 So As you know, hopefully We have our first midterm exam on on Thursday That's what it says. So it's going to be right here from 340 to 5. Please arrive early Because there's not enough space for everyone. No, I'm kidding but First come first serve But you as you see we are we are pretty tight on space And as I told you I tried to get a second room, but my request was denied unfortunately But I think we'll be we'll manage but but just you realize there's so many students So there are a lot of logistical problems as it is. So just to make these problems a little bit Easier to handle please arrive as early as you can. Well, obviously you can't come before 330 because there is not a class But if you can arrive shortly after 330 and take your seats, I'll be great because then we'll distribute the exams So we can start start exactly at 340 Okay now all the information about About the exam is available online and I will not go over it now But I just want to mention a couple of things one is about About what's called a cheat sheet You will be allowed to have one page one page of formulas Standard size not This is standard size It has to be handwritten Number one has to be handwritten. It cannot be typed. It cannot be a photocopy. It has to be done by you That's the point of this And it has to be on one side only not on two sides But on one side only if we see that somebody has two sides or a much bigger sheet of paper or something else We'll have to confiscate it. I'm sorry But we have to be fair to everyone. Okay, so we'll just apply the same rules If we see that there is one which does not conform to the standards, we'll have to take it from you Okay No other materials are allowed no scratch paper if you need scratch paper later on an exam You'll ask me and I'll give you some paper But there will be just one page of formulas on one side that you will be allowed to have and then I will Distribute with with the ta's will distribute the exams to you. So you'll have the exam on which you will write your solutions. Okay Is that clear? Okay, no calculators. No books. No nothing. No other no scratch paper Okay Are there any questions about the logistical aspects of the exam? Yes You can you can write anything you want on the GG as long as it's handwritten as long as written by you Okay, handwritten in one side and on the standard side, please sheet of paper Okay Any other questions hi, what is going to be curved? It's a it's more of a it's more of a terminological question. I guess So it's a question not so much about the exam about this midterm exam but about the grade of for the for the for the course right because on this exam We're not going to give you a Grade a letter grade necessarily we'll see but the point is that we'll just give you some ranges Maybe for the for the grades just to give you a ballpark ballpark figure for the grade Okay, you'll get the score so we say five or six problems So it'll be say 50 or 60 points maximum and you'll know what your score is and we'll give you the idea of where You are in them, you know in the class. Okay, but of course This score will be taken as 20% of your of your final score For this course, right and so when we get the final score then at the end of the semester We'll we'll look at the final scores for everybody and we will then derive the grade and it will be curved in the sense That there are no predetermined preset ranges that the ranges will depend on how everybody's doing and how the scores break Okay, so in that sense it is correct Any other questions about yes, I'm sorry No, if you need scratch paper, you just ask me big out And I should tell you at the end of the at the end of the exam Please put the your your piece of paper the sheet of paper in inside the exam. So we have it. Okay I'm sorry No, all right. Well, you will If you if you absolutely have to have it back just right I need I want to get it back I'm gonna keep it for posterity Alright, so now we'll talk about Now we'll talk about the material for this meter. So today is a review lecture. So what I'm going to do is I'm going to review the material And then I will you can ask me questions so there are There is there is there are various resources for your preparation for the for the exam which are available online there is There is a mock me term there are review problems, okay And then there are office hours or also which are organized by your by your TAs and this is another resource that you can ask me questions Now there will also be office hours right after this this class right here where you can also continue asking me questions at that time But first I want to I want to summarize what we've done so far and it's been five weeks Yes, if you really have to ask me now Okay, when tonight tonight the answers for the previous homework Set right and for the review problems will be posted tonight around six six around six thirty seven And mark me tongue with no solutions will not be posted All right, but you can ask me about problems from up now, for example so So I want to summarize what we've done I Want to summarize what we've done in this course up to now Okay, not necessarily in chronological order, but I want to kind of give you a kind of a bird's-eye view of the of the first five weeks of this class and So the most important point from my perspective to realize here is what exactly are we doing here? Okay? So we are studying various objects on the plane and in space so in other words the ambient space There is ambient space That's a space where all of those objects live Okay, and that could be this that's either plane Which we also denote by R2 or It's a space and we denote by R3 and in this ambient space We study we study Objects which are one-dimensional Or two-dimensional that's the essence of this course and we've done quite a quite a bit in this in this vein already So let me summarize what we've done First I want to talk about the one-dimensional objects The one-dimensional objects are also known as curves So the first the first topic that we have studied is is it is curves on the plane and in space In other words curves in R2 and R3 So what what did we talk about? What did we how did we study them? Well, we figured out that there were two major ways of representing curves Okay, one is we can represent them in parametric form or in other words parametric representation Because the curve is one-dimensional we need one extra