 Thank you Welcome back to our little math theater So you guys doing all right? Yeah, okay so last time Last time we started to talk about parametric curves and Today we will continue We will continue with this topic and We will learn various very important things about parametric curves such as tangent lines arc length and various areas of Surfaces which are obtained from parametric curves So the first thing we will discuss today is tangent lines and This is actually very easy to introduce this topic is very easy to introduce because this is something familiar from single variable calculus in single variable calculus in single variable calculus we study graphs of functions, right and so Graph of a function is something which we draw on the plane with two coordinates x and y and It looks something like this and we usually write Y equals f of x Where f is a function f is a function in one variable So this is a graph. This is a graph of the function so now Often times often times in mathematics or in science and engineering You want to know various approximate results about your objects your object may be too complicated And you want to get sort of the first order approximation as they say or it might be that just you want to understand it Qualitatively for example, let's say if this is the graph of the temperature in Berkeley, California Where x is a time? So let's say over the last week, you know, actually would be more like this Very hot last weekend, right? So then maybe you don't want to know exactly what the graph looks like what you want to know For example the trends does the temperature increase does it decrease things like that and for this It's very useful to find some approximate tools which would you know, you'll be able to say a lot of things about your object without really You know getting hundred percent of the information about it and tangent lines is the first thing that you you can use for that And the reason is very simple The reason is that out of all the curves that you can draw out of all the curves So this is what this is an example of a curve But of course you can draw many more very complicated examples I mean circle and so on goes without saying but even kind of really wacky curves you can draw like anything you want to draw Like a Picasso, you know, it's also param. It's also curve right oftentimes. He oftentimes just drew it in one stroke So out of all of these curves, there's a whole variety of those curves and very complicated But there is a very simple class. There's a class of the simplest ones and those are the lines The lines are the simple curves, okay? so lines are the simplest curves and Equation of a line If a lines can be graphs of functions and those functions usually look like this There will be something like k times x minus x zero plus y zero where k is called a slope he's called a slope and It's a line with a slope k and also line which Passes through Through the point x zero y zero. So let's let's draw. Let's draw it actually Say here. So let's say this is x zero y zero on the plane and the line is going to look something like this, okay so To draw a line what you need to know is a point Through which it passes and also the slope what what do I mean by the slope the slope is the tangent of this angle? so in other words you need to know the angle between this line and the x axis that's this angle theta and The tangent of this angle is called the slope and that's what we call k in that formula The slope of this line So this is the simplest this is the simplest graph and the simplest curve really I Mean what could be simpler than this see the point is You are considering on the right hand side You have a function of x and the simplest function of x that you can write is a constant function and Then the next simplest function is a constant function plus a linear function, which is what we've written but the constant function Could also be thought of a special case of this namely if k is equal to 0 slope 0 This will just disappear and then you would have this so in other words The the case of the function constant function is included because you're allowed to have arbitrary k So k equals 0 would give you the constant function. So that would be Just a special case of this when y just equals y 0. It's a it's also a line It's a horizontal line. It has slope 0. It's just a special case of this In some sense, there is no point. There is no point of distinguishing this case from this more general case of lines So that's why I'm saying that the simplest curves that you can draw other lines Because dependence on x dependence of the function f of x is the simplest possible It only contains a constant term and a linear term in x in other words degree one It doesn't have x square doesn't have x cube not to mention, you know logarithms cosines and all those other complicated functions Okay, so that's the first fact which you have to remember that out of all the curves There are the simplest ones. These are the lines and we know the equations of the lines. They're given by this by this formula And so the next thing the next idea of calculus and we really the most important idea Maybe of all of calculus is that for many many functions namely the so-called smooth functions or differentiable functions You can approximate your function very well on a small scale by by a line or more precisely Approximately the graph of the function by a line two things two ways to think about it a Smooth function can be approximated by a linear function like this which geometrically means that the graph Can be very nicely and usefully approximated by a line and In practice the way it works is that if we pick a point on This curve and let's call it again x0 and y0 Then we can we can think of a whole variety of lines which pass through this point There are many of them there infinitely many in fact infinitely many lines, but out of all of those lines There will be one line Which will be the closest to this graph and that's the tangent line So I forgot to bring my color chalk again, but I guess I hope you get the idea So this is this is this is a tangent line And what's so special about this is that it is the closest one to the graph of the function In the following sense that if if you slightly if you move it just slightly It will intersect It will intersect the graph at two different points, you know If if we blow this up If you blow this up it's going to look like this I just blow up a very small neighborhood of this point It's going to look like this if I change is just change it just slightly You know like this. It's it's already intersects at two different points So you just that you adjust your line just so that it touches the graph at one point That's intuitively what the tangent line is the closest one the closest of all lines To this graph To this graph at this particular point if you change your reference point Of course, you're going to get a different one in other words here I'm talking about the tangent to this particular point which is that point x0 y0 on the big picture But if you if you go on to a different place You go to different place Here for example, of course you will get an entire different line So when you