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Felicia Sanders
If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$
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Masoumeh Amirshekari
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Scott Neske
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Okay, so now that we've reviewed some of the key concepts from pre calculus ready to jump in and talk about calculus and I kind of want to give you just a quick overview of what calculus is, at least in one particular example now I told you already that calculus studies functions on the real line. I'm gonna be a little bit more specific and tell you what calculus is in one sentence. So calculus, what it really does, is it studies how functions. So these functions We've been talking about functions on the real line change. If I could summarize calculus in one sentence, this will be it. It's really studying how functions change up till now. We were looking at functions, and we're looking at snapshots of functions. What was going on at some specific time? Calculus is going to give us a way to study how functions change and move and bend and curve. And it's an extremely powerful technique. So you've already seen a little bit about of this and pre calculus, so recall that if you look at the graph of a line, so let's just plot a really simple line. We actually had something that told us about how the lined changed and that was the slope. So the slope of the line recall was kind of informally, it was the rise over run. So it was saying how much I had to go up every time I go over one unit. So this was kind of like changing. Why, right here This is changing X. So then the slope was just change. And why over change in X so you can see that the slope is really telling you how the line is changing, going back to this idea of change. So every time I go a step forward, how much do I have to increase or decrease to stay on the line? That's what the slope is telling me. But as we've seen, there are a lot more complicated functions than just lines. So in general, a function is not going to look like a line. The function is gonna look well, just like some function. Maybe it looks something like this. And now the idea here is we can still talk about how this function is changing. It's just more complicated because how the function is changing is actually changing and the idea of calculus and really sort of in this introductory topic. The tool we need to develop is we need to develop a way to zoom in and look really, really close at how a function is changing. And now why do we want to do that? Because if I zoom in on this piece of the function, so if I blow this up right here and zoom in on a really small scale, the function actually looks like a line. If I zoom in close enough now, assuming the function is relatively smooth, which we'll talk about what that means later. So I can actually just kind of give a similar definition. I could just say What is my rise overrun? Changing whatever change in X for this particular really, really small piece of the line. So the first thing we need to do before we even really get into talking about how functions change is we need a way to zoom in and talk about functions on a very, very small scale, and that leads to the idea of what's called a limit. So a limit is nothing more than zooming in on a function and seeing how it's changing when I take a small little step in some direction, so that's what we're going to jump into now.
Derivatives
Differentiation
Applications of the Derivative
Integrals
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