parameter one extra parameter say t And then we write We write Represent the curve in the following way we write each coordinate as a function of that variable that additional variable t So x would be some function f of t and y would be some function g of t If we work on the plane in R2, that's all because you only have two coordinates Right, but if we work in space in three-dimensional space, then we have a third coordinate So I'll put it in brackets That would be in case we have a third coordinate that would be a third function. So that's parametric representation so an example of this Example of this would be say x is equal to Cosine t and y is equal to sine t It's one of the first examples. We looked at this is I'm looking at the two-dimensional case So there is no third coordinate. So in this case, this is a sure this presents a circle in R2 of radius one and You see here that it's important also to to keep track of the range of the parameter t So if you really want to get the circle, you have to specify the t is between 0 and 2 pi, so then it's really a circle. Otherwise In principle you get if you don't specify this and as if you allow this variable To go beyond this range that it means that you'll have to wrap around the circle many times So when you do parametric representation, keep track of what the limits are for this variable t Okay, so that's the first one The second way to represent a curve is by means of of an equation I'll just first say Cartesian equation in R2 in the case when the curve is in is on the plane It can be represented by one equation with respect to the two coordinates x and y Cartesian Cartesian refers to the Standard coordinate system which we which we draw x and y Because of the French mathematician and philosopher Descartes We call it Cartesian So Cartesian equation in R2 Example Circle square plus y square equals one at the same circle as We as the one we represented over there in the parametric form can also be represented by this one equation on the plane Okay, note that in R3 In R3 if we are in three-dimensional space we would need two equations two equations Because the dimension of the ambient space is now three and so to get to a curve We have to drop the dimension by two so we have to impose two constraints or two conditions two equations and For this reason we don't we usually do not use Do not use this this form this presentation two equations, but sometimes sometimes it happens so an example is Intersection of two surfaces of two surfaces This is in the homework. There are a couple of exercises Like that where you are given two surfaces and they are each surface Each is given by an equation Each is given by one equation and so if you have an intersection of two surfaces It means that you have to impose these two equations simultaneously When you impose these two equations simultaneously you will describe the intersection an intersection of two surfaces is usually a curve So that's why a curve would be then described by two equations So that's that's essentially how the this kind of pairs of equations show up showed up in our studies so far because we talked about intersections of two surfaces and That intersection being a curve But we'll I'll talk about this later when now when we talk about More general surfaces and curves Okay, so that's Cartesian so usually Cartesian equations are in R2 because it's just one In R3 it will be two equations and normally we would just prefer to write a parametric form where you just need one parameter It's really a matter of convenience Okay Now on the plane We also have another standard coordinate system which is called polar coordinates and oftentimes Curves can be represented nicely by using polar coordinates. This is also in R2 on the plane So example, I mean representing curves in polar coordinates So an example of such a curve would be something like r equals cosine theta Yeah, you know, it's going to be a circle which is centered at one half So that's an equation that's an equation in Cartesian core not in Cartesian coordinates But in polar coordinates, so it represents a curve, but using a different system of coordinates And surely we we have much more complicated examples For example, you could put one plus cosine theta and it will look very different Or you can put cosine 2 theta or cosine 3 theta and there are there were multiple exercises in this direction in in a homework We also discussed them in class Okay, so that's another way of representing curves on the plane Now the next we look at the V class of curves, which are the simplest curves both in on the plane and in space And those are the lines simplest curves In other words up to now I'm we're talking about the general methods how to represent curves now I'm talking about a particular particular classes of curves and the simplest class of Curves is lines and lines we learned how to represent in different ways and The standard way is a parametric way parametric representation Which we write like this r is equal to r0 plus vt so t here is a parameter and r0 is Is a position vector of a particular point on this on this line, right? So here's a line. So that's our zero Position vector of a particular point on this On this line and V is a vector along this line Okay, so these are the two pieces of data that you need to to Give a coordinate representation of parametric representation for For a line you need the point and you need a vector which goes along this line Now Here already we are using vectors. So this was I Don't want to separate this as a separate topic vectors in a way It's it's really some technique that we learned to deal with the two-dimensional and three-dimensional space and objects in In two-dimensional and three-dimensional spaces It's very convenient because if you use vectors you can you can add them up and We saw how by using this addition of vectors and also multiplication of vector by a scalar We can represent all points along this line in one stroke. So to speak with by by this formula r0 plus vt But if you wish you can write everything in components you can write our zero as some point as x0 y0 and z0 Remember this is an notation we use for vectors this angle brackets as opposed to the round brackets for For the for the for the point and it's not just you know to be pedantic and to make Everyone's life complicated, but there is a as I explained on multiple occasions There