talk about tangent line you have to say tangent at which point it otherwise It doesn't make any sense Each point on the graph has its own tangent line and a priority. They are all different and most in Almost all cases. They will be different So this is not to say that this line can replace the graph of the function because you see they diverge If you go sufficiently far from this from this point, they become different You know this expression going off on the tangent Yeah, sometimes you'll catch me doing that I suppose Anyone is capable of doing that and what it means precisely is that if you go off on a tangent You will sooner soon enough you'll be far away from the object itself But the good news is that as long as you are in a very small neighborhood of this point the difference is almost negligible and this is a key idea of calculus really and Of calculus of single variable now because we are talking now about functions in one variable But you will see that the same idea will be applied very usefully also for multi variable functions For example, if you have function two variables, you'll be approximating graphs by planes instead of lines and so on Okay So so I hope I convinced you of importance of this because what does it give what does it give you? For example, you see that the function is Your function is increasing at this point right if if the slope is is positive like this, right and The function is decreasing if the slope is negative So already you can learn a lot of things about your function by studying the tangent line Now the next next come the question is how to find the equation of the tangent line We know that it's going to look like this, right? because Because all lines look like this more precisely all lines which are graphs of functions They all look like this. So which one is it in other words? We have to find the coefficient k and we have to find this number y 0 and we have to find the number x 0 That's what determines the line as we discussed we need x 0 y 0 in the slope But we already know x 0 and y 0 because that's that is the reference point That's the point on the curve to which we look for the tangent line so we already know no x 0 and y 0 and The question is how to find the slope of the tangent line K of the tangent line and the answer was given Before in single variable calculus and the answer is is very nice. One might say beautiful Okay, the answer is K is equal to the derivative of this function at the point of zero In other words, you don't need to draw anything. You don't need to make any complicated calculations All you need to know is a derivative of your function and usually your function is described in a very explicit way Like you know say very normal function or cosine or sine or exponential function For which you know what the derivatives look like because you've learned them from you know By by calculating them once and then you make a list and you remember them so so some taking derivative is something which which we Know quite well, right? So all you need to know to find the slope is to find the derivative It's to know how to find the derivative and once you know the derivative You know the slope and so then you can put these things together and you get the equation of the tangent line So the final result is that the equation of the fine of the tangent line It's just obtained by combining all this information, right into this formula So instead of k, you will put f prime over x zero, which is what I wrote here Right, and then you have x minus x zero Plus y zero that's the equation of this tangent line No matter how complicated your function is The equation of the tangent line is going to always look like this now Of course, I am cheating a little bit because All of this is applicable to functions for which the derivative exists in the first place Not every function has a derivative. These are the so-called differentiable functions But the functions that we are going to study in this course. They are going to be differentiable And so so this method will apply So you have to realize that this is something which works in some sense for the nicest possible functions Namely differentiable functions or smooth functions if you will The ones for which the graph is sort of a smooth curve as opposed to a curve Which has angles which has sharp angles because if you have a sharp angle, it's not clear how to make a tangent line, right? It's not it's not going to touch the graph because there is a corner. There's a sharp corner. So this case we don't consider We only consider the smooth ones But the smooth the class of smooth functions is a lot is very large and and we are focusing in this course on the Smooth functions, so so we are fine our method is applicable here and once it is smooth It has a derivative and so you can write easily the equation of this line That's what we've learned in single variable calculus now In single variable calculus you study graphs of functions in one variable which are curves on the plane and Last week we talked about more general curves. We said, okay, there is a there are many curves Which you can get as graphs of functions, but not all There are more general curves which are not graphs of functions And there are two ways to represent them one is by an equation like x square plus y square equals one like a circle radius one Or in a parametric form Okay, so that that means that we have now a larger class of curves Which includes but not is equal is not equal is bigger than the class of graphs So now we'd like to ask the same question about this parametric curves In other words, we want to learn how to compute the tangent line to such a such a curve at a given point So that's the question that we are we are going to ask so in other words now We have a parametric curve parametric curve And the parametric curve is given by a pair of functions f of t And g of t as we discussed last time right where t is an auxiliary variable the parameter on the curve and we want to learn the equation that what is we want to find out what is the tangent line to To this curve at a given point x 0 y 0 where of course For this point x 0 y 0 to belong to this to this curve Both of these values x 0 and y 0 have to be values over f and g at the given parameter value t 0 Right, so that would have to be f of t 0 and that would have to be y 0 would have to be g of t 0 for some t 0 some value We'll we'll look at examples in a bit. You'll see what I'm talking about Okay, so what is the tangent line? To this curve in other words, we would like to extrapolate this formula We'd like to generalize this formula Which by the way is the way mathematics is done, you know You you don't immediately get the answer in all cases you may you work out the simplest case and then you try to generalize So if you look at this formula, it may be the answer may not be so obvious because the here this answer involves this function f which You know because we're talking about the graph of a function, which is really the special case special case of the general parametric curves in which Parameterization is x equal t and y is equal to f of this this shows you right away How special this case is how special? Well, it's just