Transcript for:
Understanding Vector Functions in Calculus 3

all right so like I said uh we're opening up a brand new chapter kind of taking another step uh you see we we have not done a whole lot of calculus yet should have done zero calculus but we we learned all about 3D we learned about what it what it takes to be in this kind of world uh of of Cal 3 uh we learned about vectors but we haven't done a lot with them so today we we're going to learn about something called Vector functions what we're working with with this whole chapter how we start to think about calculus how we start to think about what does a derivative even mean in 3D what what is that idea because if a derivative is a slope of a curve at a point we don't have slope anymore we have vectors now we have these these things called vectors and we're going to talk about Vector functions so here's the few things I want you to know about Vector functions right off the bat number one vector functions and I I'll show you why but uh Vector functions are man they're just an extension of parametric equations they they literally are a parametric equation do you remember anything about parametric equations like at all yes I remember they got a whole lot of t's in them like all the little T's well yeah that's going to be in Vector functions also so they're an extension of that they are a ve they are a parametric equation in fact I'm going to show it to you right now so with our parametric equations we normally saw them like this we had X and we had Y and they were defined independently but they were defined with the same parameter that that idea of parametric equation says here's a parameter it's one variable that's used to define two other dependent variables like X and Y so we'd have X is a function of T and we think of it normally in terms of time even though it doesn't really relate exactly we think of it time and then we would have y as a different function of T okay here's my question what's the only independent variable that we actually have here t x and y even though they are variables they they depend on what the value of T is that's the way that parametric equations work so this uh this T let's talk more about it this T was called a parameter and it was on some interval that gave a common domain uh for for both of these other functions it means that we can pick whatever T we want as long as it works in both of these functions so if we have denominators they still can't be equal to zero if we have square roots we still can't plug in numbers that are negative inside of square roots so they have this this common domain as well so this T is the parameter for some interval uh I on a common unit on the comain if you want like the English translation of this it says hey you know what X and Y are now no longer independent VAR Ables either one they're they're both dependent they they depend on T this T is the only variable that you can plug stuff in on the board right now so it's called the parameter and and it's going to work for a certain interval so that both common so that both of these functions are defined that that's what that common domain means it means that we got this interval of t for which these T's both work in both of our functions right now show F feel okay with that that's from parametric but I wanted to kind of clue you in so okay now how do we extend the picture well how many dimensions do you have here two two variables two axes two Dimensions if we want to extend this to 3D we just got to Simply add another another variable instead of just X and Y now let's talk about X Y and Z you know we're going to keep X X is f of T that looks pretty good and Y as G of T and then well what's the only other variable that we're missing question I was just wonder what symbol was what the symbol for the 3D is what oh you know what uh a lot I'm going to use three several different ones we can do 3D just to mean 3D but R3 like this is uh real numbers as you draw this box letter R so the real number system in three dimensions is what that means so I'm have I never talked about that before good thing I brought it up now thank you you're welcome for three space for R3 for 3D same stuff on a three-dimensional coordinate system same stuff we' need another another dimension we need a z so let's see u a b c d e f g why not okay real quick how many independent variables do we have right here how many independent variables there's still only one how many dependent three three in fact it's really weird right because this is still parametric but it says for every value of T I'm giving you an X and A Y and a z coordinate I'm still giving you a point so for every single value of T I get a little point in 3D you can see how we could move around every space in 3D according to this sort of a system it's parametric the variable is T it's called a parameter our dependent variables are X Y and Z we can get a point by plugging in a value of T had not you're okay with me on that stuff all right now let's let's put it together T is still our parameter by the way I'm not going to write that out but T is still our parameter it's still all this stuff still works it's it's literally parametric uh but but wait a second vectors are also defined like this a vector R if I had this just random Vector R I said you know what how do vectors look well Vector vectors look this way vectors look like this they have um what comes first x y z x they have an X Co coordinate they have an X component in fact like honestly this x is literally the x coordinate of the point because vectors are always defined by DET terminal points for position vectors and then that little I do you remember talking about that you'd have an x coordinate and then I and then you'd have uh oh what's next what a y coordinate with j and you'd have a z coordinate with K I want to refresh your memory on this one because I'm GNA we're almost done with the whole proof of what Vector function not even a real proof just put stuff together but this is what vectors do vectors are always position vectors are always defined by the terminal points we start where where do we start with Vector vectors where origin and they terminate at whatever this this this says you remember talking about that it was 3 2 1 it'd be terminated 3 21 so literally if we had this format the JK the standard basis Vector we go well yeah that makes sense we have the x coordinate and I telling us how far we move along the X Direction along the x- axis we have oh wait vector addition we can add vectors together right we have the y coordinate J how far do we move along the Y and then the Z coordinate K how far do we move along the Z and we put those together we get a vector but these are literally the coordinates of the point where the Vector terminant show F feel okay with that one but but wait there but wait there's more uh if we look back at this what if we don't have an actual number x what if we have this function that gives us a number X dependent on what T is what if we don't have the the number y the coordinate y we have a function that gives us the number y depend on what T is what if we don't have the number Z we have a function that gives us what the Z is depending on the the value of T well then we can take this parametric expression for it and plug it into this vector and get a vector function that's literally what a vector function is it's a parametric definition of your XY coordinates plug into a vector that's that's all it is so if we put these ideas together um by the way what what is this Vector what variable is this Vector function going to be based on so this is going to be a function Vector function but it's based on T because that's the only independent variable that we have okay well yeah you know what it's usually an x coordinate but we don't have an x coordinate the way we think like a number itself we have something that's giving us the X we have a function for X we have a different function for y and we have a different function for Z and that right there is called a vector function I want you to think see how see how it plays out I hope that you're you're with me on this one is it true that I can plug in a value of T and get an x value a y- value and a zv value and if I have i j k then every single value of T gives me a specific Vector show fan if you guys see that that's what a vector function is um so but what's it man what's it do do well before we get there I want you be thinking about that before we get there just always keep in mind that's X that's Y and that's Z before we get to what it is there's only one other way we can kind of see this uh besides having the standard basis Vector j j k we can also have that Vector bracket notation you remember dealing with that it's the same stuff so what would come first X would normally come first so in this case it's yeah just a function and then Y and then Z XYZ I'm not going to write all of this out just a couple statements um keep in mind that that for this to work you have to have the same domain for what you're plugging in here and here and here you don't get to pick some t's for here and different set of t's for here you have the common domain it works just like parametric does that make sense to you so if you can't plug in zero for the Y you can't plug it in for anywhere also keep this in mind when you're finding out a specific Vector you need to use the same value of T So if I want to figure out what's the vector at time let's call T time time of 3 seconds you'd have to plug in three in all three spots to get that specific Vector it's kind of common sense but I want to make sure that that actually makes sense to you does that make sense so uh the common domain has to be the same domain for all of these sub look call them sub functions your your coordinate functions got to be the same domain also what value are you using for your X you got to use for your Y and your Z to get that specific uh Vector so what's it give you what's it going to do think about that man what's that Vector a vector it's a vector it's a it's a function that gives you vectors all right so it it says Hey plug in a whole bunch of values of T you're going to get a whole bunch of vectors does that make sense you a whole bunch of vectors all of those vectors are you listening start the origin yeah but they all have terminal points and if we let this value of T go through it domain it's going to give me a whole bunch of vectors that point somewhere like that point that point that point and those points listen carefully okay those points it's like Connect the Dots here the terminal points of our vectors are creating this little Dot Plot that gives us a curve a curve in space so what Vector fores give you are curves called space curves because you're going through space they're giving you space curves so what's happening here basically is we let U we let R of T be a vector from the origin to some terminal point on a curve it's not a surface right now that's not what's happening we can't just say hey and make this the surface that's not what this does what vectors do is they have one point and if I let that Vector move let's say let the vector move it can it can go longer go shorter and it can move through space but it's creating this this curve for us through space the terminal points of