WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about properties of functions.
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Functions are extremely important to math; we keep talking about them, because we are going to use them a lot; they are really, really useful.
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To help us investigate and describe behaviors of functions, we can talk about properties that a function has.
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There are a wide variety of various properties that a function can or cannot have.
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This lesson is going to go over some of the most important ones.
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While there are many possible properties out there that we won't be talking about in this lesson,
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this lesson is still going to give us a great start for being able to describe functions,
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being able to talk about how they behave and how they work.
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So, this is going to give us the foundation for being able to talk about other functions in a more rigorous way,
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where we can describe exactly what they are doing and really understand what is going on; great.
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All right, the first one: increasing/decreasing/constant: over an interval of x-values,
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a given function can be increasing, decreasing, or constant--that is, always going up, always going down, or not changing.
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Its number will always be increasing; its number will always be decreasing; or its number will be not changing.
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And by number, I mean to say the output from the inputs as we move through those x-values.
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This is really much easier to understand visually, so let's look at it that way.
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So, let's consider a function whose graph is this one right here: this function is increasing on -3 to -1.
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We have -3 to -1, because from -3 to -1, it is going up; but it stops right around here.
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So, it stops increasing on -3; it stops increasing after -1, but from -3 to -1, we see that it is increasing.
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It is probably increasing before -3, but since all we have been given is this specific viewing window to look through,
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all we can be guaranteed of is that, from -3 to -1, it is increasing.
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Then, it is constant on -1 to 1; it doesn't change as we go from -1 to 1--it stays the exact same, so it is constant on -1 to 1.
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However, it is increasing before -1, and it is decreasing after 1; so it is decreasing on 1 to 3, because we are now going down.
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So, it continues to go down from 1 on to 3, because we can only be guaranteed up until 3.
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It might do something right after the edge of the viewing window, so we can only be sure of what is there.
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It is decreasing from 1 to 3--great; that is what we are seeing visually.
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It is either going up, straight, or down; it is either horizontal, it is going up, or it is down.
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Increasing means going up; constant means flat; decreasing means down.
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Formally, we say a function is increasing on an interval if for any a and b in the interval where a < b, then f(a) < f(b).
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Now, that seems kind of confusing; so let's see it in a picture version.
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Let's say we have an interval a to b, and this graph is above it.
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If we have some interval--it is any interval, so let's just say we have some interval--that is what was between those two bars--
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and then we decide to grab two random points: we choose here as a and here as b;
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a is less than b, so that means a is always on the left side; a is on the left, because a is less than b.
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That is not to say b over c; that is because--I will just rewrite that--we might get that confused in math.
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a is on the left because a is less than b; if we then look at what they evaluate to, this is the height at f(a), and then this is the height at f(b).
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And notice: f(b) is above f(a); f(b) is greater than f(a).
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So, it is saying that any point on the left is going to end up being lower than points on the right, in the interval when it is increasing.
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In other words, the graph is going up in the interval, as we read from left to right.
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Remember, we always read graphs from left to right.
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So, during the interval, we are going from left to right; we are going up as we go from left to right.
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We have a similar thing for decreasing; we have some interval, some chunk, and we have some decreasing graph.
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And if we pull up two points, a and b (a has to be on the left of b, because we have a < b), then f(a) > f(b).
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f(a) > f(b); so decreasing means we are going down--the graph is going down from left to right.
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We don't want to get too caught up in this formal idea; there is some interval, some place,
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where if we were to pull out any two points, the one on the left will either be below the one on the right,
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if it is increasing; or if it is decreasing, it will be above the one on the right.
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We don't want to get too caught up in this; we want to think more in terms of going up and going down, in terms of reading from left to right.
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Finally, constant: if we have some interval, then within that interval, our function is nice and flat,
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because if we choose any a and b, they end up being at the exact same height.
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There is no difference: f(a) = f(b); the graph's height does not change in that interval.
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While the definitions on the previous slide give us formal definitions--they give us something that we can really understand
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if we want to talk really analytically--we don't really need to talk analytically that often in this course.
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It is going to be easiest to find these intervals by analyzing the graph of the function.
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We just look at the graph and say, "Well, when is it going up? When is it going down? And when is it flat?"
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That is how we will figure out our intervals.
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We won't necessarily be able to find precise intervals; since we are looking at a graph, we might be off by a decimal place or two.
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But mostly, we are going to be pretty close; so we can get a really good idea of what these things are--
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what these intervals of increasing, decreasing, or constant are.
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So, we get a pretty good approximation by looking at a graph.
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And if you go on to study calculus, one of the things you will learn is how to find increasing, decreasing, and constant intervals precisely.
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That is one of the major fields, one of the major uses of calculus.
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You won't even need to look at a graph; you will be able to do it all from just knowing what the function is.
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Knowing the function, you will be able to turn that into figuring out when it is increasing, when it is decreasing, and when it is flat.
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You will even be able to know how fast it is increasing and how fast it is decreasing.
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So, there is pretty cool stuff in calculus.
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All right, intervals show x-values: for our intervals of increasing, decreasing, and constant,
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remember, we are giving intervals in terms of the x-values; it is not, *not* the points.
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We describe a function's behavior by saying how it acts within two horizontal locations.
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We are saying between -5 and -3, horizontally; it is not the point (-5,-3); it is between the locations -5 and -3.
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And don't forget, we always read from left to right; it is reading from left to right, as we read from -5 up until -3.
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So, the other thing that we need to be able to do is: we need to always put it in parentheses.
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Parentheses is how we always talk about increasing, decreasing, and constant intervals.
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Why do we use parentheses instead of brackets?
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Well, think about this: a bracket indicates that we are keeping that point;
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a parenthesis indicates that we are dropping that point, not including that in the interval.
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But the places where we change over, the very end of an interval, is where we are flipping
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from either increasing to decreasing or increasing to constant; we are changing from one type of interval to another.
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So, those end points are going to be changes; they are going to be places where we are changing from one type of interval to another.
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So, we can't actually include them, because they are switchover points.
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We want to only have the things that are actually doing what we are talking about.
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The switchovers will be switching into something new; so we end up using parentheses.
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All right, a really quick example: if we have f(x) = x² - 2x, that graph on the left,
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then we see that it is decreasing until it bottoms out here; where does it bottom out?
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It bottoms out at 1, the horizontal location 1; and it is decreasing all the way from negative infinity, out until it bottoms out at 1.