that x is t and and y is some function because If you if you have this parameterization, then the first equation tells you that x is just t so you can just substitute x instead of t and you get y equals f of x you get the graph of the function So in other words in this special case the function f f small f small is just the function t And the function g small is another function f capital And now we have a more general case where both f small and g small are some complicated functions Which are not necessarily equal to t or anything on anything given an advance So we need to generalize this formula and it's not obvious immediately what the answer should be Because it really appeals to this particular case to this very special case But it's actually not so hard to guess the answer But to guess the answer we have to remember how we derive this formula in a single variable calculus in the first place Okay, and actually for that I will use I will use this picture So you see What is the slope? The slope is the ratio The slope is the ratio of delta y Delta x the increment in y over increment in x that's because that's how Let's look at the graph of the line and let's just recall Let's just recall The definition of the tangent because remember k. I said k is a tangent of the angle So what is tangent? Tangent when you draw this triangle is that is the ratio of the change in y in this triangle over the change in x so So k Is equal to delta y over delta x on this line? So for this tangent line we we have the same the same thing There is a delta y and there is delta x Now so this is delta y over delta x on the line on the tangent line But the point is that this is approximately but this increment in y For very small on a very small scale is almost equal to the increment in y on the graph itself and The change in x of course is the same for both So you should look at this picture and see that even though the tangent line that it goes off on the tangent They diverge a little bit, but not so much and the closer you are to the point actually the difference is less and less So in fact this ratio is almost the same as delta y over delta x but now on the graph itself and So the slope can be computed as the ratio of the increment of the function delta y this one To the increment in x So in other words this is delta y on on the tangent Right, this is delta y on the tangent and this is delta y on the graph and they're almost the same So the slope for this computing the slope we might as well take delta y of the graph and divided by delta x And when this becomes very small When delta x becomes smaller and smaller, so you are getting closer and closer to the point This becomes what we call the derivative dy dx over dx which is f prime So that's the reason why where you actually get the derivative because what you get is dy over dx and Y is f of x so dy dx is f prime At your reference point f zero x zero So that's how you derive this formula in other words this this calculation is what gives you this So now we can use the same the same formula because we have worked out now the formula for For the slope and we see that the slope k again is dy over dx but Remember y and x are given by this formula in other words y is g of t and x is f of t So we can use that to write dx is f prime of t dt and dy Is g prime of t dt? And so what you get here is G prime Divide by f prime of t Dt and now it's tempting to just cancel cancel out these two Dts And actually you can you can you are allowed to do this under some mild restrictions They are not going to get too much into details But I'm just giving you an intuitive Derivation of the formula, but what I'm saying now can actually be made rigorous and precise And it took some you know it took centuries to really work out all the details and to really explain what this dt really means We'll talk more about this when we talk about Differentials of functions in two and three variables and you will see why this kind of calculation is legitimate because The way it is now dt is kind of a mysterious object and people never people don't explain in In the book. It's not really explained what dt is and I'm not going to explain it now I'll just I'll explain it later because for now We just we just have to sort of take it for granted the fact that we are allowed to cancel them out The only condition which needs to be satisfied is that f prime is non-zero If f prime is non-zero for a good reason because f prime is zero you are in this formula Dividing by it so you are not allowed to divide by zero if f prime is not So this formula makes sense as long as f prime is non-zero and the formula again reads just like this More precisely we have to say at which value of t but remember That's why I was careful here when we talked about the question I said what is a tangent line to this curve at a given point x zero y zero and x zero y zero was a point Which corresponded to a particular value of the parameter, which I call t zero So to be absolutely precise This is a formula for the slope of that tangent line Where both of these derivatives are evaluated at t zero any questions? why is it at t zero because We are calculating the slope at a given point and the point on a parametric curve point is determined by the value of the parameter So because here I'm sort of writing all over the place But here this is the answer to a question which is written in the opposite board Where t zero is introduced so you have to you have to look at both of them. All right, so let's see What do we want to do with this Okay, let's let's look out. Let's do an example. Let's do an example. So find tangent line To the parametric curve given by by this equations at the point t equal one So this means that t zero is one this infamous t zero which appears Appears in this formula and this exercise is one Because well here's it in t equal one my point is I'm trying to use notation in the following way that when I say t I kind of view it as an independent variable it can take any values When I talk about a specific point then I want to say that t equals equal to some specific value In a general formula, I don't want to say which value it is one two three I don't want to say it so that's why I use the notation t zero So t zero just means a particular value of t as opposed to a variable itself It's a subtle difference. So if you if it's lost on you don't worry about it too much. All right Find the tangent find the equation of the tangent line the equation of the tangent line In the case of a graph of a function Is written here? Now in the case of curves parametric curves the equation is going to be y equals I Look at I take this. I mean the equation is always like this and now k is equal to this So I just substitute this k into this formula. So I get three prime of T zero over f prime of t zero x minus x zero plus y zero Where x zero and y zero are the values of the function to begin with so that's going to be f So it's a very straightforward exercise Because all you need to do is just to calculate each of these numbers would show up here So let's first calculate x zero and y zero Okay, x zero is going to be the value of x when t is equal to one Right, it's just this is f and this is