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RC
Numerade Educator

# A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The notion of a function is one of the most fundamental concepts in mathematics. An example is the function that relates each real number x to its square. The output of a function f corresponding to an input x is denoted by f(x), which is read as "f of x" or "f at x", or simply "f of x", when the context makes it clear. Functions of various kinds appear in many areas of mathematics, and their study is one of the central topics of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, or it could be described explicitly, as a formula or as a graph. The input and output of a function could be real numbers, the integers, a subset of the rational numbers, a set of real numbers, or more general objects such as vectors. The set used to define a function is called the domain of the function. The set of permissible outputs

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### Video Transcript

Welcome to our math review video on calculus. In this video, we're going to talk a little bit about the basics of derivatives and integral. We're not going to get into the nitty gritty details of calculating them. We'll save that for the example videos later on. Also, we're going to talk about dot products and cross products really quickly here at the beginning. Now, with dot products and cross products, you need to be able to understand something about vector notation where, if I have to generic vectors that I'm going to write out like this. If you have not watched the vector videos yet thin, this notation will not make much sense to you. And that's okay. Um, you can go and watch those videos at your leisure and then refer back to this when you need help remembering how to calculate dot products and cross products. But I'd like thes to be here just for reference so you can come back and find these equations quickly. So if we're taking adopt product, which is sometimes also called a scaler product because it ends up being a scaler value, it's gonna look something like this where we have the magnitude of eight times the magnitude of B times, the cosine of the angle between the vectors A and B. Remember that a magnitude is simply gonna look like this. We have a X squared plus B X squared. Okay. And then our sorry plus a y squared and then be would be equal to the square root of B X squared plus b y squared. Okay, so pretty easy to calculate. But there is another way to calculate it, which is when we got these two things together, we can take the some of the products of the similar components. That is to say, we have a X Times BX plus a Y times b y. And we'll get the exact same result for these two equations, which is a really handy thing to know for dot products. As for cross products, it's a little more complicated. We can find the magnitude pretty easily, and it looks a lot like the dot product equation, the first product equation, except we're going to use the sign instead of the cosine. Though this does not tell you the direction, remember across product should you give you a direction now to find the direction. You can either use the right hand rule, which we'll talk about later. Or you can use this matrix where we have X hat. Why Hat Z hat. So you'll actually need a Z component here. So plus ese see hat and busy is he hat and you'll have a X A y and a Z B X B y and busy, and you'll need to be able to take the determinant of this three by three matrix. If you don't remember how to do that, you can watch one of the later videos. I'll cover it, Um, and I'll also cover it again when we go when we run into the cross product in one of some of our equations. Um, in the meantime, let's go ahead and start talking a little bit about derivatives and integral. Remember, derivatives and inter girls are basic operators in calculus, just like how we had addition and subtraction before, when we were talking about Ah, mathematics, uh, or I should say arithmetic rather, um, so we have derivatives and integral, which operate inversely to each other, meaning if I have a function f of X, and I decided I'm going to take the derivative of this function. But then I decided to take the integral of that whole function that I took the that I had before. What I'll get back is simply the original function that I started with. Okay, this means that whatever the derivative did to ffx the integral undid it. Likewise, you could take the integral first and then the derivative and the same thing would happen. Um, now, this is a really helpful thing to remember and it because it will be it will help you as you try to figure out what the integral Czar of various functions derivatives have some pretty straightforward rules. Integral is you kind of have to think in reverse. Now, what are these things physically? Well, we remember that a derivative simply is the slope of a function. So if I were to have a horizontal line is my function, the slope would be zero. That means the derivative is zero. If I were tohave it like this, then the slope would be some constant positive value. If the function were to vary a lot, then we would have a varying derivative. On the other hand, if I were to look at f of X versus X and again with this horizontal function, I want to know the integral. Remember, with integral, I actually want to pick two endpoints to get a definite integral. And then the definite integral would be the area under the curve between those two points. Okay, now, there are also indefinite integral, uh, though, because this is physics and not mathematics. Generally speaking, we will be working with definite integral. Okay, um, now slope area under curve. What does that mean physically? Well, it depends a little on the situation. The first time we're going to run into this is when we're looking at position. If I have position as a function of time and I decide I'm going to take the derivative of that with respect to time, what I'm actually going to find is that this is equal to velocity as a function of time. Okay, so, by taking the time derivative of position, I came up with velocities. I went from something in meters to something in meters per second. Okay. Similarly, if I were to take the integral of velocity with respect to time, then I would come back with position so you can see how these things are operating inverse to each other. The derivative of X of T creates V of tea while the integral of E f t returns x of T. Um, understanding this relationship will make it a lot easier to understand the relationships between the different quantities that we're gonna look at later on in this, You know, in this course on drily, in physics one and two and physics, one of three is, Well, um, you can understand those things, Algebraic Lee, But having calculus in your back pocket understand what's happening can really be a fantastic help. Now, before we get into the nitty gritty of remembering how to operate with the various derivatives the and integral, we want to make sure that we remember what the rules are. Remember that there's some basic things like the derivative of a constant is equal to zero, where the integral of a constant is equal to the constant times X. Now, this is an indefinite integral. So I should technically ad a secondary constant. We'll call it see not, um, but again, in physics generally will be dealing with definite integral. Uh, meanwhile, if I were to take the derivative of the sum of two functions, say ffx plus G of X, I can actually distribute the derivative operator through. So I'll end up with DDX of FX plus D d x of G of x. Okay, and then evaluate these two terms separately. If there were subtraction sign, I could do the same thing there. Meanwhile, if I take the integral of that same set of functions f of X plus g of X, I could do the same thing where I can separate it into separate terms. So I have the integral of ffx the X plus the integral of G of X dx separately. And this could make a life a lot easier, particularly when you're trying to find integral. Um, Now there's a number of other rules. For example, we have the product rule, which is if we have ffx times g of X, then what we're going to find is that we have ffx times derivative suspect, X g of X plus g of x times derivative with respect to X of ffx. So we have the product rule and then we also have the quotient rule which is to say if we have DDX of ffx over g f X, then we can write this out as g of x times DDX of ffx minus f of x times DDX G of X, all divided by G of X squared. So that's a bit of a long rule. It could be derived from other rules as well. Um, but it's helpful toe have here written out. Now the other thing we really need to remember is something called the chain Rule. Now, the chain rule is to say, If I have a function that has another function inside of it and we'll give some examples of this later, then the way that I'm going to deal with that is I'm actually just going to take the derivative of f of G of X, all by itself isn't g of X will be unchanged, But then I'm gonna multiply that by DDX of whatever was inside of ffx, and we'll practice all of these in later videos

RC
University of North Carolina at Chapel Hill
##### Top Physics 101 Mechanics Educators  ##### Christina K.

Rutgers, The State University of New Jersey ##### Farnaz M.

Simon Fraser University ##### Jared E.

University of Winnipeg