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And then, it is increasing after that 1; it just keeps increasing forever and ever and ever.
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So, it will continue to increase out until infinity.
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So, parenthesis; -∞ to 1 decreasing; and increasing is (1,∞).
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We don't actually include the 1, because it is a switchover.
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At that very instant of the 1, what is it--is it increasing? Is it decreasing?
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It is flat technically; but we can't really talk about that yet, until we talk about calculus.
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So for now, we are just not going to talk about those switchovers.
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All right, the next idea is maximums and minimums.
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Sometimes we want to talk about the maximum or the minimum of a function, the place where a function attains its highest or lowest value.
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We call c a maximum if, for all of the x (all of the possible x that can go into the function), f(x) ≤ f(c).
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That is to say, when we plug in c, it is always going to be bigger than everything else that can come out of that function,
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or at the very least equal to everything else that can come out of the function.
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A minimum is the flip of that idea; a minimum is f(c) is going to be smaller or equal to everything else that can be coming out of that function.
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So, a maximum is the highest location a function can attain, and a minimum is the lowest location a function can attain.
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On this graph, the function achieves its maximum at x = -2; notice, it has no minimum.
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So, if we go to -2 and we bring this up, look: the highest point it manages to hit is right here at -2.
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Why does it not have a minimum? Well, if we were to say any point was its minimum--look, there is another point that goes below it.
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So, since every point has some point that is even farther below it, there is no actual minimum,
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because the minimum has to be lower than everything else.
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There is a maximum, because from this height of 3, we never manage to get any higher than 3, so we have achieved a maximum.
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And that occurs at x = -2; great.
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We can also talk about something else; first, let's consider this graph,
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this monster of a function, -x⁴ + 2x³ + 5x² - 5x.
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Technically, this function only has one maximum; you can only have one maximum,
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and it is going to be here, because it is the highest point it manages to achieve; it would be x = 2.
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But it actually has no minimum; why does it not have any minimum?
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Well, it kind of looks like this is the low point; but over here, we managed to get even lower.
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Over here, we managed to get even lower; and because it is just going to keep dropping off to the sides,
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forever and ever and ever, we are going to end up having no minimums whatsoever in this function,
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because it can always go lower; there is no lowest point it hits; it always keeps digging farther down.
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But nonetheless, even though there is technically only one maximum and no minimums at all,
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we can look at this and say, "Well, yes...but even if that is true, that there isn't anything else,
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this point here is kind of interesting; and this point here is kind of interesting,
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in that they are high locations and low locations for that area."
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This is the idea of the relative minimum and maximum; we call such places--these places--
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the highest or lowest location (I will switch colors...blue...oh my, with yellow, it has managed...blue here; green here)...
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relative minimums are the ones in green, and the relative maximums are the ones in blue.
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And sometimes the word "local" is also used instead; so you might hear somebody flip between relative or local, or local or relative.
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These places are not necessarily a maximum or a minimum for the entire function, for every single place.
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But they are such a maximum or minimum in their neighborhood; there is some little place around them where they are "king of their hill."
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So, this one is the maximum in this interval, and this one is the minimum in this interval; and this one is the maximum in this interval.
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But if we were to look at a different interval, there would be no maximum or minimum in this interval,
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because it just keeps going down and down and down.
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And if we were to look at even in here, it is clearly right next to them--if we were to put a neighborhood around this, it would keep going down.
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It is not the shortest one around; it is not the highest one around.
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But these places are the highest or lowest in their place.
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OK, so this gives us the idea of a relative maximum or a relative minimum.
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Formally, a location, c, on the x-axis is a **relative maximum** if there is some interval,
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some little place around that, some ball around that, that will contain c, such that,
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for all x in that interval, f(x) ≤ f(c)--in its neighborhood, c is the highest thing around.
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It is greater than all of the other ones.
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Similarly, for a **relative minimum**, there is some interval such that f(x) is going to be less than or equal to f(c).
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In its neighborhood, it is the lowest one around; lowest one around makes you a minimum--highest one around makes you a maximum--
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that is to say, a relative maximum or a relative minimum.
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Once again, this is sort of like what we talked about before with the previous formal definition for maximum and minimum,
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and also for the formal definition of intervals of increasing, decreasing, and constant.
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Don't get too caught up on what this definition means precisely.
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The important thing is that we have this graphical picture in our mind that relative maximum just means the high point in that area.
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And relative minimum just means the low point in that area; that is enough for us to really understand what is going on here.
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Getting caught up in these precise things is really something for a late, high-level college course to really get worried about.
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For now, it is enough to just get an idea of "it is the high place" or "it is the low place."
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Don't forget: the terms relative and local mean basically the same thing--actually, they mean exactly the same thing.
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They can be used totally interchangeably; and some people prefer to use one; some people prefer to use another.
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Some people will flip between the two; so don't get confused if you hear one or you hear another one; they just mean the same thing.
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To distinguish relative local maximums and minimums from a maximum and minimum over the entire function,
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we can use the terms "absolute" or "global" to denote the latter.
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If we want to say it is the maximum over the entire function, we could call it the absolute maximum or the global maximum.
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So, an absolute, global maximum/minimum is where the function is highest/lowest over the entire function,
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which is exactly how we defined maximum/minimum at first, before we started to talk about the idea of relative maximum/relative minimum.
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So, absolute or global maximum/minimum is over the entire function--the function's highest/lowest over everywhere in the domain.
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If we want to talk about all of the relative or absolute maximums/minimums in the functions, we can call the them the extrema (or the "ex-tray-ma").
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Why? Because they are the function's extreme values: they are the extreme high points
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and the extreme low points that the function manages to go through, so we can call them the extrema.
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So, there we are; there is just something for us: extrema.
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If we want to talk about relative or absolute maximums/minimums in general, we use this word to do it.
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And absolute or global talks about the single highest or single lowest;
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relative just talks about one that is high or low in its neighborhood, in the area around that point.
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Just like find increasing/decreasing/constant intervals, we want to do this from the graph.
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We don't want to really get too worried or too caught up on these very specific definitions,
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the formal definitions we were talking about on the previous slide.
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We just want to say, "OK, yes, we see that that is a high point on the graph; that is a low point on the graph."
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So, find your minimums; find your maximums by looking at the graph.
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And once again, if you go on to study calculus, you will learn how to find extrema precisely, without even needing to look at a graph.
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You will be able to find them exactly; you won't have to be doing approximations because you are looking at a graph.
