Calculus 1 Lecture 2.1: Introduction to the Derivative of a Function

two point one is called the derivative an ominous sounding title this thing should look familiar jeez I hope that looks familiar to you good because if it doesn't you haven't been here the last 45 minutes and that looks familiar we've used this for a few things we've used this to find the instantaneous rate of change we use it to find in stays velocity we've used it to find the slope of a curve at a point you're easier to find the equation of a tangent line by finding the slope of that curve then making that equation out of it so this is I'll just call this because it's what all those things mean this is the slope of the curve at a point so primitive point this thing which I have already told you stands for the slope right this thing says as you approach one single point so this is the slope of a curve that's f at a single point you got the idea what this idea represents is something called the derivative so what is the derivative there you go my job is done I won't touch of that I've taught you a derivative I just didn't tell you what's a derivative this thing is a rivet this idea is a true well here's the thing about it all right you're going to be taken lots of derivatives and like I told you before people in calculus that usually they get so involved in doing the computation they forget what you're actually doing what are you doing when you're finding a derivative what are you finding that's what you're doing what are you doing everybody you're finding that that's what a derivative represents this thing is a slope of a curve at a point that's the derivative function it's telling you how to find the slope of a curve at a point so we're taking derivatives that's all we're doing and there's a special notation for the derivative looks like this that's the function right that we're talking about here how you say go from a function to a derivative it's very easy I'll give you some more notation later on but you go okay this is a function if that's a function that's a drew the little apostrophe thing up it's not a 1 it's not a 1 it's a like that some people have a little - it's not a 1 the derivative of the function that's what it says the derivative of function or the slope of this function at any point that I want is the limit as H approaches 0 of the difference quotient we've already talked about and you know what in general we're not going to have a specific point so I'm going to raise the X of 0 I'm gonna say we're not going to do this at specific point anymore maybe we'll do that later what we care about is the derivative function we want the equation like we did here we want the equation for the slope of the curve at a point you would do that we're going to stop doing this for a specific point we might plug in some afterwards well I want the function itself because what you found out in the last example of a fear was if I told you to plug in more than one point that gets pretty cumbersome and tedious right if you do it for every single point that sucks we want to find out this thing the function and then be able to plug in any amount of time or any x value that I want that's better so I'm going to leave off the X of 0 signifying a specific fixed point I'm going to say just find me in the general function for the derivative the derivative function and that's that this is the derivative of f with respect to X the derivative of F with respect to X says that rid of that function a death taking X as our variable also you can say this as Prime like Optimus Prime cool but F Prime so this would be like the math for Jionni another of F prime F prime of X say that with me f prime of X that means the derivative of F with respect to X so f prime of X I'd say that let's see we get time for a couple examples would you like to see a couple more on how to do these derivatives now we get column derivatives they do imply that it is a instantaneous rate of change an instant ace bilasa T if your view of the pacific position curve the slope of curve at a point if you're just dealing with a basic function you want to see some examples let's do this thing okay so I want you and I want us to find the derivative of this function so that would be wait now do this squared find the derivative of this function and then I want you to find the equation of the tangent line to that function at a certain point can hit that's a great test question that's code word for word that's most likely gonna be on your test pretty close I'm not very original Scott so I figured you can do it you can show me anyway the derivative of that function and then find the equation tangent line so first thing I want to do I want you to find f prime of X now this is nothing new because all this is is the limit f of X plus h minus f of X all over H I showed you several times on how to do that yeah I showed you that I need you to label out f of X plus h and label out f of X and if you can do those things and you can substitute them into those expressions and you can work the algebra this actually works itself not too bad the calculus is very easy now it's really you just have to manipulate tell me what is f of X in this case everybody what is it yes that's definite f of X now I need to be able to find f of X plus h X plus h that means X plus 8 should show up in parentheses in your problem tell me what it is - ok I like it - then one okay definitely put the C's and do I have a -3 or not let's stop right there real quick pause make sure you're okay are you okay I'm getting f of X and the X f of X plus h in our graph no yes let's try this again if you are no then don't raise your hand if you're a yes then raise your hand are you okay getting these two things that's how I check okay over here okay cool let's take those expressions plug in so instead of f of X plus h I got two X plus h squared minus three - here's my f of X plus h I've already got that now I want to subtract f of X which is two x squared minus three tell me the gigantic flaw that I have on my board right now very good yeah that's got to be there because that's going to change your sign you're subtracting that entire at the back set that will expression so we'll keep on going here we're going to distribute we'll get to X plus h squared is x squared plus 2xh plus h squared minus three minus two x squared plus