This is G So what you need to do is you need to calculate the value of of this function first at equal one That's logarithm at one and logarithm at one Is zero exactly? Very good. So now y zero is you calculate one times e to the one so e to the one is e and if you don't remember what e is You should go back and read It's a it's a particular constant which is defined by the property that the derivative of this function e to the t At t equals zero is equal to one So it's a it's a base of the natural logarithm. So it's a particular number. It's like 2.7. It's like it's like pi It's a very important universal constant So I on purpose chose this example to Because you know in the homework you will you have to deal with this Constance like e and things like that. So you have to get you have to remember them and get used to them Okay, so it's a particular e is a particular number. It's not a variable. It's a particular number Which is equal to one. I don't remember exactly, but I think some like 7, 8, 1 whatever Maybe I shouldn't write it because if it's not correct, I Kind of make fool myself All right So, okay, so we found x zero and y zero that's great and next we need to find a slope So for that we need to find the derivatives of this so f prime of t Is 1 over t and g prime of t So again, if you don't remember how to differentiate logarithm, you have to remember it because this is something This is something from single variable calculus and we are going to use these results freely We are using everything we've learned so far which means single variable calculus in particular We have to know derivatives of all these functions and all the rules how to calculate derivatives like this So g prime you use the derivative of the product. So that's going to be e to the t Plus t I'm sorry plus And now if you if I wanted the value At t equals zero at t zero which is one I'm going to get one for this one And I'm going to get Right and now I need to calculate the ratio between them So let me just write it above So I have to take the ratio of g prime over f prime and g prime is 2 e and this is one So that's going to be 2 times 2 e then I have x minus x zero which we found is zero So actually it's going to be 2 e times x Plus y zero and y zero we found to be e So that's the answer That's the equation of the tangent line Okay So so next what else can we learn about this? Uh amongst all the lines there are special ones. There are ones which are vertical and the ones which are horizontal So how to find out when it's vertical or horizontal? Um You just look at the slope so the slope is as again as a tangent, right? So the slope is zero The slope is zero if and only if the tangent line is horizontal If the tangent line tangent Is horizontal this we learned uh in single variable calculus But we never talked about when tangent line is vertical And there's a good reason for this because the tangent line for a graph to a graph of a function is never going to be vertical You kind of if you have to think about it a little bit and then you'll see that it's not possible So the tangent line can only be horizontal but not vertical in the case of a Of a graph of a function, which is our special case like this But in the most general case it surely can be vertical or horizontal for one thing you could switch x and y And when you switch x and y what I mean is switching f of t and g of t, right? We are allowed to do that because x and y are now completely unequal footing and so If you switch x and y a vertical becomes horizontal and vice versa So clearly vertical vertical tangent lines will something will be something which is will show up as well So the way to see that then is Is better to look at this formula at the at the more general formula which we have just found for the slope And so we see that if g prime Of t zero is zero it means that the tangent is horizontal Right, I would like to say that But the problem is I have to make sure that this formula is valid and the formula is valid if f prime is non-zero So you have to have two conditions satisfied g prime is zero, but f prime of t zero Is non-zero and and this by this I mean I mean the end the both conditions are satisfied Once again g prime is zero, but f prime is non-zero then it's horizontal And if you want to if you want to understand when it's vertical you just switch x and y So when you switch x and y f starts playing a role of g and g starts playing the role of f So we just without thinking just switch them and you will you will get the condition for the vertical one for the vertical tangent lines F prime is zero, but g prime is non-zero Is non-zero that's none is vertical so So let's see Let's see some some example Um when tangent lines are vertical or horizontal How to find out See the problem is if both of them are zero You're kind of dividing zero by over zero and that's really not Well defined so it really depends on the situation. It could be Um, it could be anything so it really depends you have to study it in more detail But if if just one of them is zero by the other one is non-zero Then you can say for sure that it's vertical or horizontal depending on which which one is zero and which one is non-zero So example two X is t times t squared Minus three and y is Three t squared minus three I'm sorry three times Times t squared minus So let's compute So this is f right again. This is f of t and this is g of t So what is f prime? Well, we can write this as t cube minus three t So that's going to be three t squared minus three G prime This is this is now Three t squared minus nine. So that's going to be what 60 right and To find out when it's vertical when it's horizontal We have to find the values of t for which one of these two functions is zero so f prime of t Equals zero means that three t squared minus three is zero Or in other words t squared is equal to three which is the same as t squared is equal to one So there are two solutions t equal one or t equal negative one Okay, so for those values f prime is zero But to be able to say conclusively whether the tangent is vertical We also have to check the values of the other function to do or more precisely the derivative of the other function, right? So we have to substitute these two values Into the other derivative. So we get g prime of one is six So non-zero good G prime of negative one is negative six non-zero again good. Well good in the sense that we we caught the Point or two points in this case at which the tangent is vertical So what are these points? points where tangent line tangent Sometimes I'll just write the word tangent just to make it short but That means tangent lines the same thing the tangent line is vertical are The points for for corresponding to the value t equal one Which is the point if we put substitute equal one we get what? One minus three. So which is negative two, right? So it's negative two And y if you substitute one we get one minus three is negative two times three negative six And The second one with when t is equal to The negative one So if it is negative one we get negative one Right negative one minus three times negative one which means plus three. So that's two That's two and here we put negative here. It