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And you won't even have to look at a graph to find them.
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So once again, calculus is pretty cool stuff.
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Average rate of change: this also can be called average slope.
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When we talked about slope in the introductory lessons, we discussed
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how it can be interpreted as the rate of change, how fast up or down the line is moving.
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If we have a line like this, it is not moving very fast up; but if we have another line like this, it is moving pretty quickly up.
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So, it is a rate of change; the slope is how fast it is changing--the rate of change; how fast are we going up?
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Now, most of the functions we are going to work with aren't lines; but we can still use this idea.
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We can discuss a function's average rate of change between two points.
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So, if an imaginary line is drawn between two points on a graph, its slope is the average rate of change.
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Say we take two points, this point here and this point here; and we draw an imaginary line between them.
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Then, the slope of that imaginary line is the average rate of change,
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because what it took to get from this point to the second point is that we had to travel along this way.
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And while we actually went through this curve here--we actually went through this curve,
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but on the whole, what we managed to do, on average, is: we really just kind of went along on that line.
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We could forget about everything we went through, and we could just ask, "Well, what is the average thing that happened between these two points?"
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And that would be our average rate of change--how fast we were moving up from our first point to our second point.
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So, if we want to find the average rate of change, how do we do this?
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Let's say we have two locations, x₁ and x₂,
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and we want to find the slope of that imaginary line between those two points on the function graph.
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So, that line is sometimes called the secant line; for the most part, you probably won't hear that word too often.
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But in case it comes up, you know it now.
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Remember, if we want to find what the slope of this imaginary line is, the slope of this secant line, we know what slope is.
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How do we find slope? Remember, slope is the rise over the run, so it is the difference between our heights
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y₂ and y₁, our second height and our first height--what did our height change by,
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and what did our horizontal location change by--our second location minus our first location?
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So, our horizontal distance is x₂ - x₁; and our vertical distance is y₂ - y₁.
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So, y₁ is the height over here; y₂ is the height over here.
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y₂ - y₁ over x₂ - x₁ is the rise, divided by the run.
00:18:47.000 --> 00:18:51.800
But what are y₁ and y₂?--if we want to figure out what y₁ and y₂ are,
00:18:51.800 --> 00:18:54.500
well, we just need to look at what x₁ and x₂ are.
00:18:54.500 --> 00:19:01.100
So, since x₁ and x₂ are coming to get placed by the function,
00:19:01.100 --> 00:19:06.400
then y₂'s height is really just f(x₂), because that is how the graph gets built.
00:19:06.400 --> 00:19:11.400
The input gets dropped to an output; we map an input to an output.
00:19:11.400 --> 00:19:15.700
And y₁ over here is from f(x₁).
00:19:15.700 --> 00:19:20.000
So, since our original slope formula is y₂ - y₁ over x₂ - x₁,
00:19:20.000 --> 00:19:24.600
and we know that y₁ is the same thing as f(x₁) and y₂ is the same thing as f(x₂),
00:19:24.600 --> 00:19:31.000
we can just plug those in, and we get the change in our function outputs, f(x₂) - f(x₁),
00:19:31.000 --> 00:19:33.900
divided by our horizontal distance, x₂ - x₁.
00:19:33.900 --> 00:19:40.000
For our average rate of change, we just look at how much our function changed by between those horizontal locations.
00:19:40.000 --> 00:19:46.000
How much did its output change by? Divide that by how much our distance changed by.
00:19:46.000 --> 00:19:51.400
It is often really useful and important to find what inputs cause a function to output 0.
00:19:51.400 --> 00:19:57.200
So, if we have some function f, we might want to know what we can put into f that will give out 0.
00:19:57.200 --> 00:20:00.100
That is the values of x such that f(x) = 0.
00:20:00.100 --> 00:20:08.100
Graphically, since f(x)...remember, f(x) is always the vertical component; the outputs come to the vertical;
00:20:08.100 --> 00:20:12.700
so, if our outputs are coming from the vertical, then 0 is going to be the x-axis.
00:20:12.700 --> 00:20:19.200
We have a height of 0 here; so graphically, we see that this is where the function crosses the x-axis.
00:20:19.200 --> 00:20:24.000
Our crossing of the x-axis is where f(x) = 0.
00:20:24.000 --> 00:20:28.700
This idea of f(x) = 0 is so important that it is going to go by a bunch of different names.
00:20:28.700 --> 00:20:36.600
It can be called the zeroes of a function; it can be called the roots of a function; and it can be called the x-intercepts.
00:20:36.600 --> 00:20:40.700
x-intercepts--that makes sense, because it is where it crosses the x-axis.
00:20:40.700 --> 00:20:45.500
Zeroes make sense, because it is where we have the zeroes showing up.
00:20:45.500 --> 00:20:48.900
But how can we remember roots--why is roots coming out?
00:20:48.900 --> 00:20:54.200
Well, one way to think about it--and actually where this word's origin is coming from--
00:20:54.200 --> 00:20:58.600
is because it is the roots that the function is planted in.
00:20:58.600 --> 00:21:05.100
The function we can think of as being planted in the ground (not literally the ground, but we can think of it as being the ground of the x-axis).
00:21:05.100 --> 00:21:08.700
So, it is like the function has put down roots in the soil.
00:21:08.700 --> 00:21:13.000
It is not exactly perfect, but that is one good mnemonic to help us remember.
00:21:13.000 --> 00:21:17.000
"Roots" means where we are stuck in the soil; it is where we are stuck in the x-axis;
00:21:17.000 --> 00:21:22.600
it is where we have f(x) equal to 0, or where we have an equation equal to 0.
00:21:22.600 --> 00:21:26.900
But all of these things--zeroes, roots, x-intercepts--they all mean the same thing.
00:21:26.900 --> 00:21:30.700
They are the x such that f(x) = 0; we can also use these for equations--
00:21:30.700 --> 00:21:37.000
we might hear it as the zeroes of an equation, the roots of an equation, or the x-intercepts of an equation.
00:21:37.000 --> 00:21:41.100
There is no one way to find zeroes for all functions.
00:21:41.100 --> 00:21:47.700
We are going to learn, for some functions, foolproof formulas to find zeroes, to tell us if there are zeroes and what those zeroes are.
00:21:47.700 --> 00:21:52.000
But for other functions, it can be very difficult--very, very difficult, in fact--to find the zeroes.
