okay understanding calculus in 10 minutes okay so that's the topic of this video now notice the the video isn't saying or my topic isn't um or title is learn calculus in 10 minutes or master calculus or totally you know understand it completely just basically understanding calculus 10 minutes like understanding what it is you know that's what the whole point of this and really My kind of main goal here is to kind of maybe demystify what calculus is and lessen the intimidation factor. Because there's a lot of students out there that say, oh, I could never do calculus. It's too complicated, etc. You know, there is a lot to learn in this topic, but hopefully you'll kind of see the big picture here. But let me just before we get into it real fast, let's just talk about where you would take calculus as a math student.

So generally in high school. OK. So in high school for most students you start off with Algebra 1 then you go to Geometry you go to Algebra 2 And then you go to Pre-Calculus in your last year of high school now some students Accelerate this and they'll actually end up in Calculus in their last year now when you go to college I would say maybe about 70% of the majors Degrees in college right so at the university level many of them Maybe well definitely over half are going to require you to at least have one semester of calculus So we don't take it in high school.

You'll you'll see a little bit of it in college now If you're some sort of science or technical major or finance or whatnot, you'll probably take even more of it So it's probably going to be in a lot of your futures If you're going going the college route, okay, so that's just basically where where it lies in the spectrum of learning all right calculus in 10 minutes let's get right to it so calculus basically helps us figure out kind of two problems two kind of things in math i'm going to start with one that's kind of easy that would be like the area and volume problem okay so i'm just going to sketch out here real quick so this is like a rectangle right So if I said find the area of this rectangle, we have a formula for this, if you recall. So it would be like length times width, right? So the area of this rectangle is going to be the length times the width. No problem, okay? So we know this because we were given a formula.

Let's do a circle, all right? What if we had a circle? And this is not a perfect circle, obviously. But if I said find the area of this circle, Hopefully some of you out there, or maybe most of you, remember there's a particular formula for this.

Okay, it's pi r squared, r being the radius, which is like from the center out this distance. And pi is a number of 3.14 approximately. So I can use this formula to figure out the area of this circle. And for this rectangle, I can use this formula.

Now, let's just do one more example. What if I had a triangle? Okay, so again if I wanted to know the area this would be One half base times height right? So this would be the base The triangle this would be the height.

Okay, so Formulas help us find the area and by the way, this goes for volume as well so these formulas come in handy with these kind of basic figures like triangles and You know circles and etc So let's take a look at where calculus helps us out, okay? Where it's a tremendous value. And by the way, I can't overstate the importance of calculus in mathematics, engineering, science.

I mean, it's huge. Really is as powerful as you may think it is, probably more so. All right, so what if we had some crazy figure like this, for example? I'm just trying to draw something, okay?

so what if I said find the area of this particular object or this figure well probably you're gonna be looking for a formula right you're gonna be okay yeah I could find the area of this just give me the formula well guess what there is no formula for something like this this is very challenging okay so calculus helps us figure out the area of a crazy-looking figures like this okay and volume they don't have to be as You know, uh... you know abstract as this but but this is what calculus can do for us this is the power of it because without calculus we would have to go through and just kind of like maybe try to estimate the area of this particular thing but calculus can give us the exact area okay now let's suppose that i took this um this figure and let's let's kind of put this like on a on an axis okay and i circled it around spun it around So maybe it's kind of doing this business. I don't know what it would exactly look like, but it might look like an object, maybe something like this, kind of maybe sketching this out.

And then it looked like it would have some sort of hole in the middle, right? Maybe like this type of deal. Very difficult to kind of maybe imagine, but you can see if you did have this. Let's kind of fill this in a little bit.

All right. And then you just kind of rotated this around. You would create some sort of weird looking figure like this. So this figure, you would be looking for the volume, right?

Like how much water would be able to fill this particular, you know, let's say it's a maybe it's a vase or something you could you could put in. OK, once again, calculus will tell us exactly how to do this. And. You may think, wow, you must have to go through all kinds of crazy advanced calculations in order to do that.

And surprisingly, it's really not as bad as you think. Once you understand the steps, you can do it. But you can see the power of being able to figure this stuff out. Let's go ahead and take a quick look, a real basic example on how we will find area. and volume in calculus.

All right. So maybe some of you out there have already taken algebra and whatnot. I'm just making a real quick xy-axis.

All right. Let's kind of maybe draw a little figure like so. OK.