is a big difference between vectors and points right for example points. We cannot add up and vectors We can add up so but once you have a point you can draw a vector from the origin To this point and that's that's the position vector So essentially just means changing the round core round break is by the angle break And if we write V as a BC Then this formula can also be written in components just like this so the upshot of all this is that All you need to know to write down the equation of Of a line is a point on that line in a vector along this line That's all and once you know them you put them in this very simple formulas, and that's that's their parametric representation So these are lines general curves are more complicated The in that this formulas would show up on the right-hand sides are more complicated than this. These are the simplest Simplest Functions in one variable linear functions plus constant function, but in general you'll have more complicated expressions like t squared or higher powers of t or trigonometric functions or whatever okay Nevertheless no matter what your curve is you can always approximate this curve by By a line in the neighborhood of a given point and that's that's sort of the major One of the major ideas that we've discussed up to now Which is linear approximation? linear approximation in other words you can approximate complicated curvy objects by simple linear ones In a neighborhood of a given point not everywhere simultaneously not everywhere at once but in the neighborhood of a given point you can do that And what it what this amounts to in the case of curves is? finding tangent Tangent lines tangent vectors and tangent lines So of course this means to finding tangent lines tangent vectors and tangent lines The general curves and I want to emphasize at a given point Because if you change the point of course the tangent line is going to be different. So let's say here a curve could be Be complicated like this then this would be tangent line So the curve itself could be given by these equations X equals f of t y equals g of t and z equals h of t and There will be some value t zero Which will correspond to x to point x zero y zero and z zero That's a point on this curve and then you can be asked to to write down the equation for tangent for the tangent line at this point and the way you do this is by taking the derivative of The vector function, which is obtained by combining these three functions so the tangent vector v would be just x prime of t At t zero y prime of t zero and z prime at t zero or in other words What we can call our prime of t zero and once you have this v Okay, then you can write down the equation of this line So this is a tangent vector This is a tangent vector and the tangent line will then be given by the equation R is equal to R zero R zero being again as before being Point being the position vector of x zero y zero and z zero Plus this v Times a parameter and this at this point You just I want to emphasize that there is no reason to use the same parameter t in fact It's better not to use the same parameter t because if you use it it kind of you kind of Implicitly suggesting as though this were the same parameter as a parameter for the original curve Which it is not the tangent line and the curve are unrelated except that Just in a very small neighborhood of this point. They are very close to each other So here it is much better to Emphasize the fact that they are unrelated and to use a different parameter For example can use a letter s So there it's a different but I want to emphasize I want to emphasize that this is a different parameter Is that clear? Question. Oh, yes Yeah, well, so so what this means is this means the derivative evaluated at t zero, right? That's what I mean after that I take this v This particular v and I I write the equation in this form I can I want to call it s you can call it something anything else any other letter you like But I want to emphasize that it shouldn't be the same as t because then it's confusing because then it looks like there's some connection between the two curves the curve itself and the tangent line, right, but there isn't the Think of this parameter t say as a time time which you know the time along the curve So let's say there is a there's a bug which is which is traveling along this curve and at this function this position functions f g and h at Would correspond to the position of that bug at the point at the time t, right? Well, then there is another bug which is traveling along The tangent line and these bugs don't know each other. The only thing is that they kind of they don't even necessarily come in contact with each other right because because you see at this point Because when one of them is here the other one could be somewhere else, right? Because it just leave they have different time scales and different time, you know different clocks, so So on this curve this point corresponds to t equal t zero Right, but on the tangent line This place this point corresponds to s equals zero Because when s is equal to zero the second term drops out V s drops out. So what you have is just our zero. So the second bug is here when one his or her time shows Zero right whereas the other one is here at the time t zero. So there's no there's no connection between the two The point of tangent line is to have a line which is the simplest curve simplest possible curve Approximating in the best possible way our original curve in the neighborhood of that point But it doesn't mean that when we parameterize them that the parameterizations themselves should be related to each other Okay, any other questions so so these are These are tangent lines and by the way, they're this includes for example the issue of the slope when we talked about Parametric curves at the beginning when we talked about parametric curves on the plane We didn't talk about tangent lines We talked about the slope of the tangent line But of course a slope of the tangent line can be easily found in the case of a curve in the case of a line on the plane from these equations Okay, so that's more or less the outline of Of the material that we learned about curves There is one more sort of a subtopic here, which is applications of curves So there are various kind of quantities we Learned how to compute Related to curves so there are various integrals more precisely various integrals, so