[Music] hello and welcome to the match 2 component of the online bac program on data science and programming in this video we are going to do a review of the some of the ideas you have learnt in math 1. so you have seen some ideas of calculus already in maths one at least you have seen the ideas of functions and a function of one variable and then the ideas of tangents and so on so you have seen some of the geometry and you have seen some examples of functions so the first couple of videos including this one we are going to do a little bit of a review of what you probably should have seen in maths one so if you do not remember this will help you to remember and also we may end up doing a little bit extra in each of those videos and then once we have refreshed ourselves with some of the basic concepts of related to functions and the geometry corresponding to those the graph and so on ah we will go on from there and study the idea of the idea how to capture these ideas in terms of calculus okay so let's quickly look at some topics that you have seen in maths one so you have seen what is a straight line in maths one we will be reviewing it i am just sort of quoting the topics right now you have seen what are quadratic equations you have seen what are polynomials in general you have seen what is a function so in particular this is this includes functions from real numbers to real numbers they also seen other kinds of functions but those will not show up in this particular course ah well i i should take that back what i meant was you have seen other functions which are which have which are on natural numbers and so on those kinds of functions will not be studied here but we will study other kinds of functions on rn and so on ok and then i think you have also seen what are exponentials and logarithms so um this is again something that we will review in in this in these couple of videos that we are going to do okay so let us start with what is a function of one variable so a function is defined to be a relation between relation from a set of inputs to a set of possible outputs where each input is related to exactly one output okay this is a slightly cumbersome definition but in a minute we'll make more sense of that in fact this is not exactly a definition but a heuristic about what a function is this means that if the object x is in the set of inputs which is called the domain okay this now you may remember then a function f will map the object x or take the object x or change the object x or translate the object x you can use any of these words or transform the object x to exactly one object f x in the set of possible outputs this is called the codomain okay so this is a very general idea of a function and so yeah maybe at this point we should say not function of one variable but what is a function so this is about what is a function so if you have f x to y so pictorially this this set of sentences these sentences are represented by this picture here f x 2y which says that f is a function from x to y x is the domain y is the codomain and then we have something called the range of a function this means it is the set of values that f takes in other words it is a set of values it is the set the subset of y of the co domain such that there is some x such that f of x is equal to that particular value okay so now what is the function of one variable here we have we have said something about one variable okay so function of one variable means we are thinking so for for the purpose of this video and the next few videos when we say function of one variable we mean a function f from r or some subset of r need not be the entire r so a subset of r two r ok so maybe i should draw this better so a subset d to r where d is a real subset of the real numbers so why is this a function of one variable because typically we represent this by f of x where x is an x is a real number hence one variable if it was a function of in where d is in r two then we would say it's a function in two variables and so on so those are things we will study in the next coming few weeks but for now we have the domain which is a subset of r and we have the codomain which is again the set r so we know examples of such functions so for example we we have seen in match 1 the idea of a linear function so linear function looks like m x plus c so ah m and c are real numbers and then when you evaluate this for every x it gives you some real number so for example if you have f of x is ah 5 x plus 2 then f of 0 when you evaluate this at 0 you get f of 0 is equal to 5 times 0 plus 2 so that's 2 and then f of 20 is equal to 5 times 20 plus 2 so that's 100 plus 202 so i i hope you remember what is a linear function and there is a picture associated with it this is the graph of that function so you plot the point x and you plot the corresponding point f of x on the y axis so you plot the points x comma f of x right and in other words this is the line y is equal to f of x which is a line the bold line in ah in black which is drawn here so this is called the graph of the function so this is the graph of the function and one reason to call it a linear function is that the graph is a line yeah so linear has something to do with the fact that this is a line so for a linear function we have f x is f of x is m x plus c and here m is called the slope of this line and it it calc it has something to do with this angle which you may have seen in maths 1 and this is supposed to represent the y-intercept so if you if you put