math 10 -12 .Calculus lesson 4

hi everyone how are you today hope you're doing fine so you all come to today's lesson who continue with our topic on Mentos tip which is calculus now today our main focus will be on stationary points which are also called the turning point so now what is this point so the definition of a stationary point for a turning point is simply that this is the point where the graduate of the curve is zero so a stationary point which is also called a turning point this is a point where dy by DX is equal to zero okay so now just giving a summary of it all we have three types of training points so even when you drain the graphs you are going to obtain three types of curves so we have a curve which is called advanced map so a maximum right here where the graph pens is what we are finding the maximum value which is a white value okay then also we have another type of the stationary point which is a minimum so at the bottom right here we have a minimum Oh fine then the third type is an inflection so for an inflection this is how it looks like it can face in any direction but it has a certain point where it is inflating okay so these are the three types of stationary points now what are we learning about stationary points we are going to see how questions in your exam can come so for example the simplest question of them can come find the turning point for them for the equation y is equal to 3x squared plus 6x plus 1 so we want to find the tipping point for this cap now this is an equation on the curve so we want to find the taking point now by definition we know that dy by DX is equal to 0 so at any point is the point where dy by DX is equal to 0 so what you're going to do is step number one so the first step that you're going to do here is you are going to differentiate the given equation so you differentiate that is the first thing that you're going to do so differentiating now we have our equation which is y is equal to 3x squared plus 6x plus 1 so differentiating this one we are going to say dy by DX is equal to zero which is equal to this one this one comes down here times 3t to be 6x plus 6 because one comes there used to be 6x we subtract 1 minus 1 is 0 any number to the power 0 is a 1 so 16 wants to give us 6 then the derivative of a constant number is 0 so we ignore it so now we can solve this one because we have an equation 6x plus 6 is equal to 0 so we mean 6x is equal to negative 6 we go that side so you divide by 6 you also divide by 6 you have now X being equal to negative 1 so now you have found the value of it okay the question say find the taking point for this graph now the taking point has to be a coordinate which should be in the phone X comma Y okay so now for such questions also they actually specify so it's taking point find the tipping point for this one so having found the value of x the next thing I'm going to do is you're not going to substitute into this one now the same equation so step number two step two here you do a substitute so substitution into the given equation what are you substituting the value of x which were found so you are going to say Y is equal to 3x squared plus 6x plus 1 so this one here to be three open brackets there you put your negative ones where's plus six you put your negative one plus one so this one here it is three was one square near to 1 squared is 1 times 3 is 3 plus so this Plus now before your 8 plus this plus and negative giving us negative 6 plus 1 so 3 minus 6 is definitely negative 3 plus 1 here negative 2 so having found this one we are now going to say ok we can now find a retaining point or the stationary point to be equal to since the X is negative 1 so it will be negative 1 comma negative 2 and we have found come at any point ok let's see now how best we can also see how it comes so in faith Allah and mostly mostly the power of your exams it's always such a question like this one which doesn't miss in you exam it always appears so this have a few oh how it can be applied in an exam situation ok so using the same knowledge an extract of the question we have been given here reads find the coordinates so the diagram below shows a sketch of the graph there is a sketch of our graph okay so for the sketch of our graph now we are saying we should find the coordinates of E and G so this is basic thinking look at how this graph is cutting the x axis so definitely here this point here the point X is the first point that is here you can really tell that this is a common Co because the Y value here right here is e 0 also here you can also tell something this is G Komen 0 so a point is 0 so now how do I find the even here it was in value is X and the G value mistakes but in both scenarios as you can see the value of y is zero and we have been given an equation there so since we have been given an equation we can use the same equation because we know that Y is zero in both instances so what we as if you same is well in why we put zero so we are going to have now an equation 2 plus X minus x squared is equal to 0 so the fact that the highest power is 2 there we know that this is a quadratic so we are going to say a is equals 2 okay that is negative 1 they have been our B is equal to 1 then our C is equal to 2 so we find our product which is a times C so our product to be negative 2 then we're going to find our factors to factors such that when we add them to be equal to 1 but we multiply when we multiply those to give us back the negative 2 so what we are going to do here is very simple we are going to check if you get negative 2 if you get 2 minus 1 here you are going to find one so this one we can relate so we say we have 2 and negative 1 so these are the factors because when you say 2 minus 1 e to give us 1 but when you said to 1181 to give us negative 2 okay we all have knowledge on these things so we rewrite this one it is this they will replace this guy here so instead of a trait 2 plus 2x minus x minus x squared okay so it is to have replaced that one then we categorize this like that because it's a its factorization so what is common in here is to open brackets we mean with 1 plus X the negative here what is common there is X in here our main we see one negative Plus so I mean with it it's there then this and this are the same which is enough evidence that I'm doing the correct thing so finally the factorized equation now he'll be 2 minus X and I get 1 this is the exam which would be X plus 1 so this is equal to 0 then the next thing that you're going to do is you solve this one so 2 minus X is equal to 0 so this one be negative x is equal to 0-2 because goes there so multiplied by a negative 1 to be X is equal to also here by negative 1 X is equal to 2 so you are found you will find where X is equal to 2 you also solve this one which is X plus 1 is equal to 0 so X will be equal to 0 minus 1 so X is equal to minus 1 now you have found two values of X the other one easy negative 1 the other one is 2 which makes sense inside the x axis is positive this side is negative so definitely our he now here it could be because a negative side it could be negative 1 comma 0 and 2 comma 0 and you have found the coordinates ok so the second question here it says something that is very simple now here we are trying to apply calculus so something that is very simple here they are saying find the maximum value over Y so to find the maximum value of y we are going to use because the maximum value of y ETA I is tending and we know that at the telling point dy by DX is equal to 0 so we are simply going to say okay dy by DX is equal to 0 okay which is equal to in this case now so dy by DX is goes to 0 so the first thing that we will do now is we are going to differentiate this equation here so dy by DX of this equation here this is a constant to be 0 so we know it this one is just X so it will just be 1 minus 2 x that is dy by DX should be equal to zero because it is 10 so now we have my question there so this one me negative 2x is equal to zero then this process that side to be minus 1 so / - 2 / - me this and that was this and that was so our X is equal to 1/2 okay so that is how you find the file now before calculus was introduced into the syllabus there were some crooks method that we are using but it works because there was no popular since diversity so what you can do is because you know that between this one and this one this graph is going to ten right at the line of symmetry so you can easily say you act this - and this one here and then you divide it by two so you find the average of these points so two plus this one here been divided by two hands were used to find the same five but maximum value Y which is here we have found our X so we substitute into the equation direct Y is equal to two plus X minus x squared so this one be equal to two plus our X here we know it's have been - this is 1/2 squared okay so we solve this one we are going to have this is as good as 2 plus 1 over 2 and 6 over 1 then plus 4 squared this is 1 over 4 because 1 squared is 1 and 2 squared is 4 so there we can sort it out such doubts Micah seems to be failing due to gravity get back it wet so we have 4 then 1 into 4 in sperm so 4 times 2 this one is 8 plus 2 into 4 is 2 times 1 is 2 plus 4 into 4 is 1 and 1 is 1 so this one be equal to 8 plus 2 that is 10 10 plus 1 that is 11 so 11 over 4 and this one equals to 2 remainder 3 over 4 and we have found our answer and this is how we determine the value ok so I think we'll end here for now from do more practice on the sample question like a fee as a new key for an exam so please revise on this and we'll do the other questions later thank you