Learn about Maxima Minima Slope Intuition Video

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Learn about Maxima Minima Slope Intuition

Well, I've been requested to do some videos on concativity and maxima and minimum points and inflection points and all the like that you normally see in your first semester of your calculus course, so I thought I would start doing them. But before we kind of go into the actual problems that you might see in your book, I just want to give you a broad intuition because in calculus class people have their these rules when the second derivative is positive your concave this way, when the second derivative is negative, your concave this way and people memorize the rules and then they take the test and if they memorize it correctly they do well in the test and they forget it maybe they memorized it again by the time of the final exam. But by the time they're 30 years old they will have forgotten it and that frankly is useless unless your whole goal is just to pass exam. So I want to give you some intuition and hopefully you will never forget this intuition. So when you're first learning calculus there’s much more to functions than what I'm going to do in this video but there's kind of three types of interesting points on most of the curves will talk about, so let's say let me draw a curve. Let's say the curve looks something like this, if this is the x axis, well that’s too thick. That’s the y axis. The curve could look something like this. I don’t know let me see, it could look like this and so on, right and if you have a curve like this and you’ll see all sorts of curves when you do your calculus and because you have multiple degree polynomials etcetera what are the interesting points? Well, the interesting points tend to be when the slope is equal to zero, so in this where does the slope equal zero? What are the interesting points here? Let me make sure I'm using the right tool right there the slope is zero because if you either take the tangent right there it would look something like that. Right there the slope is zero as well because if you took the tangent it would look something like that and then right there as well the slope is zero and the line it really isn’t the tangent. It crosses the graph but as we could see the graph kind of flattens out there so these are the three points that you tend to deal with in your first semester of calculus class when you're graphing and looking for extreme up and this is called a maximum and in this case it's a local maxima. If this was the highest point for the entire function it would be a global maximum. This is a local minima and if it was the lowest point of the entire graph which we actually drew maybe this graph keep going up like that and keeps going up like this, so this would be the global minimum point. And that this is called an inflection point because it's neither a maximum, minimum but you’ve kind of see that the graph kind of inflects. So let's study them a little bit more and see if we can get a little intuition on what happens to the graph, the first derivative and the second derivative on each of these points. Well actually let's start with the minimum point because that’s what we see the most. You know and we just studied y = x2 or regular parabolas that was the most common thing we saw and then of course if you have a negative of a parabola it comes like that but anyway let's study a minimum point where the graph is concave upward. So let's say we have a point like this. So that’s the graph of the function right? I don’t know what that is, it's f (x). So what is happening to the slope as we go roughly from here to there? Well then let's try a few points and see if we can by drawing the tangent lines we get an intuition for the slope. So let's see, if I pick this point just randomly. The slope looks something like that if I have to do a tangent. If I take this point I'm just moving to the right a little bit. I'm just increasing my x value. The slope will look something like that so it's negative here but if you look here the slope is less negative, right? The slope is less negative there and then if we go to the minimum point right there the slope flattens out, right and so of course the derivative there would be zero. And then as we pass that point what happens? Well here the slope is increasing a little bit more and now the slope is actually positive, right? So the slope was negative, less negative, zero, a little positive and then when you go here it goes more positive so let's see if we can graph that, just to get an intuition of obviously I haven’t written any numbers down, so we don’t know what the exact value of the slope is. So what does the slope look like? So here it's really negative. So the slope is going to be something really negative here. I'm just picking a random point and then at this point right here it was less negative so the slope is going to be less negative and then roughly here the slope is at zero, and then the slope gets a little bit more positive and then over there it gets a little bit more positive. So the slope if we were to actually graph the derivative of the function at that point it would be upward slope and so I'm assuming this is the second degree polynomial. This is third degree and then this would actually be a curve. So the slope will actually look something like this, right and does that make sense? Well sure, the graph of this could be y is equal to I don’t know ax2 + b and then the derivative or f (x) = x2 + b and so the derivative of that of course would be 2ax + 0 right. So this would just be aligned and it actually goes through zero-zero but it didn’t have to but hopefully that gives you a little intuition. But now let's move to the second derivative. So the slope of the slope, let me draw it down here, the second derivative. My wife is coming home so I have to finish this video before she gets here otherwise I have to continue it tomorrow, see she called five minutes ago so I have to hurry. Anyway, so this is f1-1(x) or the second derivative. So what's happening to the second derivative here? Well, the second derivative is just a slope of the derivative. So what's the slope? Well it's just a constant. In this case it's just a constant upward moving slope, right. So the second derivative, actually if we were to just evaluate that it would be 2a so it's going to look something like this and it's just going to be a constant number and it's going to be positive. All right, so in general when you study your extreme of points and say whether it's all minimum or a maximum etcetera, etcetera. The way that you think about it is well if it's a minimum point like this, what is happening to the slope around it? Well, the slope has to be constantly increasing even though as we go from here to here the value of f (x) is decreasing. The rate in which it's decreasing is getting less so the slope is actually increasing. It's getting less and less negative and it increases even more. So you know, you're at the minimum point when the slope is increasing and similarly you know that’s this concave upwards when the second derivative is positive. So hopefully that helps you a little bit get the intuition of what happens when you're at the minimum point in your concave upwards or what happens when you have the opposite well pretty much everything becomes the opposite. Let's say that this is your curve, I know and this is the X axis if that’s your curve, well here you have a very high slope. Then your slope becomes a little bit less and then it becomes zero and then it becomes a little bit more negative and then it becomes more negative, right and then it goes more negative. The rate of the slope becomes more and more negative the whole time that were kind of going through this maximum point or this concave down root. So what is the slope look like? Well in this example if I were to draw the derivative the slope is going to look something like this, right because back here the slope is actually a really high number. It's going to be up here and then as we get close, as we move further and further increasing values of x the slope decreases. It becomes zero right around here and then it just keeps decreasing and then similarly if we were to draw the second derivative what will that look like. Well, the second derivative is going to be slope of this function. So it’s just going to be a negative number if we assume that this is constant. So if this is the x axis the second derivative is going to be something like that. It's going to be something like that. It's going to be some negative number. I'm assuming this is zero-zero. Anyway I am about to run out of, well I’m not about to run out time but I think I don’t want to rush it. So that’s the maximum and minimum points and in the next video I’ll give you a little intuition on what happens with inflection points and then we will do a bunch of examples that you will normally see in a calculus textbooks. But I just wanted to do this for you because there’s nothing magical about a lot of the rules that you will learn when they say “Oh, you take the test second derivative and if the second derivative is positive it's concave upwards and if the second derivative is negative it's concave downwards and you might fill your brain with all of this memorizations but it's hopefully if you think about it this way fairly intuitive. And it hopefully should be fairly intuitive that whenever the first derivative is zero your slope is zero and you're at a point that is interesting at least in this context. But anyway I will do the next video in inflection points and then I will do a bunch of actual problems. See you soon.

