Math Review

Hello, everyone and welcome to Pearson Plus Four Physics. My name is Patrick and I'll be your main guy throughout our content. It's my job to help you actually understand the concepts you see from lecture to help you prepare for your exams. I have a bachelor's degree in physics and I've been helping college level physics students since 2012. I love to make science interesting, but more importantly, I love giving students the skills and tools that they need to be successful in their courses. Now, throughout our content, we'll dive deep and explore topics that are commonly taught throughout most undergraduate physics courses such as kinematics courses in energy, thermodynamics, electricity, and magnetism circuits, optics, and many more our courses and our content is also designed with you in mind so that you can use whichever textbook that you're using for your class. So here's how it works. You'll watch our content videos where I'll go into great detail on each topic. I'll explain all the concepts, equations and problem solving strategies that you need to know. We also love to solve real problems in our lessons which help you gain the skills and the confidence to tackle the next set of problems. Our content is also built to be engaging. So rather than just simply watching a video, you'll have your own copy of our lessons to write on and follow along. So I'll be right alongside with you through these examples to help you prepare for homeworks and exam day. And once you're done with our concept videos, you can also test your knowledge on our practice problems, which will always have video solutions showing you how to get the right answer. Now, if you ever need additional help, feel free to reach out to our team by leaving a comment on one of our videos and we'll get back to you. So once again, welcome to Pearson plus Four Physics. Let's get started.

Everyone. I'm super excited to have you in our physics course here on Pearson channels. Before we get started, I want to briefly review some important math skills things from algebra and trigonometry that we'll use pretty frequently throughout the course. Knowing these will give you the best chance of success as we navigate through physics. So let's get started.

So generally speaking, it's gonna be really important that you have a really good handle on algebra and algebraic expressions in physics. So the first thing we're gonna talk about is how to simplify expressions. So for example, what I'm gonna show you is that we can take expressions like this and through subtraction and multiplication. In addition to all those things, we can actually just simplify this to something like X. And I'm gonna show you how this works. Basically, we take long expressions and we make them simpler by reducing the number of terms, we just write them in fewer terms. So for example, if I have expression over here, basically, what I'm gonna do is I'm gonna distribute anything that's on the outside of parentheses, then I can group together the terms that are similar and then I can combine them by adding and subtracting. So the way this works is that I have two X plus three, don't change anything there, but I have to distribute the four into everything that's inside of the parentheses. So I get four X plus eight. Now, what I do is I just group the, the things that are like each other like the two X and four X and the three and the eights. So this is two X plus four X which just becomes six X and then I have three and eight, which just combine down to 11. So this whole entire expression here really just becomes six X plus 11. And that is the simplest way I can write that. All right. Now, let's move on now. Nothing you're gonna have to be really good at. Um is, or just familiar with is expressions or with exponents. So I'm just gonna go over really quickly what exponents are? Um basically exponents just represent repeated multiplication. We'll do this a lot when we're doing talking about scientific notation. Um So just some basics here, if you have like a number like four multiplied by itself five times, it's really sort of tedious to write out. So what we do is we write this with a sort of shorthand notation four with a little superscript of five. It's a little tiny five in the top right corner. The way we say this is, it's four to the fifth power, the four is the base, it's the number or variable that you're multiplying. Um So that's the four and the exponent of the power is how many times you're multiplying that base by. So base is four and the power is five. Generally, the way that we write any exponent is if you have something, it could be a number or a variable A and if you're multiplying it by itself n times, then we just write it as a to the end power right now. So that's just how generally exponents work. Um, what you're also gonna have to be pretty, sort of a pretty, you know, have a good to handle on is how to manipulate exponents. Um So there's a couple of rules that are, you're gonna probably use, uh, more frequently than others and to sort of summarize them in a little table here. So this is important ones like the product and quotient rule. These come up all the time. Uh This is when you have something like four squared times four to the first power, you're multiplying things of the same base. Basically what you're gonna do here is you're gonna add the exponent. So this is like four to the two plus one power, which is four to the third power. So when you're multiplying things that are the same base, you add their exponents. And then when you're dividing things of the same base like four to the third over divided by four to the one, you're actually just subtracting the exponents. So this becomes four squared. So when you're dividing, then you subtract the exponents, couple of other really quick ones here. Anything to the zero power always just equals one. That's just a rule in math that you should know um whenever you have negative exponents like four to the negative two power. Uh Basically the way, the way this works is you're gonna sort of flip it to the bottom of a fraction. So now this becomes 1/4 to the second power and you drop the minus sign. So generally what happens is when you have negative exponents on the numerator, you flip them to the bottom, the denominator write them with a positive exponent, but you also can have something like a negative exponent in the denominator. In that case, you flip it to the top and you write it with a positive exponent. So you always do the reciprocal and then rewrite it with a positive exponents, right? So another one is called the power rule. And this is basically where you have a power that's on top of another power on the outside of a parentheses. Basically what you're gonna do here is you're gonna multiply their exponents. So it's kind of similar to what we do with the product in quotient, but now you're multiplying the exponents. So this becomes fourth to the sixth power. So you multiply them and then finally powerful products. This basically just means if you have something like three times four squared, you distribute the exponent to everything that's on the inside. So this becomes three squared times four squared. Same things happens with quotients. You basically just distribute the exponent to everything that's inside the parentheses. This becomes 12 squared over four squared. So really what you're doing here is you're just distributing the exponent to every term that's in the parentheses. And in the case where you have a fraction, you just distribute it to the numerator and denominator. All right.

