Understand Calculus in 10 Minutes

okay understanding calculus in ten minutes okay so that's the topic of this video now notice the video is insane or my topic isn't our title is learn calculus in 10 minutes or master calculus or totally you know understand it completely just basically understanding calculus ten 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 there's a lot of students out there that's all I could never do calculus it's too complicated etc you know there's 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 a real fast let's just talk about where you would take calculus as a math student so generally in high school okay so in high school for most students you start off with Algebra one then you go to geometry you go to algebra two and then you go to precalculus 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 but a degrees in college right so at the university level many of them may be well definitely over half we're 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 count if you're some sort of science or technical major or finance whatnot you'll probably take even more of it so it's probably going to be and a lot of your futures if you're going going the college route okay so that's this basically where where it lies in the spectrum of learning all right calculus in ten minutes let's get right to it so calculus basically helps us figure out kind of two problems to kind of things in the 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 you 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 width no problem okay so we know this because we are given a formula let's do a circle all right what if we have a circle and this is on 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 M 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'm learning to know the area this would be one-half base times height right so this would be the base of 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 word calculus helps us out okay where it's the tremendous value and by the way I can't overstate the importance of calculus and mathematics engineering science I mean it's it's huge really is as powerful as you you know may think it is probably more so all right so what have 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 going to be looking for a formula right you're going to 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 crazy-looking figures like this okay and volume they don't have to be as you know 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 this figure and it's let's kind of put the SEC on a on axis okay and I circled it around spun it around so maybe it's kind of doing this business now I don't know what it would exactly look like but it might look like a an object maybe something like this kind of maybe sketching this out all right and then it looked like it would have some sort of hole in the middle right maybe like like this to have to deal very difficult to kind of maybe a magic but you can see if you did have this hmm it's kind of filled this in a little bit 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 oh maybe it's a vase or something you could you could put in okay once again calculus will tell us exactly how to do this and you may think wow this is it once you must have to go through all kinds of crazy advanced calculations in order to do that and surprisingly it's really not as as you think okay once you understand the steps you can do it but you can see the power of being able to figure this this stuff out let's go ahead and take a quick look real basic example on how we find area and volume in calculus alright so maybe some of you out there already taking algebra and whatnot I'm just making a real quick XY axis alright let's kind of maybe a little figure like so okay alright so let's suppose I wanted to find the area from here to here underneath this curve alright let's save this area right here all right let's say this part underneath this curve I'm wanting to find area so that that shape if you look at it there it kind of curves on top right it goes down so you would be asked to find the area 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 then 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 what the purpose of this video but this this little rule right here just described this curve this is the rule that tells us that this is the shape of this is described at this particular rule okay we call these things functions so as long as you have this this function or the description of what of this curve okay then we can use calculus to find the area so let's say on our little graph this is it's a 1 this is 2 let's say this is 3 4 & 5 ok so what would the way we would do this is we would go ahead and and have this little crazy symbol okay might seem as good for it's called the elongated s but it's basically means the Sun okay and I want to get too far into it because that's not the purpose of this video but we're going to say is a hey we want to find the area underneath this particular curve and and I want you to start from two and go to five 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 to this video here I'm gonna get to that in a second okay but but this is the set up 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 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 and elongated s is fine find the area like find me the area underneath this particular curve all right that's all it means this means 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 okay so what we do now okay is is a little rule that says take this number this exponent in this power we just add one to it okay 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 okay 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 you can kind of 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 and 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 you know mathematics involved I mean if you take in basic maybe pre-algebra you know middle school math you could figure this out right I would just say I'm telling you the steps now what gets calculus what makes calculus more complicated for people is that these curved descriptions these little curves than 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 matter just learning the rules to to get to this part okay so that's the big deal this is like part one of calculus okay now calculus like I said there's like two big proms it solves for us so that's the first problem area in volume then we 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 alright and what do I mean by this um 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 that 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 the rate of maybe cancer in a particular city that's going up a time so when these 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 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 alright we call this we call this steepness I kind of like to use the word as I'm using step but it's really a steepness okay there my word is but actually the technical word is slope all right that's probably better who are free anyways so it's the slope of the curve like hey now 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 at the end 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 I mean just I'm going to draw a new curve here to make this a bit more pronounced okay maybe something like this okay 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 the 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 I put another point right here it's kind of going like this so you know I'm getting different estimates but I don't want to estimate I want to know the exact slope right on one point of a curve okay and you can think of this as the exact rate of change in one precise moment all right so let's think of this as time okay and maybe this is population growth I want to know the exact rate of population at this exact time maybe it's a May eighth of what every year okay what a particular time you want to know exactly you don't want to know an estimate kind of goes back to our area blind palm 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 mean is the function okay let's say and this for those you're out there this is not an actual 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 rule okay now remember we use these function descriptions as when we're trying to find an area and 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 if any exact point along the curve and that's that symbol looks like this it's also looks like so all right this is another symbol in calculus DX over dy and there's even another one F prime etcetera 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 alright so far 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 for 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 we do the same thing so that's X here there's actually a little one we don't write it but there's a 1 up there so 1 times 2 is 2 and then X here to the zero so that's going to be just X to the 0 power is just 1 so this is our first derivative ok of this particular function this curve description and this is a rule to tell us the slope of the curve anywhere along the curve ok 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 precise slope at this moment of time ok and this helps us solve tremendous problems in mathematics ok here let me um let's let's kind of maybe kind of first of all to be kind of clean this graph up let's say you wanted to know that's how 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 or 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 let's say cancer you know or reducing you know blab bad blood cells in your body or whatnot you know here you know you're testing you're testing your test and you want to know wow this curve is telling me right in this area that 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 would want by finding the slope of the curve right here this is what allows us to answer these things precisely okay I call like maximum and minimum problems so anyway I'm sure this video went over ten minutes but hopefully you got something out of it right this is the big deal of calculus it is a big deal it's a huge deal okay but it's not 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 math you can get through calculus all 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 studied calculus for is to help us with these area and falling in problems it's a huge okay we use the integral for that all right and then the other thing is to is define 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 symbol doing the same thing okay all right so hopefully you got something out of this video for those who are out there taking calculus know a lot about this and you're saying oh well this is not technically correct or this Anette well listen trust me I die I'm a math back I'm 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 interests in it but anyways thanks for watching and if you like this video please subscribe to my channel have a great day

Channel: TabletClass

Views: 2,516,404

Rating: 4.6342416 out of 5

Keywords: calculus, learn calculus, introduction to calculus, calculus review, calculus help, intergral, derivative, math

Id: WjJ-kpgps1c

Length: 21min 58sec (1318 seconds)

Published: Fri May 19 2017