these vector vectors created by the vector function create a curve called a space curve so hence if you understand that concept it's kind of a weird idea we're not traveling along this way anymore we're traveling by going we're pointing where we're traveling I'm I'm going to draw you a picture in a second the only thing I'm going to write is um the terminal points of these vectors will be on some curve through space for the entire domain of t for for did you guys write that down here's here's what all of this says says in English all right it says and if you only write out I'll write out too so a vector function here's the whole statement a vector function is a parametrically defined function where the terminal points of these vectors Trace out a curve in 3D that is all that is happening so the terminal points of the vectors Trace out a curve in 3D because we have a domain for the independent variable t for our parameter because we have a domain that goes from smallest to largest it also has to have an orientation it's got to be traveling a certain direction through space does that make sense to you so if going hey tal 1 tal 2 the word time has passed the the variable T is is moving along also in that in its own domain there which means that the vector points somewhere and then later it points somewhere else that always gives us an orientation of travel head not if you're okay with that one really should make sense right should make sense because vectors are basically defined by their their terminal points so do you want me to write that statement out by the way would that be helpful for you yes okay so basically a whole punch line to everything for for well I'm tired of writing I'm tired of writing I'm stop there all right good my shoulder hurts got do more Delta lights uh so a vector function is parametrically defined yeah yeah the parameter T hello pluging and T's that's going to give you vectors that's what it's called a vector function those vectors point to points their terminal points are points those points Trace out a curve that that's what's going on the fact that t has a certain domain like it starts somewhere it goes somewhere uh makes certain that our Vector function has orientation it also goes in a certain direction that that's the whole whole idea uh the only other thing that we um that we want to talk about is that this really shouldn't be a problem for us because every Vector like ever let's say a certain Vector R sub t r of T Sub 0 let's say it's -35 2 stick with me here Point Vector which one both both it's a vector yes but it has that terminal point what's the terminal point of this Vector what point is it terminate at so the terminal point here our Vector - 352 yeah it starts at origin It points to35 2 where that is but its terminal point where it's stopping is at 352 if I take all of these little vectors and I put them all on my graph from the origin to the termina points it's going to create a curve what a curve is is this connection of an infinitely many points right let those vectors point to the points and we end up tracing at a curve in 3D I think I've over talked it I'm going to stop with that one um the the only thing I want you to know is that every single every single Vector itself gives another point on that curve the picture looks something like this so this is coming around wrapping over top the Z coming around the Y and then and then going on you guys see the picture I'm trying to draw for you here's how Vector functions really work it says if I wanted to Define this with a vector function what happens is that I start I start with some just a few examples here what's happening is that a vector is pointing to these points let's call this R sub of T1 I know if you can't read that that's that's okay what's going on I'm not even to write this anymore my first Vector points or a vector points to some point on my curve and I start going what's the next one let's say it's all of these things and what's going on is then pointing to terminal points on this curve and that's that's what my Vector function actually does it says hey where am I at this point at this time I'm at this point what about this time I'm at this point what about this time I'm at that point and I'm pointing to my curve instead of traveling along it it's kind of a weird idea but I'm pointing at where I'm at in space that's that's the plan here so if should be okay with the plan so this is the picture of vector functions really pointing where the points are on the curve it allows me to travel it but not going through the curve but standing at the origin saying here here here here here that's the that's the idea now what's nice about them after all that that talk is they're not actually that hard to work with you're going to end up probably liking Vector functions uh we'll talk about one ex just one example of vector functions we'll spend most of our time sketching them because they are a little weird to sketch uh but here's our first very first example exciting it's what the ex stands for it's not example it's exciting that you're not as excited as you should be this this this it's cool okay you know what I know it looks brand new to you because it is it is brand new to you unless you've had this class before um can you can you see that this is a vector function can you see it how can you tell it's a vector function bracket say vector and look inside the vector there's T's in there right you go oh yeah functions Vector given by functions a a vector function that's it or if I would have put T's with like I's J's and K's well that would also be a vector function there's a couple ways you can write it here's what I want to want you to understand when we have this yes the x coordinates and Y coordinates are going to be here but they're given by functions of T you're not going to see a whole lot of X's y's and Z's you can see a whole lot of the parameter so what I want to do right now is is threefold I want to find the expressions for X Y and Z cuz this going to be very important in a minute I want to find the common domain for the parameter T and I want to find uh just one vector given by a point so let's start with this one let's find X Y and Z can you'all tell me what is the function that is giving us the x coordinates here or the X component of the vector what is that perfect hard no it's just the first one it's just like every Vector works man uh the x is given by T what's the Y given by and what's the z component or the Z coordinate of our terminal point what that what's that given by feel okay with that so far let's talk start talking about the domain you know what I'm going to erase this and move because I want to write the domain underneath each one hang up if I asked you to could you tell me the domain of t for by the way X Y and Z those don't have domains you're not plugging anything in for X Y and Z those have ranges of anything what we can get out of X Y and Z they don't even have domains our domain is always based on the only independent variable that we have which is T so let's see if we figure it out can you tell me what T has to be just for the x coordinate everybody come on what's T got to be uh yeah probably we can do that probably easier to do this for them so T's got to be greater than or equal to zero yes no we know we can't have negatives in there zero is fine though how about here could you tell me this could you tell me that one what's the what's the domain for the y coordinate y component what can't T be sure why not yeah but anything else is fine for that one component how about the Z what about oh do you remember what can I plug in for t for Ln of T do you remember one can I plug in zero no can I plug in negatives no do you remember what the graph looks like no it's looks like that pretty much like that so anything positive I can plug in but not zero because as I go zero from the right I get Negative Infinity so this says T has to be strictly greater than zero here's how I do all of these domain problems I write out what and this is what you should do too write out the X write out the Y y r the Z find the natural domain the stuff that actually works for X for y and for Z and take the most restrictive one take whatever is restricting you for the common domain so you go hey you know what I can be greater than or equal to zero here that's that's fantastic but this one says I've got to be strictly greater than zero that knock this one off the board this do even matter anymore you go oh don't even care because that one's more restrictive does that make sense and then we go but wait a second also can't be equal to one so when we write our domain we have to take all of that stuff into account we say I what's the smallest value I can have two zero but not there not bracket parentheses you remember talking about that so I can start almost at zero What's the where do I go what's the next point that I have to consider here I can almost get to one can I include one no that's what that means but then I anything after one is fine so so when we write out these domains for our parameter for our independent variable we go okay well write out what you can't do write out what you can do I can be greater than equal to zero but this one says I got to be greater than zero strictly so I know that the smallest value is zero with parentheses I know I can't have one so we exclude the value of one everything else is good so hence feel okay with with that one that that's how you do this common domain now the last one could you figure out could you figure out two specific vectors what happens at T = 2 and T = 4 the only reason why I want you to do is so they believe me that we actually get vectors out of these things so if we want to find the value or sorry the vector at tals 2 what number we plugging in where are we plugging it into yeah that's that's it so at tals 2 it says uh hey this is all based on on T's right uh plug plug in your T So plug it in so if I wanted to find the vector at T = 2 okay PL plug in two what's the sare root of two I don't know leave it Square Ro of two can I plug in two here yes of course it's on my domain you can plug anything you want as long listen listen there's only two two little side notes okay you can plug in anything you want provided you plug it into every the same value of t for every single T and provided that whatever T you're plugging in is in your domain it has to be in the common domain you can't plug in one here and here even though it lets you because you can't plug in one there that's the nature of the common domain you don't get to plug in anything that works for all of this stuff you plug in anything that works for all this stuff provided it's in your common domain it's why we do that first does that make sense so if I plug if I asked you to plug in one you couldn't do it because like a third of your vectors are going to be undefined you can't do that so we'd have of two we'd have looks like we got what what's the next one 1 - one is one one okay and then ln2 I don't even know what that is just leave it ln2 now what is that what did we just find literally just a vector yeah it's also G so it's giv us this it's saying hey at the time tals 2 I'm pointing at this I'm I'm going along the specific Vector I'm pointing at this point that's my terminal point