All right. And let's move on here. So solving equations is gonna be a super important part of the course throughout the course, we're going to be solving lots of different variables and lots of different equations. So it's good to have a good handle on that. Basically, the way we're going to do that is we're gonna use different operations like addition, subtraction, multiplication, stuff like that. And the whole thing we're trying to do is isolate variables. Um Sometimes it's X, sometimes it's going to be other variables. We're gonna use all different kinds of letters throughout the course. And basically what happens is whenever you do an operation to some kind of an equation, you actually have to always have to do operation to both sides of the equation. So for example, if I got two X minus three equals zero, I I need to solve for X, what I have to do is I want to get X by itself and I have to do a bunch of operations. So the first thing I have to do here is I have to distribute the two to everything that's inside the parentheses. And you end up with two X minus six is zero. Now, I want to solve for X in this equation. So I want to get X by itself and I have to isolate it by moving stuff to the other side and whatever you do to one side, if I add six, then I have to add it to the other side. You always have to do that. So I get uh just two X equals six over here, but I'm not done yet. I have one last thing to do and I have to divide by two. But remember whatever I do to one side, I have to do the other. So always just, you know, be really, really careful that you do that. When you work the sandwich, you're gonna get is X equals three. So the general steps to do this is you're gonna sort of distribute any constant sided parentheses. You're gonna combine like terms that we've already seen how to do that. Then you're gonna group the X terms and constants on opposite sides, isolate and solve for X. And then you can always just check your solution by replacing this there, this a solution that you get back into the original equation and make sure that you get the right answer. All right. So let's just do the second example. Again, I've got 1/4 X plus five equals negative three. Again, I want X by itself. So I want to sort of sort of move everything that's a constant over to the other side. What I end up with is 1/4 X and then when I move the five over, I have to subtract five and then subtract five and this actually ends up becoming negative eight. Now, what happens is in order to get rid of the 1/4 what I'm gonna have to do is I'm gonna have to move multiply by four on this side of the equation. So I have to multiply by four. But then I also have to multiply by four here as well. What you'll end up getting here is X equals negative 32. And now you've solved for this equation and if you want to check, you can always just plug this negative 32 back inside of this equation just to make sure that it makes sense, right? So that's how to solve equations for different variables. Super important skill will be using it a lot throughout the course. Let's move on here. Now, a lot of the course will also just involve graphing. So graphing was usually gonna involve something like plotting points or equation on a rectangular coordinate system. This is basically just sort of like the two dimensional plane that we have over here. And the way that we do this is you're going to be given some kind of an equation and we're going to have to just plot a bunch of points. So the way we do this is we isolate Y to the left side. So for example, if we have something an equation like this, you want Y by itself on the left side because then what you're gonna do is you're just gonna plug in values of X anywhere you see X inside and you're going to calculate a bunch of values. And then you can basically just get ordered pairs. So for, for example, if you plug zero into this equation, you get zero squared minus three times zero, both those things go away. And then you end up with just two. So your Y coordinate is two. So the ordered pair is zero comma two and we can plot that here on the grid. So zero comma two is going to be this point over here. Remember ordered pairs are always X comma Y, you do the X value first and then the Y value. So just a couple of more, if you plug in 11 squared minus three times one plus two, you actually just get zero. So the ordered pair is one comma zero. And actually the same thing happens if you plug two into this equation, you're gonna get two squared which is four and this is minus six plus two also works out to just zero. So this is actually two comma zero. So it means we can plug the next two points in one comma zero, which is right over here and then two comma zero, which is right over here. And then finally, if you plug in three, which you're gonna give us two. So the coordinate is three comma two and then four comma six. If you plug all of these things in together, what you're gonna see here is you're gonna get these points, three comma two and then four comma six, which is all the way up here. So basically what you're gonna do is you're just gonna isolate and then when you calculate a bunch of Y values, um you're just gonna pick a bunch of X values to sort of plug into the equation and then you're gonna plot those points that you've gotten from step two. And then finally, you're just going to connect those points with a line or a curve or something like that. In this case, this isn't a line. In fact, it's actually gonna look something like looks like this, which you may remember is actually the shape of a parabola or a quadratic equation. All right. So that's basically how to plot equations and plot points and things like that. All right, let's move on.

