Math Review - Online Tutor, Practice Problems & Exam Prep

Understanding algebraic expressions, exponents, and solving equations is crucial in physics. Simplifying expressions involves distributing and combining like terms. Exponents represent repeated multiplication, with key rules for manipulation, such as the product and quotient rules. Solving equations requires isolating variables through operations on both sides. Graphing involves plotting points based on equations, while systems of equations can be solved using substitution. Additionally, recognizing direct, inverse, and jointly proportional relationships is essential for analyzing variable interactions in physics.

Introduction to Physics Channel

Hello everyone, and welcome to Pearson Plus for physics. My name is Patrick, and I'll be your main guide 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, forces and energy, thermodynamics, electricity and magnetism, circuits, optics, and many more. Our courses and our content are also designed with you in mind so that you can use whichever textbook you are using for your class.

So here's how it works. You'll watch our concept videos where I'll go into great detail about each topic. I'll explain all the concepts, equations, and problem-solving strategies that you need to know. We also love solving real problems in our lessons, which helps 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 homework 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 for physics. Let's get started.

Introduction to Math Review

Hey 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.

Simplifying Algebraic Expression

So generally speaking, it's going to be really important that you have a really good handle on algebra and algebraic expressions in physics. So the first thing we're going to talk about is how to simplify expressions. For example, what I'm going to show you is that we can take expressions like this, and through subtraction, multiplication, and addition, all those things, we can actually just simplify this to something like x. And I’m going to 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. For example, if I have this expression over here, basically, what I'm going to do is I’m going to 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 \(2x+3\), don't change anything there, but I have to distribute the 4 into everything that's inside the parenthesis, so I get \(4x + 8\). Now what I do is I just group the things that are like each other, like the \(2x\) and \(4x\) and the \(3\) and the eights. So this is \(2x + 4x\), which just becomes \(6x\). And then I have 38, which just combined down to 11. So this whole entire expression here really just becomes \(6x + 11\), and that is the simplest way I can write that. Alright? Now let's move on.

Another thing you're going to have to be really good at, or just familiar with, is expressions with exponents. So I'm just going to go over really quickly what exponents are. Basically, exponents just represent repeated multiplication. We'll do this a lot when we're talking about scientific notation. So just some basics here. If you have, like, a number like 4 multiplied by itself 5 times, it's really sort of tedious to write out. So what we do is we write this with a sort of shorthand notation 4 with a little superscript of 5. It's a little tiny 5 in the top right corner. The way we say this is it's 4 to the fifth power. The 4 is the base. It's the number or variable that you're multiplying, So that's the 4, and the exponent or the power is how many times you're multiplying that base by. So base is 4 and the power is 5. 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 \(a^n\). Alright? Now so that's just how generally exponents work.

What you're also going to have to be pretty sort of pretty you know, have a good handle on is how to manipulate exponents. So there's a couple of rules that you're going to probably use, more frequently than others, and I sort of summarize them in a little table here. So there's some important ones like the product and quotient rule. These come up all the time. This is when you have something like \(4^2 \times 4^1\). You're multiplying things of the same base. Basically, what you're going to do here is you're going to add the exponents. So this is like \(4^{2+1}\), which is \(4^3\). 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 \(4^3\) divided by \(4^1\), you're actually just subtracting the exponents. This becomes \(4^2\). So when you're dividing, then you subtract the exponents. A couple of other really quick ones here, anything to the 0 power always just equals 1. That's just a rule in math that you should know. Whenever you have negative exponents like \(4^{-2}\), basically, the way this works is you're going to sort of flip it to the bottom of a fraction. So now this becomes \(1/4^2\) 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 could have something like a negative exponent in the denominator. In that case, you flip it to the top and you write with a positive exponent. So you always do the reciprocal and then rewrite it with a positive exponent. Alright.