and it's going to be on the curve that this thing defines that's what's going on right now show Answer feel okay with with that one all right find uh find the four if you haven't done already easy medium hard what do you think so far yeah looks like 4 1/3 L four did you guys get the same thing I got oh no hopefully you didn't yeah yeah whatever you say I'm not even listening anymore I have to sleep whatever it's late tired going 35 minutes yet goodness so F be okay with the idea of vector functions so if you want like a 20 second recap here they are uh Vector functions is a a vector vectors that are given you by equations for X Y and Z independently that when you put those components together gives you Vector points to a point on a curve that that's what's going on we take all the vectors together it gives us all the points points in the curve together it's just pointing at them that's what that's what's happening in order to plug in points though you have to be defined for all three of those components at the same time it's got to be defined uh that's basically it now you can we're going to move on a little bit we're going to talk about how to sketch them because you need to know what these things look like so we're going to move on to sketching these these Vector functions I'll give you a piece of advice before we we keep going Vector functions can we're can do with a whole lot of them but they can have only two components if they do it's called a parametric equation graph them on a plane okay don't worry about 3D because it's not 3D if you only have two components it's going to be on a plane does that make sense to you so I'll write a little B about that but we're going to be we're going to be sketching I'm going to give you the general rundown some of this is not going to make sense yet because we have not done an example yet but I'm going to give you the general rundown about how you sketch pretty much every Vector function like ever provided it's on a a normal type of surface so here's our general General idea number one oh my gosh wake up oh good okay now you'll get number one um number one am I really that boring it kind of we just pretend like we're not right thanks for pretending you do a great job I can't even tell at all number one OMG is that better for you people like toads uh OMG you're my like BFF on the Twitter sphere hash brown tag LOL just throwing on the line I don't even know face space my book I don't care listen uh number one write down what x and what Y and what Z are independently do not try to do in your head write literally write them down identify X identify Y and identify Z do that first number one brought the bat be time number two use one for two D for 2D it's a it's uh use both but for for 3D for these actual curves identify one or two components to make up a curve or a surface uh trust me it's going to be vague right now because we're not doing an example but this is what we are going to be doing so number two use one or more components to find a curve or Surface in order to do this we're going to have to get rid of t in okay here here's the deal if you only have two components like X and Y or Y and Z or x and z if you only have two components the curve that you find will be a curve so that be a surface just a curve and it's going to be on a plane so for two components sketch it on a plane for three components for three components you're going to be sketching that curve please please listen I don't want you to get lost here listen Vector functions they are curves they are not surfaces they are curves so this I'm going to I've lost people before on this like what do you mean talking about surfaces now yeah but there's a caveat to this okay these are all curves they're pointing to points man it's you're going to create a a curve that's what's going to happen but what happens here if we if we use this model these curves are going to be sketched on the surface of some of these surfaces they're going to be tra like a c if it's a cylinder it's going to be traveling along the cylinder if it's a cone it's going to be traveling along the cone if it's a hyperbolic paraboloid it's going to be traveling along the hyperbolic paraboloid the thing that's what's happening here so the idea is identify X Y and Z yeah use one or more of those to give yourself one of those surfaces that you know about paraboloid cylinders and then the curve that we have for these three components the curve is going to be laying on top of that surface so these things are not surfaces but they can travel along surfaces that's the difference here head not if you understand that the concept here so what are they are these Vector functions surfaces or Curves curv but they can be traveling long surfaces so for three components the curve the vector function will be on a surface and lastly the last thing that we do uh man you never do it this way you never just plug in values of T get a whole bunch of points and start plotting them it ain't going to work out so hot for you okay but once we understand that we're 2D or 3D once we understand that we're on a plane or on a Surface then at the very end when we get a good idea of what the surface is then we start using values of T to get at least two points because what those two points are going to give you keep in mind these things all have orientation correct so we're going to be able to travel along that surface with a given path oh we're going this way or we're going this way so we're going to use at least two values of T to find out our orientation and to figure out at least a little bit of of um how the curve looks so use some values of T to find some points and the orientation just keep in mind man you got to find points that are this is why even we did this you got to use points that are actually on the domain you have that seem obvious everything got to be defined so from here in number three use values of T to find points and orientation tell you what we're going to do two examples and we'll pause for a break we'll come back we we don't have that many honestly uh just a couple then we'll talk about limits and continuity two of the easier things that you'll ever do in this class and then we'll be done with our section it's a quick section so let's start with um with just some 2D ones they I know they're param well one 2d1 then a 3d1 they're just parametric but I want to show you exactly how to work with them so here's our first example okay hey first idea can you tell me that that is a vector function is that just readily apparent to you that that thing is a vector function at this point yes it says Hey Vector given by a variable that's a vector function that's what that thing is what is the one variable that we have up here but it's also given us some components that t should give us X Y maybe Z so right now number one I want you to write it explicitly don't do in your head trust me it it's going to be a lot easier if you write this out trust me trust the Leonard has the Leonard ever steered you wrong only a couple times okay uh trust him on this one all right write this stuff out because it's going to be a lot easier for you to do substitutions and make equations if you do so what I want to do is I write want to write out X and Y and Z what is our X what's it equal beautiful you understand it what's our y what's our Z oh there's not one H how many how many dependent variables do you have two how many independent one so what that means is that we can put these together to get a curve out of this if you have two components sketching on a plane what plane the XY plane if you had x and z you'd sketch it on the Z it look exactly the same okay it look exactly the same it's just your ver your your axes would be different does that make sense to you you YZ sketch on a ly plane let sketch it on a plane that's what's going to happen here um so first us we go okay hey here's X here's y I know I'm going to get a curve I know it's going to be on a plane it's two components two Dimensions so start doing all that math that you know how to get rid of T so use substitutions use whatever you need so for instance here I go okay um I know that X isot of T I know Y is 4 - t what if I let's work here try to do a substitution or you can go this way it doesn't really matter but this would be a little bit nicer can I get this T so I can make a substitution what would I do over here Square yeah you do stuff like that so X2 = T if X2 = T then yal 4 - x^2 or if you like it better this way I like it better that way x^2 + 4 here's the whole whole point identify x y and z we did that we only X and Y use one or more components I used one component to get a curve or a surface that's a curve and I got rid of T if you can do that what that says is this is a curve it's actually going to be on a plane because I only have those two very I only have two components right we're sketch on a plane I'm going to sketch on the XY plane question why do we do substition because what we're going to try to do is get some sort of curve we understand out of this all right and so we go well you know what I can't sketch this and this very well for for myself so let's do a substitution to get rid of that t just like parametric from Cal 2 when when you graph the every time you did parametric if you think that look at notes look at the videos every time we did it we basically got rid of the parameter like every single time why because we don't know how to we don't know what that is in our head that's really hard to think about X and Y independent of each other and then put it together when we put them in one function however that makes a lot of sense because now we're rectangular watch carefully though can you graph that this is come on people what is that opening downward love it and on the y axis that one so this is I'm going to give you a bit of advice with these two dimensional stuff when you're graphing the curve itself graph it in a dotted line would you do that please here's the reason why when we graph this in a solid line this listen this this is nature of parametric equations and nature of vector functions this gives you way more than what we have in this graph okay so this is going to give you the the curve that our curve is on I know that's weird it gives you the curve that our curve is on okay they gives you the pathway that we're going to our vectors are going to point to uh where the actual point to could be less than this so I'm going to graph it like like this this isx2 + 4 that's that's what that thing is listen this can be it's not necessarily super hard but it can be confusing um are you guys okay on identify X1 Z show hands if you are if you okay with that one sbly the easiest part are you understanding that we're going to use one or more compon components to basically get rid of T to figure out some sort of rectangular sort of equation that we can graph head now if you're okay with that if you have two components graph it on a plane if you have three we're not there yet we'll graph it in 3D we're not there yet um and then when we sketch this curve that you find please listen carefully to the verbage okay this right here is just normal curve the vector function will lay on top of this curve somewhere or on top of the surface if we're in 3D the vector fun this is not the vector