RC

# 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

#### Topics

No Related Subtopics

### Discussion

You must be signed in to discuss.
##### Top Educators ##### Christina K.

Rutgers, The State University of New Jersey ##### Andy C.

University of Michigan - Ann Arbor ##### Marshall S.

University of Washington ##### Aspen F.

University of Sheffield

### Recommended Quiz

#### Physics 101 Mechanics

Create your own quiz or take a quiz that has been automatically generated based on what you have been learning. Expose yourself to new questions and test your abilities with different levels of difficulty.

### Video Transcript

Welcome to our math review video, where we'll look at the applications of Integral in physics now for the same reason that derivatives are important. Physics integral will also be important in physics and will need to be able thio operate on the same types of functions. If you remember those functions where polynomial you're gonna metric functions and exponential functions. If we're able to work on these, you'll be able to solve most of the problems thrown at you in an introductory physics course. Now, um, looking at these, then, uh, let's jump right in. If I were to take the integral of a of a polynomial function, let's just give us a yourselves a generic when they x squared plus BX plus c take the integral with respect to X here. Then what am I going to obtain? Well, uh, remember using the power rule in reverse. What we're gonna find is Owen. First of all, we should distribute thes to make it easier on ourselves so we can have think about them one at a time. Now, when I take the integral of a constant, as I have shown before, what we're going to get is a C X. Now there will be some additional constant that I'll call see, not here because we're using Indefinite. Integral. Generally, we won't have that when we're doing applied problems because we'll use definite integral. Meanwhile, when I come here to be ex well, I know I'm gonna have an X squared and my be stays around. But what comes here will remember I should be able to take the derivative of this and get back BX, which means I need a one half because when that two falls down, it will cancel out and I'll be left with B X all by itself. Similarly, I'll have an X cubed here and I'll need a one third in front of a. So this is going to be my integral of the function a X squared plus bx plus c Remember, Integral Zahra, little tougher. You gotta think backwards for all the rules that you learn with derivatives. Similarly, if I'm looking at my trig functions and I say okay, I want to take the integral of sign of X, the X well, we know we're going to end up with cosine. The question is, will we have a negative or a positive. Now I recommend that you go back and you look at the actual plots of this. But if you just want a simple rule to recommend the Thio to think about, then remember the plot that I the picture I wrote before where I had sign of x co sign of acts. And when I took the derivatives, I came up with co sin of X and negative sign of X. Well, in this case, what I'm gonna do is I'm going to get rid of the arrows going that way and instead I want to go backwards. Okay, so this will be when I'm applying an integral okay, we'll go backwards across. So when I take the integral of sine of X, I'll get a cosine but this negative still hanging around. So it's going to show up here. Meanwhile, when I take the integral of cosine of X, I'll just get sign of X. I recommend you write this down a few times for me having this picture in my head. This table is really helpful and has always helped me to remember uh, where the negative sign goes Now. These will also have constants for the same reason as the polynomial. Take a quick look, then at Exponential Justus how, before we had the derivative of X was equal just to e x, We're going to come up with the same thing here except, well, I e x plus a constant. Now, when we look at this again and we say, Well, what if we have the integral of e to the X d x? Well, we know that the A falls down when we get it a derivative, so we'll have to pull it down like this so that when we take the derivative of this function, we'll get the same thing back and we can check that quickly. If we were to take the derivative of one over a e d d a x plus c not the sea not disappears because it's a constant. The A falls down, cancels with this A and we're left with just e d. A X, which is correct. Always remember, if you're worried about whether or not you're your anti derivative was correct, just take the derivative, and if you get the same function you had back then, you have the correct answer

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

Rutgers, The State University of New Jersey ##### Andy C.

University of Michigan - Ann Arbor ##### Marshall S.

University of Washington ##### Aspen F.

University of Sheffield