doesn't matter because we square So it's going to be the same two and negative six Okay, so that's how you find and horizontal for the horizontal one you have to To do the same but with g prime say g prime equals zero, which means t equal zero right t equals zero and then you substitute here and you see that f prime Is negative three which is non zero. So that means great. So that's at this point We also get a horizontal tangent line And then you find that point in the same way. I'm not going to do it. I think it's clear Simply substitute in in x and y substitute the value t equals zero Okay, any questions? Yes Yes, right Okay, that's that's a good question actually Let me give you one examples kind of to to show you that it could be anything. Okay, very simple example Oh, yes, I have to repeat and also for our worldwide audience. I hope you're being filmed. So um The question is give an example Example when Uh both f prime and g prime Are equal to zero What does it mean geometrically? Okay So So let's do this. Um So I want to find two functions, which at a particular value of a point Value of the parameter have zero derivatives Okay, the simplest function which has which could have derivative zero is t squared Not t because t the derivative of t is one. So it's a constant cannot be zero right that t square already has Derivative equal to two times t and if t is zero, that's zero So let's say this is t squared uh t t squared And let's say that this one is also t squared So in other words what I mean to say is that x is t squared and y is equal to t squared right So what does it look like? Actually in this case x is equal to y So it looks like a line Right, which is kind of diagonal in other words the slope the angle here is 45 degrees the slope is one right Because it's a funny thing. Yeah, you say this is t squared. This is t squared, which means that x is equal to y It's like it's almost like we are eliminating the parameter But there is a catch and the catch is We have to be careful what are the ranges of the variables something which I mentioned at the end of last lecture Because the point is that for both positive and negative values of t This is going to be positive or more precisely. No negative could be zero or positive Unlikewise here. So that's why on purpose. I didn't draw the entire line But only half of it So the image which represents this parameter curve is this half a line And actually then you have to be careful. What are the ranges? I didn't say anything about the ranges If you take just the positive ranges positive values of t is going to be this And if you take negative values of t is going to be the same thing So it's actually if you don't say anything about the range of t and kind of implicitly say that it's from negative infinity to positive infinity Then it's going to be this curve twice you kind of come from you You you come from here and then you come back If you say say for example t from zero to infinity then you are going to to get Just this this half a line once Okay, so now suppose we So we see very clearly graphically we see very clearly what the object is which makes it much more much more easy to analyze But now let's compute the derivative. So f prime of t is 2t and g prime of t Is 2t right So at t equals zero So there's a point t equals zero which is here This is a point t equals zero So when t is equal to zero Both are zero And so the slope as I said you cannot use this formula for the slope because you're dividing zero by zero But then of course the question what is a slope in this case? Well, the slope is one right It's sort of half a line. So you you may feel a little bit uneasy because There's no other end But you can still think about about the slope of this right of the tangent line in this case The tangent is going to be Parallel it's going to coincide with the curve itself at least on this part, right? And so the the slope is going to be just the slope of this curve which is which is one So you have sort of zero by zero, but What happens is that the derivative Is zero just at this point, but outside of this point is non-zero So you can approximate the ratio of two derivatives by the ratio By the ratio of these two functions And then take t to zero because I mean what you see what you get you see is two t over two t And that's one, right So in this particular case, even though you cannot apply the formula, you know that the answer is one But to show you that you are sort of really on a slippery slope No pun intended Let's suppose you put let's suppose you put put something like Five Okay, let's two let's put two, okay So then you have this and so instead is going to be a more sharper sharper line steeper line okay, so then y is going to be two times x This is y equal x but this is y equal two times x so for this one the slope is two, right? By the derivatives now are going to be two t and four t So you see both again are zero Right and so and you cannot use the formula Because zero over zero is undetermined and the point is that even even Even though it's again zero by zero, but now the answer is not one but the answer is two So that sort of illustrates that zero over zero could actually be anything. So in these two examples, it's one or two That's right. That's right. So so that's right So to repeat the he's saying that so it looks like we're just applying the l'opital's rule Which I hope you remember what l'opital's rule is Which is to say that we are actually looking at these two derivatives. So now it's going to be two t over 40 And we don't substitute t equals zero immediately in the numerator and the denominator But we look at this function for t very close to zero and we see what is it? Well, if t is non-zero, this is this makes perfect sense. It's going to be two over four, which is one half, right Except I'm I'm taking the ratio in the opposite order Sorry, so we have to do f g prime over f prime. So it's four over two And that's going to be so four over two and that's two So the ratio itself is well defined even though the If we do it if we substitute too quickly too soon then it will be zero for zero But and then I I don't want to go too much off on the tangent here. Okay, so I'll just uh, sorry, I couldn't resist but I just I will let you play with other examples for example try to to do say t square here But here put t cube or something like this and see what will happen It's really it's it's a very nice Example to consider but let's go back to our curve so So now we know about Tangent lines when they are vertical when they are horizontal. What else do we need to know? Well, in the single variable calculus, we also talked about second derivatives. So You see the point is that first derivative But we're not here to listen to music Should avoid this Now at least not during the during the lecture The first derivative is a slope dy over dx And it tells you the general direction of the function if the slope is positive It means that y is increasing as x is increasing If if the slope is negative it means that y is decreasing as x is increasing But and oftentimes we we call this the first order approximation First order now you can see why it's first order because it's the first derivative