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Math 104: Calculus13 chapters | 105 lessons
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I really like roller coasters. I really like it, in particular, when you get right up to the top of the roller coaster just before you're going to go plummeting down to the bottom. I also like that point at the bottom where you're suddenly pulled back up, and your innards go all the way down to your feet.
Let's see if we can analyze a roller coaster. Let's take its position across the Earth as x and its height as y. So let's draw out the roller coaster height as a function of its location, x. I really like, in particular, the point at the top and the point at the bottom of this roller coaster function. So what are these points? These are what we call extrema. Extrema are extreme values, like a maximum or a minimum value. On a roller coaster, on this particular roller coaster, there's a maximum height, so it's a maximum y value, and there's a minimum height, a minimum y value. But what about these other two points, close to the beginning and close to the end? If I ignore the rest of this graph, that's a minimum value. If I ignore the rest of this graph, that's a maximum value. So what are those? Extrema can be both. Extrema can be global, or what we call absolute extrema. This is a maximum or a minimum value for the entire domain. That's the very top of the roller coaster and the very bottom of the roller coaster. But you can also have local extrema. We also call them relative extrema, and these are the maximum or minimum values in a small region. So going back to my roller coaster, I have the global maximum and the global minimum, as well as local maximum and local minimum places. You can think of these kind of like mountain ranges and valleys. The global maximum on earth is Mt. Everest. So I've got a local maximum as well. That's that little hill in the backyard where I throw all my trash.
Let's take a look at this example. In this example, we have, very obviously, a global minimum. It's at the very bottom of this graph. We also have two maximum values. We have this local maximum on the right-hand side and this global maximum on the left-hand side. Now keep in mind that every global maximum or minimum point is also going to be a local maximum or minimum point. The trash heap in my backyard is a local maximum, but Mt. Everest is also a local maximum. It's just local to that region.
Let's do an actual example, like y=sin(x) between 0 and 3pi. If I draw this out, how many maximum and minimum values do I have? I definitely have one global minimum value, down here at -1. What about a global maximum? In this case I have two global maximums. There are two points that are both equal to 1. That's okay because I can't pick one over the other. They're both the same value, so I've got two global maximum values. I also have local minimum values at the beginning and end of my range, so I can't forget those.
Let's make this a little bit more formal. A global maximum is at some x value, like x max where y, or f, at that point is greater than any other value in your region. So this is like saying the height on the top of Mt. Everest is greater than the height at any other point on Earth. The global minimum is at the x value (x minimum, let's say) where the value at that point is less than every other point on your domain. For example, the Dead Sea is lower than any other point on Earth, at least above ground. A local max is just the locally largest point, or the locally tallest point. It's where f(x) is greater than f(x) for any point around it. This is the local trash heap. The top of that trash heap is higher than any other point around that trash heap. The local minimum is the same way. It's the minimum value in some nearby area.
Let's do another example. Let's say we have a function, and it looks a little bit like this. How many maximum and minimum values are there on this graph? Well, there's one global maximum value, here, and there's one global minimum value, here. Then, we have a local max here, here, here and here. So we have one global maximum value and five local maximum values, because we've got these four maximum values that are only local maximum values but we also have the global max, so we have five. Similarly, we have one global minimum, but we have six local minimums. Now, you might have just said we have four, but remember you have to include the ends.
Let's do another example. Let's say f(x)=x for -1 < x < 1. So this is between -1 and 1, but I'm not including the points at 1 or -1. So I can graph it out, and I've got two open holes at both ends. How many maximum and minimum values are there? Well, it's kind of a trick question. There are no maximum or minimum values. That's because I can't pick a point here that has no points that are larger than it. You would like to say x=1. That's not in our region. You can say just less than x=1, but we can't actually define that, so there is no point where there is a global or even a local maximum value. The same thing happens at -1. So in this case there are no extreme values.
Let's review. Extreme values are your highest and lowest points. You can have maximum values, such as tall mountains and you can have minimum values, like your local valley. You can also have a local maximum or minimum and a global maximum or minimum. An example of a local maximum would be the trash heap behind my house. Its height is taller than anything around it, but not if you start to include all of Earth. The global maximum would be Mt. Everest. There's nothing on earth that is taller than Mt. Everest. Second, whenever you're trying to find extreme values on some domain, make sure you consider the edges of your domain. Lastly, discontinuities are not maximum or minimum values. So we had that graph where we had holes at both ends. Those don't count as maximum or minimum values.
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