Another one's 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 parenthesis. Basically, what you're going to do here is you're going to multiply their exponents. So it's kind of similar to what we do with product and quotient, but now you're multiplying the exponents, so this becomes \(4^6\). So you multiply them. And then finally, power of a product. This basically just means if you have something like \(3 \times 4^2\), you distribute the exponent to everything that's on the inside. So this becomes \(3^2 \times 4^2\). Same things happens with quotients. You basically just distribute the exponent to everything that's inside the parenthesis. This becomes \(12^2/4^2\). 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. Alright.

Solving Equations & Graphing

Alright, everyone. Let's move on here. So solving equations is going to be a super important part of the course. Throughout the course, you'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 going to use different operations like addition, subtraction, multiplication, stuff like that. And the whole thing we're trying to do is isolate variables. Sometimes it's x. Sometimes it's going to be other variables. We're going to 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 always have to perform operations to both sides of the equation. So, for example, if I've got 2x-3=0, I know I 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 2 to everything that's inside the parentheses, and you end up with 2x-6=0. 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 6, then I have to add it to the other side. You always have to do that. So I get, just 2x6 over here, but I'm not done yet. I have one last thing to do and I have to divide by 2. But remember, whatever I do to one side, I have to do to the other. So always just, you know, be really, really careful that you do that. When you work this out, what you're going to get is x equals 3. So the general steps to do this is you're going to sort of distribute any constant inside of parentheses. You're going to combine like terms, and we've already seen how to do that. Then you're going to 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 solution that you get back into the original equation and make sure that you get the right answer. Alright? So let's just do the second example again. I've got 1x4 + 5 = -3. Again, I want x by itself, so I want to sort of move everything that's a constant over to the other side. What I end up with is 1x4, and then when I move the 5 over, I have to subtract 5 and then subtract 5. And this actually ends up becoming negative 8. Now what happens is, in order to get rid of the 14, what I'm going to have to do is I'm going to have to multiply by 4 on this side of the equation. So I have to multiply by 4, but then I also have to multiply by 4 here as well. What you'll end up getting here is x equals negative 32, and now you've solved for this equation. 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. Alright? That's how to solve equations for different variables. Super important skill. We'll 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 going to involve something like plotting points or equations on a rectangular coordinate system. This is basically just sort of like the 2-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 an equation like this, you want y by itself on the left side because then what you're going to do is you're just going to 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 example, if you plug 0 into this equation, 0x2-3∙0+2, both those things go away, and you end up with just 2, so your y-coordinate is 2. So the ordered pair is 0 comma 2, and we can plot that here on the grid. So 0 comma 2 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 1 1x2-3∙1+2, you actually just get 0, so the ordered pair is 1 comma 0. And actually, the same thing happens if you plug 2 into this equation. You're going to get 22x,4-6+2, also works out to just 0. So this is actually 2 comma 0. So it means we can plug the next two points in. 1 comma 0, which is right over here, and then 2 comma 0, which is right over here. And then finally, if you plug in 3, what you're going to get is 2, so the coordinate is 3 comma 2, and then 4 comma 6. If you plug all these things in together, what you're going to see here is you're going to get these points, 3 comma 2 and then 4 comma 6, which is all the way up here. So, basically, what you're going to do is you're just going to isolate. And then when you calculate a bunch of y values, you're just going to pick a bunch of x values to sort of plug into the equation, and then you're going to plot those points that you've gotten from step 2. 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 going to look something like it looks like this, which you may remember is actually the shape of a parabola or a quadratic equation. Alright? So that's basically how to plot equations and plot points and things like that. Alright? Let's move on.

Solving Systems of Equations

So a lot of times in physics, you're going to end up having an equation in which you don't have enough information to solve it, so you're going to 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. Basically, what we're going to do is we're going to substitute one equation into the other to reduce the number of variables. I'm going to show you how this works. The first thing we're going to do here is if you have two equations, you're going to solve one equation for, let's say, the y variable or whatever the easiest one is to solve. In this case, I've got y equals 3x minus 6, so I've got already y by itself.