x equals 0 you get f of 0 is equal to c yeah so this is supposed to represent so f of 0 is over here right this point here is 0 comma f of 0 and we just saw that f of 0 is c so this is the point 0 comma c so c is exactly this length here so this m is supposed to have something to do with the angle so this is theta and m is supposed to be tangent of theta and if you haven't seen this before we will be studying this in the next video the point i wanted to make is that the m and the c have something to do with the geometry of this line so one represents the slope by the slope we mean the angle at which it is inclined to the x axis and the other ah determines the distance where it hits the y axis from zero okay ah well i so when i say distance i i mean it could be a positive distance which is in the positive side of the y axis or it could be a negative distance which is in the negative side of the y axis okay so so you have seen this idea of a linear function and this is the graph of such a function it is a line okay let us recall what are quadratic functions so a quadratic function is a function of the type a times x minus b squared plus c so we could expand this and i will write it in a minute how to how it looks like and here is an example of a quadratic function so the quadratic function typically looks like it's been drawn here this is called a parabola and that's the figure in red it's called a parabola maybe i should use red and this is the graph of the function f which means it is the set of values x comma f of x as x runs over the real number of course we have not plotted all the values if you plot all the values mean or these two arms of the parabola will shoot off so this is an example of a curve so this is not ah this is not a line so typically things which are not lines but have this kind of shape of a single you can think of it as being drawn by by without raising your your pen or pencil such things are called curves ah so line is a particular case of a curve but if you have a line then you usually do not use the word curve you use the word line so when you say curve typically you mean that it's something which is not a line but which you can draw by by a stroke of your pencil or pen and of course when you draw such a thing in principle you could draw more complicated figures so curves are more general than so graphs are particular kinds of curves so ah curves this could also be a curve but this is not a graph ok fine so this is the graph of f of x ah with some particular values for a b and c so again how do i how do i read of what the values of a b and c are from the picture or can i say something about this picture well you can see that the uh smallest value of this this f of x is taken when x is equal to b that's when this this part contributes the least which is 0. and i mean i so when you have drawn okay so the first thing is when the graph is like this this a is going to be positive okay so this is positive if the graph is a parabola like this and a is negative if the graph is a parabola like this and then further um the so once it's greater than 0 then the smallest value of this this function is taken when x is equal to b so in this particular case here b is equal to 2 and then once we know b is equal to 2 you substitute x equal to b then you get f of b is equal to a times b minus b squared ah plus c so b minus b squared is 0 so that gives you f of b is equal to c so in other words this distance here maybe i will draw this some other color this distance here is exactly f of b is c so in this case it is f of two so ah again we we see that these b and c and a have something to do with the geometry of of the parabola or the geometry of the graph and we can go back and forth from the graph we can determine the a b and c and from the a b and c once we know a b and c we know the function and hence we can draw the graph so there is an interplay here between the algebra which is the functional form a times x minus b squared plus c and the geometry which is the actual picture that you can see and what we are going to do more and more is try to understand how these two relate to each other ok that is the sort of goal of of this this part of what we are doing so along the way we are going to so the way to do this or one way to do this is to study calculus which is what we are going to do ok so let me expand this and also see what we get so this is a times x squared minus 2 bx plus b squared plus c which if you write is a x squared minus 2 a b x plus a b squared plus ok so i can use some other notations for a b and a b square plus c maybe i will call this a prime sorry b prime and maybe i will call this c prime so i can write it as ax squared plus b prime x plus c prime and so the other way of thinking of this is that um this is a polynomial in in x of degree 2. so this is something that you may have seen earlier and we are going to see very shortly so what are polynomial functions so a function f from r to r you define f of x to be a n times x to the power n plus a n minus 1 times x to the n minus 1 all the way up to a a0 so this is considered this is a polynomial of degree n and these ais are the coefficients of the polynomial and they belong to the real numbers ah so the example is so the function here you can look at x cube minus 4 x and here is the graph so this is where you are plotting the graph x comma f of x so in other words where x runs over r is that said so in this case this example this is x comma x cube minus 4 x as x runs over r