14.1: Functions of Several Variables

it's functions of several variables so let's start out with the basics notation you've seen before is f of X f of X represents a function in terms of X example you might see something like y equals 4x squared Y in this case is the same as f of X this is a function of one variable what we're gonna do now which I think you've not seen before is Z is going to be a function of x and y it's gonna be a function of two variables so some examples area of a triangle so area is 1/2 base times height that's a function of two variables area is a function of the base and the height to write this in function notation you would write F of B comma H is 1/2 base times height so that Z equals f of XY Z we call the dependent variable X and y we call the independent variables so you guys have seen that notation before and definitely heard those terms we're looking at functions of more than two variables or more than one variable rather so it could be two three variables etc so if you have a function of three variables maybe you call it W that could be f of X Y Z we are mainly going to focus on functions of two and three variables we're not going to go much beyond that important piece to know is that the restrictions of the independent variables determine the domain of f so if I ask you to find the domain of the function you're gonna look at the domains of each of the independent variables or the restrictions on the independent variables here's our example we are going to find the domain with f of XY is equal to natural log of the quantity XY it's two different ways that you can express the domain one is in words and then one is in a picture natural log we only take the natural log of quantities bigger than zero so we need XY to be greater than zero so like I said there's three different ways that we can show that one isn't a picture so if we look at a coordinate plane where does our domain fall so where are allowable points so where's XY first and third quadrant what about the axes no that's one way to show the domain in other ways in words so the domain is all ordered pairs and in quadrants 1 & 3 but not on the axis that's the second way to express the domain third way is with a little bit of mathematics shorthand so D is all the XY points such that XY is greater than zero so all of those are different ways to express the domain WebAssign is gonna ask you to do multiple different versions just please use whatever form you are asked on a test either I will tell you I want a graph I want it in words or I'll let you choose does that make sense okay great we're gonna do another example then we're gonna move on and talk about graphing okay so the second example we are looking at f of X Y Z is equal to x divided by the quantity square root of 9 minus x squared minus y squared minus Z squared okay that numerator we don't need to worry about denominators what we need to worry about few things denominator cannot be zero and then everything underneath the square root has to be positive so we know that nine minus x squared minus y squared minus Z squared has to be greater than zero so we don't want to divide by zero if I move the nine over and divide everything by negative 1 I get x squared plus y squared plus Z squared is less than 9 so if we're expressing the domain two different ways one is that math shorthand the domain is all X Y Z such that x squared plus y squared plus Z squared is less than 9 one way to express the domain other ways in words what is this in words like if we were to graph that what would it look like it's a sphere everything inside the sphere that has a radius of three so the set of all X Y Z inside the sphere of radius three which is centered at the origin okay also with problems like this you might ask to be to find a specific value like maybe find f of 2 1 1 this should be pretty intuitive plug in 2 1 & 1 if you do that you get 2 divided by the square root of 3 let's see okay questions on domain before we move on to graphing great we're gonna do some graphing then so let's say that Z is our function of two variables functions of x and y that is going to represent a surface in 3-space so for this next example we're gonna do two different graphs first one f of XY is equal to the square root of 4 minus x squared minus y squared so my suggestion to you would be to first replace f of X Y by Z just cuz it's easier to work with okay this is not anything that we've seen before so we might have to do a little manipulation to get an equation that we've seen or that is familiar to us so we're gonna square everything so we add Z squared equals 4 subtract x squared subtract Y squared which gives us x squared plus y squared add Z squared equals 4 now that's a sphere centered at origin with a radius of 2 that is not what our initial surface is though why hmm yeah so how does that affect the graph ok so what I wanted us to notice is that that f of X Y has to be at least 0 because it's a square root so that is telling us that Z has to be at least 0 so rather than a sphere we have a hemisphere so that is my best interpretation of a hemisphere second one f of XY is equal to one subtract X subtract one half Y anyone know what this is going to be i graph this one is it going to be if nothing's squared all linear terms what it's gonna be a plane my best suggestion for planes is to find all the intercepts so make ymz zero find X make X and Z 0 and find y x and y 0 and find Z so if I make Z 0 and they make y 0 I'm gonna get X to be 1 and make Z 0 and X 0 I'm gonna get Y to be 2 I'll make these to 0 I get C to be 1 have you guys graphed planes before we're there any like this one you're on your WebAssign ok one person nodded yes so I'm gonna say maybe this is generally what we do we're gonna plot those 3 intercepts and then you sketch out the triangle so obviously your plane continues but that gives the position of the plane okie-dokie any questions on graphing before we move on to our last topic our last two topics no we're okay okay next one we're going to talk about level curves and contour plots okay so let's say we have some figure like this and that is the graph of Z equals f of X Y so basically it's like a big blob guy's picture that okay so here's what we're gonna do we are gonna take a plane and we're gonna slice it through horizontally so we're going to take a plane maybe and slice it through here so that is the plane Z equals K okay do we recognize that when we do that there's gonna be an intersection of the plane and the surface might be something like this that's where they intersect okay so the intersection of the plane and the surface that's called a level curve so a level curve is when you slice a plane through the figure or through the surface rather it's the figure that results so for example if we look at Z equals x squared plus y squared my first question for you is if we were to graph this what would this be it's from last chapter well really the first chapter the paraboloid yes paraboloid yeah spiracle paraboloid remember we made Z a whole bunch of different values you always got a circle but Z is always positive so look something like that okay if we wanted to find it a level curve we would make Z different values so maybe we would plug in 1 that would