Now what we're going to 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 going to do here is I'm going to take 3x minus 6, and everywhere I see y in equation b, I'm going to replace it with 3x minus 6. So what this looks like over here is it looks like 2x plus 3x minus 6. This ends up just being 3x minus 6. It's not y anymore. I'm just going to replace it with 3x minus 6, and this equals 4. 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 2x plus 3x just becomes 5x minus 6 is 4, and then all I have to do is just move the 6 over to the other side, and basically, I end up with 5x equals 10. So now all I have to do is divide each side by 5, and what I end up with is x equals 2, and that's the answer.

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 of the equations. I need to also figure out what the y value is, since that's the fourth step. You're going to 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 going to be y equals 3, and now we don't have just x because we know what x is. X is just equal to 2, so we plug this back in. So 3 times 2 minus 6. This ends up just giving us 6 minus 6 over here, and so this gives us a y value of 0. Alright? So in other words, the solution to both of these equations is actually when x is 2 and y is equal to 0. So this represents the xy solution that satisfies both of the equations. Alright? And that's the y value. So that's how to solve a system of equations by substitution.

Slope

So in physics, some of the most common graphs that you'll see are graphs of lines, and one of the most important variables you'll have to solve for is the slope of lines. 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. The green line kind of looks like this. The red line is a little bit steeper, so it's going to have a higher slope. Mathematically, what's going on is just 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 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 just 2, and then you're going from 1 to 2 over here, so the change in x is just 1. So when you calculate the slope, it's just ΔyΔx, 2 over 1, and the slope is just a value of 2. Alright?

Let's do the same thing for the red line. All we're going to have to do here is just pick 2 points. I'm going to 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 going to end up going 5 units. That means my delta y is 5, then you have to go over by 1 unit over here. So delta x is 1. The slope in this case is 5 over 1, and this is 5. So just as we said before, this slope here, which is 5, is a higher number, and it just represents that the slope is steeper than this green line over here, which has a slope of 2. Alright? So that's all there is to slopes.

Graphing Lines

Now if you have to graph equations in physics, a lot of time, they'll end up being lines, and so line equations in slope intercept form will tell you everything that you need to graph it. So, usually, you're going to see something like y=mx+b, which, remember, is the form of a line in slope intercept form. So, for example, something like y=2x-1 is in slope intercept form. So really quickly in this example here, the y-intercept is just the constant that goes 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 2. So that's the slope and the intercept. To graph it, all you have to do is basically just plot the y-intercept, which in this case is going to be (0, -1). That's where it crosses the y-axis, and you're going to plot the next line or the next set of points by using the slope. The slope tells you that you're going to rise 2 over 1. So in this case, you have to go up 2 and over 1 to get to the next point. So that's the next point over here. You could also go down 2 and then to the left one to get your other point over here. Once you do that a few times, you will see that the sort of trend of this line is it's going to look something like this. And so this is the equation of your line in slope intercept form, and that's how to graph it.

Quadratic Equations

Okay. 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 raised to the second power instead of the first power. So this is an equation in a standard form of a quadratic. There are basically just two different ways that we're going to use in physics to solve. We're going to take square roots, or we're just going to use the quadratic formula. And it really just depends on how the quadratic looks. So, for example, if you have something like x+2 equals 4, that looks like where you have x+number that's squared, and then you have a constant on the other side. If you have something that looks like this in a little bit more of a standard form, that kind of looks like ax2+bx+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 equation written in different ways. We're going to get the same exact solutions for this. So let me show you how to take the square root. Basically, the way this works is if you have something looks like this, in order to get x by itself, I'm going to have to get rid of the squared over here, so I'm going to 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 going to 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+1=±4. So what this works out to is you have x=-1±2. And so what this is saying is that there are two solutions. There's negative 1 plus 2, which ends up being just 1, and there's negative one minus 2, which ends up being negative 3. So those are your two solutions to this quadratic. Usually, quadratics will give you two solutions.