So a lot of times in physics, you're gonna end up having an equation in which you don't have enough information to solve it. So you're gonna get another equation and now you have to deal with these two equations over here. This is called a system of equations. And we're just going to review how to solve these really quickly. And basically, what we're gonna do is we're gonna substitute one equation into the other to reduce the number of variables. I'm gonna show you how this works. The first thing we're gonna do here is if you have two equations, you're gonna solve one equation for uh let's say Y variable or whatever the easiest one is to solve. In this case, I've got Y equals three X minus six. So I've got already Y by itself. Now, what we're gonna do here is we're going to plug the right side of this equation A in, for whenever we see Y in equation B. So basically what I'm gonna do here is I'm gonna take three X minus six and everywhere I see Y in equation B, I'm going to replace it with three X minus six. So what this looks like over here is, it looks like two X plus this, this ends up just being three X minus six. It's not Y anymore. I'm just gonna replace it with three X minus six and this equals four. Notice that what I've done with this equation is now, instead of having two variables, I only have one and now I can just go ahead and solve for X. And that's what my third step is. You just solve this for X and we know exactly how to do this. So this two X plus three, X just becomes five, X minus six is four. And then all I have to do is just move the six over to the other side and basically end up with five X equals 10. So now, all I have to do is divide each side by five. And what I end up with is X equals two. And that's the answer. So this is the answer. That's my X value that works for both of these equations. But I'm not quite done yet because this is only one of the values that works for both the equations. I need to also figure out what the Y value is. And that's the fourth step. You're gonna take this X value that you've already figured out. And you can actually plug it into either one of these two equations to solve for Y. So you could plug it into A or B it actually doesn't matter. So let's go ahead and do this. This is gonna be Y equals three. And now we don't have just X because we know what X is X is just equal to two. So we plug this back in so three times two minus six, this ends up just giving us six minus six over here. And so this gives us a Y value of zero. All right. So in other words, the solution to both of these equations is actually when X is two and Y is equal to zero. So this represents the XY solution that satisfies both of the equations. All right. And that's the Y value. So that's how to solve the system of equations by substitution.

So in physics, the some of the most common graphs that you'll see are gonna be graphs of lines. And one of the most important variables you'll have to solve for is the slope of lines. So let's go ahead and talk about that real quick. The slope remember is just a number that represents how steep the line is sort of how vertical it is. So the green line kind of looks like this. The red line is a little bit steeper. So it's gonna have a higher slope mathematically what's going on? Is it the change in the Y value divided by the change in the X value? One of the things you've probably heard before in some classes is rise over run, which is basically just the change change in Y over the change in X, we use this little delta symbol which you will see a lot in physics to describe the change in a variable. So real quickly here, if we want to calculate the slopes of these two lines on the graph, you just have to figure out the rise of the run. How much does it rise? Well, you're going from 2 to 4. So the change in Y is, is two and then you're going from 1 to 2 over here. So the change in X is just one. So when you calculate the slope, it's just delta Y over delta X 2/1 and the slope is just a value of two. All right, let's do the same thing for the red line. All we're gonna have to do here is just pick two points. I'm gonna pick this point over here and this point and figure out the rise of the run. So if I have to go from this point to this point, first, I have to go up. And if you count up the number of units, you're actually gonna get end up going five units. That means my delta Y is five and you have to go over by one unit over here. So delta X is one. So the slope in this case is five of one and this is five. So just as we said before, this slope here, which is five is a higher number and it just represents that the slope is steeper than this green line over here where it has a slope of two. All right. So that's all. There is two slopes.