function right now this is the curve you guys see what I'm talking about it's the curve the vector function is going to be on this yeah but it doesn't have to be the whole thing it's going to be part of it later on we're going to have a surface the vector function is not the surface it's on the surface just like the vector function is not this curve it's on the curve I know that's a small point but let's see what actually happens here the last one is use values of T um let's plug in -7 should I do that what should I plug in probably zero to see where it's starts because this is going to have an initial Point as an orientation we to start somewhere for for a defined domain like that so if I plug in if I plug in tal Z if I plug in tal Z listen though where we plug in where we're pluging points in they have to be values of T they're not values of X they're not values of Y they're values of T so we go back and we use this thing can you plug in zero into my Vector function of course you can what is it 4 4 I'm sorry what 4 4J I thought I said 4T I was like no guys we plug it in for t 4J hey 4J tell you what it's got a terminal point what is it z z four did you notice how you got Z four out of that is check your is zero is 04 somewhere on this curve yes if it is not you have the wrong freaking curve okay it has to be 04 this this is the vector here here it is this is the vector 0 or 4J it starts the origin it goes up for that that literally is 4 J the terminal point is 04 that's literally the starting point that's the initial point of this Vector function right here all we got to do is figure out which way it goes so it's greater than zero is it going to go this way or this way plug in another Point plug in anything you want as long as it's greater than zero plug in one plug in two I don't care which plug in in four I'd probably plug in four why cuz it's nice and because I can show you the vector real nice if I plug in four here's a of 4 is 2 i 4 - 4 is 0 J I have 2 I can you tell me is 2 I is the terminal point of 2 I somewhere on this curve so you guys aren't even are you guys with me cuz 2 I 2 I does this goes out here here's two 2 I terminates at at two 0 that's literally on my curve and it tells you something it says time is passing from 0 to four here's zero here's four so time is going like this my vectors are doing they're pointing at these points going this way what that tells me is that my curve starts here with my initial point it's traveling this way along the curve I found and it's going forever and my vectors are doing this it's pointing at all these points my Orient you don't have to show this you don't have to show the the little Vector functions I'm just showing you what it's actually doing what you do have to show me is if it has an initial point you got to show me that if it's got a terminal point you got to show me that and you got to show me the orientation since we start here and later call later timelin we hit this I know the orientation is that you did the same thing with parametric it literally is parametric but I know that a lot of teachers in calc just go through it very very fast I personally don't uh but a lot of teachers do they go oh yeah parametric you get a later well you're now getting it later here's my next question this was part of the curve is it part of the vector function no no the vector function lays on top of the curve or the surface that you find that was the big deal we don't even need that that's the vector function show should be okay without one okay tell you what um you guys want take a a break now or you want to do one more and take a break I don't hear either way one more okay it'll be pretty quick actually it's going to be very fast uh well F Leonard fast it'll only be like 25 minutes all right I know Leonard fast is slow I'm sorry I like to talk and hear my own voice I mean did that come out you know what we're going to go a little bit quicker though uh leard quicker because we we we've done this already so I want to outline the the basics but this one's going to will be faster I promise first question I got for you is that a vector function in your eyes yes no Vector function yes or no absolutely it's a vector function if it's a vector function you need to immediately identify what x y and possibly Z are can you identify the X Y and the Z components right now go ahead and write down would you left Siders can you tell me what the X component is Middle people y component and right side people Z did you make it through Section 11.5 what up huh what is that you need know right now what is that come on what is it what is it line it's a line do you guys see the line do you guys remember lines this is where lines came from they are vector functions they are parametrically defined Vector functions this is literally a line do you guys see the line it's exciting that's where it came from I wasn't making it up I I did it twice now we proved it twice that's a line that's it it's just a vector function but defined as a vector function you should be able to tell me right now a point on oh come on you need to know it your test is coming up too you need to tell me a point on that do you see the line it's cool this is a parametric oh wait a minute do you remember the symmetric equations and the what was the other one besides symmetric equations parametric equation for a line do you remember that you're not as excited as you should be parametric equation of a line can you tell me right now one point on this line give me a point on this line one two and that's a point on line how could I find it plug in t equal 0 and you could find that that point plug in t equal 0 seriously here's 0 0 0 you'd have 1 2 and three you'd have X the Y and the Z that's the vector I know that's the vector 1 2 3 but the terminal point of that Vector is the point 1 2 3 and not you're okay with with that one so far could you give me so okay um if you're like wait a minute what about all this other crap well all this of the crap says says this we're done you already got the mine you know this thing is a line does that make sense to you so we said hey can you identify X Y and Z yes here's X here's y here's Z can you use one or more components to identify a curve or Surface yes here's the surface or sorry the curve here's the curve it's it's literally a line I know we have a line can you find okay we know it's a line need can you find some values of T to give the points and the orientation if it's a line the points are easy the orientation says I'm traveling along the vector the direction Vector would be 1 -1 -2 you guys with me on that one okay can you give me another point on this line so at tal Z notice that but you have to know that though right that that point happens at T equals 0 that's where we we even got that idea was 1 2 3 plug inal Z you got that point on that line what's another good value this one might be um little non-intuitive here counterintuitive what's another value that you would use I wouldn't use one we're plugging in for T correct I wouldn't use one I would use the value of -1 or I would use the value of two which is what I'm going to use or I would use the value oh I don't want to do that one three halves why they got sub because I I want I want things that are easy to plot and things that are easy to plot are right around axes i' plug in something that makes this zero or this zero which is what I'm going to do or this zero which is what I'm not going to do plug in a value of T that makes one of your components coordinates zero use that use value of T to make at least one one component equals zero so I'm going to choose two because I don't want to go negative because that's going to be it's also because it's yeah it doesn't really matter you could use negative one or you could use two it's about the same I'm going to use two here I'm going to get three zero that's what I want that's why I chose the two and then let's see so I want to take it back 20 second recap make sure we're okay on this stuff uh did you did you see the vector function right from the start you go yep Vector function can you identify the X Y and the Z is that that okay for you yes yes no do you see that sometimes we get some nice stuff sometimes we don't even have to work for the curve it's given to us you got a line that's just a line don't overthink it it's literally just the parametric equation no D it's got a t in there of a line I just pulled out no D like sixth grade welome sixth grade well we know it's a line all you need to graph a line is how many points two you need two points for a line to graph it on a graph you need two points because the vector that's really hard to use right we could tell the direction Vector one point is given to us but you need to know it's at tal Z the other point you have to find that point plug in a value of T that's going to give you a component equal to zero I chose two we get three 01 head now if you're okay with with that so far and then we graph it if I would I graph this in 2D or 3D what do you think three components 3D it's a line through space now of course that means that you know how to graph or plot those points if you remember how to do this even if you don't here it is again because we we didn't spend a whole lot of time doing it if I wanted to plot the point 1 2 3 that's one on the x- axis two on the Y AIS 3 on the Z here's how to do it remember drawing did you remember redrawing the x axis and the y- axis and the Z remember doing that with with your curve sketch you know I showed you in class if you do that if you redraw the Y but right from your x intercept if you redraw the X but right from the Y intercept you see how these things are parallel to the X and it's just redrawing this it's just redrawing your X and your y but at the appropriate coordinates heading now if you're okay with that one here's your X here's your y it's at Y = 2 yes Y = 2 and x = 1 yes and where they intersect right there that is the impression or project ction of this point on the XY plane like looking down from a bird's eye view that's what would it look like it's shooting through okay we just need to translate this three units up there's two ways to do it probably the easiest way is this way just watch carefully I can't do it a whole bunch of times okay it's this one wherever that projection is take that redraw something parallel to that from the height of that point and then just like we drew the parallel to the X just like we do the parallel to the Y we're also going to draw the parallel to the Z but you're going to do it from your projection erase a little bit that is where the point 2 3 is I I Know It's Tricky until you get it uh but the idea is a lot of parallel do the parallel of the Y where your X chord is parallel your X y coordinate is draw that line Draw Something parallel and then where the parallel to your Z is and where the intersect your diagonal line that we drew that's where your Point's at I know it's it's hard to actually do uh but did it make sense on how to find that took a little bit more time there doing that I know we did it the first time we did 3D but I did it fast now what why do we make one of these zeros because this is a whole lot easier now because when your X or your y or your Z coord is z you're actually on a plane so when I do my next