Learn about Mean Value Theorem Video

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Learn about Mean Value Theorem

I've got on several requests to explain or teach the Mean Value of Theorem, so let’s do that in this video. This is the Mean Value Theorem. I have mixed feelings about the Mean Value Theorem. That’s kind of neat but what you’ll see is kind of—it might not be obvious to prove but in the intuition behind is very obvious. The reason why I have mixed feelings about it, is that even though as you’ll hopefully see a theorem. As you’ll hopefully see that the intuition is pretty obvious but they stick in the math books and people are just trying to learn calculus and get to what matters and then they put the mean value theorem in there, and they’re and have all this function notation, and they have all of this words, and it just confuses people. Hopefully, this video will clarify that a little bit and I am curious to see what you think of it. So let’s see. What is the mean value of theorems say? Let me draw some axis? I'll do a visual explanation first, The Mean Value Theorem. I think colors for magenta. So that’s my X axis. This is my Y axis. Let’s say, I have some function on F of X. So let me draw my F of X. That’s as good as any. And this is some function on F of X and I'm going to put a few conditions on F of X. F of X has to be continuous and differentiable. I know a lot of you probably get intimidated when you hear these words and sounds like you know, what a mathematician would say and it sounds very abstract. All continuous means is that the curve is connected to itself as you go along it. In here, the conditions are over a closed interval. This is another very mathy term you’ll see. So there are often they had a close interval from A to B. All that means is an interval. Let’s say A is the low point. Let’s say this is A. We don’t know what number that is. That can be minus five or who knows. Let’s say this is B right here. Let’s say that’s B. So when people talk about a Closed Interval, to find on the close interval, that means that the function that this to be defined at every number between A and B and the function is to be defined at A and at B. If they said over an open interval between A and B, that means that it’s only defined at every value between A and B but not necessarily at A and B. So that’s has to be continuous, differentiable, and let’s say it is defined over the close interval and this is just the notation for it A, B. That means it has to be defined at all of the X values from A to B including A and B. If it was an open interval, you would write it like this. You’d would write A and B. That means it interval for all the numbers between A and B but not including those, so let’s ignore that. So back to the mean valued there. So you know hopefully what continuous means. Let me draw here a function that’s not continuous. So a function that’s not continuous would look like this. It would go like this and then it would start up here and go like that. This would be an example of a function. Let’s say the same axis. Let me draw in a different color. That was our Y. If that was our Y axis and that was our X axis, just to give you a reference when I drew. So if the function is continuous, continuous, continuous and then it jumps, that disconnect— that would make this function discontinuous or would not be a continuous function. A function has to be continuous and what does differentiable mean? Differentiable means that at every point over the interval that we care about, you have to be able to find the derivative. That means you can take the derivative of it. It’s differentiable. What else does that mean? That means that if you are to graph the derivative of this function that it is also continuous and I’ll let you think about that for a second. Actually, in this video I'm going to show you an example of a function that is continuous but not differentiable and because of that, the mean value theorem breaks down. Anyway, let’s get back to the Mean Value theorem. Most of the functions we deal with satisfy all three of these things. Almost you know, you are doing limit problems and they’ve tried to make these things break down. Anyway, back to the functions. This function meets all of these requirements so all it says is—if I were to take the average slope between point A and point B, so what is the slope, the average slope between point A and point B? The slope is just rise over run, right? So what is it? Let me see if I can draw the average slope. The run would be this; distance. That would be the run and this would be the rise. This is the point right here. That’s the point A, F of A. Over here, this is the point B, F of B. What’s the average slope between A and B? What’s rise over run? So what is the rise? What is this distance? How much have we gone up from F of A to F of B? Well the rise will just be F of B, this height, F of B minus F of A, F of B minus F of A and what’s the run? What’s this distance? Well, it’s just B minus A. And if I were to draw a line that has that average slope, it would look something like this. We can make it go through those two points but it really doesn’t have to. So let me do it in a—let me just do it in a blue. So that’s the average slope between those two points right. What is the mean value theorem to tell us? It says if F of X is defined over this closed interval from A to B and the F of X is continuous and is differentiable that we can take the derivative at any point, then there must be some point C where F prime of C is equal to this thing so it’s equal to F prime of C. So what does that telling us? So all that’s telling us is, if it were continuous differentiable defined over the close interval that there’s some point C, and C has to be between A and A. There’s some point between A and B and it could be at one of the points but there’s some point C where the derivative at C or the slope at C or the instantaneous slope at C is exactly equal to the average slope over that interval. So what does that mean? We can look at it visually. Is there any point along this curve where the slope looks very similar to this average slope we calculated? Well sure, let’s see. It looks like maybe this point right here, just the way I do it. This is very inexact with that point. It looks like the slope—I could say that the slope is something like that right there. We don’t know what analytically this function is but visually you can see that at this point C, the derivative so I just picked that point. So this could be our point C. And how do we say that? Well because F prime of C is this slope and it’s equal to the average slope. F prime of C is this thing and is going to be equal to the average slope over the whole thing. This curve actually probably has another point