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Well, I've been requested to do some videos on concativity and maxima and minimum points and inflection points and all the like that you normally see in your first semester of your calculus course, so I thought I would start doing them. But before we kind of go into the actual problems that you might see in your book, I just want to give you a broad intuition because in calculus class people have their these rules when the second derivative is positive your concave this way, when the second derivative is negative, your concave this way and people memorize the rules and then they take the test and if they memorize it correctly they do well in the test and they forget it maybe they memorized it again by the time of the final exam. But by the time they're 30 years old they will have forgotten it and that frankly is useless unless your whole goal is just to pass exam. So I want to give you some intuition and hopefully you will never forget this intuition.

So when you're first learning calculus there’s much more to functions than what I'm going to do in this video but there's kind of three types of interesting points on most of the curves will talk about, so let's say let me draw a curve. Let's say the curve looks something like this, if this is the x axis, well that’s too thick. That’s the y axis. The curve could look something like this. I don’t know let me see, it could look like this and so on, right and if you have a curve like this and you’ll see all sorts of curves when you do your calculus and because you have multiple degree polynomials etcetera what are the interesting points? Well, the interesting points tend to be when the slope is equal to zero, so in this where does the slope equal zero? What are the interesting points here?

Let me make sure I'm using the right tool right there the slope is zero because if you either take the tangent right there it would look something like that. Right there the slope is zero as well because if you took the tangent it would look something like that and then right there as well the slope is zero and the line it really isn’t the tangent. It crosses the graph but as we could see the graph kind of flattens out there so these are the three points that you tend to deal with in your first semester of calculus class when you're graphing and looking for extreme up and this is called a maximum and in this case it's a local maxima. If this was the highest point for the entire function it would be a global maximum. This is a local minima and if it was the lowest point of the entire graph which we actually drew maybe this graph keep going up like that and keeps going up like this, so this would be the global minimum point.

And that this is called an inflection point because it's neither a maximum, minimum but you’ve kind of see that the graph kind of inflects. So let's study them a little bit more and see if we can get a little intuition on what happens to the graph, the first derivative and the second derivative on each of these points. Well actually let's start with the minimum point because that’s what we see the most. You know and we just studied y = x2 or regular parabolas that was the most common thing we saw and then of course if you have a negative of a parabola it comes like that but anyway let's study a minimum point where the graph is concave upward. So let's say we have a point like this. So that’s the graph of the function right? I don’t know what that is, it's f (x). So what is happening to the slope as we go roughly from here to there?