Now, 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, you can always pop it into the quadratic formula that will always give you your correct answers. Alright? 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 going to be x=-b±b2-4ac/2a, which in this case is just 2 times 1. So just to simplify a little bit, this is going to be x=2±4--12/2. And if you simplify this even further, what this turns out to be is it turns out to be x=-1±2, which is exactly what we got over here. Remember, this just means two things. It means negative 1 plus 2, which in this case is 1, negative 1 minus 2, which in this case is negative 3. 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 convert it to standard form, these things are the same.

Alright? So that's how to use quadratics and square roots to solve quadratics. Alright. So now that we know how to solve quadratics algebraically, a lot of times you're going to see quadratics graphed, and you'll have to identify some key information from graphed 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 going to be in vertex form because it's easy to plot them. And basically, it looks like a(x-h)2+k. The vertex, meaning the little points where the parabola opens up, is going to be the coordinate h,k. So in this case, we have -x-12+4. The coordinate of the vertex is actually just going to be 1,4. And because of this negative sign, it's actually going to look like a frowny face. So what we can see here is that the coordinate 1,4 is going to be the sort of top of our parabola. It's going to be the maximum point over here, and it's basically going to go, like, looking like this. In order to get those points, we're going to have to solve for the x and y intercepts. And basically, to do that, you're just going to solve for when y is equal to 0 for the x. And then for the y intercept, you're going to set x equal to 0 and solve for that variable over there. So I'm just going to actually go ahead and do this for you. It's actually just the of the quadratics that we solved, up above. And so what this is going to look like, it's going to be like negative 13. That's the point where it crosses 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 0, so you basically just set this equal to 0 and solve. What you're going to see is that this quadratic hits the y intercept or the y axis at the point 3. So once you solved all your points, you're going to connect them with a smooth curve just to see what your parabola looks like. We 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 going to 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 1. And then as you're going left to right, it starts decreasing everywhere after positive one. Alright? So that's how to graph and understand key information from quadratic equations.

Proportional Reasoning

So very commonly throughout all of physics, you'll see questions that ask you how a 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 are 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 = 2x. Basically, just go ahead and plug in a bunch of values. So if you have x = 2, you'll get y = 4; if x = 10, y = 20, and so on. 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 do the y values. As x gets smaller, so do the y values. These things are sort of directly correlated.

Let's take a look at another example. We have y = \frac{1}{x}. Let's just plug in a bunch of numbers. For x = 1, y = 1; for x = 2, y = \frac{1}{2}; for x = 3, y = \frac{1}{3}; and then y = \frac{1}{4}, y = \frac{1}{5} as x continues to increase. So these are actually fractions, and we can see here that the fractions actually get smaller with higher xs. These relationships are inversely proportional because as the x values increase, the y values decrease, and vice versa. As x values go down, the numbers get bigger for y. So these relationships 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 = ma, and these are actually two variables. We actually say that these are jointly proportional, and this is what happens here. So m and a could be any numbers, and we're just going to plug in a bunch of numbers here. But f = ma is one of the most important equations that you'll see in physics, and it's just a multiplication of two things. So for example, 5 \times 2 = 10, 4 \times 1 = 4, 3 \times 0 = 0, and so forth. There's a little bit of a pattern that emerges here because as m gets bigger and a gets bigger, we can see that f gets bigger. And then as m and a both get smaller, so does f. But there's also another kind of problem which you're going to see, which is that because it's two variables that are multiplied together, you can also have situations where you want a constant f. So for example, I want f to just be 20. There are actually just a lot of different combinations of m and a that will make that work. So for example, 1 \times 20 = 20, 2 \times 10 = 20, and 4 \times 5 = 20. And so what you're going to see here is that for constant f's, as the m gets bigger over here, the a actually has to get smaller. So as m gets larger, a has to be smaller in order to compensate so that they both make f = 20, and also the opposite happens. As m gets smaller over here, the a actually has to get bigger, so these relationships are also 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.

