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Rubab S.

April 27, 2021

What is definition of calculus

Campbell University

Oregon State University

Harvey Mudd College

Baylor University

01:09

Felicia Sanders

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

0:00

00:06

Jeffery Wang

Jsdfio Klsfjwjf

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Okay, so we're ready to move into the final topic of this course in calculus. And so up till now we've talked exclusively about derivatives. Now, of course, we looked at limits first, but we really just looked at limits to define the derivative. And then we also looked at applications of the derivative. So we looked at curve sketching, understanding how functions change, understanding extreme values of functions, optimization, problems, finding anti derivatives. So all of these questions air just applications of the idea of differentiation. Okay, so we're going to completely switched gears, and we're going to talk about a seemingly unrelated problem. And the problem we're going to discuss is finding area. Now, of course, this problem really goes all the way back to just basic geometry. You know, if I want to find the area of the triangle, then it's something like one half base times height, Okay, but we're gonna look at regions that are more complicated than triangles. Okay, So for instance, what if I wanted to find the area underneath the graph of a function? So if I look at, say, the function X squared and then I want to find let's say it. Cut this off right here. And I want to find this area underneath the curve. Now notice. It's not exactly a triangle anymore. It's sort of like a triangle with a curved top. Okay, but that's a typical region that you might want to know the area of. So what is the area here? Well, that's the idea that we're going to tackle first. Actually, how do we compute the area of regions that are more complicated than just simple polygons? Okay. And so why are we doing this? I mean, well, one, it's interesting. Okay, It is an interesting question, and we will initially use three idea of limits. Okay, So what will basically do is we'll take a region like this and define the area. What we do is we break it up into small pieces like this. And so each one of these small little pieces is basically a rectangle, so it's area is just length, times width, and then somehow we're just gonna add up all of these small little pieces. And now to do that precisely. And to get an exact value, we're really going to have to look at a limit. Okay? And that's going to allow us to find the area of certain regions. But the real reason why this question is relevant in our conversation about calculus is that there is actually in unexpected connection between finding area. So, like we just talked about finding the area underneath curves or in general regions, so between finding areas and derivatives of functions in this unexpected connection is sort of the crescendo of calculus. Okay, so it's really exciting. So we need to talk first about this problem of finding areas. And then what we're gonna do is we're going to see this amazing relationship between finding areas and the derivative of functions, and this has a very special name. Appropriate name. We're going to discuss what's called the fundamental serum of, of course, calculus. So I'm sure you're familiar with the fundamental theorem of arithmetic. That every number can be expressed uniquely is a product of primes, the fundamental theorem of algebra that every polynomial could be factored over the complex numbers. Well, this is the fundamental theorem of calculus. It explains the connection between finding areas and the derivatives of functions. So this final topic really is forming like a crescendo of everything that we've talked about so far, so I hope you're excited. It's super interesting and also extremely helpful and solving real life problems, and we'll see examples of this as we move forward.

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