so do you think calculus is hard well what if I told you that anyone can understand the basics of calculus in just a few minutes now if you don't believe me stick around and I think this video May blow your mind if you are ever curious about advanced mathematics like calculus but before we get started let me quickly introduce myself my name is John and I have been teaching middle and high school math for decades and if you need help learning math come on over to my site TCM academy.com you can find a link to
that in the description below and if this video helps you out or if you just enjoy this content make sure to like And subscribe as that definitely helps me out all right now uh before we get started with this video you don't need to know algebra or geometry I'll explain everything so if all you know is basic mathematics I think you will understand the gist of what I'm going to be talking about all right so let's go ahead and learn a thing or two about calculus all right so let's start here and I got some
uh shapes right I'm like I got a circle I got a rectangle and a triangle and I said okay let's find the area okay how can I find the area well I could find the area of a circle by using this formula pi r 2 right so R is the radius it's the distance from the center to the edge but if I knew this information I could plug it into this formula and I could find area okay like oh all right yeah I remember that and here I have a rectangle this is the length and
this is the width so how could to find the area of this guy well again I can use this formula area equals length time width right you're like oh yep I remember that and here I have a triangle you're saying oh the area of a triangle is 1 12 base times height so this would be like the height right here and this would be the base so I could find the area of a circle a rectangle or triangle not a problem right so what's the secret though the secret is knowing the formula for those respective
shapes okay all right that's all fine and dandy but what happens when we encounter a situation like this you're like what's going on here what if I said find the area of this shape okay now this shape is kind of Ed by this little squiggly curve it has this little border and this border and this border so now what do we even call this shape it's not a it's not a triangle it's not a rectangle it's not a circle who knows maybe we'll call this thing a squiggly find the area of a squiggly and you
might be saying all right no problem um I know how to find the area of a circle I know how to find the area of a rectangle and a triangle and a few other things I just need to know the area of a squiggly okay what's the area or the formula for the area of a squiggly and I would tell come back and tell you I'm sorry there is no formula there is no formula like we were able to use formulas you know those nice formulas I just showed you to find area of these basic
shapes but what about this crazy little looking figure here well we're going to have to come out uh come at this at a different approach so the first thing that calculus does for us it allows us to find the precise exact area of crazy shapes any shape that you want you can find the area of um of it and this goes for volume as well all right so if I said find the volume of a cube all right so most of you might know how to do that or like a cylinder we also have formulas
for that but what if I had like some crazy look looking like like weird shape like this well we there is no formula we're going to have to use other mathematics and that would be calculus and that's going to be our superhero here in terms of being able to find the area of things like this now how does calculus work what's the main kind of concept you know terms of being able to uh find the area of something like this well here's how it works so if I said well just use what you know let's
just focus on the rectangle okay we know how to find the air of a rectangle maybe we could just find a good estimate of this little squiggly shape so you could say okay let me kind of draw one rectangle here another rectangle like this and kind of like some other rectangles like so you kind of see what I'm doing right I can just find the area of this rectangle this rectangle this one this one this one and add them all up and I would have a a decent estimate uh of the area of this little
squiggly figure okay so calculus is going to be uh conceptually using uh the area of these basic shapes precisely the rectangle okay we're going to use the rectangle but we're going to use it in a little bit different way than this but the main concept I'm trying to get you to understand is that we can get estimates of the area of anything we want by using things that we do know okay we do know how to find the area of these guys so I can use little little tiny rectangles little little things or I can
use bigger ones and I could just kind of uh you know kind of create that shape or estimate that uh shape roughly and then add up the area of all those little um little rectangles but we're going to use the rectangles more in this manner okay so we're going to kind of do it like this all right so we're going to add up rectangles kind of vertically okay for the purposes of calculus all right so you can see here these are all rectangles right now I have uh information down here I can get I get
these dimensions and I can get the height and I can get the width so all I would have to do now this is a better estimate than what I just showed you okay I would just have to add up this rectangle this rectangle this rectangle that one all these rectangles if I add up add up the area of all the little rectangles all right I will get a really good estimation of my little squiggly here now this is going this is where um calculus uh is kind of heading here right that we're developing or we're