you've got you've got the arc length Right, you've got area under the group under the curve or an area enclosed by by some by some curves And close the area Area and also surface Revolution surface area, so these are there are various integrals which you can set up Related to curves and this and of course this is something you need to know that there's a various formulas Involved here. Okay, but that that takes care of of the one-dimensional objects On the plane and in space as and as you see I kind of organized all this material in under one umbrella topic Whereas in fact we studied that we studied this material in slightly different ways, so not necessarily exactly the same way Because we first talked about planes and then three-dimensional space But I want to emphasize the fact that actually there is not so much of a difference that The way curves are on the plane and in space is actually very similar Revolution surface I didn't want to say the word revolution the big brother is watching So But you know what I mean Okay Now next next is surfaces, right? So these are one-dimensional objects and next we talk about surfaces surfaces are two-dimensional objects And surfaces live in R3 Because we can't fit you can't feel you know basketball in a on the plane in a plane in a two-dimensional space It has to be it has to live in a three-dimensional space dimension one higher So what about surfaces so surfaces as far as the presentation of surfaces is They're given by one Cartesian equation They're given by one Cartesian equation So remember for for curves. They are given by one parameter parametric form or they we have to use We have to do two Equations in R3, but now for surfaces is the other way around we need the one equation But if you if you want to do a parametric form, we'd have to choose two parameters So that means that it's more economical to use Cartesian equations To describe the surface by an equation as opposed to using parameters So but I live it as I leave this as a little note We need two parameters Two parameters now and in fact this is something we will do later on in this course when we talk about Various double and triple integrals. We will have to parameterize curves But for now, we don't really we don't really use this method. We exclusively represent Surfaces by equations by a single equation instead of writing points on the surface by in terms of two parameters and so what comes next is Different examples different examples of of surfaces, right And once again once again the simplest one Simplest class is plain consist of linear ones, which are planes simple as class Okay, so what do? What do we need to know to represent a plane? to represent a plane we need we need a point and And the and the normal vector as I remember we saw in one of the most memorable Images from this course so far You have a normal vector to this plane and you have a point and this is sufficient information to describe all the points on the plane So let me draw it here There's a plane and Let's say this is a This is a normal vector you can't see blue. Oh Oh So now I know when I when I want to leave a sublime message on the board I know which which chalk to use Alright, so which which do you see the white? I hope you see although today the Today the board is very white. So it's let's see. I've got yellow and I got green green Yeah, I think we are all concerned about the environment, right? So Let's use green is that better? Okay, all right, so we we exile the blue we exhale the blue chop no more Only when I only when I want to write something that you don't see Now so we have a normal vector. This is a normal vector and again We have a point. We need we need to choose a point in both cases lines and and planes Now this n this n is usually written as a bc But I don't want to write it as a bc because we already used a bc for for for lines I just want to emphasize that this is a different type of vector. Okay, so let me write okay, and so then the equation the equation of the plane is is simply D times x minus x 0 Plus e times y minus y 0 0 So that's the equation which actually we derived by using dot product in the first place But I don't want to repeat this this now It's there is a very simple way to explain why this is precisely represents all the points on the plane So this is an equation on the variables x y and z where where All other quantities are given. This is d e f our coordinates or components of the normal vector So they should be given and likewise x 0 y 0 and z 0 are the coordinates of a point and they should also be given You can rewrite this By opening opening the brackets Okay, so you can rewrite this by opening the brackets So what you can do is can isolate the terms with x y and z so you will get dx plus E y plus f z equal something D e f g Okay, where where g is x e That's our D x 0 plus E y 0 plus f Z 0 Right, so this is a number. This is a particular number So the reason I'm writing this is sometimes you can be asked the following question You're suppose you're given a plane Let's say you have a plane 4x plus y Plus 3z plus 5 so and you can be asked. What is a normal vector to this? What is a normal vector? So you just look at the equation and you see right away what the normal vector is Because these are precisely the coefficients in front of the variables x yz These are the components of that vector. So for instance, you could be asked here is the equation of a plane Write down the equation of parametric equations for for the line Which is which is normal to this to this plane in which passes through a particular point So then what should you be thinking you should be thinking? What do I need to know to write down equation of a line to write down equation of a line? I need a point and I need a vector the direction vector Okay, so the point let's say will be given So then how do I find out what the direction vector is? Well, the line is supposed to be in this setting supposed to be Perpendicular so the direction vector of the line is a normal vector of the plane And now the point is that you can find you can see it right away when you look at the equation. It's four Four one three is a normal vector Right, so the the equation contains all the information that you need and that's how you can always That's how you can you should approach this kind of problems To write down the equation of a plane you need to know the normal