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RC
Numerade Educator

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Calculus - Example 1

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 review video where we're gonna look at dot products and cross products in the later videos will look at some calculus operations before I do this. I want to do Ah, quick review of vector notation here. So remember, uh, if you watch the last video then you saw that I wrote vectors like this X ex hat, plus a y. Why Hat plus a see. See hat. So what this means is that I have a vector that has an X component, a y component and a Z component. Now, if I were to do this in the two dimensional direction in two dimensional plane, I'd say x and a y. And then this would be my total vector A where X hat means thistles, the amount I'm going in the X. Why hat means this is the amount I'm going in. The Y and Z hat would mean that's the amount I'm going in the sea. The magnitude of such a vector would be equal to the square of a X squared, plus a y squared, plus a Z squared. Now let's give ourselves a second vector again exactly as we had before. B x X hat plus B y. Why Hat plus Easy Z had now we already discussed dot products and it was pretty simple affair. Where we have a dot product gives us a scaler quantity, a quantity that is not a vector anymore, and we can write it in two ways. One is we could have the product of the two magnitudes of the vectors multiplied by the cosine of the angle between them or identically. We can have the some of the products of the similar components, so we have X times, bx plus a Y Times B y plus a Z times BZ. Okay, And what the dot products doing is it's giving us some evaluation of of how these things are multipliers, taking an amount of B that's in the direction of a and then multiplying those two quantities together. This is very helpful when we come across things like work later on and energy, um, and it shows up a couple other places in the introductory physics classes as well. Now the cross product, which again is written like this, is a little more finicky. We can quickly come up with the magnitude with a very similar equation, except we're going to use the sine of the angle between them but coming up with the direction because, ah, cross product vector product actually produces a vector as the result, Um, coming up with the direction is a little more difficult. Generally speaking will use what's called the right hand rule, which is where you're going to curl your fingers and have your thumb pointed. Now it's a little difficult to illustrate this without having video, so I definitely recommend if you want to see the right hand video, you can look it up on YouTube or I'll have a video later where I'll illustrate it with my own hand. Um, but personally, the way I like to think about it, when all I'm doing is is calculating across product is with the Matrix that I showed before. In fact, let's get a clean sheet here and well, do the determinant of this matrix very quickly. So if I have a cross be that's going to be equal to X hat. Why Hat Z hat? You have a X a y and a Z B x b y, and busy, and I'm going to take the determinant of this matrix. Okay, so there's two ways Thio to write this out. The simplest one to remember might be this that you're going to say. Okay, Um Well, first of all, I'm gonna have the X hat direction that's gonna be multiplied by this determinant, which is just a two by two determinant, which is a little easier to calculate. The most challenging thing about this is remembering to put the negative in here. So now the determined for this matrix will be the X's and disease. So I'll have a X b X and a see busy plus z hat times the times, the determinant of X a Y over bx e y. Now, if you remember, a two by two matrix can be calculated like this. And as long as you plug all those in, you'll get exactly the right answer again. We'll review this when we come to it. It's not gonna happen until physics 102 Really? And with a very small application here in physics 101

RC
University of North Carolina at Chapel Hill
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