Now, if you have to graph equations in physics, a lot of time, they'll end up being lines. Um And so line equations in slope intercept form will tell you everything that you need to graph it. So usually you're gonna see something like Y equals MX plus B which remember is the form of a line and slope intercept form. So for example, something like Y equals two X plus minus one is in slope intercept form. So really quickly in this example, here, the Y intercept is just the constant on the outside, which is just the negative one that's at the end. And then the m the slope is just the number that goes in front of the X which in this case is two. So that's the slope in the intercept. So to graphic all you have to do is basically just plot the Y intercept, which in this case is gonna be zero comma negative one. That's where it crosses the Y axis. And they're gonna plot the next line or the next set of points. By using the slope, the slope tells you that you're gonna size 2/1. So in this case you have to go up to and over one to get to the next point. So that's the, the next point over here you could also go down to and then to the left one to get your other points over here. Once you do that a few times, you'll see that the sort of trend of this line is, it's gonna look something that looks like this. And so this is the equation of your line in slope intercept form. And that's how to graph it.

OK. So now that we've reviewed lines, some of the other common equations that you'll get in physics will be quadratic equations. Remember a quadratic is when you have something like a variable like X raced to the second power instead of the first power. So this is an equation in a standard form of a quadratic. There's basically just two different ways that we're gonna use in physics to solve. We're gonna take square roots or we're just gonna use a quadratic formula. And it really just depends on how the quadratic looks. So for example, if you have something like X plus one squared equals four, that's looks like where you have X plus a number that's squared and then you have a constant on the other side. But if you have something that looks like this in a little bit more of like a standard form, that kind of looks like A X squared plus BX plus C, then you could always just use the quadratic formula, but it's a little bit more tedious. I'm actually going to show you that these two things are the exact same equations written in different ways. We're gonna get the same exact solutions for this. So let me show you how to take the square root. Basically, the way that this works is if you have something looks like this, in order to get X by itself, I'm gonna have to get rid of the squared over here. So I'm gonna have to isolate the squared expression and then I have to take the square root of both sides. And when I do that, I'm gonna have to take the positive and negative roots. That's always how you take square roots. So what ends up happening is the squared and square root cancel and you're left with X plus one equals positive and negative square root of four. So what this works out to is you have X equals negative one, you're going to move this to the other side plus or minus two. And so what this is saying is that there's two solutions, there's negative one plus two, which ends up being just one and there's negative one minus two which ends up being negative three. So those are your two solutions to this quadratic, usually quadratics will give you two solutions. Now, um that's how to take square roots. But if you're ever given a quadratic that looks like this in which you can't factor it or you don't know what method to use, then you can always pop it into the quadratic formula that will always give you your correct answers. All right. And this is just a little bit more tedious. But basically, just remember here that the coefficients of your quadratic are A B and C and you just pop the appropriate expressions and letters into this formula over here. So in other words, this is gonna be X equals negative B which in this case is two. So this is negative two plus or minus the square roots of B squared. So this is gonna be two squared uh minus and then this is gonna be four times a times C which is negative three all over two A which in this case is just two times one. So just to simplify a little bit, this is going to be X equals negative two plus or minus. And this whole mess actually over here just becomes four minus a negative 12. So it actually, I'm just gonna skip a couple of steps, but this ends up being negative or sorry squared of 16 all divided by two. And if you simplify this even further, what this turns out to be is it turns out to be negative two plus or minus the squared of 16 just ends up being four. And this is over two. So really what happens is that your solutions ends up just being an X equals uh negative one plus or minus two, which is exactly what we got over here. Remember this just means two things, it means negative one plus two which in this case is one negative one minus two, which in this case is negative three. So it makes sense that we got the exact same answers whether we use the square root or the quadratic. Because again, these two equations are the exact same thing. You can just check that if you distribute and sort of foil this out and uh convert it to standard form. These things are the same. All right. So that's how to use quadratics and square roots to solve quadratics. All right. So now that we know how to solve quadratics algebraically, a lot of times you're gonna see