vector But conversely if you already have the equation of the plane you can immediately find out what the normal vector is by just looking at the coefficients Yes What else would you need to do other than find a parallel? line Right, how would you find the line parallel to? well, so to To find a line just to find a line parallel to a plane is not a well-defined question right because There are many parallel lines to a plane Passing through a given point right? let's say If you have a point somewhere here, and you want to look there is a there is a parallel plane To this plane passing through this point So let's say this piece of paper would be represent would be part of that plane But inside inside that plane there are many many lines which you can always say a parallel to this plane So you so you can't be asked Write down the equation of a parallel line you can be asked right down the equation of a parallel plane You see now that is actually a good question. Let's suppose Let's suppose you're asked to write down the equation of a plane which is parallel to this one and which passes through parallel plane passing through through the point one two three Doesn't matter. I'm just taking random numbers Okay What is the equation of that plane well? Since this plane is parallel to the original one they share normal vectors right the normal vectors are the same So I might as well use the same normal vectors as for the original plane which would be four One And three right so these two planes will have to share the left-hand sides Of this equation we'll have to have the same left-hand side. The only thing that could be different is the right-hand side And how do I find the right-hand side? Well for that I use the second piece of information second piece of information is that this plane contains this particular point So for this particular point The right-hand side should be equal to the expression I get when I substitute the coordinates of that point right if I substitute the coordinates of that That point I get four times one plus two plus three times Which is what four plus six Four plus two six plus nine fifty So the equation is just 4x plus y plus 3z is equal 15 whereas the original one it was equal to five In fact, so all the planes which have the same left-hand side But different right-hand side will represent planes which are all parallel to each other But they will pass through different points only one of them will pass through a particular point For example the point one two three But it's very easy to find the equation of the plane which passes through that point by simply substituting the coordinates of that point In the left-hand side, yes So again Finding there is not a single perpendicular plane to a plane right The condition of being perpendicular would would uniquely define a line a Line there is a say through a given point. There is a unique through a given point on the plane There is a unique line which is perpendicular And we just discussed to find that line you just need to know the normal vector But the normal vector you can find out right away Right from this Now there is another question that you can be asked is suppose you intersect two planes and find out the equation of the line Right, so that's a little bit more tricky. So here you have to take the cross product of the normal vectors So there are all kinds of yes If you are given a line Okay, so that's very good. So the There's a question also another question that can be asked is relative positions of line and planes, right? So here so so this particular question I'm asked is about relative position of a line in the plane So let's suppose you have you're given an equation of a plane Dx plus Let's actually let's take this one again. So 4x plus y plus 3z Goes five and then suppose you have a line. So it's like one. Let's say one plus two t And then here will be negative one minus t is 2 plus t and so So this are typical these are typical equations for a line in a plane and you can be asked say Do this line and plane intersect or are they parallel to each other? Because these are really the only options, right? If you have a plane and you have some line the line either is going to intersect somewhere this plane Or it's going to be parallel to it or actually there is one more option Where it actually may be part of this plane How do you find out? Well, it's very simple you simply substitute this Parametric equations into this formula and you see whether you can find a solution for t for this equation If there are no solutions, it means that they never intersect, right? If there is a solution if there is a unique solution means they intersect at one point in if the equation is satisfied for all values of t That means it just belongs to it So in this case we'll have 4 times 1 plus 2t plus y plus sorry plus negative 1 minus 2 plus 3 times 2 plus t equals 5 So we open the brackets we get 4 plus 8t minus 1 minus t plus 6 plus 3t So we get Did I make a mistake? 4 plus 8t, that's right Yeah, I hope you do it better than I do on Thursday 4 plus 8t so okay now we So this for t we get these three terms so that's that gives us 10 4 minus 1 Plus 6 gives us 9 right plus 5 so 10t is equal to Negative 6 so there is a unique solution which is 6 minus negative 6 6 over 10. What did I do? Okay? It's one of those days So my minus negative 4 over 10, but I'm glad that you're paying attention. That's a good sign negative 2 over 5 so What could happen is that it could happen that The all the terms with t disappear it could happen that they all cancel each other out and then you get a number on the left-hand side and number on the Right-hand side and then two things could happen They're either the same in which case it means that the equation is satisfied for all values of t Which means that the line belongs to the plane or it may be say some like 5 equals 6 Which is wrong which is false and therefore the equation is not satisfied for any value of t that means that they're parallel Someone was asked wanted to ask me question. Yes. I answered your question. Okay, good so But as you see the most likely Generic case in a more generic case there will be some non-zero coefficients in front of t And then there will be unique solutions so generically a line and a plane will intersect But sometimes it could happen that they cancel out All right, so where were we? I talked about simplest class of of surfaces