three sure the plus three is coming from just ripping that negative out there that's going to give us that plus three all over H a couple more steps well we still got to distribute that to so we'll get our 2x squared plus 4x h plus 2h squared minus three minus two x squared plus three all orig now I know it looks nasty but is this really hard to do I really it's mostly just algebra but supposed to combine like terms and descriptive distribution just keeping your signs correct at this point they're not even hard limits shoot I've given you some really hard limits in this class haven't I trig functions all sorts of crap like that this is easy compared to those this you just got a factor stuff and that's not that at all so we got a limit as H goes to zero if I combine all the stuff a lot of things should and must cross out and otherwise you basically here's the point if you don't cross out everything with an 8 if everything that doesn't have an H in it here's not your work out for you because you're going to have to factor this H out at some point and be able to work with it in some way so the X Squared's are gone the threes are gone I get 4x h plus 2h squared all over age if we continue of course you will notice I'm still writing limit because we haven't taken the limit yet when we factor out the H we get 4x plus 2h all over H tell me what happens now this is a good part that's what we wanted to have happen can you insert that H equals zero yeah we let H go to zero that means this term goes to zero as well that means we still get a 4x filling it with that one there's an assess what you just done you found the derivative so if f of X equals 2x squared minus three F prime of X that's first derivative the first derivative of F with respect to X is 4x okay what did you just fun so this slope of what curve source Oulu tangent line sure at some point slope of which curve this curve does this change for other curves yes the derivative is different for every curve like what you found is the slope of f the slope of this curve at which point end point now here's a second for this question find the equation of the tangent line to f of X at 2/5 what two things do you need to make the equation of a line do you have a point points have to be given to you do you have a slope kind of you have a formula for the slope right get a formula so here's your slope and here's your point how do you find your slope okay I definitely need voice up on it okay great y minus y1 equals M X minus x1 okay great what is my y1 I need more participation this what's my x1 2 so would you agree I'm going to do Y minus 5 equals x minus 2 would agree with that the question is and this is what I asked you originally what goes there some people give me a number some people give me 4x listen if I want the equation of a tangent line I want the equation of tangent line specific to this point if you put 4x in here this gets nasty right this is nasty don't do that if I want the equation of tangent line I want y equals something X plus something I want an equation of a tangent line Angela it's a straight line but for us here that's what I'm talking about the sort of tangent lines are straight lines I don't want for X that was what I'm trying to say that's the formula for your slope that's not the slope itself that's the formula for slope at any point tell me what point am I talking about good what's the x-value for my point plugin - so the slope will be look at F prime of 2 F prime of 2 this says this it says this is the slope at the point x equals 2d do you see the notation slope that to slope of 2 is going to be 4 times 2 or 8 that is my slope that's how you find your slope you deal with your slope function slope formula you just plug in your x value how many will solve that feel okay with that one cuz I have a lot of people get stuck here they go well what do I do I just have the derivative what do I do the derivative if they sound like that but you plug in your x value that's how you find the slope this is only a formula for the slope you got to use it to find slope the slope is at well your x value - in this case if it was a different point it'd be a different X right I could just say at x equals 3 you plug in 3 just plug in the value that will give yourself after this very Elementary add 5 and not too long at a time we've just found the equation of a tangent line that's the line that will touch this specific curve at one point and that point is 2/5 so we found that laser lines can touch that curve at 2pi but you have to realize here today well we put a lot of things together damn glad you were here so from last time we're still working on derivatives and we've already done the derivative of 2 x squared minus 3 let's look at fine driven something a little bit more complicated remember personally what the derivative does for you sure slope tangent line or in other words the slope of a curve at a point right and we've gone past the the stage of just buying at a specific point what we're doing is we're finding the formula for the slope that way we can plug in any point that we want if I the slope at any point remember that that was the last thing we were working on so let's go ahead let's find the derivative which is so conservative point you remember how to write the derivative of f of X by the way good F prime of X that looked like a little - so find F prime of X or f of X equals two X cubed minus X basically here's a lot of derivatives set instead then you can find the derivative or the slope of a tangent line to occur but slope of a curve at a point at any point by taking a limit our fun limits that we have fortunate these are a little bit easier than some trig limits we know as H approaches 0 f of X plus h minus f of X all over H that's the idea if we can fill those things out and work with it that we'll have our function on will have a derivative of what I told you to do is find f of X plus h and find f of X before you fill it out find it off to the side so my question is can you tell me what is f of X in our situation right here let's start easy good are you okay with the two x cubed minus x where that's coming from so let's just start our function that's all gonna list down now we've got to go ahead and find the f of X plus h f of X plus h says do I just add H at the very end is that the appropriate thing to do no it's like