that is what this red line red curve is giving you so this is a function and it's a very nice symmetric function i mean not symmetric about the x or the y axis but symmetric in some axis ah and yeah you can see that as x becomes large this function is shooting off to infinity and as x becomes small the function is becoming smaller and smaller and shooting off towards what is called minus infinity okay well what else do we get from the picture ah we can see that there are three points where it becomes zero so how do we find those three points well you solve this equation yeah so here's where the algebra kicks in so to get those points for which f of x is zero so f of x is zero that means x cube minus four x is zero that means x times x squared minus four is zero that means x times x minus two times x plus 2 is 0 so this this happens so f of x is 0 exactly when x is minus 2 2 or zero or maybe i should just minus two zero or two just to be ah just for the symmetry to sort of be clear so minus two zero and two um give you give you the zeros of this polynomial and so in general of course a polynomial may not have zeros and we will see this such a polynomial later on maybe not in this video but in the next one and then we can use this the form of the polynomial to get some interesting properties of the graph so this is something that we saw for linear and and quadratic functions that the coefficients had some role in the geometry of the graph and similarly over here also the coefficients have a role in the geometry of the graph so such things are going to come on come in and this is one of the starting points of calculus we will study more as we go ahead okay well let us look at the exponential and the logarithmic functions ah so this is maybe something that that was touched on in math one so um ah i have mr g so f is a function from r to r and g is also a function from r to r so f is a function from r to r and g is also functions from r to r and what is f f is defined as a to the power x so what is the meaning of a to the power x this is a question in itself and when when a and x are both well at least when x is an integer the value is clear right if i if i say ah 2 square then the value is clear its 4 or even if i say 0.5 square then the we know how to compute this so this is 0.5 times 0.5 which is five by ten multiplied by five by ten or half multiplied by half which is one fourth which is point two five right but i could i could if i do something like pi square well this is just pi square it's it means you take the number pi and multiply it by itself so this is pi times pi okay and we if we um want to know what it is we can approximate it to as as good a value as we want so we still have some understanding of what what are numbers like pi squared but if you change the exponent so here this is the exponent so if you change the exponent to to a number which is not an integer so if if you have something like 2 to the power minus 2 i still know what this is 1 by 2 to the power which is 1 4 or if i have 0.5 to the power minus 2 then this is one by point five to the power two which we know is one by one fourth which is four and similarly if i do pi to the minus two this is one by pi squared which again we can approximate to whatever degree of precision we would like ok so we understand how to do this process when we have integers what happens when we do not have integers for example if i take a rational number so if i take 2 to the power 1 3 what does this mean well ah so the idea here is that this is the cube root of 2 okay so what this means is this is some number such that when you take the third power you get 2 and does such a number exist indeed it does that something that needs proof we are not going to get into such things but something i think you can believe ah similarly on similar grounds lines we can now answer things like what is 0.5 to the power 5 6 right because we know what is 0.5 to the power 5 and then we have to evaluate that to the power 1 by 6. so it is that number such that when you take the 6th power you get 0.5 to the power five so for rational numbers also we know how to how to uh how to understand e to the power such a rational number ok so but now we are saying you look at the function f of x is a to the power x where x is running over all real numbers so we are we also have to deal with things which are not rational for example if you take 2 to the power root 2 what is the meaning of such a thing so this this is a little tricky we have to really scratch our brains about what what is the meaning of of this and still i think with a little bit of thought which i leave to you you should be able to figure out what is 2 to the power root 2. what may be really not so easy is what is 2 to the power something which is even worse something which is irrational so sorry not just the rational but not even of this form like root 2 what is a 2 to the power pi okay so this you have to understand by understanding that when you have things like these in the exponent what you do is you take so suppose i want to define 2 to the power root 2 what i do is i take some numbers which are rational numbers so take so take rational numbers which approximate root 2 as closely as you would like or maybe maybe let me change this to say take a rational number take a rational number which approximates root 2 as closely as we desire let us call it ah n or maybe thats a bad notation ah rational number um q which approximates root 2 as closely as we desire so for example