where the slope is equal to the average slope. See, this one looks like right around on there. Just the way I drew it. It looks like the slope there could look something like could be a parallel as well. These lines should be parallel. The tangent line should be parallel. So hopefully, that makes a little sense to you. Another way to think about it is that your average—let’s actually let me draw a graph just to make sure that we had the point home. Let’s draw my position as a function of time. This is something and this will make it applicable to the real world. That’s my X axis or the time axis. That’s my position axis and this is going back to our original intuition of what a derivative is. This is time and I call this position or distance or it doesn’t matter. If I was moving at a constant velocity, my position as a function of time would just be a straight line and the velocity is actually your slope. Let’s say I had a varying velocity and in reality if you are driving a car, you are always at a variable velocity. Let’s say I started to stand still at time to equal zero and then I accelerate and then I decelerate a little bit. I keep decelerating and then I come to a stand. So my position stay still and then I accelerate again. Decelerate, accelerate, etcetera. I have a variable velocity and this could be my position as a function of time. So all this says. Let’s say that after—you know this is time zero, position zero. Let’s say after one hour, Lets say that, that is one hour. That time equals one hour. Let’s say that I have gone 60-miles. So what can you say? You can say that my average velocity, so velocity average, it equals just change in distance divided by change in time so it equals 60-miles per hour. So what the Mean Value Theorem says is “okay your average velocity”. So you could almost do it as the average slope between this point and this point with 60. If your average velocity was 60-miles per hour, there were some point in time maybe more but there is at least one point in time where you are going exactly 60-miles per hour. That makes sense right? If you averaged is 60-miles per hour and maybe you are going 40-miles per hour on some of the point. But at some point went 80 and in between you have to be going 60 miles per hour. Let me see if I could draw that graphically. This slope is my average velocity. The way I draw it there’s probably two points, let’s see probably right around here. I was probably going 60-miles per hour. The slope is probably 60 there. The instantaneous velocity probably there as well. Before I leave, let’s do this analytically just to—I guess work with the numbers. The reason why I have mixed feelings about the mean value theorem is it’s useful. Later on probably if you become a math major you’ll maybe use it to prove sone theorems or maybe you’ll prove it itself. But if you are just applying calculus for the most part, you’re not going to be using the mean value theorem too much. Anyway, if you got to know it you got to know it. It does tell you something else about the world, so t is interesting that way. Let’s say we have the function F of X is equal to X squared minus four X. And the interval that I care about here is between, is the close interval, so I am including two from two to four. Now, the mean value theorem tells us that if this function is defined on this interval and it is right, we could put any number. The domain of this is actually all real numbers. I could put any number here so obviously it’s going to be defined over this interval. So it’s defined over the interval. This is continuous, this is differentiable. You could take the derivative and the derivative is continuous. So the mean value theorem should apply here. Let’s see what value of C is equal to the average slope between two and four. What’s the average slope between two and four? That’s going to be F of four so the change in the function F of four minus F of two divided by the change in X, so it’s four minus two. This is equal to the average slopes. F of four is 16 minus 16, so that’s a zero. Let me make sure that four times four – 16, minus four times four, 16 minus F of two. F of two is two squared is four and then minus four times two so minus eight. Let me show that right. All that over two and so this equals minus four so it equals four over two. The average slope from X is equal to two to X is equal to four is two. Now, that mean value theorem tells us that there must be some point that’s between these two maybe including one of those where the slope at that point is exactly equal to two. Let’s figure out what point that is. That is C. Let’s take the derivative because the derivative at C is going to be equal to two so we just take the derivative so let’s say F prime of X is equal to two X minus four. We want to figure out at what X value does this equal to. So we say two X minus four is equal to two. Where does the slope equal to? You get two X is equal to six, X is equal to three. If X is equal to three, the derivative is exactly equal to the average slope. Let me see if a can do the graphing calculator here. Let me see what I can do. Here is the graph of X squared minus four X. let me see if I can make a little bit bigger X squared minus four X. the interval that we care about is from here to here so the average slope, the average slope over that interval was two. If we were to draw the slope, it is like that. The slope would look like that and at the point three, the slope is exactly two. Let me actually to draw it. This isn’t too hard to draw for myself. Let me see. If that’s the X axis, I wan that that graph out of the way. That’s the Y axis. So the graph goes to the point zero, zero. I’m going to draw this as neatly as possible. Nope, that’s not neat. The graph goes something like this and it tips up and it goes like that. Actually it keeps going straight up like that. It’s a parabola. So this is the point four. The point two is here and at two, we’re at negative four. So the vertex is at the point two minus four. So what we said, the average slope over—This close interval that we care about between two and four it is from two here to four here. That’s the interval two to four. The average slope is two. It doesn’t look like it only because I’ve kind of compress the Y axis. We are saying at the point X is equal to three. The slope is equal to exactly that. So at X is equal to three, the slope is equal to the same thing. That’s all the mean to the value of theorem is. I know it sounds complicated. People talk about continuity and differentiability and F prime of C and all of this. But all it says this— there are some point between these two points where the instantaneous slope or the slope exactly that point is equal to the slope between these two points. Hope I didn’t confuse you.

