Understanding Taylor Series for Two Variables

Hi students, this is Alex here. In this video we are going to see the introduction for the Taylor series for the function of two variables. Whenever there is a function f of x, y is given, we can expand this using the Taylor series about a point a, b or this can also be told as in the powers of x minus a and y minus b. So we will be using the Taylor series formula and we are going to discuss how to write that formula. The formula is f of x comma y is equal to first term is f of a comma b. The value of a and b is substituted for x and y. We get a value and that will be substituted here. Then second term is plus 1 by 1 factorial open bracket x minus a into partial differentiation of the function with respect to x at a comma b plus y minus b into partial differentiation of y at a comma b. Once. if we know how to write this term using this term we can write the rest of the terms this we will treat it like a and this we will treat it like b so the next term will be of the form a plus b whole square so a plus b whole square we know it is a square plus 2ab plus b square and next to that we will be using a plus b whole cube form which is a cube plus 3a square b plus 3ab square plus bq. So after this we have to put plus. Then continuation of this formula the next term will be 1 by 2 factorial. So we had 1 by 1 factorial now it is 1 by 2 factorial. I told it is of the a plus b whole square. Now a square this square. This square will be in the form of x minus a whole square but we do not put f square. Instead it is double derivative. Partial differentiation of the x and it is the second order. About a comma b. Then plus 2ab. So 2 into you have to write x minus a then y minus b. x minus a into. y minus b. Then fx fy. So that can be written as fxy of a comma b. Then b square which is nothing but y minus b whole square into second differentiation of f with respect to y at a comma b. So we completed the Next term. Then further as I told we will be using a plus b whole cube form. So plus now 1 by 3 factorial it is a cube first. So cube of this that will be x minus a whole cube. But f x x x above the point a comma b plus 3 a square b. That is. 3. a square is x minus a whole square and b is y minus b and f x x y will come because the first term is square and this is power 1. So it is x x y about the point a comma b. Then next comes 3 a b square. So this will be 3 a and b square. y minus b whole square. Now f xyy because this is power 1, this is power 2 about the point a comma b. And the last term is we have b cube. So b cube once again we are going to use this form y minus b whole cube into f yyy above. the point a comma b So plus dot dot dot. This is a way to write the Taylor series formula. So the term starts from f of a comma b then continuously these terms will come. After writing this formula whatever the term we needed that is first differentiation with respect to x and y partial differentiation In second differentiation whatever the term we needed we have to find out separately and substitute in this formula and this gives the Taylor series expansion of a function with two variables.

So we will be using the Taylor series formula and we are going to discuss how to write that formula. The formula is f of x comma y is equal to first term is f of a comma b. The value of a and b is substituted for x and y. We get a value and that will be substituted here. Then second term is plus 1 by 1 factorial open bracket x minus a into partial differentiation of the function with respect to x at a comma b plus y minus b into partial differentiation of y at a comma b.

Once. if we know how to write this term using this term we can write the rest of the terms this we will treat it like a and this we will treat it like b so the next term will be of the form a plus b whole square so a plus b whole square we know it is a square plus 2ab plus b square and next to that we will be using a plus b whole cube form which is a cube plus 3a square b plus 3ab square plus bq. So after this we have to put plus.

Then continuation of this formula the next term will be 1 by 2 factorial. So we had 1 by 1 factorial now it is 1 by 2 factorial. I told it is of the a plus b whole square. Now a square this square.

This square will be in the form of x minus a whole square but we do not put f square. Instead it is double derivative. Partial differentiation of the x and it is the second order. About a comma b. Then plus 2ab.

So 2 into you have to write x minus a then y minus b. x minus a into. y minus b. Then fx fy. So that can be written as fxy of a comma b.

Then b square which is nothing but y minus b whole square into second differentiation of f with respect to y at a comma b. So we completed the Next term. Then further as I told we will be using a plus b whole cube form. So plus now 1 by 3 factorial it is a cube first. So cube of this that will be x minus a whole cube.

But f x x x above the point a comma b plus 3 a square b. That is. 3. a square is x minus a whole square and b is y minus b and f x x y will come because the first term is square and this is power 1. So it is x x y about the point a comma b. Then next comes 3 a b square. So this will be 3 a and b square.

y minus b whole square. Now f xyy because this is power 1, this is power 2 about the point a comma b. And the last term is we have b cube. So b cube once again we are going to use this form y minus b whole cube into f yyy above. the point a comma b So plus dot dot dot.

This is a way to write the Taylor series formula. So the term starts from f of a comma b then continuously these terms will come. After writing this formula whatever the term we needed that is first differentiation with respect to x and y partial differentiation In second differentiation whatever the term we needed we have to find out separately and substitute in this formula and this gives the Taylor series expansion of a function with two variables.

Transcript for:Understanding Taylor Series for Two Variables

Transcript for:
Understanding Taylor Series for Two Variables