00:21:52.000 --> 00:21:56.500
And although we are going to learn some techniques to help us on the harder ones, there are some that we won't even see
00:21:56.500 --> 00:21:59.000
in this course, because they are so hard to figure out.
00:21:59.000 --> 00:22:03.300
But right now, the important thing isn't being able to find them, but just knowing
00:22:03.300 --> 00:22:11.500
that, when we say zeroes, roots, x-intercepts of a function, or an equation, we are just talking about where f(x) = 0.
00:22:11.500 --> 00:22:17.200
So, don't get too caught up right now in being able to figure out how to get those x-values such that f(x) = 0.
00:22:17.200 --> 00:22:23.700
Just really focus on the fact that when we say zeroes, roots, or x-intercepts, all of these equivalent terms,
00:22:23.700 --> 00:22:31.500
we are just saying where the function is equal to 0--what are the places that will output 0?
00:22:31.500 --> 00:22:37.000
Even functions: this is a slightly odd idea (that was an accidental joke).
00:22:37.000 --> 00:22:43.000
Even functions: some functions behave the same whether you look left or right of the y-axis.
00:22:43.000 --> 00:22:47.900
For example, let's consider f(x) = x²: it is symmetric around the y-axis.
00:22:47.900 --> 00:22:55.300
What do I mean by this? Well, if we plug in f(-3), that is going to end up being (-3)², so we get 9.
00:22:55.300 --> 00:23:03.500
But we could also plug in the opposite version to -3; if we flip to the positive side, -3 flips to positive 3.
00:23:03.500 --> 00:23:10.200
If we plugged in positive 3, then f(3) is 3², so we get 9, as well.
00:23:10.200 --> 00:23:15.100
It turns out that plugging in the negative version of a number or the positive version of a number,
00:23:15.100 --> 00:23:19.700
-3 or 3, we end up getting the same thing; for -2 and 2, we end up getting the same thing.
00:23:19.700 --> 00:23:24.800
For -47 and 47, we end up getting the same thing.
00:23:24.800 --> 00:23:29.100
So, whatever we plug in, as long as they are exact opposites horizontally--
00:23:29.100 --> 00:23:33.900
they are the same distance from the y-axis--the points are symmetric around the y-axis--
00:23:33.900 --> 00:23:38.700
they are going to come out to the same height; they are going to have the same output.
00:23:38.700 --> 00:23:43.500
We call this property even; and I want to point out that it is totally different from being an even number.
00:23:43.500 --> 00:23:47.000
It is different from an even number--not the same thing as that.
00:23:47.000 --> 00:23:52.400
But we call this property even for a function.
00:23:52.400 --> 00:23:57.100
A function is **even** if all of the x for its domain, for any x that we plug in...
00:23:57.100 --> 00:24:01.900
if we plug in the negative x, that is the same thing as the positive x.
00:24:01.900 --> 00:24:11.100
Plugging in f(-x) is equal to plugging in f(x); so we plug in -x into the function, and we get the same thing as if we had plugged in positive x.
00:24:11.100 --> 00:24:16.800
We can flip the signs, and it won't matter, as long as it is just negative versus positive.
00:24:16.800 --> 00:24:21.900
Why do we call it even? It has something to do with the fact that all polynomials where all of the exponents
00:24:21.900 --> 00:24:25.400
end up being even exponents--they end up exhibiting this property.
00:24:25.400 --> 00:24:30.000
But then, this property can be used on other things; so don't worry too much about where the name is coming from.
00:24:30.000 --> 00:24:36.300
But just know what the property is: f(-x) = f(x).
00:24:36.300 --> 00:24:40.800
Odd functions are the reverse of this idea: other functions will behave in the exact reverse.
00:24:40.800 --> 00:24:47.700
The left side is the exact opposite of the right side; for example, f(x) = x³ behaves like this.
00:24:47.700 --> 00:25:00.400
If we plug in -3, we get -3 cubed, so we get -27; but if we had plugged in positive 3, we would get positive 3 cubed, so we would get positive 27.
00:25:00.400 --> 00:25:08.000
So, you see, you plug in the negative version of a number, and you plug in the positive version of a number;
00:25:08.000 --> 00:25:10.400
and you are going to get totally opposite answers.
00:25:10.400 --> 00:25:15.300
However, they are only flipped by sign; -27 and 27 are still somewhat related.
00:25:15.300 --> 00:25:20.600
They are very different from one another--they are opposites, in a way; but we can also think of them as being perfect opposites.
00:25:20.600 --> 00:25:27.800
-27's opposite is positive 27; so an odd function is one that behaves like this everywhere.
00:25:27.800 --> 00:25:32.300
We call this property odd; it is totally different, once again, from being an odd number.
00:25:32.300 --> 00:25:38.600
A function is **odd** if, for all x in its domain, f(-x) is equal to -f(x).
00:25:38.600 --> 00:25:43.200
And that is a little confusing to read; but what that means is that, if we plug in -x,
00:25:43.200 --> 00:25:48.900
then that is going to give us the negative version of if we had plugged in positive x.
00:25:48.900 --> 00:25:55.300
So, if we plug in a negative number, and then we plug in a positive number, the outputs
00:25:55.300 --> 00:25:59.000
that come out of them will be positive-negative opposites.
00:25:59.000 --> 00:26:01.500
One of them will be positive; the other one will be negative.
00:26:01.500 --> 00:26:07.800
So, negative on one side and positive on one side means that the outputs will also be negative on one side and positive on the other side.
00:26:07.800 --> 00:26:12.000
It is not necessarily going to be the case that the negative side will always put out negative outputs.
00:26:12.000 --> 00:26:14.800
But it will be the case that it will be flipped if it is odd.
00:26:14.800 --> 00:26:17.200
This will make a little more sense when we look at some examples.
00:26:17.200 --> 00:26:18.800
And once again, why are we calling it odd?
00:26:18.800 --> 00:26:24.800
Once again, don't worry too much about it, but it because it is connected to polynomials where all of the exponents are odd numbers.
00:26:24.800 --> 00:26:29.200
But don't really worry about it; just know what the property is.
00:26:29.200 --> 00:26:33.200
Even/odd functions and graphs: we can see these properties in the graphs of functions.
00:26:33.200 --> 00:26:39.900
An even function is symmetric around the y-axis: it mirrors left/right, because when we plug in a positive number,
00:26:39.900 --> 00:26:44.900
and we plug in a negative number, as long as they are the same number, they end up getting put to the same location.