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I've got on several requests to explain or teach the Mean Value of Theorem, so let’s do that in this video. This is the Mean Value Theorem.

I have mixed feelings about the Mean Value Theorem. That’s kind of neat but what you’ll see is kind of—it might not be obvious to prove but in the intuition behind is very obvious. The reason why I have mixed feelings about it, is that even though as you’ll hopefully see a theorem. As you’ll hopefully see that the intuition is pretty obvious but they stick in the math books and people are just trying to learn calculus and get to what matters and then they put the mean value theorem in there, and they’re and have all this function notation, and they have all of this words, and it just confuses people. Hopefully, this video will clarify that a little bit and I am curious to see what you think of it. So let’s see.

What is the mean value of theorems say? Let me draw some axis? I'll do a visual explanation first, The Mean Value Theorem. I think colors for magenta. So that’s my X axis. This is my Y axis. Let’s say, I have some function on F of X. So let me draw my F of X. That’s as good as any. And this is some function on F of X and I'm going to put a few conditions on F of X. F of X has to be continuous and differentiable. I know a lot of you probably get intimidated when you hear these words and sounds like you know, what a mathematician would say and it sounds very abstract.

All continuous means is that the curve is connected to itself as you go along it. In here, the conditions are over a closed interval. This is another very mathy term you’ll see. So there are often they had a close interval from A to B. All that means is an interval. Let’s say A is the low point. Let’s say this is A. We don’t know what number that is. That can be minus five or who knows. Let’s say this is B right here. Let’s say that’s B.

So when people talk about a Closed Interval, to find on the close interval, that means that the function that this to be defined at every number between A and B and the function is to be defined at A and at B. If they said over an open interval between A and B, that means that it’s only defined at every value between A and B but not necessarily at A and B. So that’s has to be continuous, differentiable, and let’s say it is defined over the close interval and this is just the notation for it A, B.

That means it has to be defined at all of the X values from A to B including A and B. If it was an open interval, you would write it like this. You’d would write A and B. That means it interval for all the numbers between A and B but not including those, so let’s ignore that.