quadratics graft and you'll have to identify some key information from graph quadratics. Remember that quadratics kind of look like parabolas, they can either look like smiley faces or frowny faces and it just depends on their equation. Now, the equation that you'll see most of the time for quadratics is gonna be in vertex form because it's easy to plot them. And basically it looks like A X minus H squared plus K, the vertex X meaning the little points where the parabola opens up is gonna be the coordinate H comma K. So in this case, we have negative X minus one squared plus four, the coordinate of the vertex is actually just gonna be one comma four. And because of this negative sign, it's actually gonna look like a frowny face. So we can see here is that the coordinate one comma four is gonna be the sort of top of our Parabola. It's gonna be the maximum points over here and it's basically gonna go look looking like this. But in order to get those points, we're gonna have to solve for the X and Y intercepts. And basically to do that, you're just gonna solve for when Y is equal to zero for the X and then for the Y intercept, you're gonna set X equal to zero and solve for that's variable over there. So I'm just gonna actually go ahead and do this for you. It's actually just the of the uh quadratics that we solved uh up above. And so what this is gonna look like, it's gonna be like negative one and three. That's the point where is the X axis. So it's going to look something like this. Whereas the Y intercepts are where you set the X coordinate equal to zero. So you basically just set this equal to zero and solve what you're gonna see is that this quadratic hits the Y intercept or the Y axis at the 0.3. So once you solved all your points, you're gonna connect them with a curve just to see what your parabola looks like. So you basically just connect all these things over here with a little curve. And so our parabola looks like a little frowny face that looks like that. And from this graph, we can tell when the graph is increasing and decreasing. Those are gonna be important things. So for example, we can see here that as we're moving left to right, the graph is increasing up until we get to the X equals one point. So it's increasing when X is less than one and then as you're going left to right, it starts decreasing everywhere after positive one. All right. So that's how to graph and understand key information from quadratic equations.

So very commonly throughout all of physics, you'll see questions that ask you how variable changes when another variable in the equation changes as well. And this is an area of math called proportional reasoning, which basically just analyzes how one thing increases or decreases with another. There's a couple of different situations that you'll commonly see. So I just want to review them really quickly. So first, let's say I had something like Y equals two X basically just go, go, let's go ahead and plug in a bunch of values. So if you have X equals 210 and so on, would you to see here is that Y is 2420, negative two and negative four? So as we can see here, there's a direct relationship between Y and X, we say they're directly proportional as the X values get bigger. So does the Y values as X gets smaller, so do the Y values. So these things are sort of directly correlated. Let's take a look at another example, we have Y equals one over X. Let's just plug in a bunch of numbers in for one, I get one for two I actually get one half for three, I get one over third and then I, 1/4 and 1/5. So these actually are fractions. And we can see here that the fractions actually get smaller with higher XS. So these things are inversely proportional because as the X values increase, the Y decreases and then vice versa, as X values go down, they actually the numbers get bigger for Y. So these things are inversely proportional. Now, let's take a look at the last situation which is actually going to be the more common one, let's say you have something like F equals MA. And these are actually two variables, we say that these are jointly proportional. Um And this is what happens here. So M and A could be any numbers and we just gonna plug in a bunch of numbers in here but F equals MA is one of the most important variables or sorry equations that you'll see in physics and it's just a multiplication of two things. So for example, um five times two is 10, 4 times one is 43 times 00. And then both of these things here are negative too. So there's a little bit of a pattern that emerges here because as M gets bigger and A gets bigger, we can see that the F gets bigger and then as M and A both gets smaller. So does F but there's also another kind of problem which you're gonna see. Which is that because it's two variables that are multiplied together, you could also have situations where you want a constant F. So for example, I want F to just be 20 there's actually just a lot of different combinations of M and A that will make that work. So for example, one times 20 is 22 times 10 is 24 times five is also 20. And so what you're gonna see here is that for constant FS as the M gets bigger over here, the A actually has to get smaller. So A as M gets larger, A has to be smaller in order to compensate. And so that they both get 20 then also the opposite happens as M gets smaller over here, the M actually has to get bigger. Um So these things are all sort of like inversely proportional as well. So this is called jointly proportional. It's one of the more common things that you'll see in physics.