point my 3 01 everybody real quick what's the x coordinate of 3 01 what's the X three so I'd come out here I know I'm at that x coordinate head not what's your y-coordinate ah that means that from here I don't go out or I don't go back all I can do now is go up or down how far do I go up or down that's easier you don't have to go sideways now you go okay I'm going down one which means that I'm going to draw parallel to my Z I'm going to draw parallel to my X cuz I'm all I'm dealing with is x and z right now correct me if I'm wrong but my Y is zero right so x and z is the only two things I want to draw parallels of and where those guys intersect that's my other point you might want to watch that a couple times to see how that I don't have time to do this all day long okay but this is how it works we went out one over two up three that's one point on this line hey we had it a long time ago we're I'm showing you how to do it uh plug in something makes one of these zeroes way easier this is three 3 zero I didn't move down one that's the other point how how many points do you to make a line two so now we go hey this line in purple is simply going to go through these two points it's hard to visualize that because that line is going through 3D uh but that's what's happening this is coming out of the board going down that that's what that's that's do that's the first line we've ever grafted 3D because it takes Vector functions to actually think about it to actually do it now um the orientation does this think hard about it does this have an initial Point like this one did does that have an initial Point what's the domain of T go forever because it's a line right it goes forever so it doesn't start anywhere it doesn't stop anywhere but it definitely has a Direction that's why we choose some good points and we put them in order so we plugged in zero yeah that's that's this point we plugged in two is two happening before or after zero so I'm traveling from here to here not the other way around so my orientation I'm erase that so I'm not confused my orientation is going from this point downward so we're traveling in that direction it certainly has a direction to it have I made this make sense to you can you understand the idea yes okay we're going to practice a couple more complicated ones that are action of surfaces when we okay so back at it uh some real real life Vector functions stuff that's not just lines stuff that's not just 2D stuff that's real like 3D I'm going to show you exactly how to do that right now so the stuff I've given you now it's going to be a lot less vague those instructions were on the board a lot less vague right now if you read through those or you have memorized firstly oh man it's obviously a vector function what's the first thing you always do with Vector functions like ever what do you do let's identify X Y and Z go ahead and do that now this is not the hard part right this is not the hard part is the surface and then sketching on the surface that's the hard part so what is your X everybody please t y t z t you guys are so Geniuses love it okay that's fantastic and then we have a parameter T and it's going from zero positive 0 to Infinity so that we keep that in the back of our head that we are going to start somewhere on this curve which is going to be on a Surface now how many components do we have that is not a trick question how many components do we have yeah three it's going to be in 3D this is not going to be on a plane it's going to be on a Surface what's the next instruction that I gave you was identifying then then what I know I wrote it down come on it was number one identify X Y and Z we have done that number two it said use one use one or more components to make up a curve or a surface we're going to do that now you can't graph this right now do does anybody know what that looks like the answer is no I guarantee you don't know what it looks like I don't even know what it looks like all right we're going to have to ident the surface that this is going to be sketched on top of how we do that listen use a couple components that are easier to work with u the idea here is I TQ I don't know but I see xal T and that's a direct substitution into either of these guys does that make sense I'm going to do the easier one I'm going to make this as easy as possible to try to figure out a surface so right now what I know is that because I've identified X Y and Z our Vector function our curve is going to be on a Surface not on a plane on a Surface right now the idea is find the surface and not you're okay on that one find the how we do it I'm going to use these two make it as easy as possible I'll give you some notes on that in a second if x = t and y = t^2 y = x^2 and now if you're okay with that one are you sure you're okay with that okay now before you answer you got to think about it we are in 3D right now right we're in 3D right now right what is that in 3D what is that it's a how many variables do you have it's a cylinder along the z-axis with a trace ofab what's this look like on the XY plane a parabola this is a parabola projected along the Z AIS that's all it is is this making sense to you we've literally done this before that's why I said if you've done 116 if you know how to graph especially cylinders this is easier if you look at they go I don't know this is impossible you need to know okay there's there's two variables I am in 3D though because I have three components that means that's a cylinder along the missing variable Z with a trace of a parabola on the XY plane that's what it is write it down at least that much it's a cylinder along z y because we're in 3D and there's only two variables that means a cylinder it's along the z- axis cuz there is no Z variable right now this doesn't it's not even there and it's a trace on the X XY plane a a parabola right there so right now what I'm going to do is I'm going to trace out by the way we are we are sketching here okay we are not being exact we just need to know that this is a parabola on the XY plane and it's projected along the z-axis it looks about like that can you guys see the cylinder that I tried to create there for you guys had not if you can see that so this is a cylinder of that parabolic shape that's traveling along this now here's the whole big kicker firstly if you do this differently there are some other surfaces that you could get uh so a little side note here you can do this by substituting here obviously the surface is going to be different it's going to look completely different so how do you know what to do use the surfaces that you are familiar with so if I'm familiar with cylinders I'm going to try to make cylinders as often as I can and cylinders that are really easy to sketch as often as I can so even though there's more surfaces possible and you can do it differently all it takes is one you just need one surface am I making sense to you so side not here is sometimes more than one surface is possible you just need one surface any surface any surface will work but stick to ones that you know stick to ones that are easy to visualize more than one surface is possible stick to ones you're familiar with cylinders cylinders are easy graph those sort of things other little side note what two components did you use first right off the bat what two components did you use to give yourself this surface which one did you not use that is the one that will give you your curve the component you don't use will give you the curve on that surface or I'll say it this way because the next example two examples from now was a little weird the component you don't use First I'm going to pause right there there's a little bit more to that that I'm going to say in just a second but I want to make sure you're absolutely certain on what is going on firstly look up here at the board do you understand how to get this surface you understand that is a put it together it is a cylinder along the z-axis with the shape of a parabola on XY plane traveling right along do you understand that is this the vector function is this the curve that I'm looking at no the unused component is going to give you the curve that curve is going to be on this cylinder it's going to be right on the the the surface of that that cylinder that's what's going on so the unused component gives the curve on the surface that you just made in this case it's a CER on the cylinder's surface it's a pretty cool idea of what we're what we're doing it's kind of neat if you need it even more explicit this gives the cylinder or the surface this is going to give you the curve how well let's let's think back to those directions that I gave you did we identify X Y and Z did we use one or more components to give ourselves the surface the last thing says if you want to read that again the last thing says um find some points find some points on this thing and if we know that this curve is going to be on the surface if we find two points we can estimate we can sketch it we're just sketching here we can sketch what that curve is so what's the first value of T that you're going to plug in zero zero yeah zero absolutely why because that's what your domain says start at zero that's my initial Point okay now listen where are you going to plug it in are you going to plug it in here here or at the very beginning that that's what you're going to plug it into that's the vector function that's going to give you your terminal points it's going to give you your point on your on your curve so let's plug in tal 0 and we'll do only one more okay you know what I know it's a tricky one but uh plug in t equals z you tell me the point what is that hopefully you're pluging in zero okay zero so the vector is the zero Vector it's 0 0 0 the terminal point now so I'm cheating here okay I'm going to just write down terminal points all I care about is a point right now the terminal point is 0 0 0 so even though this is technically a vector I'm talking about the terminal point of said Vector head notk on that one where does my Vector function start and that's literally where it starts so right now in my purple pan I know my curve starts right there I know for sure now the reason why we draw our surface is because it makes it a lot easier to plot points and a lot easier to estimate what that curve is going to look like so let's plug in another value uh you can plug in one but it's not very interesting it doesn't give you a lot of shape to this does that make sense I'd plug in something like two I won't plug in anything bigger because it's going to get big really really fast but I plug in two and when I plug in t equals 2 the vector is 248 the vector 248 has a terminal point at the point 248 heading off you with on this one listen 248 has got to be somewhere on the surface just like 0 0 is somewhere on that surface you guys get it just like the the curve had to be on the initial curve the vector function had to be on the curve the vector function has to be on the surface that Point's got to be there so we're going to estimate Here's 2 4 let's make it and now I did that for convenience okay so that I know this point is 248 why cuz I did it ahead of time all right but uh 2 48 do you see how the point 2 48 is going to be on this cylinder right up here it's on this cylinder it's not inside of it it's