Trigonometry

So throughout most of physics, we're going to be talking about sines, cosines, and tangents. It's going to 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 side sine, cosines, and tangents basically just relate all those three things together. Always just remember SOHCAHTOA. That's probably something you've heard before. It basically just helps you understand which things are getting divided. So for sine, it's going to be opposite over hypotenuse. That's the 'so' part. So in this case, the sine of this angle over here is going to be the opposite side divided by the hypotenuse. It's 35. So the cosine is going to be the adjacent side over our hypotenuse. That's the 'kah' part of SOH CAH TOA. And so in this case, the cosine of this is going to be the adjacent side over the hypotenuse, that's 45. And then the tangent is actually going to be the opposite over the adjacent side. That's the 'TOA' parts. And so in this case, the tangent of this angle is going to be the, sorry, the opposite side divided by the adjacent side. So this is going to be 34. Alright? So, that's how to use SOHCAHTOA. Some of the other equations that you might have to know are just how to get the opposite side, which is going to be the hypotenuse times sine or cosine. And some other helpful formulas are going to be, you know, things like the Pythagorean theorem or just this, sort of, like, basic identity of sines and cosines where sine squared and cosine squared is just 1. And then also just this one over here where tangent is sine over cosine. All these things are going to be very, very helpful for you to understand as you get into physics. Alright.

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 0. It's going to be really helpful to sort of memorize them because we'll be using them a lot in physics. So we're going to go over them really quickly here. So remember that the unit circle is really just a circle with a radius of 1, 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 hypotenuses of those triangles will always have a value of 1. And so, for example, this triangle over here has an angle of 30. This one has 45. This one has 60, and the opposite and the adjacent sides will always have different values, depending on what those angles are. I've got a table here that kinda summarizes this. So really quickly here, for 90 degrees, you've got and the tangent actually just doesn't exist. But, basically, what you're also going to see here is that as you go down on this table for sine values, it's actually, like, sort of going up on the cosine values, and these are things that are almost like their exact mirror opposites of each other. So really, these are the only ones that you actually have to memorize because the tangent, remember, is always just equal to sine 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 what the tangent value is. Right? So I'm not going to go through them. You can just look at them, just sort of make sure that you understand and memorize these things. The only other thing I have to point out is that these things will act

Derivatives & Integrals

Now if you happen to be taking a calculus-based physics course, there are a few extra calculus concepts you'll need to know. 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.

Graphically, what the derivative represents is just the instantaneous rate of change or the slope of a tangent line at a certain point. So if I have a parabola like this, and if I wanted the slope of a tangent line that's a line that's tangent to this point, I'm basically going to draw a line that touches the graph only once. That is what a tangent line means. 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 derivative is.

Mathematically, what happens is that in physics, there are going to be common types of functions that you're going to see, and we're going to have to mathematically calculate the derivatives. We're going to use these following rules: for a constant function like f(x)=3, it looks like a perfectly flat line. Derivatives of constant functions are always 0, as a flat line always has a slope of 0, which is what the derivative represents. Now if you have something like a constant times an x value, like, for example, -2x, the derivative is always just whatever the coefficient is, so in this case, it's -2.

To take derivatives using the power rule: if you have a variable like x raised to some power, you drop that exponent down in front of the x, so n×x, and then subtract the exponent by 1. For example, with x2, we make it 2x. For a polynomial, such as a combination of multiple functions like x2+3x, you will take derivatives of each of the terms independently.

Now that we've reviewed derivatives, occasionally in physics, we'll also have to use integrals, though they won't be super complicated. Graphically, the integral represents the area under a curve for a specific function. For example, the integral of a line graph under the curve from 0 to 4 is everything inside this big triangle. To approximate integrals, especially when doing graphs, you can add the areas of a bunch of rectangles or squares under the curve, or count a bunch of boxes to estimate the area.

Mathematically, integrals are like the opposite of derivatives. Here's the general form of a definite integral. If you have a constant function, you attach an x to it, so 3 becomes 3x. For xn, there's a power rule for integrals: increase the power by 1, and divide by the new exponent, so x2 becomes x33. If you have a polynomial, like x2+2x, integrate each term separately.

That's just a brief review of integrals.