if you if you look up root 2 in a table you will be able to see its 1 point something something and then you can write it as approximately equal to something by 100 right so that that is a rational number so one point something something is a rational number and then you can do one two to the power that that rational number q which we know how to do right this was exactly what we discussed over here okay so so we understand how to do 2 to the power q so 2 to the root 2 is very closely approximated by 2 to the power q and if you want a better approximation you choose a rational number which is closer to it okay and that is how you define things like 2 to the power root 2 or even worse things like 2 to the power pi so using this you can make sense of f of x is equal to a to the power x that that was the real point how do you define for every x what is a to the power x you do it in this way if x is an integer we know what it means ah it just if x is 2 then it's a squared so that's a times a if x is 10 its e to the power 10 which means you multiply a to itself 10 times if x is minus 5 that means it's 1 over a to the power 5 and so on and if now if x x is a rational number you raise a to the power whatever there is in the numerator and in the denominator you are taking that root that its root right so if x is m by n where m and n are integers then you are looking really at a to the power m which we know what it is the nth root of that so it is the nth root of a to the power m so it makes sense and now for any other number we can define it by getting rational numbers as close as we want to that okay so that is how we define this ok and what is the logarithm logarithm is the reverse process okay when you reverse out the exponential function you get log rl okay so i am not defining these carefully in in terms of the mathematics involved but i i want you to understand that even the definitions here they are a little involved when you when you have to actually formally define them so there is some some work to do okay so here's two examples uh y is 2 to the power x so that is how this function looks like so the exponential is a very rapidly growing function okay so this is the exponential function with respect to 2 and this is the logarithm function so the logarithm function is also increasing but it increases very slowly yeah they are exactly sort of they play the opposite role to each other ok so i hope these pictures ah shed some light on what these functions are and i have to point out one one thing ah a is in in whatever i said this a here better be a positive number because if you have a negative number then we are going to really struggle when we take roots and ah when we when if it 0 then when you divide yeah you will have trouble so for this to make sense on the entire domain r a better be a positive number so here a is greater than 0 similarly the logarithm is defined only on that part because it is the ah sort of reverse function of of e to the power x that means and you can see a to the power x takes values which are only positive ah a is positive that means the logarithm takes positive values and gives you values across the real line that's that's what it means so um here is the function y is log x to the base 2. so when we do it with the with a particular base when a is a particular number that it's with respect to that base and what is the relation between these two the relation between these two is that if you take logarithm of with respect to two of two to the power x then you exactly get back x this is the relation between the two so this is called composing the exponential and the logarithm and we will see what composing means in the next video ok so i hope this picture is again a little ah helps you to to understand what's happening i am not defining the exponential and the logarithm in a technically sound manner ok a function f is said to be monotone increasing if ah well it is increasing i mean you i hope increasing should make sense as x increases f of x increases so it says if x 1 is less than or equal to x 2 then f of x 1 is less than or equal f of x 2 and similarly its monotone decreasing if the opposite happens that as x increases f x should decrease so sometimes we also say monotonically increasing or monotonically decreasing yeah they mean the same thing okay so here's an example of a function which is monotonically increasing so this is uh similar to the line that we saw in the pre one of the previous slides so this line is increasing an increasing function namely as x is increasing the value of f x is increasing right that is what is demanded by this this definition here in this definition here okay and the line in red f 2 of x the graph of f 2 of x that's a monotonically decreasing function so here as x increases the corresponding f value decreases okay so these are definitions that are worth keeping in mind because eventually we are going to study maximum values and minimum values and so on so we we should know where functions are increasing and decreasing so in particular what what are increasing and for decreasing functions ok so let us compare various functions so here is a comparison of of functions which are like powers and an exponent function so i i ah point out that this function here is the exponential function and all the others are monomial functions y is x y is x squared y is x cube y is x to the power 4 and as you can see by by the way even before i go there ah the axis here the x axis here and the y axis the um