X plus h it says you can't separate that thing so tell me what is f of X plus h ok good so even where we see the X we're going to insert that entire explanation remember doing that also so it'll be x + H to the O cubed - so it does have to be X plus h in that as well so everywhere we saw an X we're now putting X plus a 2 in parentheses ok so hands how many will feel ok with that so far now we're about ready to go ahead and substitute this into our our function just make sure you do one thing remember that what this says is you're subtracting f of X right subtracting the whole thing so if you're subtracting that whole thing what do I need along with that 2x cubed minus X I need something with it right because that minus is going to distribute change of signs are you with me on that if I do it just like this this is good this great this bad expert bad why is that bad yeah this says I'm subtracting 2x cubed but I'm not subtracting the minus X this would say the appropriate thing for us that would say do you see the difference there what would you see why we need that we're subtracting the entire function f of X subtracting the entire function f of X and then putting over H now we're going to do some fancy algebra you need to know how to distribute X plus h to the third power if you ever heard of Pascal's triangle this will work for you have you run faster before that's the way that you can expand this to term or binomial expansion to any power that you want I guarantee and this is going to be without doing much work because you need to foil it out right sometimes I will get into it I'll do the X cubed plus 3x squared H plus 3x H squared plus base to third power guarantee is like that if you want to learn how to do that with high stats guys trying to come and see me or look that up it gives you the coefficients 1 3 3 1 for every row of Pascal's triangle that's how you do that minus 2x cubed plus X Oh so I've just done this little part that's all I've done what's that going to be when I'm done with that okay and then minus two X cubed and once you see where the plus X is coming from the very end minus and negative we're going to be adding that ultimately okay last a little bit if we distribute that 2 X cubed we'll do that next time we'll go ahead and stop there I'll talk about how to finish this up later okay so we're working on a problem we're trying to find the derivative which is what's driven via thank you one of you where's my boss now yeah slope of a curve at a point folks is what a derivative is what you need to know so every time I asked you that question what's a derivative it is a slope of a curve at a point that is what you're finding here you get me we're finding the function of the slope right now yes we're inviting in general so we've made it down this far we have our difference quotient and that means the slope difference quotient basically just gives you the slope at 22 points we're just allowing those two points to come really really close together that's why the H goes to 0 says that distance between the points is shrinking it's collapsing to 0 and that's going to give us a slope not at two points but at one point which is in fact a tangent line we've done the difference quotient f of X plus h there you go f of X we plug that in we've worked it out and now we're going to see that everything except for terms that have an H of them hmm pretty much have to go away because if they don't you're not gonna be able to factor that H and you're not going to cancel out that H which is exactly what you want to do let's see if that happens here do all my terms that don't have h in them disappear jeez I hope so cause let's see does the 2x cubed go but yes gone how about the X yes yes that's everything that doesn't have an H what that means is that you should be able to factor out an H from the rest of it if you can factor out an H from the rest of it you're going to have H over H that's something we can simplify you get me on this so I'm going to work over here this gives us the limit H to 0 of would you mind if I factored out an H as we went you okay with that so I'm going to factor in H this is our many terms this negative H to H cubed 6h 6 X squared 6x squared H so I'm going to factor out the H and get six x squared plus six x do I still have an H here yeah okay good you need to see that plus two H squared - one Frigidaire fueler ever done so far with factorization cool reason why this works now you have the H over H as soon as you can factor you can simplify that out you can't do it before then but now we can simplify those nature's away are you now able to plug in H equals zero and get something out of it sure let's do it what happens do I still have a 6x squared and that doesn't have an H attached to it so 6x squared right there how about the plus 6x H that's zero why is it zero yeah you're really doing this 6x times zero because 8 is 0 plus 2 times 0 squared because well H is 0 how about the minus 1 goes to of that yes absolutely so this is going to be your first derivative called the first derivative 6x squared minus 1 okay now test you on what I said two seconds ago what do we just fine right exactly whatever you mumble you're right sure yeah we found this I think I think I've heard from like 10 of you maybe all of you if you're doing ESP or something we have the slope of this curve where whatever I want yeah wherever I want if I say at x equals 3 you can find it x equals 3 huh I'm not going to I'm not going to because but we could do it right if I said fine at x equals 3 how would you go about doing that you plug in 3 to this that's going to give you the slope just the slope are you with me you plug in 3 to this to actually find the point you follow me on that one so if I ask you for at find the equation at x equals 3 the slope is going to be f prime of 3 the point is going to be 3 comma whatever you get when you plug in that point in our case I plug in 3 tell me what I get when I plug in 3 51 the slope would be when I plug in 3 here if I do that I'm going to get 5351 is right right that would be my do you see how we're getting the quickly through that do you see where we're getting the point from the point comes from my original function that that's what gives you points on your graph the slope comes from your derivative function so you plug that in we find the slope you