not out it's on this cylinder head no if you're okay with that one and if it didn't work out like that if you if you didn't make your I just kind of made it a scale so I 1 2 3 4 5 I made sure that was eight therefore this curve does this it starts here for sure after 2 seconds it's going through that point it's traveling along the outside of the surface it's going to continue on it doesn't stop here but it's going to look something like this and we we use this to go well you know what that t Cub that t cube is climbing faster and faster and faster in it so when I sketch this I'm going to draw a straight line I'm draw something that is an exponential look to it so it's going to climb faster and faster and faster but it's climbing outside of this so it's going to look something like that I always do it better fast about like that and the orientation is that way I want to know if that one makes sense to you show fans if it does you feel okay with that so we had initial Point yeah it's got to be on the surface find the surface first find an initial point if you if you can at least find two points sketch along the surface not inside of it generally not straight lines along that surface are you sure it makes sense let's try another one uh we're going to go through this one about the same pace so hey firstly do we get a vector function is this Vector function going to go forever or are we going to have a terminal point terminal for sure sure why because it gives me a definite domain for that now with every single Vector function ever you should be able to identify X Y and Z I want you to do that right now go for it x y and z did you get those as Xyz you know with the easy one like like the line it was pretty easy to see it was a line I don't know what that looks like unless I can find a surface to draw it on I definitely don't know what that looks like unless it do you know what that looks like here's what x is doing uh 2 cosine T here's what Y is doing 4 sin T go and then z t i I don't know what that does I know that Z's climbing at a constant rate that's that's okay but what's it climbing around what's it on what's it look like that's what all this stuff does this says put together a surface it's a surface it's G to be on the surface we got to do the same thing with that I don't know why my voice did that sory it's a surface I can get all squeaky like that too sweet um it's going to be on that surface I probably should because my voice is the same harmonics as this fan so when I try to lower the fan out of the video it also lowers my voice it's very annoying to try to edit that so ice you just talk like this it's way higher but then you would all walk out so number one you identified X Y and Z yeah number two use at least one probably two components probably not all three sometimes we have two but probably not all three to try to make up some sort of surface now it's counterintuitive you're going to use these two why because you think back to parametrics and if you think back to parametrics you know that xal cosine and Y it's going to give you a circle or a lipse that is what it's going to give you why because cosine s sin s that whole cosine Square sin Square identity equals 1 we can substitute for that so I go this one that that doesn't do much for me especially if I go hey y = 4 sin z i i i can't even graph that so I don't even know what that means but these two these two I know that if I solve for cosine and for sign if ever you see signs and cosiness to the first Power not squar just to the first Power equaling X and Y if you see that you can use the Pythagorean identity solve for S and cosine I know you're familiar with that but it's you got to see to solve for cosine and sign to be able to use that so I know if I solve for cosine and S I got it that's X2 that's y 4 respectively but I also know that cosine squ plus sin S as long as you have the same parameter that always equals one which means that if that always equals one I can take my cosine and substitute X over2 for it I can take my sign and substitute y 4 4 now I've done some math here I actually squared it but you see where x 4 y^ 16 comes from show fany you do you feel okay with that one yeah put it together come on think about this this is in let's think about it how many components do you have two two total components it's going to be 2D or 3D three components 3D for sure well we use two of them yeah but what is it think 3D that's why we think first is it 2D or 3D because we're this is going to change right it matters what is this thing what is that cylinder it's a cylinder along what axis and the shape of the what is it ellipse is on what planey yeah that's exactly right so do you guys see that that's a cylinder two variables but 3D right two variables it's cylinder along Z it's going to have a shape of an ellipse a trace of ellipse on the XY plane I want you to practice right now I want you to practice drawing this ellipse on the XY plane try to draw that cylinder try seriously I'm going to give you a minute uh one minute try to draw that cylinder I'm going to draw it as you're doing it but I want to make sure you know where this thing is it's got to be in the XY plane I need you to know your X intercepts your y intercepts what axis is along try it I got something like that did you get this this ellipse on the XY plane and then we know it's traveling along what axis everybody what what one something like that did you get a picture at least something like that now here's the whole big shebang is this the vector function no this is the surface that we're going to draw the vector function on that's pretty that's pretty cool um the Z okay W gave it away gosh started what's the one component you haven't used yet or at least the one you didn't use first yeah zal t is the curve that's going to be on that cylinder so so right now this is why we do it this way we do the cylinder so we don't get to worry about how that curve is taking shape we know that that curve is going to travel along this cylinder for a certain period of time now how we do it man just plug some points in plug in some points uh what's one point that you would definitely plug in right now Z okay everybody right now what's where where are you going to plug that that point in where you going to do it the very very original part so right here you're going to plug that in it's going to give you a vector but vectors always have terminal points which is really nice it's going to give you a point hey true or false true or false that point is going to be at the origin yes or no let me ask you a question see if you're really paying attention some of you are some of you are not is this can this curve possibly go through the origin no does this cylinder pass through the origin then the curve can't pass through the origin the curve if you're getting it the curve has to be on this soda can it's got to be on there if you get off that soda can you're wrong okay it's if you find any point that is not somewhere on the cylinder you have made a mistake fix your jump because your your math is off all right I can't I don't care what I cannot go through there because my soda can does not it doesn't go through there it's on it's on the edge of that soda can that's what's going on so if I plug in zero I get two I get zero and I get zero so two 0 0 head now if you're with me on the two 0 can you please put the point two 0 0 on this graph somewhere and see where it is can you do that what's the x coordinate what's the y coordinate Z coordinate Here's 2 0 0 oh my goodness is it on the soda can is it on the cylinder yes kind of squash soda can but yeah it's right on there oh wow um what's the next value that you'd plug in don't say two I'd probably plug PL in 2 because that's an important one where things do interesting things I'd probably plug in / 2 now the Z's not going to be very nice but the X and the Y are going to be very nice let's plug in Pi / 2 and see what happens if we plug in pi/ 2 here cosine Pi 2 is zero s of 2 is 1 pi/ 2 is pi 2 you guys okay with that one hey question quo is that somewhere on your cylinder the point Z Zer okay here's zero for x four one to oh my gosh and then up pi over 2 how much is pi over 2 well half of 3 one 1.6 roughly what's the next value that you might want to plug in pi would work what pi is going to do coine PI what is that one good zero hi are these values making sense to you let's see if that's on the cylinder here is -2 y I'm not moving Z I'm on the back side of that cylinder you guys see what's happening what's going to go on and the last one I'd plug in uh I'd probably just plug in 2 pi just see what happens I know I'm missing 3i over 2 but I I already see what's going on here I know I'm starting here I know I'm traveling along on this outside of the soda can I'm going to hit here I'm going to hit back here somewhere roughly there roughly there so I'm going to come up here I'm going to hit here I'm going to hit here I'm going to travel around and then at 2 pi I've got two 02 five 1 2 3 4 5 six half so I'm kind of missing my point over here but do you see what this graph is going to do I want to I want to this is a weird one for people to see do you see where the cylinder comes from do you know the curve is going to be on the edge of the cylinder so if it is this point man that point that's on the cylinder this point that's on the cylinder this point that's on the back side of the cylinder if I miss if I did 3 pi2 it be on the edge this side of the cylinder if I do 2 pi it's right above the two it's I have to draw my parallel lines so remember that I've got to do that so it's right above there it's on the the lip of that cylinder that's that's where that thing is does that make sense to you so okay well let's um let right I missed it make a point bigger it's going to go on the back side it's going to come up here and then it's going to swing up so it's on the front it's going on the back and then it's coming back on the front where you can see it can you visualize that I know it's hard to draw like 3D can you give me the orientation it's going that way that way that way so this is traveling up the edge of a soda can at a constant rate that's what's happening to me it's pretty amazing that we can even draw this I think that's that's pretty cool um do you understand how it should draw it it takes a surface if you try to draw that from scratch with just points you are not going to get the spiral of the soda can you're not going to get it you got to got to do in the order I'm telling you identify X Y and Z use a couple components you know it's 3D it's going to be typically a cylinder if you're just using two it's going to be a cylinder if you got to use all three it's going to be a surface identify a couple variables you can make a cylinder out of hopefully a cylinder it's way easier the one you don't use gives you the curve on the outside of the cylinder how you find it just plug in values of T and verify they're on the surface that you just made show H feel okay with that one I want to change this just a couple couple notes on it I want to change it just slightly uh two times going change two times number one what if I change this to not t but what if the T were a three what if the T were a three I hope you're paying attention because I don't