namely planes and various questions that could be asked about planes But these are not the only surface that we have studied. We have also studied quadric surfaces surfaces Which are given by quadratic equations Okay, so the next the next example is Here he is quadric surfaces and for quadratic surfaces. We have Equation like this except now we allow Second powers Squares x square y square also mixed products like x y or y z Okay, so what you need to know here is That we can we can break that all of those quadratic surface break into several major groups Okay, and what and how to tell whether a given equation describes a surface in that group And what are the sort of the quality features of that of that group? So you you've got here an ellipsoids and you've got hyperboloids of two different types And you've got paraboloids of two different types Okay, so you have to be able to tell by looking at the equation as to what it represents Qualitatively you don't have to necessarily to you know, you will not be asked to draw What is called? Hyperbolic paraboloid, okay, that's That's not that's not we don't that's not the goal to have the test your drawing skills The goal is to see that you realize you understand the difference between difference different equations that describe quadric surfaces And what are the salient features of different quadric surfaces? all right by the way when I talked about When I talked about quadric surfaces, I did not Mention two important class two important groups of quadric surfaces So in addition to the ones before in addition to the ones discussed before We have this Cylinders we have these two classes one is called cylinders And cylinders are our surfaces which in which which are described by equations in which one of the variables Is not involved so you could have for example x squared plus y squared equals one so the variable z is not present in the equation and It's very easy to represent this you just look you just draw the corresponding curve on The plane which is which corresponds to these two coordinates which are involved in this particular case It's a circle and then you sort you take this you think of this as a frame And then you kind of you move that frame parallel to the z-axis and the surface You get by sweet by by sweeping You know which will be swept by by this by this frame by this curve will be your surface So that's in this case. That's exactly the cylinder So that's why they're called cylinders even though they the original frame doesn't necessarily have to be a circle It could be hyperbole for example or a parabola, so you you'll get sort of a parabolic Surfaces you think of them as kind of cookie cutters You know that you just have a certain shape that you want to cut and then you make a cylinder out of that shape So that's what those cylinders are And this and the second one is Second sort of class of surfaces which we didn't talk about in class But I want to mention because they're also important are the cones Okay, so the simplest example of a cone is given by this equation z squared equals x squared plus y squared that's That's given by this By this picture Well, it's sort of self-explanatory because it is called a cone and it looks like a cone except it's sort of a double cone when you draw it like this as a double cone in the sense that Z could be both positive and negative and There is a basic symmetry between the upper half of this cone and the lower half of this point of this cone if you flip the sign of Z The equation will not change because the square will kill that sign anyway, right? So that's why it has to has these two parts So these are the cones Okay, so planes and quadric surfaces are the two classes of surfaces We have studied in in greater detail than the more general surfaces and for more general surfaces We have discussed linear approximation So again, I want you to see I would like you to know to see that I appreciate this analogy for curves we talk about linear approximation of Curves by tangent lines and for surfaces we talk about linear approximation of general surfaces by tangent planes and just like in the case of just like in the case of Tangent lines we have a very efficient method for writing down the equation of the tangent plane So tangent planes So what you need to know here is that if you have a graph of a function in two variables Z equals f of x y and you have a point x zero y zero and f of x zero y zero Which we'll call z zero Then you should be able to write down the equation of the tangent plane to this to the surface at this point, okay And the way you do it is so that it's actually very similar to the equation For the tangent lines we talked about this last time So the equation specifically looks like this Where so we have two partial two partial derivatives of the function f f sub x and f sub y evaluated at this point This is the equation of this is the tangent plane of the tangent plane through the graph Okay, you can also be asked to write down the equation of a normal line Of a normal line to the to the to the graph. What do I mean by a normal line? That's the line which passes through the same point and which is perpendicular to this tangent plane and We just talked about how to write down equations of normal lines You see and for normal lines what you need to do is you need to keep track Of the coefficients in front of x y and z, right? So what are these coefficients? let me Actually Do it right here So you see in this if I this is a nice way to write it, but If I want to write it in a way We usually write I have to put all variables on the same side So instead of writing like this I would have to write f sub x x minus x zero plus f sub y y minus y zero and then I would have to Take the other guy to this side also So that means the coefficient here is negative one so what So what is the normal vector then? That's this green That this green vector that we talked about earlier. It's just f sub x x zero y zero f sub y at x zero y zero negative one So that's a normal vector And now you can write down the equation of a normal line By using this as a direction vector for that normal line, right? What do we need to write down equation of a line? We need to know the starting point And we need to know the direction vector Well, the starting point is given it's x zero y zero And z zero where z zero is again just the value of the function at x zero y