All right. So let's suppose I wanted to find the area from here. to here underneath this curve all right let's say this area right here all right let's say this part underneath this curve i wanted to find area so that that shape if you look at it there's it's kind of curves on top right it goes down so you would be asked to find the area of of this object okay once again you're looking for formula there is no formula that exists now in calculus or in mathematics in general okay what we have When we have curves and things, we have little things called functions, all right? They basically just describe the curve. So this curve could be described by this, x squared, okay?

You don't really need to know that right now because that's not the purpose of this video, but this little rule right here just describes this curve. This is the rule that tells us that the shape of this is described by this particular rule, okay? we call these things functions so as long as you have this uh this function or the description of what of this uh curve okay then we can use calculus to find the area so let's say on our little graph this is let's say one this is two let's say this is three four and five okay so the way we would do this is we would go ahead and and have this little crazy symbol okay you might have seen this before It's called the elongated s, but it basically means the sum.

Okay, and I don't want to get too far into it Because that's not the purpose of this video, but we're gonna say is hey we want to find the area Underneath this particular curve and and I want you to start from 2 and go to 5 All right, so that's how we write that. Okay, then we put a little tiny little DX here That will be kind of like in my part 2 This video here, I'm gonna get to that in a second. Okay, but but this is the setup All right to find the area underneath this curve. Okay, you're saying well, yeah, that looks pretty complicated.

That's fine but really in the The steps to actually find the area are not complex at all What we do in calculus once we write this out this long thing You can kind of think of this long symbol this elongated s is fine find the area like find me the area underneath this particular curve all right that's all it means this mean this is from the left from the left side of the curve and this is to the right side Not difficult, right? This is the description of the curve. Once again, if we're given it, not difficult. OK, so what we do now, OK, is there's a little rule that says take this number, this exponent and this power. You just add one to it.

OK, so we're going to keep that same variable, add one. So two plus one is what? Three. Pretty simple.

Then whatever that result is, we're going to divide by that number. OK. so all we're going to do is take this right here and we're going to use this to find the area the way we do that is the following okay we're going to subtract we're going to use this thing all right and we're going to subtract the following okay we're going to plug in five over here okay we're going to start from the bigger side so we'll plug in five cubed right from our right side and i'm not going to get too far into it then we're going to plug in from the left side like so and when we do this little calculation okay we're going to actually get the area so it's really not difficult in terms of the uh you know mathematics involved i mean if you take in basic maybe even pre-algebra you know middle school math you could figure this out right i would just obviously i'm telling you the steps now what gets calculus what makes calculus more complicated for people is that these curved descriptions these little um curves and these functions they you can get more complicated okay so when these get more complicated you have to learn more rules but it's really a matter of just learning the rules to to get to this part okay so that's the big deal this is like part one of uh of calculus okay now calculus like i said there's like two big problems it solves for us. So that's the first problem, area and volume. Let me erase this and then we'll get into the second cool problem. All right.

Right. So the second thing calculus does for us, and let me go ahead and draw a little xy plot here, is calculus helps us determine How steep, okay, something is. What its slope is, all right?

And what do I mean by this? Let's suppose from here to here, I wanted to know generally how steep this line is going from these two points. So I could say, well, it's kind of going this direction. Now, I'll get into why this is important here in a second. We want to know the direction of curves, okay?

We want to be able to get their steepness because they're always changing if you study this curve this curve is always changing this could be for example just imagine this can be anything you want it to be it could be population growth right over time all right let's think of it that way this could be a stock that's going up in time this can be maybe the rate of maybe cancer in a particular city that's going up in time so When we have what we call like rates of change and things are constantly kind of maybe growing or they're fluctuating, we want to be able to estimate between time periods or between blocks of the curve, it's the steepness. Okay, because this gives us an indication of where things are going. You can tell here the steepness is different here versus where, let's say at this point in this particular graph.

Okay, so the steepness is always changing. all right we call this we call this steepness i kind of like to use the word that i'm using step but it's really uh uh steepness okay there my word is but actually the technical word is slope all right that's probably a better word for you anyways so it's the slope of the curve like hey now how where is it kind of so what's its actual slope now you can figure out the slope if you have two points that are on the curve it's pretty easy okay because Effectively what you're going to do is just determine like this these two points I can just figure out the rise and that's just this amount right here over the run Okay, so the slope is defined by the rise over the run. So if I can get those two measurements, it's no big deal I can just figure it out and there there I go. But here's here's where Calculus really becomes powerful.