It's the first order approximation Oftentimes it's also good to look at the sort of second order approximation. In other words to look at the second derivative in the second derivative Is d squared y over dx squared? Or if you will d dx of dy dx And that's something which tells us If you think of this as the velocity, this is acceleration. It tells you the trend in other words whether Say this function is increasing, but is it increasing faster As time goes by or is it increasing more slowly? and the way This is this can be seen geometrically is from the concavity of the function if the function is like this It means that the second derivative is positive We will call this concave upward. That's what it's called in the book. I'm actually I'm not sure Maybe I would call it concave downward, but that's the terminology. So we'll stick to it So this means concave upward Upward and it means that this is greater than zero and If it's negative It's it's concave downward So it looks like this So it gives you more qualitative features of the of the of the curve So even if you cannot draw the curve right away, it tells you by calculating these derivatives, you will know The ranges of parameters for which the curve looks like this approximately or like this And there is a simple formula for this in terms of f and g but I will let you read about it in the book. It's very straightforward I don't want to waste time on this Okay, so that's the that's the other thing you need to know in addition to the first derivative Which gives you the equation on the tangent line You can also have the second derivative tell you about the qualitative behavior of the graph Kind of a second order approximation Okay So what are we going to do next? Are we going to do next is we are going to use to talk about other features Um Of parametric curves And so far we talked about so far we talked about differentiation Right, so this information about the tangent lines has to do with the derivatives And now we'll talk about integrals And integrals are not about the local behavior of the function like the derivatives Which tell us about Kind of behavior of the graph on a very small scale But integrals are about the global behavior about averaging of the function In other words about areas various areas which are related to the graphs So So here again, we use as a sort of a guiding principle A guiding principle the material that we learned in single variable calculus And then we generalize it to the more general parametric curves namely We have to remember the formula About The area under the graph of a function So again, I go back to the single variable situation And I'm I'm drawing you I'm drawing graph of a function f of x And suppose on the x line on the x axis I mark two points a and b Okay, and I look at the graph above it. Let's assume that the graph In this range from a to b that the graph is entirely above the the axis like like shown and not below or not like going from Upper half to the plane to the lower half plane. Let's assume for simplicity is like this Then we can ask what is the area which is enclosed between The x axis the graph and the vertical lines Which are x equal a And x equal b And one of the triumphs so to speak of the single variable calculus was a formula for this Which I actually kind of alluded to in my first lecture and which is that This is what's called the integral So let me say area under the graph Is given by the integral f of x tx Okay, and this we can write as g of x b to a That is to say g of b minus g of a Where g is the antiderivative of x antiderivative of f So in other words Finding areas involves integration And this formula shows that integration is really the procedure which is which is inverse or opposite to Differentiation because to find the integral you have to not differentiate this function But rather find a function whose derivative f is that's what we call antiderivative. So in other words Find g such that g prime is f Finding antiderivative So that was the story And now we would like to generalize it because again with you now This graph as a special parametric curve How special the curve given by by that parameterization And now we want to generalize it to the case over Over parametric curve for which the functions f and g are arbitrary functions I mean the small f and g Not the big ones which I use here Okay So So the question becomes Suppose we are given a parametric curve Same kind of same For parametric curve Suppose you have a parametric curve now, you know, well Bad a bad drawing because it looks exactly the same as that one So let me make it That's the only one I can draw I don't know just the natural kind of a natural impulse Okay, so let's say and again we We pick some interval here from a to b and we're going to ask what is What is this area? So in other words now again f is f of t and y is g of t And let's say t is between some alpha and beta So that a this is x right and this is y So I call them a and b. So what are they if t is from alpha to beta? It means that a is f of alpha and and b is g of beta But somehow a and b are not so important now. What is important Is the values of the parameter t goes from alpha to beta? So now the Again, you have to when this kind of question is asked you have to be careful to make sure that That the graph is indeed above the Above the x-axis After I write down the formula we'll discuss briefly about what happens if it's not the case Okay now The question is to generalize now this formula and so the and it's actually very easy because Another way to look at this formula is to say that it is given by the formula There is a way to rewrite this formula. We just have to remember that f of x is actually y So another way to think about this formula is to just say that the integral of y dx From a to b and so We can now this makes sense even for parametric curves because x and y still make sense so the area between Let's just say the area of this of this figure Is going to be equal to the integral again of y dx but the problem is that The x and y are given by this as functions of a variable t So at first it looks a little bit like a nuisance But but then you have to remember that actually is something we've learned before When we studied the integrals we oftentimes saw that it was beneficial to Substitute a different variable instead of x oftentimes in single variable calculus To actually technically evaluate the integral to find the antiderivative It was oftentimes useful to make substitution and to say that x Is some function of t of some other variable So let's just call that other variable t and say that x is equal to f of t And then we had a formula for this For calculating the integral in terms of this new variable t And what was this formula actually it's very easy to to write it if we just remember How to compute the differential dx So the the main formula here is dx is equal to f prime Of t dt that's what dx is Yes, yes, you're right. Thank you very good Yeah, I should I should develop some system of prices for Those people who find this Yeah, extra points on I'll think about this But so you should be on the lookout because you know I can make mistakes sometimes on purpose and