give us a circle that's one level curve maybe we would make Z 4 that's another circle it's another level curve so every time we put a plane through here the intersection is always going to be a circle do you guys see that so if we were to graph them these are gonna be all the level curves so those are the set of level curves each one is gonna correspond to a specific Z value so that's the way you find a level curve so you might be told on WebAssign find the level curve for z equals 4 in this case it's a circles radius - okay here where we have a bunch of different level curves written that's called a contour plot so a contour plot this when you sketch the set of level curves as soon as you sketch more than one you sketch multiple of them that's called a contour plot so the contour plot for the one that we just did if we make Z one we get a circle of radius one we make Z four we get a circle of radius two if we make z 9 we get a circle of radius three and keep going now when you ever you draw a contour plot I made these kind of small but with each of these lines each of these level curves you need to label Z so this outer one is when Z is 9 this one is when Z is 4 this little one is when Z is 1 ok you will be asked questions based on contour plots so for example a lot of times what you'll be asked is there'll be a point somewhere on the contour plot and I'm going to ask you some information about that point for example what happens as I move toward the origin so imagine this point is somewhere on the figure as I move towards the origin what is happening on the figure to the point guys are making this Z is decreasing the points going down yeah that's all I'm looking for oh so also as I move away from the origin point moves up the figure now do you know what I'm asking for okay great ready for another contour plot it's gonna review some precalc skills specifically some graphing ok we are going to sketch the contour plot of f of X y equals x squared subtract 4 y squared okay so to sketch the Scott contour plot you're gonna choose different values of Z the one that is probably the best to start with is if Z equals 0 if Z equals 0 you get x squared equals 4 y squared so you take the square root and you get x equals plus or minus 2y what is that two lines probably better to think of it like this okay so we're gonna start by graphing that I would make your graph on the larger side because we have some other things that we're gonna need to graph and then remember to label which one that comes from okay I'm going to tell you the next best one to choose is four so we'll do that one next if we choose e2 before we get four equals x squared subtract 4y squared divided by four so we get 1 equals x squared over four subtract Y squared what is that come on guys what is that a graph of this is a hyperbola okay good what else do we know about this hyperbola which direction is it opening let's start with that does it open up down or left right we don't remember hyperbolas so good yeah it's whichever one comes first or whichever one is positive so X direction it's gonna open left right if you put X to be 2 you'll get Y to be 0 so that's this point here that point there your hyperbola is opening like this if we chose equals for what might be another good one to choose yeah is e equals negative 4 what do you think we're gonna get so divided by negative 4 when we do y is gonna become positive so that'll be y squared subtract x squared over 4 okay so this one is a hyperbola again if we make X to be 0 we get y to be plus or minus 1 so this will be kind of like your vertex of your hyperbola and at this point you should get the picture so for your contour plot there's gonna be a whole bunch of more hyperbolas going up down and then left right okay questions on sketching contour plots are you gonna ask me when we're gonna use this if you plug in a 0 for X here you get Y to be 1 is that enough information so then here if you make y 0 X is plus or minus 2 this is just the intercept okay if I'm at the origin and I move left what is happening if I am on the graph like if I'm walking on the graph yes he is increasing and moving up if I'm at the origin and I move along the positive or negative y-axis what's happening if I'm on the surface Z is decreasing I'm going down so what I also could ask you is I could give you this contour plot and ask you what the figure is so what is the figure it is a hyperbolic parabola that is the saddle yes okay we've got a few more things to talk about one of the things we were going to talk about is was graphene hyperbolas do we want to graph hyperbolas anymore do you think you are okay you got it okay great good I don't really want to talk about that it's easier for me last thing we need to talk about is what is a level surface okay if we have two variables if we look at the graph we can find level curves that are two dimensional so crap the graphs are surfaces we can project as level curves so here's what I mean by that if I have a function of two variables the graph is three dimensional the level curve then is two-dimensional with me okay if I have three variables it's a four dimensional graph we are not graphing that obviously we're gonna look at level curves but instead of level curves we can project level surfaces so a 3d graph has a two-dimensional level curve a four-dimensional graph has a three-dimensional level surface so here's what I mean by that let's consider the function of three variables to be x squared plus y squared plus Z squared we're gonna describe the level surfaces so instead of inputting for Z this time you're gonna input for F so maybe we make F 1 so we get 1 equals x squared plus y squared plus Z squared what's another good value to use for F in this case for eventually we're going to generalize when F is just some value K looking at all of these what are the level surfaces spheres yeah so inwards we would say the level surfaces are spheres centered at the origin what happens as we move away from the origin in any direction close F increases value so if you move away from the origin the value of F increases okay ready to do one more example and then we're done okay last example you were going to use the contour map to estimate the value of f of 2 comma 3 thank you for all of your feedback don't recall asking for it but thank you anyway okay do you understand how to read this map now a little this is a contour plot so this outer circle is when we pass through the plane Z equals 50 on the surface that was the intersection this one here is when we Plast through the plane Z equals 40 this one Z equals 30 this is Z equals 10 so those are all the level curves put together onto a contour plot or it also called a contour map okay here's the way we're gonna go about this we need to find out where F of 2/3 happens so f of 2 3 happens about here okay so it's not quite 50 kind of close to 30 kind of close to 40 I would estimate it to be maybe 47 48 because it's close to the 50 anything that you gave me that was around 50 but less than 50 I would except sound good so if you told me 30 I would not take it 47 48 49 Oh questions on the first section

Channel: Alexandra Niedden

Views: 49,507

Keywords:

Id: 4SbxluFD47U

Length: 30min 5sec (1805 seconds)

Published: Tue Sep 17 2019