00:26:44.900 --> 00:26:47.200
They get output to the same place.
00:26:47.200 --> 00:26:52.900
An odd function, on the other hand, is symmetrical around the origin, which means we mirror left/right and up/down,
00:26:52.900 --> 00:27:01.900
because when we plug in the positive version of a number, it gets flipped to the negative side, but also shows up on the opposite side.
00:27:01.900 --> 00:27:07.200
It flips to the negative height or the positive height; it flips the positive/negative in terms of height.
00:27:07.200 --> 00:27:10.000
So, let's look at some examples visually; that will help clear this up.
00:27:10.000 --> 00:27:21.800
An even one: f(-x) = f(x); let's see how this shows up; if we plug in 0.5, we get here; if we plug in -0.5, we get here.
00:27:21.800 --> 00:27:28.200
And look, beyond the fact that I am not perfect at drawing, they came out to be the same height.
00:27:28.200 --> 00:27:40.800
If we plug in 2.0, and we plug in -2.0, they came out to be the same height.
00:27:40.800 --> 00:27:46.000
You plug in the negative number and the positive number, and they end up coming out to be the same height.
00:27:46.000 --> 00:27:50.500
That is what it means to be even; and since all of the positives will be the same as the negatives,
00:27:50.500 --> 00:27:55.900
we end up getting this nice symmetry across the y-axis; it is just a perfect flip.
00:27:55.900 --> 00:28:03.900
If we took the two halves and folded them up onto each other, they would be exact perfect matches; it is just mirroring the two sides.
00:28:03.900 --> 00:28:10.200
Odd is sort of the reverse of this: f(-x) = -f(x).
00:28:10.200 --> 00:28:19.000
For example, let's plug in -1: we plug in -1, and it ends up being at this height, just a little under 2.
00:28:19.000 --> 00:28:28.300
Let's see what happens when we plug in positive 1; when we plug in positive 1, it ends up being just a little under -2.
00:28:28.300 --> 00:28:33.500
So, we flip the horizontal location; that causes our vertical location to flip.
00:28:33.500 --> 00:28:42.100
Let's try another one: we plug in 2.0, and we are practically past it; so we should be just a little bit before 2.0.
00:28:42.100 --> 00:28:47.300
And we plug in -2.0, once again, just a little past it; so we are just a little before -2.0.
00:28:47.300 --> 00:28:55.500
And look: we end up being at the same distance from the x-axis, but in totally opposite directions.
00:28:55.500 --> 00:29:04.100
2.0, positive 2.0, causes us to go to positive 4 in terms of height; but -2.0 causes us to go to -4 in terms of height.
00:29:04.100 --> 00:29:12.200
So, they are going to flip; if you flip horizontally, you also flip vertically; and that is why we mirror left/right and mirror up/down.
00:29:12.200 --> 00:29:18.700
We are not just flipping around the y-axis; we are flipping around the origin,
00:29:18.700 --> 00:29:31.300
because we are flipping the right/left and the up/down; flipping around the origin is flipping the right/left and the up/down.
00:29:31.300 --> 00:29:36.000
We mirror left/right; we mirror up/down; that is what is happening with an odd function.
00:29:36.000 --> 00:29:37.900
All right, we are finally ready for some examples.
00:29:37.900 --> 00:29:41.400
There are a bunch of different properties that we covered; now, let's see them in use.
00:29:41.400 --> 00:29:47.200
The first example: Using this graph, estimate the intervals where f is increasing and decreasing.
00:29:47.200 --> 00:29:50.100
Find the locations of any extrema/relative maximums/minimums.
00:29:50.100 --> 00:29:54.200
And our function is -1.5x⁴ + x³ + 4x² + 3.
00:29:54.200 --> 00:29:57.800
Now, that is just so we can have an idea that that is what that function looks like.
00:29:57.800 --> 00:30:02.300
But we are not really going to use this thing right here; it is not really going to be that helpful for us figuring it out.
00:30:02.300 --> 00:30:07.000
So first, let's figure out intervals where f is increasing or decreasing.
00:30:07.000 --> 00:30:12.200
First, it is increasing from all the way down (and it sounds like we can probably trust
00:30:12.200 --> 00:30:24.100
that it is going to keep going down, because we have -1.5x⁴); it is increasing up until...it looks like just after -1.0.
00:30:24.100 --> 00:30:35.000
It is increasing from negative infinity (because it is going all the way to the left--it is going up
00:30:35.000 --> 00:30:39.600
as long as we are coming from negative infinity, because it goes down as we go to the left, but we read from left to right),
00:30:39.600 --> 00:30:46.600
so it is increasing from negative infinity up until...let's say that is -0.9, because it is just after -1.0.
00:30:46.600 --> 00:30:52.000
And then, it is also going to be increasing from here...let's say it starts there...up until about this point.
00:30:52.000 --> 00:31:04.400
So, where is that? It is probably about 1.4; so it is increasing from 0 up until 1.4.
00:31:04.400 --> 00:31:15.100
Where is it decreasing? It is decreasing from this point until this point.
00:31:15.100 --> 00:31:21.300
That was -0.9 that we said before; so we will go from -0.9 up until 0.
00:31:21.300 --> 00:31:27.400
And then, it increased up until 1.4; so now it is going to be decreasing from 1.4.
00:31:27.400 --> 00:31:30.300
And it looks like it is going to just keep going down forever, and it does indeed.
00:31:30.300 --> 00:31:35.200
So, it is going to be all the way out until infinity; it is going to continue decreasing; great.
00:31:35.200 --> 00:31:42.000
Now, let's take a look at the extrema; where are the relative maximums/minimums?
00:31:42.000 --> 00:31:49.000
We have relative maximums/minimums at all of these flipovers that we have talked about, here, here, and here.
00:31:49.000 --> 00:31:56.500
So, our relative maximum/minimum, our high location, the absolute maximum/minimum, is going to be up here.
00:31:56.500 --> 00:32:11.400
Relative maximums: we have x =...we said that was 1.4, and that point is going to be 1.4.
00:32:11.400 --> 00:32:20.400
Let's take a look, according to this...and it looks like it is just a little bit under 8; let's say 7.9.
00:32:20.400 --> 00:32:29.400
And then, the other one, the lesser of them, but still a relative maximum--it is occurring at x = -0.9.