can now use point-slope and figure out that equation very easily it'll be a straight line all your tangent lines will be straight bottoms how many will feel okay with this example at least get as far as a derivative all right would you like a couple more examples yay fantastic what a good Friday all right more tables first example is this you're going to find something up kind of an interstate here now we're actually do a little critical thinking about this could we use this whole shebang this whole in a thing to find the slope of that because this stands for slope right could we do that to find the slope however look at that what is that yeah that's one of these once one of those a slope intercept they give you a linear equation could you look at this and tell me what my slope is here what's my slope then I guarantee you if you were able to if you were doing that your slope is three no matter what because that's a constant right that slope doesn't change at all that's a straight line without doing any work at all I know that the slope or my first derivative is going to be three from that function because the slope is 3 what I'll write out for here for you here is if you have any slope-intercept form of a line don't waste your time doing that you need to know what you're doing by finding a derivative what are you doing by finding a derivative you're finding the slope of a curve at a point right what's the slope of this curve at any point because lines don't change slope of that curve is n no matter why so for for any y equals MX plus B slope equals M it's a constant it doesn't change it's a constant doesn't change at all so with any line that that's pretty easy we have to do that formula at all we just know that the slope of a line is that that M that number that coefficient of x which is pretty cool so the slope of a any straight line is just and that works for our derivatives as well okay let's deal this one that's I did want to show you that so if you got that on your homework and you like Lao you can do all this stuff now if you get a straight line just take the derivative and that's just additional slope it's a shortcut I guess but a really a shortcut yarding on the slope is I'm just telling you that derivative means slope right for real so if you know the slope is automatically that is your derivative don't waste time now how about this one here we were able to use this because the slope of the line is always constant is the slope of this thing constant is that straight line you know how that looks you'll do like a dance move yeah maybe like a disco will do all the Exponential's later I didn't do this goes I'm not that old that was my best disco impersonation anyway let's go ahead let's find the derivative and the equation for the tangent line at x equals 4 new tracks first thing we got to do of course well in order to find the derivative this is not easy like our lines alright we can't just look at the angle oh the slope is we don't know right we don't know so we're going to have to use the formula for the derivative for the slope so in other words we're going to have to fill out this pain why don't you try that now go ahead and give it a try remember how we do it is we want to find f of X and f of X plus h and non-separable X plus h separately and then plug them in go ahead try that shouldn't take you long for this one by the way you can do this one of two ways you can plug in a specific value for X if you'd like to what I prefer you to do is do the derivative in general so this is a secondary question okay and the equation of the tangent line I want to find the derivative first what we're most worried about right now is finding the function for the derivative do you get me on that so please if you if you plug in for just erase that put the X I want the formula for the derivative and then we'll be plugging in that point do you understand the difference there so we're finding that in general that way if I give you a different point like x equals 5 you don't the reach of all your work because that would that be length but as you've like 10 different tangent lines you don't want to do this 10 different times you wanna do it once and then just plug in 10 different numbers it's way quicker you see the difference that's what I want from so here for f of X plus h I'm going to have the square root of what square root of x plus h or x plus h hopefully like that f of X well that's a square root of x did you get those two things correct now we just have to substitute those into their respective places so this is going to give us the limit as H goes to 0 of the square root of x plus h minus square root of x all over H ah ah so all those limits that I gave you where you had square roots minus something else wasn't just arbitrary because now you're seeing some of the techniques that we use for our limits from the last couple chapters remember that how do you get rid of that how'd you get it so we did that are you going to distribute the top yes should be the bottom heck no you're trying to get rid of the H so this wasn't just useless practice we were doing it's why I made such a point of this when I went over a couple things with you that these parentheses you're about to put on here are meaningless never be talking about those parentheses oh please not your head did you ever talk about parentheses good so I hopefully wasn't just talking to a wall and we need parentheses because we're multiplying this factor times that entire expression which lets us ultimately cross us up out simplified okay now you can help me out with this I know I still have a limit as H goes to zero when I distribute the numerator which I do have to do because listen listen what you're trying to do is get rid of the radicals up here right wherever you're trying to get rid of the radicals that's what you distribute should i distribute the H no no I'm not trying to remove any radicals from right there so whatever you're trying to rationalize that's what you distribute what if you're not trying to rationalize that's what you're going to try to cross out ultimately if you were to distribute this now you'd have to refactor it later you don't want to do that you won't leave it the way it is so you can cross the H out that's your whole goal across the H out so we're going to get help me out here okay I get the X plus h I got