So throughout most of physics, we're gonna be talking about sines, cosines and tangents. It's gonna be really helpful for you to refresh on your trigonometry. So remember that sines cosines and tangents all basically relate angles and sides of a right triangle if I have a right triangle like this, and we've got the angle that's on my left side over here. I've got the adjacent opposite and hypotenuse sides of a right triangle and uh sine cosines and tangents basically just relate all those three things together. So always just remember SOCA to, that's probably something you've heard before basically just helps you understand which things are getting, getting divided. So for sign, it's gonna be opposite over hypotenuse. That's the so parts. So in this case, the sign of this angle over here is gonna be the opposite side divided by the hypotenuse, it's three or five. So the cosine is gonna be the adjacent side over Hyotan, that's the part of SOTO. And so in this case, the theta or the, the cosine of this is gonna be the adjacent side over the hypotenuse that's 4/5. And then the tangent is actually gonna be the opposite over the adjacent side, that's the to parts. And so in this case, the tangent of this angle is gonna be the uh sorry, the opposite side divided by the adjacent side. So this is gonna be 3/4 all right. So um that's just how to use soto uh some of the other equations that you might have to know are just how to get the opposite side, which is gonna be the hypotenuse time sign or cosine. Um And some other helpful formulas are gonna be, you know, things like the Pythagorean theorem or just this uh sort of like basic identity of signs and cosines where sine squared and cosine squared is just one. And then also just this one over here where tangent is signed over cosine, all these things are going to be very, very helpful for you to understand as you get into physics. All right. So now that we've got a basic understanding of sine cosines and tangents, those three things will all have special values for very common angles that pop up in physics like 30 60 90 45 and zero. It's gonna be really helpful to sort of memorize them because we'll be using them a lot in physics. So we're gonna go over them really quickly here. So remember that the unit circle is really just a circle with a radius of one. And the basic idea here is that you can basically just sort of create a bunch of right triangles by sweeping out angles of different values. And so you can just create a bunch of triangles everywhere. And the hypothesis of those triangles will always have a value of one. And so, for example, this uh triangle over here has an angle of 30 this one has 45 this one has 60 the opposite and the adjacent sides will always have different values. Uh Basically, depending on what those angles are. And I've got a table here that kind of summarizes this. So really quickly here, uh for 90 degrees, you've got 10 and the tangent actually just doesn't exist. But basically, what you're also gonna see here is that as you go down on this, on this graph here for or this table for sign values, it's actually like sort of going up on the cosine value thing. These are things are almost like their exact sort of mirror opposites of each other. Um So really, these are the only ones that you actually have to memorize because the tangent remember is always just equal to sign divided by cosine. So if you ever forget the tangent values, all you have to do is just if you remember these values over here, you can just divide them and you could always get to what the tangent value is, right? So I'm not gonna go through them, you can just look at them. Um just sort of make sure that you understand and memorize these things. Um The only other thing that I have to point out is that these things will actually have positive and negative values depending on which quadrant of the graph that you're in. Um So for example, everything is positive here, whereas the only thing that's positive over here is sign tangents and then cosign for the different quadrants of the unit circle. All right. That's just a hopefully a pretty, pretty quick review.