So back to the mean valued there. So you know hopefully what continuous means. Let me draw here a function that’s not continuous. So a function that’s not continuous would look like this. It would go like this and then it would start up here and go like that. This would be an example of a function. Let’s say the same axis. Let me draw in a different color. That was our Y. If that was our Y axis and that was our X axis, just to give you a reference when I drew.

So if the function is continuous, continuous, continuous and then it jumps, that disconnect— that would make this function discontinuous or would not be a continuous function. A function has to be continuous and what does differentiable mean?

Differentiable means that at every point over the interval that we care about, you have to be able to find the derivative. That means you can take the derivative of it. It’s differentiable. What else does that mean? That means that if you are to graph the derivative of this function that it is also continuous and I’ll let you think about that for a second. Actually, in this video I'm going to show you an example of a function that is continuous but not differentiable and because of that, the mean value theorem breaks down.

Anyway, let’s get back to the Mean Value theorem. Most of the functions we deal with satisfy all three of these things. Almost you know, you are doing limit problems and they’ve tried to make these things break down. Anyway, back to the functions. This function meets all of these requirements so all it says is—if I were to take the average slope between point A and point B, so what is the slope, the average slope between point A and point B? The slope is just rise over run, right? So what is it? Let me see if I can draw the average slope. The run would be this; distance. That would be the run and this would be the rise. This is the point right here. That’s the point A, F of A. Over here, this is the point B, F of B.

What’s the average slope between A and B? What’s rise over run? So what is the rise? What is this distance? How much have we gone up from F of A to F of B? Well the rise will just be F of B, this height, F of B minus F of A, F of B minus F of A and what’s the run? What’s this distance? Well, it’s just B minus A. And if I were to draw a line that has that average slope, it would look something like this. We can make it go through those two points but it really doesn’t have to.

So let me do it in a—let me just do it in a blue. So that’s the average slope between those two points right. What is the mean value theorem to tell us? It says if F of X is defined over this closed interval from A to B and the F of X is continuous and is differentiable that we can take the derivative at any point, then there must be some point C where F prime of C is equal to this thing so it’s equal to F prime of C.

So what does that telling us? So all that’s telling us is, if it were continuous differentiable defined over the close interval that there’s some point C, and C has to be between A and A. There’s some point between A and B and it could be at one of the points but there’s some point C where the derivative at C or the slope at C or the instantaneous slope at C is exactly equal to the average slope over that interval. So what does that mean?

We can look at it visually. Is there any point along this curve where the slope looks very similar to this average slope we calculated? Well sure, let’s see. It looks like maybe this point right here, just the way I do it. This is very inexact with that point. It looks like the slope—I could say that the slope is something like that right there. We don’t know what analytically this function is but visually you can see that at this point C, the derivative so I just picked that point. So this could be our point C. And how do we say that? Well because F prime of C is this slope and it’s equal to the average slope. F prime of C is this thing and is going to be equal to the average slope over the whole thing. This curve actually probably has another point where the slope is equal to the average slope. See, this one looks like right around on there. Just the way I drew it. It looks like the slope there could look something like could be a parallel as well.

These lines should be parallel. The tangent line should be parallel. So hopefully, that makes a little sense to you. Another way to think about it is that your average—let’s actually let me draw a graph just to make sure that we had the point home. Let’s draw my position as a function of time. This is something and this will make it applicable to the real world. That’s my X axis or the time axis. That’s my position axis and this is going back to our original intuition of what a derivative is.

This is time and I call this position or distance or it doesn’t matter. If I was moving at a constant velocity, my position as a function of time would just be a straight line and the velocity is actually your slope. Let’s say I had a varying velocity and in reality if you are driving a car, you are always at a variable velocity. Let’s say I started to stand still at time to equal zero and then I accelerate and then I decelerate a little bit. I keep decelerating and then I come to a stand. So my position stay still and then I accelerate again. Decelerate, accelerate, etcetera.