00:32:29.400 --> 00:32:37.700
So, its point would be -0.9; and we look on the graph, and it looks like it is somewhere between 4 and 5.
00:32:37.700 --> 00:32:42.000
It looks a hair closer to 5, so let's say 4.6; great.
00:32:42.000 --> 00:32:52.200
Relative minimum--our low place: well, we can be absolutely sure of what the x is there--it is pretty clear that that is x = 0.
00:32:52.200 --> 00:32:58.300
And what is the height that it is at right there? It looks like it is exactly on top of the 3, so it is (0,3).
00:32:58.300 --> 00:33:01.300
We have all of the intervals of increasing and decreasing.
00:33:01.300 --> 00:33:07.500
And we also have all of our extrema, all of our relative maximums and minimums; great.
00:33:07.500 --> 00:33:12.700
Example 2: A ball is thrown up in the air, and its position in meters is described by location of t.
00:33:12.700 --> 00:33:18.000
Distance of t is equal to -4.9t² + 10t, where t is in seconds.
00:33:18.000 --> 00:33:24.300
OK, so we have some function that describes the height of the ball--where the ball is.
00:33:24.300 --> 00:33:27.500
What is the ball's average velocity (speed) between 0 seconds and 1 second,
00:33:27.500 --> 00:33:31.700
between 0 and 0.01 seconds, and between 0 and 2.041 seconds?
00:33:31.700 --> 00:33:37.400
OK, at first, we have some idea...if we were to figure out what this function looks like, it is a parabola.
00:33:37.400 --> 00:33:40.600
It has a negative here, so it is ultimately going to go down.
00:33:40.600 --> 00:33:44.000
And it has the 10t here; if we were to graph it, it would look something like this.
00:33:44.000 --> 00:33:50.800
And that makes a lot of sense, because if we throw a ball up, with time, the ball is going to go up and them come back down.
00:33:50.800 --> 00:33:55.900
So, that seems pretty reasonable: a ball is thrown up in the air, and its position is given by this.
00:33:55.900 --> 00:33:59.900
But how does speed connect to position? Well, we think, "What is the definition of speed?"
00:33:59.900 --> 00:34:03.800
We don't exactly know what velocity is, necessarily; maybe we haven't taken a physics course.
00:34:03.800 --> 00:34:07.400
But we probably know what speed is from before in various things.
00:34:07.400 --> 00:34:17.700
Speed is distance divided by time, so distance over time equals speed.
00:34:17.700 --> 00:34:23.400
It seems pretty reasonable that velocity is going to be the same thing.
00:34:23.400 --> 00:34:28.300
That is not exactly true, if you have actually taken a physics course; but that is actually going to work on this problem.
00:34:28.300 --> 00:34:32.000
We are going to have a good idea of what is going on with saying that that is true.
00:34:32.000 --> 00:34:34.200
All right, so what is the ball's average velocity?
00:34:34.200 --> 00:34:40.100
The average velocity is going to be the difference in its height, divided by the time that it took to make that difference in height.
00:34:40.100 --> 00:34:44.500
So, we are going to be looking for distance.
00:34:44.500 --> 00:34:55.100
If we have 2 times, time t₁ to time t₂, it is going to be the location at time t₂,
00:34:55.100 --> 00:35:02.200
minus the location at time t₁, over the difference in the time, t₂ - t₁.
00:35:02.200 --> 00:35:07.100
Oh, and that makes a lot of sense; it is going to be connected, probably, to what we learned in this lesson,
00:35:07.100 --> 00:35:12.100
since with student logic, they normally try to give us problems that are going to be based off of what we just learned.
00:35:12.100 --> 00:35:16.400
So, t₂ - t₁...this looks just like average rate of change.
00:35:16.400 --> 00:35:23.800
The average rate of something's position--that would make sense, that how fast it is going is the rate of change; the thing is changing its location.
00:35:23.800 --> 00:35:28.000
The rate at which you are changing your location is the velocity that you have; perfect.
00:35:28.000 --> 00:35:33.600
Great; so we need to figure out what it is at 0 seconds and what it is at 1 second right away.
00:35:33.600 --> 00:35:40.400
So, the location at 0 seconds; we plug that in...-4.9(0)², plus 10(0)...that is just 0, which makes a lot of sense.
00:35:40.400 --> 00:35:46.000
If we throw a ball up, at the very beginning it is going to be right at the height of the ground.
00:35:46.000 --> 00:35:59.200
Distance at time 1 is going to be -4.9 times 1², plus 10 times 1; so we get 5.1.
00:35:59.200 --> 00:36:20.600
If we want to figure out what is its average velocity between 0 seconds and 1 second, then we have d(1) - d(0)/(1 - 0), equals 5.1 - 0/1, which equals 5.1.
00:36:20.600 --> 00:36:28.700
What are our units? Well, we had distance in meters, and time in seconds; so meters divided by seconds...we get meters per second.
00:36:28.700 --> 00:36:32.200
That makes sense as a thing to measure velocity and speed.
00:36:32.200 --> 00:36:36.100
All right, next let's look at between 0 and 0.1 seconds.
00:36:36.100 --> 00:36:50.900
If we want to find 0.01 seconds, the location at 0.01 equals -4.9(0.01)² + 10(0.01).
00:36:50.900 --> 00:36:58.400
Plug that into a calculator, and that is going to end up coming out to be 0.09951; so let's just round that up
00:36:58.400 --> 00:37:02.100
to the much-more-reasonable-to-work-with 0.01.
00:37:02.100 --> 00:37:08.300
OK, so it rounds approximately to 0.01; so let's see what is the average rate of change.
00:37:08.300 --> 00:37:25.500
The average rate of change, then, between 0 and 0.01 seconds, is going to be d(0.01) - d(0) over 0.01 - 0.
00:37:25.500 --> 00:37:33.100
That equals...oops, sorry, my mistake: 0.01 is not actually what it came out to be when we put it in the calculator.
00:37:33.100 --> 00:37:45.000
I mis-rounded that just now; it was 0.09951, so if it is 0.09951, if we are going to round that
00:37:45.000 --> 00:37:51.100
to the much-more reasonable-to-work-with thing, we actually get approximately 0.1.
00:37:51.100 --> 00:38:08.500
So, it is not 0.01; 0.01 is still on the bottom, but the top is going to end up coming out to be 0.1 - 0, divided by 0.01; sorry about that.
00:38:08.500 --> 00:38:10.100
It is important to be careful with your rounding.