Now, if you happen to be taking a calculus based physics course, there's a few extra calculus concepts you'll need to know. Now, whenever we cover them in physics, we'll always give a brief refresher. But it's a good idea to understand things like derivatives and integrals before you start the course. So let's get started here. Now, graphically really what the derivative represents is the instantaneous rate of change or the slope of a tangent line at a certain point. So if I have a parabola like those like this, and if I wanted the slope of a tangent line, that's, you know, a line that's tangent to this point, I'm basically gonna draw a line that touches the graph only once. That is what a tangent line means. So for example, at this point, that's what that slope looks like at this point, the tangent line looks like this. And at this point over here, the tangent line looks a little bit different. The slopes of each one of these lines is what the derivative represents and notice how it's constantly changing throughout this graph. So that's what the is all right. Now, mathematically what happens is that in physics, there's gonna be common types of functions that you're gonna see and we're gonna have to mathematically calculate the derivatives. So we're gonna use these following rules over here. Just a couple of brief examples. So you have something like a constant function like F of X equals three that actually kind of just looks like a perfectly flat line. The derivatives of constant functions are always zero. And it's because a flat line always has a slope of zero. And that's what the derivative represents. You know, if you have something like somewhat constant times and X value like for example, negative two X, then that actually represents the graph of a line and the derivative is always basically just whatever the coefficient is. So in this case, it's negative two, that slope is always constant. Now these two are actually the most common types of functions that you'll see in uh in physics. Um And so to take derivatives, you're going to use the power rule basically, the way it works is if you have a variable that X rays to some power, you're gonna drop that exponent down in front of the X so N times X and then you're gonna subtract the exponent by one. For example, if you have X squared we're gonna do is we're gonna take the two, that's the exponent and drop it down in front of the X. So it's two X and now we subtract this exponent by one. So it's just two X to the first power. So one way you can just also write this is just two X. Now, if you have a polynomial something that's a combination of multiple functions like X squared and three X and X to higher powers. Basically, what you're gonna do is you're gonna just take derivatives of each one of the terms independently. So for example, you're going to take a derivative of X squared, that's just its own thing. So in other words, this is just two X over here and the derivative of three X just actually ends up just being three. So you basically just use this power rule for as many terms as you have in your polynomial. All right, these are the most common situations that you're going to see. That's just a brief refresher on derivative. OK. So now that we've reviewed derivatives, there are occasionally sometimes in physics where we'll have to use also integrals um but they won't be super complicated. Let's go ahead and review it real quickly. So graphically the integral represents just the area under a curve for a specific function. So for example, if I have this line like this, the integral or the area under the curve between, let's say zero and four is basically just going to be everything that's inside this big triangle over here. So the integral graph just represents all of the area that's underneath this blue sort of line over here inside this triangle. So one way you can approximate integrals, uh especially when doing graphs is just by adding the areas of a bunch of rectangles or squares under the curve. So I could try to cut this up into a bunch of rectangles and sort of add those things in here or what I can do is just count a bunch of boxes. So for example, I can count up this box um and basically just count up all the po the the squares that I can sort of count inside of this graph. Um And if you actually go ahead and do this for this shape, what you'll see is that the area is actually equal to eight, you get six squares and then you also get four of these little triangles here which are like half squares. So the area or the integral under this curve is just eight, that would be the answer. All right. Now, most of the time if you're not given a graph, you're gonna have to evaluate integrals mathematically and remember that integrals are basically just kind of like the opposite of derivatives. So here's sort of the general form of a definite integral, which are almost always going to be the kinds of integrals that you see inside of physics. So to determine exact integrals, just remember using the, the following rules over here, we're having a couple of examples if you have a constant function, like for example, just like a three without a variable, then you just attach an X to it. So for example, this would just be the integral of three, it would just be three X and then you just evaluate it from 3 to 1. And I'm not gonna plug that in um but basically just substitute three and one inside of your X in this equation. If you have something like X to the nth power, like for example, X squared, then there's also a power rule for integral where basically what you're gonna do here is you're gonna take that power and increase it by one. So X to the two becomes X to the three. But then you have to divide by whatever that new exponent is. So we're gonna have to divide by three. So the integral of X squared would be X cubed over three. And then again, you just evaluate it from uh from 1 to 3, whatever your limits are of your definite integral. All right, if you have something like a constant that's in front of that function, it's the same exact thing. You're basically just going to use the power rule except you have to multiply um that constant that's out in front of it. So for example, this negative two squared over here uh actually just becomes negative two and then we have X to the second power divided by two. Remember you're going to increase the power by one and then divide by whatever that new power ends up being. And a lot of times what's gonna happen is you're gonna have to simplify whatever you get on the top and the bottom. So this actually ends up just becoming uh this ends up just becoming negative of one X to the second power, right? So that's your integral. And again, you would just evaluate it from 3 to 1. Now, last but not least here is, if you have a bunch of terms like a polynomial, then you just integrate each one of these things separately exactly like we did derivatives for each one of these things separately. So for example, uh if you have X squared plus two X, then the first term ends up just becoming X to the third over three, we already saw that. And then the second, second term over here two X just becomes uh two X so two X squared over two. So that's what that integral ends up becoming. So this whole thing actually ends up being, being X squared over three, sorry X cubed over three plus uh X squared. And then you would just evaluate this entire thing from 3 to 1 and that would be your answer, right? So that's just a brief review of integrals.