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Problem 44 Hard Difficulty

The gravitational force exerted by the planet Earth on a unit mass at a distance $ r $ from the center of the planet is

$ F(r) = \left\{
\begin{array}{ll}
\frac{GMr}{R^3} & \mbox{if $ r < R $}\\
\frac{GM}{r^2} & \mbox{if $ r \ge R $}
\end{array} \right.$

where $ M $ is the mass of Earth, $ R $ is its radius, and $ G $ is the gravitational constant. Is $ F $ a continuous function of $ r $?

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Video Transcript

Mhm. All right. We have this piece wise function already forgot what it was forced or something. And then it's defined for the radius less than big art. And the radius greater than or equal to big R. And the question is, is F continuous at our And you have little art. But I think you mean big are because little R. Is the variable. It's like X. Okay, so to tell us some things continue as you have to check. Three things. Okay. First, does F. Of our exist? And then second, does phillips. Does the limit as Okay? Are variable is little R. Little R approaches big are of little are exist and three does No. 100, let's do that in a different color because I want to be fancy. Okay, that's the same color. It does F of are equal to the limit as our approaches big R. F. Of our so does one equal to. Okay, So what you're trying to find out is when you have this piece wise function, does it look like this? Um Like let's say this is big. Are does it come in like here and then go out here? Okay, that's not continuous. Okay, because the limit doesn't exist because from the left us one thing and from the right, it's another thing. What does it look like? This comes like this and then goes like this. Okay. No, that's not continuous because that right, there is F of our and that right? There is the limit and they are not the same. Okay, so what you're looking for is for it come together and meat and not have a break or a hole in it. That's what continuous means. Okay, so to check that this is happening, we just check these three things. Okay, So here we go. Get these out of the way. One f of big are equals. Okay, if you're gonna use big are it says to plug it into, this one says if you're using the variable that's greater than or equal to big are put it here. Yeah. So you get Gm over R squared. So number one is okay F of our does exist. Number two. Yeah. Does the limit exist? So we have to due to limits because it's piecewise. So we're going to take the limit as little our approaches big are And for numbers little are less than big are we're taking the limit of Gm Little are over big are cute. And we're coming from the left so we're gonna put it like this. Care. Okay, So when you're taking this kind of limit, any kind of limit, if you put the number that you're taking the limit as our approach is, if you can just plug it in and a number comes out, then that's the limit. So let's just plug it in and see what happens. Gm big are over R cubed, which is Gm over R square. All right, That's good. And then for the other side we have to use the bottom equation. Okay. And it says Gm over little r squared. So I'm taking the limit as our approaches big are from the right And so I'm gonna plug into this thing in place of little arm putting big are I get Gm over R squared. All right. The limit from the left exist. The limit from the right exist. They are both equal to Gm over R squared. So the answer to this is yes, limit. As our approaches are plus half of our exists that equals exists. It exists and it equals Gm over big R squared. Okay, step three. Do those two things equal to each other? Gm over R squared equals Gm over R squared. So f of our equals true. Let me have small room here. F R equals the limit as our purchase are effort. Oops big F of our. So yes, F is continuous. Uh, R equals big are came because the the point which is this is in the place where it's supposed to be. This is saying from the left, it's coming at Gm over R squared. From the right is coming at Gm over R squared. And then this is saying at our that point is colored in so yes, it is continuous at R equals R.