00:38:10.100 --> 00:38:18.000
That comes out to be 0.1 over 0.01, which comes out to be 10 meters per second.
00:38:18.000 --> 00:38:27.300
And now, you probably haven't taken physics by this point; but if you had, you would actually know that -4.9t²...
00:38:27.300 --> 00:38:31.700
that is the thing that says the amount that gravity affects where its location is.
00:38:31.700 --> 00:38:35.500
The 10t is the amount of the starting velocity of the ball.
00:38:35.500 --> 00:38:40.100
The ball gets thrown up at 10 meters per second, so it makes sense that its average speed
00:38:40.100 --> 00:38:47.300
between 0 and 0.01--hardly any time to have changed its speed--is going to be pretty much what its speed started at.
00:38:47.300 --> 00:38:49.600
That 10 meters per second is actually showing up there.
00:38:49.600 --> 00:38:55.300
So, there is a connection here between understanding what the physics going on is and the math that is connecting to it.
00:38:55.300 --> 00:39:14.100
All right, finally, between 0 and 2.041 seconds...let's plug in d(2.041) = -4.9(2.041)² + 10(2.041).
00:39:14.100 --> 00:39:25.000
So, that is going to come out to be -0.0018; so it seems pretty reasonable to just round that to a simple 0.
00:39:25.000 --> 00:39:31.100
Now, what does that mean? That means, at the moment, 2.041 seconds--that is when the ball hits the ground.
00:39:31.100 --> 00:39:34.100
It goes up at 0, and then it comes back down.
00:39:34.100 --> 00:39:40.100
And at 2.041 seconds after having been thrown up, it hits the ground precisely at 2.041 seconds.
00:39:40.100 --> 00:39:48.100
So, 2.041 seconds--then it has a 0 height; so what is its average velocity between 0 and 2.041 seconds?
00:39:48.100 --> 00:40:04.400
Location at 2.041 minus location at 0, divided by 2.041 - 0, equals 0 minus 0, over 2.041, which equals 0 meters per second,
00:40:04.400 --> 00:40:10.200
which makes sense: if we throw the ball up, and then we look at the time when it hits the ground again,
00:40:10.200 --> 00:40:15.500
well, on average, since it went up and it went down, it had no velocity,
00:40:15.500 --> 00:40:20.500
because the amount of time that it has positive velocity going up and the amount of time that it has negative velocity going down--
00:40:20.500 --> 00:40:25.000
it has cancelled itself out, because on average, between the time of its starting on the ground
00:40:25.000 --> 00:40:27.300
and ending on the ground, it didn't go anywhere.
00:40:27.300 --> 00:40:32.900
So, on average, its velocity is 0, because it didn't make any change in its location; great.
00:40:32.900 --> 00:40:39.400
The next example--Example 3: Find the zeroes of f(x) = 3 - |x + 3|.
00:40:39.400 --> 00:40:51.200
Remember: zeroes just mean when f(x) = 0; so we can just plug in 0 = 3 - |x + 3|.
00:40:51.200 --> 00:40:57.900
So, we have |x + 3| = 3; we just add the absolute value of x + 3 to both sides.
00:40:57.900 --> 00:41:02.100
We have |x + 3| = 3; that is what we want to know to figure out when the zeroes are.
00:41:02.100 --> 00:41:14.300
When is this true? Remember, absolute value of -2 is equal to 2, which is also equal to the absolute value of positive 2.
00:41:14.300 --> 00:41:20.600
So, the absolute value of x + 3...we know that, inside of it, since there is a 3 over here...
00:41:20.600 --> 00:41:24.500
there could be a 3, or there could be a -3.
00:41:24.500 --> 00:41:38.000
So, inside of that absolute value, because we know it is equal to 3, we know that there has to currently be a 3, or there has to be a -3.
00:41:38.000 --> 00:41:42.100
We aren't sure which one, though; so we split it into two different worlds.
00:41:42.100 --> 00:41:47.700
We split it into the world where there is a positive on the inside, and we split it into the world where there is a negative on the inside.
00:41:47.700 --> 00:41:58.200
In the positive world, we know that what is inside, the x + 3, is equal to a positive 3.
00:41:58.200 --> 00:42:03.300
In the negative world, we know that the x + 3 is equal to a negative 3.
00:42:03.300 --> 00:42:09.400
Now, it could be either one of these; either one of these would be true; either one of these would produce a 0 for the function.
00:42:09.400 --> 00:42:14.700
So, let's solve both of them: we subtract by 3 on both sides over here; we get x = 0.
00:42:14.700 --> 00:42:18.600
We subtract by 3 on both sides over here; we get x = -6.
00:42:18.600 --> 00:42:26.100
So, the two answers for the roots are going to be -6 and 0; that is when the zeroes of f(x) show up.
00:42:26.100 --> 00:42:29.400
The zeroes of f(x) are going to be at x = 0 and x = -6.
00:42:29.400 --> 00:42:35.900
And if we plug either one of those into that function, we will get 0 out of the function.
00:42:35.900 --> 00:42:39.600
The final example: Show that x⁶ - 4x² + 7 is even;
00:42:39.600 --> 00:42:44.500
show that -x⁵ + 2x³ - x is odd; and show that x + 2 is neither.
00:42:44.500 --> 00:42:52.000
All right, the first thing we want to do is remind ourselves of what it means to be even.
00:42:52.000 --> 00:43:01.200
To be even means that when we plug in the negative version of a number, a -x is the same thing as if we had plugged in the positive x.
00:43:01.200 --> 00:43:02.800
It doesn't have any effect.
00:43:02.800 --> 00:43:09.700
And the odd version...actually, let's put it in a different color, so we can see how all of the problems match up to each other.
00:43:09.700 --> 00:43:15.300
If we do with the odd version, then if we plug in the negative of a number,
00:43:15.300 --> 00:43:19.000
it comes out to be the negative of if we had plugged in the positive version of the number.
00:43:19.000 --> 00:43:25.000
All right, so the first one: Show that x⁶ - 4x² + 7 is even.
00:43:25.000 --> 00:43:29.200
So, that was really seeing that expression as if it were a function; so let's show this
00:43:29.200 --> 00:43:32.900
by showing that if we plug in -x, it is the same thing as if we plug in positive x.
00:43:32.900 --> 00:43:42.400
On the left, we will plug in -x; -x gets plugged in; it becomes (-x)⁶ - 4(-x)² + 7 =...