zero and then And the vector the direction vector is just this numbers f x Of x zero y zero f y Of x zero y zero And negative one And then you have to choose a coordinate along this time along this normal line And so let's again use s as before as the second As some but you can use in this case you can use t because we haven't used t in this in the setup yet So Okay, so this is the equation of the normal line Or parametric representation more precisely of the normal line So that's basically that's most of that's like 90 percent of the material we've done And then there are what we've also did last week last week. We started to discuss various aspects of the of the just general aspects of differential calculus And the first topic we discussed here were limit was limits, right? So I kind of have to It doesn't quite fit in either surfaces or lines. So I'd have to start I would I would have to start a different A different topic that would be three Right, and so that would be elements of the differential calculus This is something which we will continue to work on for the next couple of weeks And here we have limits. We talked about limits different aspects of limits and continuity And then we also talked about partial derivatives And finally we talked about the differential Although the differential the notion of differential is of course as I explained in great detail last time It's very closely related to the notion of tangent plane Tangent plane is a graph of the differential So if you understand tangent planes, you understand the differentials as well So that's that's roughly the summary. That's some summary of the material we've done so far And now we have a few minutes left. So You can ask me questions Any questions? Will there be epsilon delta proof proofs on the exam? The answer is no There will be no epsilon delta proofs, but you have to know As you can see on a on a mock meter There's a question about showing that the limit does not exist Right, so you should be able to to give an argument why it does not exist Yes What about proving that the limit does exist? That would be only in a case when it can be reduced to one-dimensional case And where you could use one of the results from one-dimensional calculus On the mock meter number four Okay So it will take So it will take me some time. So maybe I will let me ask let me see if somebody has sort of a shorter question and I will get back to this. Is okay? All right, so you were asking Oh, well, there is one problem. So the question is What what kind of problems for existence of a limit? Should we know A good example is a problem on the homework in that section where the limit can be written in polar coordinates The function could be written in terms of x squared plus y squared And then it becomes some like r squared times logarithm of r. So that's exactly kind of example. I have in mind nothing nothing else No, no not not squished here Okay Go ahead Will there be true and false questions? possibly S represents a parameter of S is a parameter it could be you could use letter t for example Okay hyperbolic Oh you hyperbolic trigonometric functions No, you don't need specifically to know them It's okay Yes I'm sorry Polar drawings would you the question is whether you would have to sketch polar drawings? Polar drawings are fair game because we We started we started them in class right and there are there have been lots of them at homework Of course, I will not ask you to do something incredibly complicated But something very basic like r equals cosine theta r equals cosine 2 theta r equals 1 plus cosine theta is something that you definitely should be able to do Right Okay Say again Yes Yeah, it's actually you know what guys I don't know. Maybe it's not a good idea. Maybe you would rather meet from me talking keep talking because I thought that you know You might be interested in some of the questions your Your colleagues have you know because some of them might open something for you as well So I but if we cannot do it if people talk to each other All right Right Oh you mean for For limits, right? So for the limits I already explained that you could have some the homework you have this function x squared plus y squared Times something like this or not maybe not exactly but something very similar. Is that what you're asking? Yeah So in this case, you should be able to explain why there is a limit when when x and y go to zero And the way you argue is by saying that in terms of polar coordinates, this is r squared times logarithm of r squared And so it effectively so this limit when x y go to zero zero means that r goes to zero So effectively answering this question is simply equivalent to answering the same question But for this function one variable and therefore You can use the methods of one variable calculus and the method of one variable calculus here is the lopital rule Right, so you can easily do it by lopital rule This is something you need to know In principle since this kind of problem was on homework It is fair game Yes What exactly is Perimeter parameter What exactly is a parameter? In the parametric curves Right, so the question is what exactly is a is a parameter in the parametric curves? well, so To answer this We look at the general curve say in r3 in space, okay, and What does it mean to parameterize it? Well for me the easiest way to think about it is as follows So think of this think of this curve as a kind is a rope Which is which is somehow hanging in the in the air, okay like Here is a here is a curve, right But it could be complicated like this but If it is if it is a rope you can always just stretch it Just stretch it out and make it into A line Right, you can always just Straighten it out not stretch not stretch straighten it out. That's what I mean Just straighten it out and make it a straight line when you make it into a straight line You can try to use measure you can try to use measurement on it You can just pick a point and say this point zero and after that you'll have you know one inch two inch and so on So you you can you can put a measurement on it But if you could do that to the To this curve when you straighten it out It means that so you can just mark mark each point each point will have a certain distance from a given point zero So that's the parameter t Then you think of then you put it back where it were but now each point has a parameter So that's what