the X because the square root times itself gives you the radicand that's the part inside the square root we're going to get plus and minus the same exact thing you see it what's left - that's right all over don't repeat that anything nice that happens they have to they have to only thing you can have left up there is something with an exit you can factor out across that itself otherwise this one falls apart so those X's are gone okay foot out I like the HS going away what do we have remaining good we don't just have this Cartwright's obits under one so it becomes a one are you okay to plug in the H wait a second H is so in the denominator is that okay sure because when I plug it in ice I get X plus zero that's fine that's still X this is something so here I have after I plug in the H I stop writing the limit I have one over the square root of x plus zero that's square root of x plus the square root of x can you make it that for ya can't put this together get to read X 1 over 2 root X show hands how many people feel okay with our derivative or with that good remember when you add root X plus root X you'll get X you get 2 root X okay that's not what its rid of it's multiplication that does now we're almost done at this point we've got to use slope intercept I'm sorry point slope we need a point and slope to find the equation of any line so I mean she takes point slope do we have a point written we have a X how do you get the Y because right now we need we need the slope which is I don't know and we need a point I can tell you the x coordinate of X corner for the point is or or okay how do you find the next part of that we plug it in here what's this again that's slope good what gives you points you're functioning where am I there that gives you points plug it in here this will give you points so plug in for here what are you so this is the point four two are you okay with that remember our function well it's simply the square root of x right square root of x says if I plug in 4 I'm going to get out to that gives me the point now this thing should give you the slope don't plug this into slope intercept point slope form that'd be ridiculous you want a number to plug in so if I were to plug in 4 here what's that going to give me clean ones 1/4 very good because square root of 4 is 2 2 times 2 is 4 that's 1 over 4 so f prime of oh it says that prime of X even that's kind of nice what's your ex your ex is 4 plug that in that's 1 over 2 times the square root of 4 ok I need to know are you ok on finding the slope not sure if you are and the point you see where those things are coming from hey plug him in and give me a slope intercept go ahead and do that hip in there ready I'm making nice points but in that sky I am today just today one day that could same old okay-y minus 10 equals 1/4 X minus 4 distribute y minus 2 equals 1/4 X minus 1 add 2 y equals 1/4 X plus 1 that's it that's it what's interesting what's kind of neat about what we're doing here this curve is kind of this curve is kind of crazy right in this it does this it's half of a parabola on its side well that line will do if you were to graph it you grab both of them on your graphing calculator that's going to make the half parabola on its side this is going to make a straight line intersecting the y axis at 1 the slope of 1/4 it will guarantee to intersect this curve now intersect it at exactly x equals 4 at the point 4 2 at one point one point only that's kind of neat right so could you tell me what's the slope of this curve at the point 4 it's 1/4 what's the equation for the tangent line it's right there so cup 1/4 I'm gonna feel arrived at that one good all right we're going to do one more example and then I'm going to tell you a little bit more about this idea of derivatives where you can take them where you can't take them and why and so a little bit more Theory for so one more example application action do you remember what instantaneous velocity was instantaneous though average velocity between two points what's someone said it what was it it involves a limit right in fact in instantaneous velocity was simply what we found out is a derivative it was a limit of the sale of the slope of the slope between two points said instantaneous said a slope of a curve at a point that's that's what it was in fact we we had basically the same thing with a tea instead of a X where F was the position curve does that look familiar to you that wasn't Stace philosophy and I gave it to you a while back well if this is true notice that all this is this is equal to the first derivative of your your math right that's what that formal is what that says to us is that if this gives you the instantaneous velocity and this is simply the formula for the first derivative velocity is just and now we can say it for real velocity is simply the first derivative of a position function now we might have known that just from this but now we can say for real so this is instantaneous velocity which is denoted V of T instantaneous velocity is simply the first derivative of a position curve instantaneous velocity is the derivative of a position curve let's do one example just to flesh it out a little bit the only thing that's 1,250 feet tall it's 5,000 feet i'ma call and find David sure let's say that 10 stories under 10 stories time say yeah that's about right so let's say this is the Empire State Building and you're up there and you with someone you really don't like my ex-girlfriend whatever age go okay that's what this is nobody's ex-girlfriends just kidding joking you will never push off Hilton right besides there's fences up there you couldn't do it if you plotted at the 1258 anyway just joking this is about the effect of gravity if you just kick something off a ledge jet so this is saying so your visit your position is starting at 1,250 feet and you're just gonna throw something off it's going to fall that's what negative 16t squared will do for you does that make sense in a vacuum you know we're not living the vacuums being a vacuum I wanna do three things I want to find the velocity the instantaneous velocity the formula for that I want to find out when this hits the ground I will find out what's the velocity when it does hit the ground so three-part question so first part find V of T second part win it round third part how fast how fast hip drum that's