looking at calculus and play okay you're like well so far you know I'm just seeing these rectangles well calculus in terms of solving the area of these crazy figures is effectively um using the concept of all these little tiny rectangles so my question to you is this what if I said hey good job you give me a pretty good estimation of this squiggly here but I want a more precise answer and you might be like oh you want a better answer than that that was a lot of work right that was like 1 2 3
4 5 6 7 8 nine 10 whatever how many little rectangles so and you had to add them all up and you're like all right listen you want a better estimation okay fine let's get a better estimation how could I get a better more precise a super precise estimation of the squigglies well you guessed it we can use skinnier little tiny rectangles okay now I'm not going to do this whole figure but you could see here what I'm doing is I'm using real super skinny thin little rectangles okay they're still rectangles and I still know
how to find the area of a rectangle all right nothing's changed but now obviously you can see I'm going to have a much better estimation of the area so um I know how to find the area of of a rectangle again right remember it's the length times the width and I can get all that information here but it's a lot more work obviously right I'm going to have to um add up a whole bunch of skinny little tiny rectangles but take a look at what's going on I mean this is getting pretty nice right I
mean like oh yeah this is going to be a pretty good estimate of the area of this squiggly uh shape okay so what's the lesson uh learned here right if I want to get a more accurate uh area Okay of a particular shape I want to use skinnier rectangles okay now what would be like the ultimate skinniest rectangle well it would be like almost infinitely small right it's super super tiny uh super skinny but if I was able to just add up all these little infinite little strips these little tinier like rectangles underneath this I
would to get the precise the precise um area of this figure and this is ladies and gentlemen the main concept of one of the uh biggest problems that calculus uh solves for us is finding the area underneath these little curves okay something that's bounded like so and so now when we take a look at uh this little calculus stuff right I'm going to go ahead and say 0 to 10 and you'll see what this means here okay now I'm this is actual calculus symbology all right you're like all right let's let's understand this what is
this saying well what this is saying in calculus is the following this little line right here this curve okay this right here can be described it can be described let me erase this uh with something called a function it's just a rule okay it's a mathematical uh description of this curve okay it says okay if you follow this rule you'll get this curve so in calculus what we're trying to do you see our little squiggly starts from zero and ends at 10 okay so in calculus I can say uh to find the area of this
squiggly start from zero go to 10 okay and we're going to be describing or we're going want to find the area or we want to add up all the strips underneath this uh function okay underneath this curve all right don't worry about this little notation that's something you know it's not trivial but we just have to write this okay but this little long thing right here this is called an integral all right and basically it's a long it means sum okay and effectively it means it's telling us add up add up from zero start at
zero go all way to 10 all these little tiny infinitely tiny um rectangles underneath this function okay that is calculus notation that's what that is saying and there's another notation that kind of goes along with that it's called a sigma notation but I don't want to kind of get into it um but anyways that's calculus this is it okay something like this is saying we can go ahead and find the area underneath curves on a graph right or we just need to know the function right we need to know the function and we need to
know where we want to uh uh start and where you want to stop so for example if I wanted to find the area between like five and 10 okay so this area right here that I highlighted so I would just go over here and I would write this differently okay I would put a five right here and a 10 right there and there you go now how we actually calculate uh this um mathematically that's a little bit more involved not much more involved but uh it's something that you can't handle if we I wanted to
make this video a little bit longer I could teach you how to we uh calculate this okay but I just want you to understand the basic concept of calculus this notation it solves up finding the area of any crazy shape that you want okay so if you change the function you got a different kind of curve or something like that or whatever or something like this doesn't make a difference we can find the area of anything we want area and volume using calculus right and we don't need a formula because obviously we don't have uh
direct formula uh for things like formulas for things like this okay so that is the first problem calculus solves is like the area problem okay let's get to the second problem before we continue on it would really mean a lot to me if you hit that subscribe button now the reason I want more subscribers is basically I look at everybody that subscribes to my channel as a new student and as a math teacher that makes me very happy so uh the best way to support this Channel and what I do is to Simply hit that