have time to do like 13 more examples but these are these are important okay if the T were a three that would be a three this would ex be exactly the same same exact cylinder but then when I got down here this would happen all these values zal 3 right what's that value what's this value what's this value I'm not going to ask you I'd still be on the cinder but what would happen is that up here at it was so pretty here's what happens if I restrict a component to to like say Z = 3 then the height cannot change which means that my points still here 203 yeah but it's not 20 0 043 yeah that's that's right here what's happening is I'm just I have a tunic can and it's on the lid of the tunic can that's what's going on so this would be this way hit here it hit sorry hit back here somewhere right there it hit over here somewhere and be back so we just float right around the lid of that tuna can so when you get a constant constant component don't let it throw you you still have the same surface it's just restricting the height you're not moving up and down you're literally just Rim rting that that tunic can that's that's the idea your F feel okay with that one okay the last one I want to talk about before we get into limits and continuity which is honestly really easy um is is this one can I change this one or you want me to do a different one you don't mind you don't care I'm not going to draw a picture of it we're just going to change and see what happens okay instead of these numbers here what if we do that I I'd like you to see how this stuff changes and the reason why I'm doing it here is because it's going to be very similar at first because the first thing you're going to look at doing is going okay X I know what x is I know what Y is I know what Z is let's try to do the same thing that we just did with with these two okay I want to see if you can make it that far you guys can identify X Y and Z head not yes or no yes and if you saw the sign and the cosine go a sign and cosine to the first Power I love that that's going to be the Pythagorean identity and you are literally still going to do that so we solve for cosine T yeah it's X over T solve for sin T it's y/ t the only difference between this example and the one we just did was we had numbers here and now those are variable listen listen if the numbers are variable then instead of getting ellipse after ellipse after ellipse after ellipse I help you're listening instead of getting that ellipse after ellipse after lips what I'm getting is this ellipse and then the denominator grows that's a bigger ellipse and the denominator grows that's a bigger elips and the denominator grow oh my gosh now I'm getting a surface do you guys see it what type of surface I'll show you how to find out I go well now now wait a second um I know that I could do cosine s and sin squ and get one that's fantastic and I know this would be x^2 over T ^2 and y^2 T ^2 = 1 verify that you can get that far and this where most people get stuck and they go what the heck Le because now I still have my T's in there and he said get rid of T's that's right get rid of the t's how do you get rid of the t's use the one you haven't used first that guy that's the only way cuz I've already gotten I've already used X and Y up right I used these two first my x's and my y's the component you didn't use now you got to bring in your your pinch hitter okay so if I use Z to represent this t we need that you guys are okay on getting that that doesn't look so good no it doesn't it doesn't look good at all so try to put it into one of the formats that you actually understand for instance If I multiply both sides by Z squared this should be a getting exciting if I put everything on one side what I mean k I'm learning Stone I think that means what in Spanish does it I got to go to Costa Rica man I got to learn it I don't even know how to say cone in Spanish how do you say con I mean oh what is this it's a it's a think about think about this this says um this says cylinder where you're expanding what's a cylinder where you're expanding well it's probably either a hyperb hyper we call those different things hyperboloid or a cone the cone is getting bigger that's the cylinder is getting bigger that's what's at it's a cone this literally a cone since that is positive it's an upper cone it's an upper cone do you guys get it's not the lower cone it's the upper cone it's like a funnel that's what's going on how can you tell it's right here this is a I hope that you remember your section 11.6 cuz this is a 3 Squares no constant one minus cone along the Z AIS just like we talked about didn't that make sense what would happen here I'm not going to draw it okay what would happen here here's your cone if you start plugging in values from 0 to 2 pi what happens is if this guy does a lot what our original thing did it starts with the origin and starts traveling along that cone and it spirals up that cone that's what's going on now of course you plug in some numbers and figure that out U to get the the spiral but that's traveling along that cone you still you still use that that component that you didn't originally use to give you the trace on the surface so everything works identifi X Y and Z use two of them at first to try to get yourself a cylinder if you can't get a cylinder use the other one to give yourself a surface one that you know like a conb then reuse that one to travel along that surface is this explained well enough for yall let tell you what let's go ahead let's take our break um I want to talk to you for a second before you all do after I pause this and then we'll come back do uh limits cut all right welcome back hey uh since we're talking about functions we get to talk about limits yay I can tell by your happy smiling faces that that's a big yay well here's a nice part about it limit yeah limits aren't the best because there there's a lot that can go wrong with them like indetermined forms and you got to use loal Rule and you got to understand what the function actually what the pieces of the function actually look like you got to know what they look like like but the good news about them is that there's nothing tricky that we're not trying to tricky here with limits uh the idea of a limit is if you have the limit of a vector function it's literally saying the limit of each component of the vector function gives you the limit of the vector function so what that means is if you can find the limit of f of T the X component and G of T the Y component and H of T the Z component that will give you the limit of a vector function now I want to say a couple things about this you can see it's incomplete firstly it should probably go without saying but I'm going to say it anyway you have to go to this you have to approach the same number for all three components so if your limit of the vector function is approaching a you got to let all these approach a furthermore if you look at it it says the limit of a vector function is uh what of those brackets mean what are you getting out of this you're getting a vector that should actually make sense what this thing says is hey what Vector is the vector fun function approaching as we're getting close to this value of T it say what Vector we getting close to we're getting close to this Vector um if if you also think about it the terminal points of the of this Vector function the terminal points are all in the curve right so from the right side and from the left side we're doing the same thing it's just we're not traveled along the curve right now we're saying hey what Vector are these two guys coming together at that that's what's going on here so um even though we're dealing with a vector function it's just asking what Vector does this Vector function approach as T approaches a and we're approaching this Vector it's just a vector head now if you're okay with with that one so far now even though you can do these independently and just put them down um typically what I like to do is is do the limit of each individual one because you're going to do different things for each component of this for for instance if I want to and I'm going to write the answer right here if I want to find the limit of this verify it's a vector function for sure it's got Vector notation it's got some T's in there T's approaching number and if you were in my Cal one or Cal 2 class you know the very first thing you do with limits every single time is plug the number in man that's the first thing you do because dealing with continuity if it's continuous the limit has to exist and it has to be the function of that or value of the limit has to be the value of the function so if we have something that's continuous plug the number in works every time only place it doesn't work is when we're not continuous and then you have some other tricks things that you do so I would find the limit of this and this and this find the limit of each individual component mash them together and you got yourself a vector so let's see is there anything tricky about this one can we just plug in two and be okay how much do we get beautiful can we just plug in two and be okay no no in this case you go well but you can do other things you can say if I get 0 over Z I know for a fact that that is a removable discontinuity otherwise known as a whole means I can Factor it and simplify out the problem boom boom can you plug in two now yes and how much do you get tell me something else you could have done with that problem since you get zero over rule would it work absolutely you're going to get 2 T over 1 hey plug in 2 you get 2 * 2 is 4 do I care no I don't care at all factoring for me was easier here I really don't care I'm not here to teach you limits right now I'm here to teach you that Vector functions still have them and you perform them the exact same way you always would plug in the number if you can't do other stuff ly talls is on the table we're still dealing with curves okay all the factoring is on the table how about this one am I going to have to do anything fancy with that one even though you got a fraction it's there's no issue here there's no discontinuity you can't even get zero here so plug the number in we get two fths so what this the only thing I I care about here is that you're able to do limits of each piece individually and then put them back together the limit of the X component y component and Z component create a vector so you need to put it in vector format that means that you have this with Vector notation or you have this either one of those is fine and I don't care which so what this says is hey uh what Vector is this getting close to as my time as my T is person two it's getting really close to this Vector that's what's happening what Vector at time equals 2 what Vector at time equals 2 there isn't one why cuz it's not even toine I can't even plug into for that Vector so this would be considered like the whole part of uh Vector functions we don't have that actual Vector what it's getting close to you remember that how you can come from both sides to a limit even though the point's not there the limit still exists that's what's going on here head not if you're okay with with that one the next one uh same exact thing we still have a limit it's obviously a vector function we have X Y and Z let's