Okay, what happens? when we want to know the slope of a curve. And let me just, I'm going to draw a new curve here. to make this a bit more pronounced. OK, maybe something like this.

OK, so we can see that the slope here is kind of going. It's kind of doing this and then it's kind of like going this way. Then it's kind of going this way and then it's going down like so.

Right. So the slope of this curve is constantly changing. Now, what if I want to know? the exact slope of this curve at this point right here.

Okay, that point. Well, by definition, the slope, we need the rise and run. We actually need two points. Like here, I can kind of estimate it, right?

If I take two points, it's kind of going in this direction. Okay, kind of something like that. But if you notice, if I put another point right here, it's kind of going like this. So, you know, I'm getting different estimates. I don't want to estimate.

I want to know the exact slope right on one point of a curve. And you can think of this as the exact rate of change in one precise moment. So let's think of this as time.

And maybe this is population growth. I want to know the exact rate of population at this exact time. Maybe it's May 8th of whatever year. at what particular time you want to know exactly you don't want to know an estimate kind of goes back to our area of volume problem you can get estimates but not the exact answer well calculus helps us determine the exact exact answer because in calculus we can determine the slope now once again what we need is the function okay let's say and this for those of you out there this is not an actual um the actual function to this curve i'm just using this for for simplicity stakes but let's say i had this function described by this uh rule okay now remember we use these function descriptions as uh when we're trying to find an area of volume in calculus but here we have something called a derivative all right and the derivative basically is it's a rule okay that allows us to find the slope at any point Any exact point along the curve and that's a that symbol looks like this. It's also looks like so all right It's another symbol in calculus DX over dy and there's even other ones f prime etc But these crazy symbols are the derivatives.

So let's take a look how I would find the derivative of a function of 2x squared plus 2x plus 1 right so fun what we call we call this actually the first derivative So all we do is we multiply this exponent 2 times this coefficient, this number. So 2 times 2 is what? 4. We keep the x and then we subtract 1 from this 2. So that's just x to the first or just x.

Now we go to the next guy and we do the same thing. So this x here is actually a little 1. We don't write it, but there's a 1 up there. So 1 times 2 is 2. And then x here to the 0. So that's going to be just... x to the zero power is just one so this is our first derivative okay of this particular function this curve description and this is a rule to tell us the slope of the curve anywhere along the curve okay but you can see here the what i just did the mechanics of finding this are really quite easy once again it's a bunch of rules so i can use this i'm not going to get into this now but i can use this here this particular rule to easily find the uh the precise slope at this moment in time okay and this helps us solve tremendous problems in mathematics okay here let me um let's let's kind of maybe kind of first of all let me kind of clean this graph up let's say you wanted to know let's say we're doing some sort of scientific study actually we do the better curve and let's let's suppose this is maybe some sort of medicine or pharmaceutical drug that you're testing okay and you want to know like hey wow this is where it's decreasing you know the symptoms of you know that's a cancer you know or reducing you know blab bad blood cells in your body or whatnot you know here you know you're you're testing your testing your test and you want to know wow this this curve is telling me right in this area at this particular say dosage okay or or combination of what you're using is what works the best and minimizes the issue so you want to know exactly what this point okay so you wouldn't by finding the slope of the curve right here this is what allows us to answer these things precisely okay They're called like maximum and minimum problems.

So anyways I'm sure this video went over 10 minutes, but hopefully you got something out of it, right? This is the big deal with calculus. It is a big deal.

It's a huge deal, okay? But it's not impossible to learn. Even if you don't have a math background where you're like super strong in math and you struggle in math, you can get through calculus, right? You can definitely get through calculus.

But it does require you to study a lot of rules. And, you know, the best way to approach it, though, is to understand the value. of calculus so remember the two big things that we study calculus for is to help us with these area and volume problems that's huge okay we use the integral for that all right and then the other thing is to um is to find the slope okay call this the derivative so we use this symbol okay all right or this symbol y prime or this symbol there's multiple symbols they mean the same thing okay all right so hopefully you got something out of this video For those of you out there taking calculus know a lot about this And you're saying no well this is not technically correct or this and that well listen trust me I I'm I have a math back. I have a degree in math, so that's not the point I'm talking to people out there who you know don't have a clue about calculus or interested in it, but anyways Thanks for watching and if you like this video, please subscribe to my channel have a great day