labeling is a bit i mean you you should be careful that here the steps are 0 to 2 2 4 2 4 to 6 and so on whereas here the steps are of order 20 okay so this y is x should i mean if if both sides were drawn in an equal way it would have been a line of with 45 degrees angle but because of the because we we wanted more values on the y side uh i i have shrunk it and that's why y is equal to x doesn't appear as 45 degrees but appears like this because it as x increases it increases so i have drawn zero to 8 and y also goes from 0 to 8 but here we have 0 to 20 okay so and 20 to 40 and so on okay so keep this in mind and why did we do that because the other functions are very rapidly increasing so the picture looks better like this so when we are x is small you can already see that as your monomials grow so the exponent in the monomial grows so y is x this is y is x and then this is y is x squared and then this is y is x cube this y is x to the power 4. so you can see that f of x is increasing as as the exponent grows after a point so initially of course something else happens y is x is actually larger up till one so up till one the others are below but at one they change ah they all become equal to one and then after that x square starts going faster than x and x cube even faster and x to the power 4 even faster and then we have this exponential function 2 to the power x which is also going quite fast as you can see it started above 0 first of all and then it was caught by almost all the functions except y is x so it always beats y is x but then after some time y is x squared beat it for a short time but after that y is 2 to the power x suddenly started taking off with respect to x squared and you can see the way it's changing it it will change and change and change and what happens if you increase your scaling further so now we have 0 to 60 on the x axis and this is 10 to the power 5 so which means 10 cube is 1000 so this is one lakh okay so here we have plotted values like 20 000 40 000 60 000 8000 and 1 lakh and now you can see how rapid this function y to the y is 2 to the power x is ah this is y is x on this scaling it almost looks like a flat line and even wise x squared looks somewhat flat right but it is not actually flat if you if you draw the scaling nicely then it looks like our parabola we have seen this parabola figure before in in our previous slide and y is x cube is fairly fast even in this scaling and then you have y is x to the power power 4 which is this brown line but now if you look at y is 2 to the power x you see that it it further tilts and it beats y is x to the power 4. so it goes faster than y is x to the power and the the point i want to make here is that if you if you have y's x or y is x to the power n then as n increases these functions sort of go towards infinity more and more rapidly but all of them all of them are eventually overtaken by the exponential function the exponential function is a terrifically fast growing function after something after after sometime it is always going to beat any polynomial so this is what this picture is trying to tell you okay so this is just intuition this is not strictly required for our course but it is good to know how these behave ok and then on the other side we have slow growing functions so here is y is x here is y is x to the power 0.8 which is the line in brown not the line sorry its y is x is actually a line but y 0 point x to the power 0.8 is actually a curve but it is a very slow glowing cuff so it might look somewhat what linear and then you have y is root x which is you can otherwise write as x to the power 0.5 or x to the power half so you can see that's below ah that's below x to the power 0.8 and then you have the logarithm function which ah towards i mean it is defined only on the positive side and it starts sort of very very far below and then slowly it comes up and then it changes its direction and it starts going very slowly and the same thing that happened for the exponential happens here but in the opposite direction namely that the logarithm goes slower than any polynomial okay so that's something you should keep in mind just for your intuition ok so let us close with some final ideas of things you may have seen in maths one namely what are tangent lines so tangent lines if you remember are lines which one obtains when you take a line which intersects the the curve that you have it may intersect in many points but then you start moving it slowly parallelly until maybe sometimes what happens is it it intersects the the points in which it intersects all sort of come together so it comes to a single point and so if a line intersects a curve at exactly one point such a line is called a tangent line so this is this is an example of a tangent line its ah so this is the parabola y is x squared and this is a tangent to that parabola at the point 1 i think and we will soon understand using calculus how to find the equation of such a line so we know that such a line is given by y is equal to m x plus c so we will ask how do i find the equation of such a line and we will use calculus to study that here is an example of y is x cube and a tangent line to that and here is an example of why is the exponential function y is 2 to the power x and a tangent line to that okay so i this was mainly to recall ideas that have been studied in some detail in maths one so if you feel somewhat uncomfortable go please do go back and check your mats 1 videos or the tutorial problems thank you for now