Well then let's try a few points and see if we can by drawing the tangent lines we get an intuition for the slope. So let's see, if I pick this point just randomly. The slope looks something like that if I have to do a tangent. If I take this point I'm just moving to the right a little bit. I'm just increasing my x value. The slope will look something like that so it's negative here but if you look here the slope is less negative, right? The slope is less negative there and then if we go to the minimum point right there the slope flattens out, right and so of course the derivative there would be zero.

And then as we pass that point what happens? Well here the slope is increasing a little bit more and now the slope is actually positive, right? So the slope was negative, less negative, zero, a little positive and then when you go here it goes more positive so let's see if we can graph that, just to get an intuition of obviously I haven’t written any numbers down, so we don’t know what the exact value of the slope is. So what does the slope look like? So here it's really negative. So the slope is going to be something really negative here. I'm just picking a random point and then at this point right here it was less negative so the slope is going to be less negative and then roughly here the slope is at zero, and then the slope gets a little bit more positive and then over there it gets a little bit more positive.

So the slope if we were to actually graph the derivative of the function at that point it would be upward slope and so I'm assuming this is the second degree polynomial. This is third degree and then this would actually be a curve. So the slope will actually look something like this, right and does that make sense? Well sure, the graph of this could be y is equal to I don’t know ax2 + b and then the derivative or f (x) = x2 + b and so the derivative of that of course would be 2ax + 0 right. So this would just be aligned and it actually goes through zero-zero but it didn’t have to but hopefully that gives you a little intuition.

But now let's move to the second derivative. So the slope of the slope, let me draw it down here, the second derivative. My wife is coming home so I have to finish this video before she gets here otherwise I have to continue it tomorrow, see she called five minutes ago so I have to hurry. Anyway, so this is f1-1(x) or the second derivative. So what's happening to the second derivative here? Well, the second derivative is just a slope of the derivative. So what's the slope? Well it's just a constant. In this case it's just a constant upward moving slope, right. So the second derivative, actually if we were to just evaluate that it would be 2a so it's going to look something like this and it's just going to be a constant number and it's going to be positive.

All right, so in general when you study your extreme of points and say whether it's all minimum or a maximum etcetera, etcetera. The way that you think about it is well if it's a minimum point like this, what is happening to the slope around it? Well, the slope has to be constantly increasing even though as we go from here to here the value of f (x) is decreasing. The rate in which it's decreasing is getting less so the slope is actually increasing. It's getting less and less negative and it increases even more. So you know, you're at the minimum point when the slope is increasing and similarly you know that’s this concave upwards when the second derivative is positive.

So hopefully that helps you a little bit get the intuition of what happens when you're at the minimum point in your concave upwards or what happens when you have the opposite well pretty much everything becomes the opposite. Let's say that this is your curve, I know and this is the X axis if that’s your curve, well here you have a very high slope. Then your slope becomes a little bit less and then it becomes zero and then it becomes a little bit more negative and then it becomes more negative, right and then it goes more negative. The rate of the slope becomes more and more negative the whole time that were kind of going through this maximum point or this concave down root.

So what is the slope look like? Well in this example if I were to draw the derivative the slope is going to look something like this, right because back here the slope is actually a really high number. It's going to be up here and then as we get close, as we move further and further increasing values of x the slope decreases. It becomes zero right around here and then it just keeps decreasing and then similarly if we were to draw the second derivative what will that look like. Well, the second derivative is going to be slope of this function. So it’s just going to be a negative number if we assume that this is constant. So if this is the x axis the second derivative is going to be something like that. It's going to be something like that. It's going to be some negative number. I'm assuming this is zero-zero.

Anyway I am about to run out of, well I’m not about to run out time but I think I don’t want to rush it. So that’s the maximum and minimum points and in the next video I’ll give you a little intuition on what happens with inflection points and then we will do a bunch of examples that you will normally see in a calculus textbooks. But I just wanted to do this for you because there’s nothing magical about a lot of the rules that you will learn when they say “Oh, you take the test second derivative and if the second derivative is positive it's concave upwards and if the second derivative is negative it's concave downwards and you might fill your brain with all of this memorizations but it's hopefully if you think about it this way fairly intuitive. And it hopefully should be fairly intuitive that whenever the first derivative is zero your slope is zero and you're at a point that is interesting at least in this context. But anyway I will do the next video in inflection points and then I will do a bunch of actual problems. See you soon.

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