sometimes Just because no one is perfect, you know So In this case, I just made a mistake. So thank you for correcting me um So this is a this is a formula we're going to use So dx is just f prime of t dt And y is g of t, okay So Substituting the variable t here simply means Using the old substitution formula, which would give us the integral from alpha to beta g of t f prime of t And that's already a very nice formula Which now does not involve x and y but only these two functions f and g and so it becomes a very nice formula because Just as as long as you as soon as you know what the g of t is and f of t is you you can calculate the area Okay, so it's it's the same formula, but just generalized to this more general context of parametric curves Now let's talk about the subtle point, which is What to do if in fact um, the picture is not like this, but It also involves some part of the lower half plane below the x axis Because this happens right So this is something we learned even in single variable calculus And the point here is That you see the point is that when you write When you write this formula if you really think of y dx If it lies above the x axis it means that y is always positive So you actually get a positive answer, which is the area, but if y is negative Then you're going to get a negative answer. So this already suggests to you That for example, if you were to consider this case Where actually it is below the axis What you're going to get is not this area, but this area was negative sign right, so minus negative the area Will be equal to this integral I want to write it like this because it includes both this case where you can just write y equals f of x and also includes this case where you simply substitute this So in other words area is equal to negative of this The integral is negative and area is always positive So to extract the area out of the integral you have to put an extra negative sign So the integral is going to be negative to begin with You put another negative sign the answer will be positive So if the graph is entirely below the x axis you just put a negative sign And so then of course there is a mixed case Where it could go like this So in this case, let's call this area a1 and let's call this area a2 So what you are going to get is The difference between the two in other words the part which lies above the axis Is going to be to contribute with positive sign and the part which Lies below the x axis will contribute with negative sign So a1 minus a2 will be the integral of yds So whenever you you fall below the x axis you are going to get negative things And so you don't you don't get the area you get negative area minus the area And that's why you get this formula So that's the most general formula which is and the same thing will be true here It's not going to be the area but rather a1 minus a2 in general Where a1 and a2 mean the same thing is here a1 is the part which is above the axis and a2 is the part below the axis, okay Is this clear? Yes the graph doubles on itself Okay, oh Okay, that's like a doomsday scenario but So the question is Right, so And now this is not like the this is not the worst case. You're right. So the There is a worse situation When it goes like this And see the before when we studied graphs of functions, this could not happen Could not possibly have happened because for each value of x we would have just one value of y But now because we're doing parametric curves anything is possible, right? So in this case The point question. So what's going to happen is Which is okay, so the here there is a different there is a different issue in my formula, which I wrote here I said t goes from alpha to beta and so I put the limits like this And I'm assuming that Alpha is less than beta So smaller value of t Corresponds to smaller value of x and bigger value of t corresponds to bigger value of x, right? So then you get this formula what could happen is that so in this picture Curve is traversed like this But if the curve were traversing like this You would get the integral in which you will the lower limit will be bigger than the higher limit And we make sense of that by saying that you switch the limits, but you put a minus sign So that's another way by which you could introduce a negative sign In in these integrals namely in the case when the left Endpoint corresponds to a larger value of t So if alpha is less than this is alpha is less than beta So alpha goes to a which is less than b beta goes to b But the other possibility Could be The other possibility could be that You could have exactly the same picture, right? But but now let's say that I could change the parameterization For example, I could just substitute instead of t. I put negative t So the what was before say a Would be Would correspond to value of alpha Which is bigger than value of beta, right? So this is x this is t So then you will end up with an integral where this formula still be correct will be integral from alpha to beta But when you when you start calculating it you will say you will have integral Let's say not from 0 to 1 but from 1 to 0 And then here you'll have You know g of t and then f prime of t So the rule is that this is the same as the integral From 0 to 1 but with negative sign You see So so you have to be careful that first of all the thing is above the axis or below the axis this kind of stuff Which I talked about before, right? The second subtlety is that you have to be careful as to in what direction The curve is going with respect to the parameterization When t is increasing are you going from left to right or you're going from right to left? Okay, so now to go back to this The interesting thing that happens here is that say on let's say it goes like this Right, so then on this segment it will go from left to right But then from this segment will go from right to left and then we again will go from left to right The fact that actually there are three different parts is not so important because The point is that in setting up the integral we will be using not a and b For general parametric course, but alpha and beta in other words we will be We will have to specify from the beginning which branch Out of these three branches we are talking about are we talking about the area under this one Or the area under this one or the area under this one because the formula really explicitly involves the end points Alpha and beta with respect to t not with respect to um Not with respect to a and b So usually we would just pick a particular branch and we'll just say In other words, we will be saying that this is this is t equal alpha And this is t equal beta and this segment will correspond to some other alpha beta and this will correspond to some other alpha beta Of course in principle you could say what what is the meaning of the integral when t will go from this value to this value And this also is very easy to figure out But at this point we are kind of losing the geometric meaning of this right So we have a clean geometric meaning when we are talking about a branch which is kind of a Which doesn't double out on itself like you said, you know But which just you have a single branch over the segment in the x in the