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RC
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Calculus - Example 3

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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Cornell University

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Simon Fraser University

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University of Winnipeg

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Video Transcript

Welcome to our math Review video. Where will consider the applications of derivatives to physics now derivative show up a lot in physics? Because very often the relationships between two different quantities to different physical quantities will be a derivative relationship. Um, meaning that we have to take a derivative of the function that we have in order to obtain the new quantity that we want. I've already given one example where if we have position and we take the derivative of that with respect to time, then we can obtain velocity. In fact, if we take the derivative of velocity with respect to time, we will then obtain acceleration. And we'll discuss that more at length when we get to the Kinnah Matics videos. Now, because of these relationships and a whole bunch of others that are are exactly the same. We come across all sorts of derivatives in physics, though not as many as you might in a math class. We're gonna mostly focus on polynomial, trig functions and exponential. The reason for this is these often show up in physical models, but then also because we're not out to get you with math. We're really just trying to make sure that you understand the relationships we're looking at here. So let's consider Polynomial is first, as those will be the first ones that you will come across when you start doing Kinnah Matics. Now, toe do polynomial is you'll want to remember one additional rule here, which is called the Power Rule. So if we're taking a derivative of something that's constant a Times X to the end, then what we're going to get as a result is n times a times X to the end minus one. So, for example, if we are going to take a derivative of a function x of T and it's equal to ex not plus v not t plus one half A t squared. Okay, So if I want to take the derivative of this with respect to time, then what I'm going to get as I'm going to get dirty with respect to time of this constant X not which is going to be zero. And then I'll take the derivative with respect to time of Vienna Times t plus the derivative with respect to time of one half a t squared. Now that all written out, then I said. As I said before this one being zero plus here the tea will disappear. Remember that the constant will come out in front of the operators, so I'll have V not left over. Plus, here I'll have one half a comes out and then I'll drop a two down using my power rule. So I'll have just a times t. So I find the velocity is a function of time is equal to the original velocity, plus the acceleration times time, which is one of the Kinnah Matics equations that we will find later. So this was this was helpful. We were able to see this and it also will if we take another derivative revealed to us a fundamental assumption about the kingdom attics that we're about to do, which is if I take the derivative of this. As I said before, this will be acceleration as a function of time. But here it just gives me a so in fact, what I see in order for these to be true, I need acceleration to be a constant, which is the basic assumption of all Kinnah Matics. So derivatives could be powerful in revealing things like that. Um Let's take a quick look at the other types of functions very briefly. If we take the derivative oven exponential, remember that. We just get back the function itself. Okay, If I threw a constant in there and we said have of E to the eight Times X, then the constant would come down. Um but that's about all that's going to change here. Meanwhile, if I have my trig functions, I have sign of x co sin of X and Tangent of X. If I were to take the derivative of all of these, then my sign would become cosign. My co sign becomes negative sign and my tangent of X becomes Seacon Square Defects Remember, seeking of X is equal to one over the cosine of X. Now, um, having seen this, then we can figure out the same thing. If I were to take the integral of co sin of X, I would get sign of X. If I were to take the integral of sine of X, I would get negative co sin of X and so on and so forth. So remembering to keep these in order and remembering with where the negative sign goes will be really helpful as we arrive at things like harmonic motion later on

RC
University of North Carolina at Chapel Hill
Top Physics 101 Mechanics Educators
Elyse G.

Cornell University

Farnaz M.

Simon Fraser University

Jared E.

University of Winnipeg

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