our questions can you please tell me how you can find the velocity of that position curve what are you going to do right now you take the derivative once you go ahead and do that take the derivative for me you already know all this stuff to take derivatives so take the derivative of that thing the loss it equals the first derivative of your position curve in this case s so s prime is velocity for you that's how you would signify that I'm going to start working out slowly keep working on your papers let's see if we come up with the same thing okay I'll go through the step-by-step but on the real stuff that's a square sorry have you notice that a lot of this is just algebra work who knows that mostly just algebra you're like dang it I pastels robot the heck letter balls dogs room okay by show of hands can you raise your hands you made it down to this far for me to make it there good okay have you made it that far you have the understanding for calculus down you have to understanding how to use that formula down the rest of it is just your algebra work distribution combine like terms things like that so in our case for us keep on words if you want to hopefully you saw why those more important right so you don't have them you're 16 T squared you're not going to go away and you need them to go away so if ever you come up with something where you're like man I'm not able to factor out an H you've probably made a mistake go back and fix your mistake it's probably what happened did you make it that far this side yes good beside just all the good stuff happen make it supposed to everything besides an H goes away okay oh that's gone that's gone I see only two terms each of them has an H let's back to the H negative 32 t minus 16 H remember I'm not so caring whether you factor out the numbers I don't care because you're going to be multiplying them anyway in just a second so all I really care about you for you right now is getting her that H that's the key here because that lets you take the limit as H goes to zero if you do that tell me what my velocity curve is equal to what happened to the negative 16 let me worry about find that good for you that's fantastic you got that you got the derivative down that is the instantaneous velocity at any point so right now we've answered vo T it is negative 32 T where T is the time that's right T's the time and we'll say I think this is in seconds so a gravity should be in say so this is in seconds next question what's in the ground when does something hit the ground when the position is zero so we just handed the ground if this was like a y-axis I started 6 feet and I drop it hits the ground after it reaches a height of 0 right what gives you height the first derivative or the actual function the function does so we want to know when does this thing make 0 that's the question you've done that since your introductory algebra class you want to find out what wins you can hit the ground when does 1250 minus 16 T squared equals 0 when's the height equals 0 that's what you're actually here do you see that that's the height right when's the height equals 0 find that one L oh come on you got this sulfur team salty there's nothing fancy about it just get T by itself which means you're going to probably either subtract 1250 or I would choose to add 16 T squared to both sides next step you're going to do 1 divided by 16 you have simple T squared so divided by 16 what's 1252 I can't do that in my head how much seventy eight point one seven eight nine eight get rid of the square how do you do that screw sure now what do you have to do when you take square root hey you typically would however we're dealing time right so as a negative square of is that going to work force negative time what happened in the past man crazy no what you do on the actual time yet technically you put a square root right but we don't do that because we're always moving forward in time so we're just going to leave the positive nothing negative take a square root of seventy eight point one two five four 8.8 for what 8.36 okay 358 so however much you want to be after here you can be even more out under here just so you know because you're going to plug in that number in just a bit so you said eight-point let's do 3 9 okay you said eight point three eight eight so eight three eight three eight eight I don't have a calculator from me eight point eight three eight let's do eight point three four that sounds good seconds cuz I'm assuming this is in seconds so when will it the ground it's going to hit the ground after eight point eight four seconds how fast is it going was hitting the ground well this is a combination of these ideas do you know when it's gonna hit the ground do you have a formula for telling you how fast something goes at an instant of time right there if you want to find out how fast it's going when it's the ground you now have a function in terms of T and you have a T plug it in it'll tell you how fast it's going when it gets the ground it'll also give you the direction that's going when it's hitting the ground not shortly after because it's going to be stuck but before so how fast that would be V of 8.8 four seconds that's negative 32 times eight point eight four keep in mind this is like neglecting the terminal velocity things to be like an adaptive okay so how fast is that going to 82 82 82 or 83 per second negative I'm sorry 288 yeah 282 283 that's pretty fast that says it's falling at a rate of 218 feet per second at the moment that it hits the ground if you were some sort of an engineer we need a little bit more complex of a thing here depend on what you're dropping you'd have wind resistance and stuff like that but you probably calculate a basic structural integrity if something's going to fall off of a 1,200 foot ledge and hit something you to calculate so it withstands a forceful impact abut up to 280 to 0.88 feet per second on top of that roof of your cab or something like that if you're working on a ledge rocks drop off you get the idea so that's our that's our idea it's kind of useful stuff pretty pretty cool can't do that normally only with calculus and limits can you actually find an instantaneous velocity so neato torpedo next up you know the next thing we got to talk about we've been taking all these derivatives we need to discuss when you can actually take a driven