangent to the curve at y equals 0.

There is a red trajectory line.

So if I were to take that point and draw a tangent line a point that just grazes our curve if I were to just draw a tangent line.

So there it comes down. There is a gecko at the end of that trajectory line.

You can imagine a tangent line that goes just like that, and it would just barely graze the curve at that point.

It means that the slope of the tangent line at x is equal to 1, to this curve, to the function, is minus 2.

Curve

So it's just the area under this curve.

I've told you multiple times that the derivative of a curve at a point is the slope of the tangent line, but our friend ?

Evolute

And I'll do it with two different parameterizations for the same curve.

This is the actual video not staged, a real research video of animal gliding down. There is a red trajectory line.

And I'm interested in what the line integral, the closed line integral, around that path of F dot dr is.

Just me just draw it here.

This is the curve for Joplin.

That's my curve.

Let me draw my curve.

And we draw the curve here.

Just around that bend.

Typically, you choose yourself a rapid transport route, and in your mind, this route is perceived as a straight line.

So x varies from the lower y curve to the upper y curve, right?

And let's say I want to figure out the area under the curve between the curve and, really, the x axis.

Evolute of this curve

You could see the curve happening.

It's a steeper upward sloping curve.

It'll spread to the entire curve.

Now, if you're on the equator here that's these two lines here, the red line and the purple line the amount of nitric oxide that's released is the area under the curve, it's the area in this space here.

But the idea here is that the area under this curve should be twice the area under this curve.

Evolute of a curve

polygonal curve

The curve could look something like this.

Now look at this curve.

You want to shift this curve up.

They're being accalerated towards the center of the curve.

They're being accelerated to the center of the curve.

And let me draw my curve again.

Look for a murderer in a spinning?

The upper y curve is x1 of y, so the upper y curve looks something like that.

Pessimistic Europeans are behind the curve.

Example tool for the curves framework

This is a test tool for the Curve Framework.

Let's say the curve looks something like this.

Let's say that this is your curve.

My end point, the curve here is c.

round a turning in the path... thank you.