write out the limit of each of those components individual why don't you go ahead and do that right now limit of cosine T limit of tangent t t limit of T Ln t s t approaches Zero from the you remember what that means from the from the right for you guys from the right can anybody tell me out there what's the limit of cosine t as T approaches zero from the right if you don't remember remember that with continuous functions right and left don't even matter just plug the number in doesn't even matter so since cosine T is continuous everywhere it has no ASM tootes no nothing just plug in zero what's the limit of cosine t as T approaches Z so from the right it must also equal one you guys okay with that one there's also a couple ideas like hey what's the limit of tangent T over T when T approaches Z from the right and you go wait a minute uh that's uh that's Z over zero lals because ly talls gives you stuff like see see squ and you go okay well could you do it why not you can do things like that you can can rely on some identities uh I've proved this in calc 1 but you can see this is equal to one why because the limit of s t over t as T approaches Z is one and it's based on that you can do things like this I'm cheating because I'm not going to write the limit which I would kill you guys for but uh you can do things like that and switch them go hey limit of sin T over T is 1 cosine T that's that's that's also one so 1 * 1 this limit is one so if you have that one you could do L you do a lot of stuff with this all said and done though did you guys catch that did limit limit that limit that's an identity it's proved by the squeeze theem in calculus one it's on the video if you want to go see it but that is one so limit of sin t t is t apprach 0 is one this is as T approach 0 1 over 1 1 * 1 is 1 now this one that's a nasty piece of fun right there because you go that's um uh oh crap how much is that when T approach zero Z zero how much is this s t approach Zero from the right the only reason why we have from the right here is it negative Infinity remember the Ln so zero times infinity that's called an indeterminate form and when we do things like that you have to change one of these to a [Music] denominator we go okay this is Ln T over 1/ T then it becomes from 0 * Infinity becomes Infinity over infinity that's the indeterminate form look it up from Cal 2 uh that's section 6.7 I believe so look that up for indeterminant forms after you change to an indeterminate form so so from an inter terminate form you should get Z over zero or like this Infinity over infinity any time you have 0 over zero or Infinity over infinity and you can't factor and you don't have identities that's when you have to use lol's rule so you use lol's rule here lal's rul is derivative the numerator over derivative denominator and then simplify plug it in then plug it in so don't forget L rule don't forget that we can do things with indeterminate forms you just can't B I'm baking Bing praying begging both okay look please don't do things like this I'm begging you I'm beg hey 0 time Infinity is zero he it causes me to drink at night please don't do that because sometimes sometimes it does and sometimes it doesn't and that's and you don't you won't know when please don't do things like 0 0 equals 0 or it sometimes does sometimes equals one you have to go through the motions you actually have to do to Lear show if hands feel okay with with these ones if you are struggling with this you can see outside of class I'll explain indeterminate forms to you all day long or you can watch C two indeterminate forms uh but that's what that one's going to be all said and done though does it give you a specific Vector yeah write down what that vector what is it simple answer for a hard problem hey what Vector is this Vector function approaching as T is approaching Zero from the right that Vector 1 1 0 that's the vector that's happening so F okay for real if you don't remember things like this uh tan inverse of t as T approaches negative Infinity this would be that here's tan inverse remember that function bottoms out of pi/ 2 don't forget things like that this would be netive pi/ 2 stuff like this exponentials when you're approach negative infinity exponentials look like this they have horizontal ASM tootes so as we're approaching negative Infinity this guy is approaching Z zero just a little Little Couple Basics but I didn't want to leave you hanging I don't have time to go over more limits I apologize for that but we're we're not in here to learn limits again we're here to learn that limits are done really straightforwardly you just you plug in the numbers if you can each individual component if not you have lows you got Factor you got lows you got indeterminate forms use those to your advantage are are there any questions at all before we we continue could you write that in standard basis form so when you see in the back of you go man I'm not seeing 1 1 Zer how could you write that Vector another way yeah so either way that's fine now last little thing in our section kind of a nice B nice little thing here continuity it's it's pretty basic uh here's what we know in order for a vector function to be continuous each component has to be continuous in order for something to be continuous it has to be defined this means you got to have a definite domain to it right so so basically finding out where a vector function is continuous is finding the common domain like 99% of the time that's all we do so with continuity just look for the domain and if it's in the it's going to be continuous on on that domain so let's find out some continuity we'll practice just two of them and be done with this section yes we'll talk about that later so continu honestly comes down to finding domain uh find the domain we we did it earlier firstly um Vector function yes no is this a vector function defined everywhere for any value of the parameter T is it defined a where is it not defined okay so we know stuff about about continuity we know that if we have somewhere we're not defined then we don't have continuity at that value which means we either have a a a hole or we have this ASM toote sort of idea that that's still that idea still works here so what we look for is okay hey um let's look at our first our first component X component cosine T minus one is cosine T defined everywhere yes anything how about the over T is this whole piece defined everywhere what problems do we have can't be how about this one how about the square root of T is that defined everywhere what do we know about t in that case what's it got to be got to be greater than or equal to zero from that piece that's what we get how about the denominator what value of T from there can't you have are you guys getting the picture here up here that value of T can that t be anything this one right here yeah that's fine how about that one can't me that's redundant but we write down all of these pieces and then we put them together so from our first component we go yeah t can't be zero obviously it's on the denominator I know know that T's can't be negative so it's greater than or equal to zero I know T can't be 1/2 cuz denominators can't equal zero I know that that t is fine that's the only good looking one up here and then I got that t cannot be zero because we can't have denominators that are zero that's the idea we're not talking about limits we're talking about being defined we're talking about continuity right now head you're okay with that one now put it together look at the most restrictive pieces this one says I can't equal - one2 okay but this one says I can't equal zero that says the same thing but that one says I got to be bigger than zero do does this even matter if I've got to be bigger than zero anyway that doesn't even matter but then this comes in and says but you can't be zero can you tell me where this function is defined and where it's continuous where is it continuous and defined they all be defined at same same same T it's called the common domain so I know it's defined and continuous on the interval T is greater than strictly greater than zero the - one2 doesn't matter it's not even in that most restrictive region of continuity with me yes no what do you think yes okay let's do one how about you try it at least parts of it you know this is clearly a vector function is this Vector function going to be defined everywhere so therefore is it going to be continuous everywhere new it's not toin in places it can't be continuous places so let's let's take a look at it I want you to look at at least the first component and the last component and write down some things about it left Siders what's that tell you about about this what values of T can't we have for the first component can't be I'll take the two but also what else you said equals z add the four to plus minus or you factor it the last one what values of T can't we have for a cube of T any can I have positives in here can I have negatives in there yes can I have zeros in there yes all values this is fine so the cube root of T you can put anything in there which means it's continuous everywhere there's no problem with the think about the cube root of t a cube root looks like this almost looks like the tan inverse that's what it looks like it's continuous everywhere this one that's going to have an ASM toote at 2 and -2 because you can't factor that out it's going to be ASM totic vertical asmt there you with number this one's weird that one's weird so you go sin inverse of T goodness how high does sign get and how low does sorry uh sin inverse works works this way it takes sign which goes piun 2 to piun / 2 And1 to one it says okay s inverse flips those so if I can get out negative one to one from sign yeah get I'm getting my invers is confused if I can get out1 to one for my sign that's what I get to plug in for my sin inverse so the the only place is even defin is if I get 1 one does that make sense to you sign inverse the inverse of inverse of sign says you can plug in anything to sign yes plug in anything to sign but what can you get out of sign you can only get out things from negative 1 to one right so the inverse says you can only plug in things from NE 1 to one that's that's all you can get out or sorry all you can plug in sign inverse because it's all you can get out of sign now what's relevant here is this relevant that's not the most restrictive thing in the world all right it says you plug in anything is this relevant you better believe it I can only plug in numbers from 1 to one here that's it that's all it's defined for is this relevant not with this one so this says I'm not even defined outside of this range this doesn't matter this doesn't matter because I can't even plug in numbers outside the neg 1 to1 so we are continuous on that interval uh one thing I do want to say this way though be sure that you can use interval notation appropriately you need to know when to use brackets and when to use parentheses the equals you have brackets not equals no brackets you use parentheses have I made this make sense for you at least limits in continuity that's the rundown it's actually not not the hardest thing in the world any comments questions at all uh about this stuff before we take a little pause not break pause you good okay