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02. The crank of prob.01 is repeated here. If OA mm = = 50 , 25 ? and ? = 55 determine the moment of the force F of magnitude F N = 20 about point O
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Have you previously defined the term "asymptote?" Important because the closed captioning never spells it properly, and typically, I believe an asymptote is usually an axis or line rather than a point of discontinuity of a function.
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why does the function move downward from negative infinity to 4
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Fill in the blanks. The graph of a quadratic function is symmetric about its ________.
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In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] $ f(x) = (x - 2)^2 $
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In Exercises 13-16, graph each function. Compare the graph of each function with the graph of $ y = x^2 $. (a) $ f(x) = (x - 1)^2 $ (b) $ g(x) = (3x)^2 + 1 $ (c) $ h(x) = \left(\frac{1}{3} x^2 \right) - 3 $ (d) $ k(x) = (x + 3)^2 $
02:02
In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] $ f(x) = 4 - (x - 2)^2 $
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Okay, So in this video, we are going to be talking about continuity. So from the name itself, I'm sure a lot of people kind of assume what this is going to be about. It's basically talking about something being continuous. So if we're talking about a function being continuous, the way that we couldn't determine that is if you can trace that function without lifting up your pen. Because that means that there were no like holes in the graph for acid trips in the graph. Because if we had, for example, of function like this, well, you don't if I'm praising over it going like this, I never had to lift up my pencil in order. Thio finish tracing over that, which means that this function is continuous. But if we had a function more like this, where there's like that point hasn't took their there's no way for me to trace it without going through. Trace it without lifting up my pen without going through that point as and which is not allowed. Another situation that you can have. Where that function would be discontinuous is if you have like an ascent toe where you have something like this. Maybe. Or maybe there's even another asset up here. And so it's something like this. There's just no way for me to trace over that without lifting of my pen and getting down to here. Because if I try to do that going through the aspecto, which is not allowed because I asked him, Job is basically like a no go zone. So then let's think about it. Well, what is How do we know when something is continuous or not? Oh, actually, before I do that, actually, ah, function. The entire function doesn't have to be continuous. Sometimes they'll even ask you about intervals of continuity, which means that for that previous example that we had with the Assen jobs, you have something like this. That means that we could say that between this interval right here it is continuous and between this interval right here it's continuous. And between this interval right here, it's continuous. It's just that the entire function is a continuous, so you can definitely specify certain areas where function is continuous as opposed to having the entire thing be continuous. That's okay. So then, if we're considering this function a function a lot of times. The questions that they'll ask you about is going to revolve around rational numbers. So if you haven't seen the rational number Siri's or the the polynomial inequality Siri's, those would be two good ones to review before you see the continuity Siri's because that's going to kind of detail. So the polynomial inequality Siri's is going to detail how to draw out a graph using a sign analysis aan den Uh, this for the rational numbers that's gonna tell you, like what kind of areas that you're not really able to go with, um, with the function. So, for example, let's just take a function. F X is equal to three over X minus four. So with this particular situation, we know that because it's like kind of like a fraction. Almost the denominator cannot equal zero. And that's just something that we've learned even when we were first learning fractions, that if a numerator is zero, that's OK, because that will just become zero. But if it doesn't, if the denominator is going to become zero, that's gonna make it undefined, which is why we cannot have that fractions denominator become zero. So then let's think about when that denominator become zero so that we understand why air that ass and took would be so then, if we set that entire thing equal to zero will be able to understand where it cannot go, which if we add 4 to 1 side to the equation, we get that X is equal to four. So that's where that's, um, that's basically where the function cannot go, meaning that there's gonna be a vertical ascent oak. At that point X is equal to four. So then, if I were to graph the out, we could say there's X is equal to four. There's that vertical ascent. So then here's where the sign analysis comes in handy. So in terms of the sign analysis, we identify our key point and then consider whether something would be positive on either positive or negative on either end of the interval. So on opposite ends of the extreme on the left side we have negative infinity, and on the right side we have positive infinity. So we need to pick a term like a value that is between the first interval of negative infinity to positive four. So I'm just gonna go out and pick zero, because that's an easy one. So if I have, um, zero minus four, that becomes negative for so this part is gonna become negative. Then I'm gonna pick value on the other interval. So between four and positive infinity, I'm gonna pick five, because that's an easy one. And you could really pick any number. Um, just pick something that's easy to work with. So if I take five minus four, that becomes positive one. So this entire thing is going to be positive. So then when drawing out the graph, this is where that comes in handy is that now I know that anything from negative infinity to positive for that assassin job is because I mean, is gonna be on the bottom half of the graph. So under the X axis, since it is negative, I'm gonna start down here close to the X axis. But then as I get towards Theus in job, I have to curve up because I don't want to touch it. So then, for the second part, we know between, um 42 positive infinity. It's going to start positive, so I have to start from way up here. So typically, what will happen is, um not always, but typically, if you ended up going downwards away from the axis, you're going to start from upwards away from the axis. So then you're gonna start here, get towards the X axis and then curved down. So this is what that graph would look like. So then, like I said, sometimes you'll be asked to define intervals of contract continuity. So you can't actually say that this interval right here is continuous. And this interval right here is continuous. So I would probably say that between negative infinity and positive for exclusive, because you could never reach negative infinity. And you also can't can't have X value of four because that's what would make that function undefined. So that's one of the intervals of continuity. And also between four and positive. Infinity is another interval of continuity. Again, we want to make sure that they are parentheses, not brackets, because it is not inclusive because we can't ever get to positive infinity. And we also can't ever touch for because that would make the function undefined.
Powers and Polynomial
Rational Numbers
Exponential Functions
Trigonometry
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