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find the links to all of this in the description below and by the way this is called integration uh using the integral all right that little fancy thing now remember this right here this little DX all right this is going to be the second kind of problem that is a derivative okay so the derivative is a little bit more a little bit more uh I want to say difficult to explain but hopefully this example Explains It All right so the derivative okay so here we have a car now the car is accelerating okay this car
is taking off here's my little car it's going faster it's going right so it starts at Z miles hour and it's um going to go uh increase its speed all the way up to 80 M hour let's say it does this in 10 seconds okay so this is an accelerating car accelerating car so if I asked you if I said all right from 0 miles hour 80 miles hour and it does this in 10 seconds said what how fast is this car going at 3 seconds okay at 3 seconds into its little acceleration how fast
what's how fast is it going okay so you could say well it's 10 seconds and you might use the uh formula rate time time equals distance but precisely at 3 seconds if it started at zero it's obviously you know less than 80 you might say I don't know maybe at 3 seconds it's going oh I don't know let's say 48 milph well how do we how can we calculate that it's if you think about it the car is continuously increasing its speed okay there's you can't take the distance it's traveled right and then divided by
its uh speed okay so rate times time equals distance this rate this speed okay that's a fixed um that's a fixed veloc vity all right in other words that's if this car was going 80 M hour down the highway let's say you know you're on the freeway and it's going 80 or the turnpike whatever may be located if it's going 80 miles hour you know all the way down the highway then it's easy to uh to obviously it's going to be going 80 miles per hour the entire time but it's easy to calculate you know
rate and distance problems because the velocity is the same but when we're accelerating how fast the car is going in this precise moment in time that's a whole different ball G okay we need calculus to be able to answer that question for us so I might ask you well what about 5.12 seconds okay into this car's acceleration how fast was it going how fast was it going okay in that moment precise moment in time well in order and this is an accelerating car not a car that's going at a constant speed so these type of
problems figure out the answer to these instantaneous um problems like get a snapshot of what's going on at that precise moment requires calculus to be able to answer a question like that we can't use um uh average uh speed and time and distance okay that doesn't work in this kind of scenario let's see if I can give you another example and maybe make this a little bit better so let's say I had like well let's say I'm doing an experiment say this is the side of a building building and I drop like a baseball or
something right and I'm doing it safely there's no one down here and when I when I let go of the baseball it's at zero miles per hour Okay now what's happening with the baseball well it's picking up speed picking up speed it's going faster faster faster until it hits the ground right let's say this is 120 ft and I don't know maybe the baseball ends up going 120 uh let's see something different Maybe ends up going I don't know 5 miles hour by the time it hits the ground now and it took oh let's say
two seconds to do that okay so I'm just totally making up numbers here but just to try to get the idea so if I let the baseball go I didn't throw I just let it go it starts to it's starting to accelerate right I'm saying well how fast is a baseball going at 78 seconds all right how fast is it going at this moment in time this moment in time this moment in time okay well again it's picking up speed it's accelerating but I want to know it's instantaneous velocity at this these little snapshots this
requires calculus okay and this requires a concept of something called the derivative okay the derivative and in calculus a derivative looks like that all right so calculus solves two big problems for us right there's it does so much more obviously but the two big problems you can think about is instantaneous velocity questions like this okay if something's accelerating we can use calculus to answer these uh questions right that's the first thing the second thing is the area problem okay we can find the area of all kinds of crazy shapes anything we want and Beyond the
area we can find the area volume Etc and of course this calculus does even more than that but if you didn't know anything about calculus okay and you understand this basic kind of of math and the uh you know the question about the car then you got a pretty good feel for what you the value of calculus uh you know does for us I hope I don't know where you're at in your math education but you know I would certainly encourage anyone out there everyone out there to explore calculus uh I know a lot of
you like I'm going to avoid it as much as I possibly can but if you could ever take this class you know pressure free you know and just enjoy the concepts of it yeah there's a lot of challenging uh things to it you know there's a lot of rules and other other stuff I'm not trying to minimize I'm not saying it's not a challenging Math course but the beauty of it is that it can solve such just awesome real world problems for us okay so that's it um again um if you need help with math
more instructional full help you know where to go just F follow those links in the description of this video also if you're new to my uh YouTube channel I'm posting stuff all the time so hopefully you consider subscribing in and of course if you like this video please smash that like button all right so with that being said I definitely wish you all the best in your mathematics Adventures thank you for your time and have a great day