00:43:42.400 --> 00:43:50.200
if we plugged in just plain x, we would have plain x⁶ - 4x² + 7; great.
00:43:50.200 --> 00:43:54.500
(-x)⁶...remember, a negative times a negative cancels out to a positive.
00:43:54.500 --> 00:44:01.900
We have a 6 up here; we are raising it to the sixth power, so we have an even number of negatives.
00:44:01.900 --> 00:44:05.500
Negative and negative cancel; negative and negative cancel; negative and negative cancel.
00:44:05.500 --> 00:44:12.300
That is a total of 6 negatives; they all cancel each other out; so we actually have (-x)⁶ being the same thing as if we just said x⁶.
00:44:12.300 --> 00:44:20.400
Minus 4...the same thing here: -x times -x cancels and just becomes plain x²...plus 7 equals
00:44:20.400 --> 00:44:25.700
x⁶ - 4x² + 7; it turns out that it has no effect.
00:44:25.700 --> 00:44:28.700
If we plug in a negative x, we get the same thing as if we had plugged in the positive x.
00:44:28.700 --> 00:44:32.500
Plugging in a negative version of a number is the same thing as plugging in the positive version of the number.
00:44:32.500 --> 00:44:37.000
So, it checks out; it is even; great.
00:44:37.000 --> 00:44:42.000
The next one: let's look at odd: -x⁵ + 2x³ - x is odd.
00:44:42.000 --> 00:44:46.200
We will do the same sort of thing: we will plug -x's in on the left side.
00:44:46.200 --> 00:44:56.400
-(-x)⁵ + 2(-x)³ - (-x); what is going to go on the right side?
00:44:56.400 --> 00:45:03.200
Well, remember: if we plug in the negative version of the number, then it is the negative of if we plugged in the positive version of the number.
00:45:03.200 --> 00:45:06.600
So, it is the negative of if we had plugged in the positive version of the number.
00:45:06.600 --> 00:45:14.000
Plugging in the positive version of the number is just if we have the normal x going in: -x⁵ + 2x³ - x.
00:45:14.000 --> 00:45:20.300
All right, so -(-x)⁵: well, what happens when we have (-x)⁵--what happens to that negative?
00:45:20.300 --> 00:45:26.800
Negative and negative cancel; negative and negative cancel; negative--that fifth one, because it is odd, gets left over.
00:45:26.800 --> 00:45:32.200
So, we have negative; and we just pull that negative out--it is the same thing as -x⁵.
00:45:32.200 --> 00:45:39.000
Plus 2...once again, it is odd; a negative and a negative cancel; we are left with one more negative, for a total of 3 negatives; we are left with a negative.
00:45:39.000 --> 00:45:47.300
So, we get 2(-x)³ minus...we can pull that negative out, as well...-x...equals...
00:45:47.300 --> 00:45:54.500
let's distribute this negative; so we get...distribute...cancellation...a negative shows up here...cancel;
00:45:54.500 --> 00:46:00.900
we get positive x⁵ minus 2x³ + x.
00:46:00.900 --> 00:46:06.700
So, let's finish up this left side and do cancellations over here as well; positive, positive; this stays negative.
00:46:06.700 --> 00:46:17.800
Positive, positive; so we get x⁵ - 2x³ + x equals the exact same thing over here on the right side.
00:46:17.800 --> 00:46:21.800
It checks out; yes, it is odd; great.
00:46:21.800 --> 00:46:31.400
Finally, let's show that x + 2 is neither; so, to be neither, we have to fail at being this and fail at being this.
00:46:31.400 --> 00:46:48.800
So, to be neither, it needs to fail being odd and being even; it needs to fail even and odd.
00:46:48.800 --> 00:46:53.800
Let's just try plugging in a number; let's try plugging in, say, -2.
00:46:53.800 --> 00:47:01.300
If we look at x = -2, then that would get us -2 + 2, which equals 0.
00:47:01.300 --> 00:47:05.100
Now, what if we plugged in the flip of -2--we plugged in positive 2?
00:47:05.100 --> 00:47:12.500
x = positive 2...we plug that into x + 2, and we will get 2 + 2, which equals 4.
00:47:12.500 --> 00:47:20.900
Now, notice: 0 is not equal to 4; we just failed being even up here,
00:47:20.900 --> 00:47:26.300
because the negative number and the positive version of that number don't produce the same output.
00:47:26.300 --> 00:47:34.200
Plug in -2; you get 0; plug in +2; you get 4; those are totally different things, so we just failed to be even; great.
00:47:34.200 --> 00:47:36.700
Next, we want to show that it is not odd.
00:47:36.700 --> 00:47:41.600
Odd was the property that, if we plug in the negative, it is going to be equal to the negative of the positive one.
00:47:41.600 --> 00:47:49.600
So, 0 is not equal to -4 either, right? If we plug in -2, we get 0, and if we plug in positive 2,
00:47:49.600 --> 00:47:53.300
it turns out that that is not -0, or just 0; it turns out that that is 4.
00:47:53.300 --> 00:48:00.900
So, we fail to be odd as well, because it isn't the case that if we plug in opposite positive/negative numbers,
00:48:00.900 --> 00:48:10.000
we don't get opposite positive/negative results, because 0 is not the opposite of -4; it is just the opposite of 0, so it fails there.
00:48:10.000 --> 00:48:15.800
So, it checks out: that one is neither; great.
00:48:15.800 --> 00:48:21.600
All right, we just learned a whole bunch of different properties; and they will each come up in different places at different times.
00:48:21.600 --> 00:48:23.500
Just remember these: keep them in the back of your mind.
00:48:23.500 --> 00:48:26.900
If you ever need a reminder, come back to this lesson and just refresh what that one meant,
00:48:26.900 --> 00:48:30.100
because they will show up in random places; but they are all really useful.
00:48:30.100 --> 00:48:32.800
And we will see them a lot more as we start getting into calculus.
00:48:32.800 --> 00:48:35.600
Once you actually get to calculus, this stuff, especially the stuff at the beginning of this,
00:48:35.600 --> 00:48:38.800
where we talked about increasing and decreasing and relative maximums and minimums--
00:48:38.800 --> 00:48:44.000
that stuff is going to become so important if you are going to understand why we are talking about it so much right now in this course.
00:48:44.000 --> 00:48:49.000
All right, I hope you understood everything; I hope you enjoyed it; and we will see you at Educator.com later--goodbye!