I have a variable velocity and this could be my position as a function of time. So all this says. Let’s say that after—you know this is time zero, position zero. Let’s say after one hour, Lets say that, that is one hour. That time equals one hour. Let’s say that I have gone 60-miles. So what can you say? You can say that my average velocity, so velocity average, it equals just change in distance divided by change in time so it equals 60-miles per hour.

So what the Mean Value Theorem says is “okay your average velocity”. So you could almost do it as the average slope between this point and this point with 60. If your average velocity was 60-miles per hour, there were some point in time maybe more but there is at least one point in time where you are going exactly 60-miles per hour. That makes sense right? If you averaged is 60-miles per hour and maybe you are going 40-miles per hour on some of the point. But at some point went 80 and in between you have to be going 60 miles per hour.

Let me see if I could draw that graphically. This slope is my average velocity. The way I draw it there’s probably two points, let’s see probably right around here. I was probably going 60-miles per hour. The slope is probably 60 there. The instantaneous velocity probably there as well.

Before I leave, let’s do this analytically just to—I guess work with the numbers. The reason why I have mixed feelings about the mean value theorem is it’s useful. Later on probably if you become a math major you’ll maybe use it to prove sone theorems or maybe you’ll prove it itself. But if you are just applying calculus for the most part, you’re not going to be using the mean value theorem too much. Anyway, if you got to know it you got to know it. It does tell you something else about the world, so t is interesting that way.

Let’s say we have the function F of X is equal to X squared minus four X. And the interval that I care about here is between, is the close interval, so I am including two from two to four. Now, the mean value theorem tells us that if this function is defined on this interval and it is right, we could put any number. The domain of this is actually all real numbers. I could put any number here so obviously it’s going to be defined over this interval. So it’s defined over the interval. This is continuous, this is differentiable. You could take the derivative and the derivative is continuous. So the mean value theorem should apply here.

Let’s see what value of C is equal to the average slope between two and four. What’s the average slope between two and four? That’s going to be F of four so the change in the function F of four minus F of two divided by the change in X, so it’s four minus two. This is equal to the average slopes. F of four is 16 minus 16, so that’s a zero. Let me make sure that four times four – 16, minus four times four, 16 minus F of two. F of two is two squared is four and then minus four times two so minus eight. Let me show that right. All that over two and so this equals minus four so it equals four over two.

The average slope from X is equal to two to X is equal to four is two. Now, that mean value theorem tells us that there must be some point that’s between these two maybe including one of those where the slope at that point is exactly equal to two. Let’s figure out what point that is. That is C.

Let’s take the derivative because the derivative at C is going to be equal to two so we just take the derivative so let’s say F prime of X is equal to two X minus four. We want to figure out at what X value does this equal to. So we say two X minus four is equal to two. Where does the slope equal to? You get two X is equal to six, X is equal to three. If X is equal to three, the derivative is exactly equal to the average slope. Let me see if a can do the graphing calculator here. Let me see what I can do.

Here is the graph of X squared minus four X. let me see if I can make a little bit bigger X squared minus four X. the interval that we care about is from here to here so the average slope, the average slope over that interval was two. If we were to draw the slope, it is like that. The slope would look like that and at the point three, the slope is exactly two. Let me actually to draw it. This isn’t too hard to draw for myself. Let me see.

If that’s the X axis, I wan that that graph out of the way. That’s the Y axis. So the graph goes to the point zero, zero. I’m going to draw this as neatly as possible. Nope, that’s not neat. The graph goes something like this and it tips up and it goes like that. Actually it keeps going straight up like that. It’s a parabola. So this is the point four. The point two is here and at two, we’re at negative four. So the vertex is at the point two minus four. So what we said, the average slope over—This close interval that we care about between two and four it is from two here to four here. That’s the interval two to four.

The average slope is two. It doesn’t look like it only because I’ve kind of compress the Y axis. We are saying at the point X is equal to three. The slope is equal to exactly that. So at X is equal to three, the slope is equal to the same thing. That’s all the mean to the value of theorem is. I know it sounds complicated. People talk about continuity and differentiability and F prime of C and all of this. But all it says this— there are some point between these two points where the instantaneous slope or the slope exactly that point is equal to the slope between these two points.

Hope I didn’t confuse you.

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