that's called differentiability can say differentiability for me very good different support so differentiability basically basically means the ability to take a derivative at a certain point it is also a great way to explore a whole bunch of point points on Scrabble words with friends it's too long for work so you can't win with that but I send Scrabble now here's the idea in order for a derivative to exist which means it's differentiable at a point this has to work in order for a derivative to exist at a point this must exist in order for a derivative to exist in a point this thing must exist well that should make sense you have to be able to take the derivative right and the derivative of means working out this formula because that is the formula for our derivative true however notice that it's also a minute limits have two limits exist if the limit from the left equals the limit from the right true that's what limits mean limit from the left equals a limit from the right those things go to the same spot but I've talked about continuity right now can we talk about the limits they match up the same spot whether it's defined at that exact spot or not doesn't matter or derivatives sorry for uh for limits it says that Kant has to match up so that must exist here's two implications that we have two implications first one let's say I'm talking about this specific point that we'll call it right now a sharp point it will call a sharp point do you understand that what this limit gives you is actually the slope of a curve true it gives you the slope but for the limit to exist in general the limit from the left has to equal the limit from the right you believe that as well what that means for us in this context is the slope from the left because this this is a slope right that's a slope keep in mind that's a slope it says for this limit to exist for this slope to exist at a point the slope from the left has to equal the slope from the right do you buy into that it's a limit the limits have to be the same but these limits commentate a slope means a slope paestum in that job perhaps what's the slope doing as we're getting to this point that's positive positive positive positive slope right but from the right hand side that's negative negative negative slope are those slopes the same then that limit doesn't exist because the limit from this way and a limit from this way according to our slope doesn't happen the limit of the function yes that does exist the limit of our slope doesn't exist the slopes are not the same remember we're not just taking the limit of the function itself we're taking the limit of the slope do you see the difference I'm not talking about whether the function itself meets the point yes of course the function meets up the limit for the function the function exists does it limit for the slope exists no that would mean the slope from the left is the same as a slope from the right is it that means that at any short point our limit will not exist therefore we can't take a derivative at that point I hope you guys see some blank lows of it are you guys okay or no be sure not really let me show you the difference between limits of slopes and limits of functions themselves this is a function right does the limit of the function exist here absolutely yes the limit of the function exists however does it limit out the slope of the function exists the slope is what we're talking about the slope this way is going here the slope here is a constant are those the same at that point then the limit of the slope doesn't exist that's the difference that we're talking about a derivative is slope not a function self it's the slope of the function so this says the slope is no those don't match up those will make the same exact slope at that point so our two rotations are number one you can't take a derivative at a short point at a short point why f prime of X is a two-sided limit it's a to side limit that gives you slope if the slopes don't match up the limit doesn't exist if the limit is this you can't take the derivative that's the idea the limit up the slope still exists other one is this one you try to try those real careful there we're talking to get at this point at that point do you see how this let's pretend it's barely a function K barely but at one point the slope would be vertical to see that at that one point look what happens to our our slopes remember talking about limit limit has to exist means the limit has some segment right and from the left here we're going to check it out positive Finity right here we're going negative infinity it's positive a negative fitting same thing no no the limit doesn't exist there as well so two implications are you can't take a derivative at a sharp point or where the slope is undefined so this vertical this one this gives at least two to two at least two different slopes for that point this one we run to find repository of infinite your slope is vertical you can't can't take it riveted with that how many okay with our two implications here all right let's see if you if you really understand it one quick example is absolute value X differentiable everywhere what do you think clear I just got a sharp point in fact if you did it with limits of the slope you'd see this the slope is negative one true the slope is positive one as you get to this point the slope is still negative one this one is still positive one right it's negative one equal to positive one then the limit of the slope can't possibly exist this is not continuous everywhere no sorry not differentiable not differentiable everywhere the question also is though could you take the derivative here and here and here here here here here here here here here here all the way up and take you to that point and all except for that line on a point yeah so what we're finding here are just the points where the function is not differentiable everywhere else will be fine so I'll make a little note for you these functions are not differentiable only at those specific points these functions are non-differentiable only at those specific points okay a real quick question a couple more piece information we're almost another section here so nice little peace permission for today firstly we're going to talk about continuity and differentiability and how they have an interplay here first question is I'm talking