I mean by by by parameter You see what I mean This is kind of it's like parameterizing We are not surprised by the fact that we have a Measurement measuring instruments That they are parameterized in other words when we measure things each point on this little ruler, you know, it has a marker, right But that's because usually we mark things on straight lines So here we deal with something which is not straight But i'm trying to explain is that there isn't so much of a difference between the two because even if you have a complicated curve You can always straighten it out and then you can view it you can apply you can apply the same Procedure to it as as to a straight line And so then you can mark each point by a certain by a certain number Right, and that's this extra that's this extra coordinate All right Any anything else, okay That's a kind of a metaphysical question philosophical question Are they going to be are the problems going to be harder than the homework? On average, I think they will be about the same as homework I mean at least I aim them to be about the same level as homework On average, but it but it's something in the eye of the beholder right what is harder than what is They're not going to be much harder in any case Same kind of problems I think that the mock the reason why I I posted the mock me term is precisely to give you an idea what kind of problems I would consider putting on the exam right, so I think that gives you an idea And speaking of which let me go back to the question about number four number four, I haven't forgotten Number four on On the mock me term I'm very I'm very proud of this of this question Because it it combines so many different things It's too bad. I cannot use it now on the midterm But maybe something similar I I'm certainly not putting this problem on the midterm now after after I explain it. Okay, so Let's calm down So do you want me do you want to hear the solution or not? Yes, okay, but then let's just Let's let's let's be let's calm down a little bit. Okay All right, so sketch we have to first sketch this surface so So see I'm going to do it now in real time because of course I don't remember I don't remember what the surfaces look like and I'm just trying to I'm going to just try to guess So first of all I see that there is one plus and two minuses, right? and so But I don't remember which one is it is it The one which with two parts or with one part. So how do I tell? The way I would tell of course you can You will have one way to do it is to write it all of this on your cheat sheet But then you then you will be using very valuable space for this Whereas in fact you can very easily deduce this by just looking at it And what I see is if you take the y squared and z squared to the right hand side So it's one plus y squared four plus z squared four Right, so what you see is that this is greater than or equal to one, right It's greater than or equal to one So that means that if x If x is between negative one and one There is no chance that I can solve this Right, so that means there is a gap There is a break the x equals zero play x equals zero plane x equals zero plane separates two pieces Two pieces of this graph and now I see that this is a hyperboloid With two parts, right? So and the x equals zero plane is the yz plane Is the yz plane so that means This is one like this Which will be exactly at the point one and there is second one which is a negative one Right, so that's the that's the way I would draw this picture now Well, there is this one over four, but just to sketch it. It doesn't really matter, right? So that's the first thing you need to do second We want to compute the surface area Of the part of the surface bounded by the planes x equal one and x equals square root of five over two So here is x equals one And x equals square root of five over two is something Which is just above one And why did I choose square root of five over two because because then If I look at the plane x equals square root of five over two I will get and I intersect that plane and this Hyperboloid I will get a circle Right and what is that circle? So it will be Square root of five over two squared equals one plus y squared over four Well, we can just write it here So this will be like five over four equals this which means y squared over four plus z squared over four Equals one quarter so that means y squared plus z squared equals one Okay, so that's this that's the circle And now to compute the area I want to represent this as a area as a surface of revolution revolution right, so Now revolution of what? It will be revolution of this of this curve in the in the x z plane Right, so I just have to draw this curve Of course you have to remember that I have to look at it not from this perspective But from the back of the of the blackboard so that x goes from left to right If I look like this x goes from right to left, but I wanted to go from left to right so actually It's going to start at one And it's going to go like this Right, so in fact what I want to do is I want to set say If you want it could be maybe it's better to do an x y plane. Let's do it an x y plane So to do it an x y plane. I have to put z equals zero in this formula So I put z equals zero Okay, and so what I get is x squared minus y squared over four equals one Right, so that's that's actually hyperbole. That's a hyperbole It's part of the hyperbole So what I'm saying is that I want to go from one to square to five over two And then this the formula for the the formula for For the for the surface area is going to be 2 pi y right 2 pi y Times square root of dx dt squared Plus dy dt squared So this will be from Well, I have to take from alpha to beta Where I would have to write some parameterization for it I would have to write some parameterization for this for this curve and As x is some f of t And y is some g of t right and so t will be from some alpha to beta And Right and then I will have to take this integral does everyone agree with this Okay, so now The only thing I think I'm afraid that I'm out of time So I'm saved by the bell But So I feel I feel as though I should let the people go but I will be happy to continue and explain the rest After a five minute break while during the office hours. Okay, so so let's stop here and Good luck on thursday, so I'll see you on thursday at 3 30 Exam starts at 3 40 And now we have office hours