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Scott Neske

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

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Jeffery Wang

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Felicia Sanders

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Okay, So a vector value to function, continuing the theme of vectors. So vector valued function is a function that outputs factors. Okay, so in general, a function just has a input output. The vector valued function is just a function that outputs vectors and for us, for now. Later we'll deal with other types of vector value functions. But for now, we really want to think about the domain of the vector valued function, just being the real numbers. And what we're going to see is that our conversations about Victor value functions are going to be almost identical to the calculus, one in calculus to that we've already learned on again. That's really because the domain of our vector value functions air gonna be subsets of the rial line. And so we've already seen examples of vector valued functions. So we saw the vector equation of a line that was an example of a vector valued function. You plugged in the value of t and you got a vector that pointed to specific points on the line. So, in general, the form of a vector valued function is gonna look like this. So we'll use our same vector notation, But we'll add in this argument T and will typically use T for the independent variable the input variable because we really want to think in the back of our mind Vector valued functions air also, uh, giving us the position of a particle as a function of time. That's a good way to think about a vector body function. So what we'll have is our output will just be, well, vectors. But each component is actually going to be just a regular function of one variable, and we'll just label them X, y and T x, y and Z. And these will be called So X Why? And see these will be called our component functions. So the components of the vector, but they're actually functions of this one. Variable t Another nice way to think about Victor Value functions is that they give So the victor valued function. Another just side note, maybe more of a technical note. We're going to assume that the component functions are a T least for now, continuously differential. So that's just so that everything I say is going to make sense. And so Victor valued function. Another nice way to think about it is that a vector valued function gives Hey parameter hi ization. So it parametric is is occurs and again here This is either in space or in the plane. And so, of course, we could have Victor value functions with just two component functions x and y It was in the plane. If it was in space, it'll have three x, y and Z and I already mentioned the vector value functions can trace out the passive particles they can also be thought of as giving paramilitaries ations of curves. So just to nail down that last point about Victor valued functions parameter rising curves I just wanted to make the point clear of what is actually the output vector. Okay, so suppose we're in the plane just to keep things visually simple. So this is the X Y plane, okay? And I'm gonna trace out some curve like this, and I'm actually going to give the curve an orientation. So what I mean by that is put simply I'm thinking about this curve is being the position of a particle moving, so his T increases three curve is going to be moving and increasing in this direction. So this is an oriented curve. And again the Orient to just means it has a direction. And so if I wanted to give a para mature ization of this curve so I would give some component functions What are the actual outputs here? Well, for any given time, T our private T is just going to be the vector that's pointing to this point right here. So this is our 50 Inspector. And so maybe over here, this is T equals zero. Maybe this is t equals wine, and then maybe this is t equals two. So you see, as time goes on, the particles gonna be moving along here and then at any given time, my vector valued function is just out putting the vector. That's pointing toe where the particle is like that. So I mentioned that because we're thinking about the domain of these vector value functions being the rial line, that, um, sort of the calculus we could do with Victor Value functions is very similar to just the calculus. Wander the calculus, too, that we did, and we'll explore that more in the next, uh, lecture. But for now, let's just note that suppose I wanted to take the limit. His T approaches a of the vector value function. Well, the nice thing about Victor value functions is that assuming all these limits exist, this is just gonna be equal to the limit. Is T approaches a of the X component function so that it's the limit is itself going to be a vector. And it's going to be the vector where each component is the limit of the component functions. And this is probably what you would expect in a situation like this, and similarly so we can define limits. This way we can say that our prime A T or R F T is continuous at a T value. It just tickles a Yes, No one that t approaches a of our vector value function is equal to just the vector that you get when you evaluate at a

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