x line In principle, you could also do you could also give interpretation to the more general integrals But you kind of lose the interpret geometric interpretation. So we will not do that So we will consider the ones which have a more clear meaning like this Okay, does it answer your question? Okay, any other questions? All right, so that's the integrals and the last thing I want to talk about Is he's arc lengths So the other thing which is very interesting is to find the length of a segment of the Of a parametric curve, okay So only less than 10 minutes and I know you'll be free to go All right, so let's just The finish line And I will be done All of the stuff is really, um, it's not so difficult If you see in each case, there is a formula So what I'm trying to do here today is to kind of give you an intuitive understanding of the formula Kind of introduce the formula for you and explain why it's true But once you know the formula you basically just have to substitute If you look at the homework exercise, most of them it's just about substituting There are a few subtle points and one of the subtle points is the way when you do the integral What exactly are you calculating? Are you calculating? So there are subtle points about the signs like the ones which I mentioned here But other than that, it's fairly straightforward And likewise what is also straightforward is Is the arc length and so so that the question really is What is the length say you have a parametric curve? What is the length of the of the part of this curve between these two points? Now of course you can say what what do I mean by the length? And for this you you can just think of the same, um Analogy which I explained last time which is a think of your curve as this wire Okay So the curve is is curvy when you when you look at it on the plane In other words, you know when I make it like this is curvy But at the end of the day I can just take it And stretch it out And measure it right so that's the length And of course the point is when I stretch it out It should not be stretchable, you know, it should be sturdy like Because otherwise I can stretch it as far as I want right so it has to be When I straight when I say I stretch it I mean just kind of making it into a straight line segment But I should assume that I'm not displacing anything that it sort of has the same It's not Its length is something which is well defined which is not does not depend on me the way I hold it right So that's exactly this so you just kind of take it and and and put like this and and then of course It's clear what it's what its length is you can measure it. So that's what we mean by arc length So in other words, even though it is curvy it has there is a notion of a length even even for curved objects And of course a great example of this is the arc length of a circle The circle is curvy right, but we know what the length is right. What is it? That's right. That seems to be disagreement on that Yes, two pi that's right So so the length of the circle is two pi and that's and that's how we sorry So conferences Two pi are that's right. I'm thinking of a circle of radius one, but if it has a radius Okay, I think now we are in agreement right Circle of radius are the length will be two pi are and in fact So this is a very important there is a very important constant showing up which is called pi I will talk about it. Have you seen the movie pi by the way? Oh, you should see it. It's cool. It's kind of cool It's one of those movies with a crazy mathematician, you know, but but it's a cool. It's a cool movie All right, so settle down now pi is a Is this universal constant which actually is defined by this property that two pi r is the circumference of a circle of radius r So this is just a good example to show you that even though the circle is curvy, but there is some length it has a length And of course the way you show you measure this length is you can think of a circle as being formed by a rope Which you kind of open up and stretch and okay, you can measure it and you can see that it's two pi r But even for more general parametric curves We have the notion of a length and we would like to calculate it by using these functions f and g And there is actually a very nice formula which involves integration for this And it's and and the point is that the formula is also very easy to derive And the reason why it's easy to derive is Because we model everything on straight lines as you know today when I started the lecture I said The simplest curves are lines and to to to really understand various characteristics of complicated curves We have to understand them first for lines And then we kind of extrapolate generalized to the more general Curves to the most general parametric curves So if we had if our curve were in fact a line Or a line segment we would be able to find the length very easily By using the Pythagoras theorem by using the Pythagoras theorem So this would be delta x and this would be delta y And the length would be the square root of delta x squared plus delta y squared So now we have a much more complicated curve here But as we discussed earlier No matter how complicated the curve is if it is smooth or differentiable It can actually be approximated on a very small scale. It can always be approximated by a line okay, so Small scale so what it means is that we should okay If we just try to approximate the whole thing by line, it's not a very good approximation But we could break it into segments like this And now each of those segments each of the small segments actually does look like an interval a segment of a line And so for each of them you can have delta x and delta y So You know, so you it will look something like this or maybe on this side it will look like this And then you would have so this is just one one of the segments which I blow up I zoom on it Right, and so I have delta x and delta y So then I have the The lengths of this line segment, which is now going to be very very close to the actual length And so I'm going to approximate the entire length by the sum of the lengths of those line segments Which is kind of like a snake like picture like you know on each of the small segments I have an interval And so the end of the day what I'm going to get Is the sum Is the integral Which looks like this dx square. So it's going to be kind of like delta x squared again Press delta y squared But on the small one in the limit when on those little pieces become smaller and smaller It's going to be dx squared plus dy squared And then what I'm going to do I'm going to To put this And this will be from alpha to beta And then I can take this under the integral And so what the result is going to be is I'm going to have dx dt squared plus dy dt squared dt So that's the answer that's the formula for the arc lengths of the Curve between the points alpha and beta t equal alpha and t equal beta And I have now given you a kind of very informal intuitive understanding of why this formula holds It essentially comes from the Pythagoras theorem So this was just a trick. I wanted to rewrite this this is in a nicer way So I just put this and I put this under Okay, so we're out of time. So see you on Thursday