about this point and this point is this function continuous at Point C is it continuous at Point C fails continuity therefore is it differentiable at Point C now the limit clearly does not exist from both sides if it's not continuous as carefully if a function is not continuous it cannot be differentiable at that point and the function is not continuous it is definitely not differentiable I'm abbreviating your continuous and differentiable okay if it's not continuous it's not there's a long words I'd like to write so it's not continuous it's not differentiable for sure you agree so the next one is this continuous everywhere continuous means you don't pick your pencil off of the papers and continuous is it differentiable everywhere yeah it's not differentiable right there so just because of functions continuous doesn't necessarily mean it is differentiable so that's not that's not always true here's what does mean if the function is differentiable it is definitely continuous differentiability is stronger than continuity business continuous but not differentiable everything is differentiable is automatically continuous so if a function is differentiable it is continuous if a function is differentiable it is continuous for certainly the opposite I've just told you is definitely not true if a function is continuous it does not necessarily mean that it is differentiable I add a point at a point okay we'll end with there we'll talk about any points next time and I'll give you some different representations of how to make the notation for derivatives sit today you okay with it yeah alright is it light so we're talking about how to do these these derivatives and we talk about continuity with that's less than we did write continuity versus differentiability we said if a function is continuous it's not necessarily differentiable that it has to be continuous and have no sharp points and then will be differentiable everywhere if a function is differentiable that certainly means is continuous though for sure so that that differentiability is stronger than continuity they can be continuous without being differentiable but you can't be differentiable without being continuous also if something is not continuous it is certainly not differentiable if they have a jump so it jumps and and I'm sorry sharp points are non differentiable now the last thing in the Italia is that I know I've given you a derivative is typically represented like this that would be the most our introductory derivative we do F prime of X you guys have seen a lot of that before right most of the time we write that however there's also some other things that we can write we'll also write to find that the derivative this DDX if you've seen that in your book that's what that stands for so DDX says I want you to differentiate the function with respect to X basically you can think of it that way so this says find the derivative or differentiate whatever function I give you with respect to that variable a derivative with respect to X of f of X it means the same thing is doing that alright same exact K okay with that one as well you can also represent it Y if you had a function y you can write read Y Prime so Y prime would also be a derivative or you can write for function y dydx so this would be a derivative of Y with respect to X same basic notation is this so we have a function notation and we have like our Y notation we have F prime of X or Y Prime we have DF DX or the DDX of f of X or F dy/dx they all mean a derivative and then the change or basically yeah you probably stick with one version for a problem so if I ask you find DDX you'll stick with D DX just a find f prime you'll just use that Prime but you need to know that when you see it it's not something confusing it's just a derivative there you okay on and so forth also we can evaluate derivatives at points we've done this to find the slope of a curve at a point you plug in numbers right and they can tell you to evaluate derivatives at certain points how that will be shown is this if you wanted to to find the slope of the derivative at a certain point it would be show like that shown like that find F prime of a where a is like three or negative two or you just plug in a number that's how that would tell you I want you to evaluate the first derivative at that specific point this one's a little different it would say I want you to evaluate the first at a specific point but they do it like this they go put this long vertical line and then put x equals a I know it looks a little weird right but it just says I want you to evaluate this is not what I'm making it long so you see this longer than normal it doesn't have to be that I'm certainly multiply right but this just says evaluate your first derivative at x equals a it's same exact thing as this they mean leave these only the same thing you could do this y prime of a or lastly you can do dy DX at x equals a just this long line just means evaluated how we will feel okay with our symbology our notation for this so this is our evaluation this is just the way we can represent our first group did and that pretty much wraps up section 2.1 that's it so if we get this step down as far as the continuity goes and find the derivatives we're good you have any questions before we move on right well I've done for you all of the theory stuff behind derivatives I've worked them out for you I've showed you how to get them I went from finding the slope to taking the limit of that slope and moving the point really close to that to another point saying this is a tangent line so we're at a point we've done velocities and rates of change right now we're going to work on techniques of differentiation how to find the derivative of any function whatsoever that we're going to start working on that stuff are you ready for it trust me you're ready for it you're right if you can take the derivative you're ready for it can't take a derivative you're ready for it do you want to do it I don't know we're ready

Channel: Professor Leonard

Views: 382,446

Rating: 4.9420133 out of 5

Keywords: Calculus (Concepts/Theories), Lecture (Type Of Public Presentation), Leonard, Math, Calculus (Field Of Study), Derivative (Literature Subject), Functional Derivative, Function (Literature Subject)

Id: 962lLfW-8Jo

Length: 76min 1sec (4561 seconds)

Published: Wed Aug 29 2012