Expand f(x, y) = 21 + x - 2y + 4x ^ 2 + xy + 6y ^ 2 in Taylor series of maximum order about the point (- 1, 2) .
Answers
the Taylor series of the function f(x, y) = 21 + x - 2y + 4x ^ 2 + xy + 6y ^ 2 in maximum order about the point (- 1, 2) is 15 + (1/2!)[8(x^2 + 2x + 1) + 2(x)(y - 2) + 2(x + 1)(y - 2) + 12(y^2 - 4y + 4)] + ...
To Find:
to expand the function f(x,y) = 21 + x - 2y + 4x^2 + xy + 6y^2 in a Taylor series of maximum order about the point (-1, 2)
Given:
the function f(x,y) = 21 + x - 2y + 4x^2 + xy + 6y^2
the point (-1, 2)
Solution:
To expand the function f(x,y) = 21 + x - 2y + 4x^2 + xy + 6y^2 in a Taylor series of maximum order about the point (-1, 2), we can use the general formula for the multi-variable Taylor series:
f(x,y) = f(-1, 2) + ∂f/∂x(-1, 2)(x + 1) + ∂f/∂y(-1, 2)(y - 2) + (1/2!)[∂^2f/∂x^2(-1, 2)(x + 1)^2 + 2∂^2f/∂x∂y(-1, 2)(x + 1)(y - 2) + ∂^2f/∂y^2(-1, 2)(y - 2)^2] + ...
where f(-1, 2) is the function value at the point (-1, 2), ∂f/∂x(-1, 2) and ∂f/∂y(-1, 2) are the partial derivatives of f with respect to x and y at the point (-1, 2), and ∂^2f/∂x^2(-1, 2), ∂^2f/∂x∂y(-1, 2), and ∂^2f/∂y^2(-1, 2) are the second partial derivatives of f with respect to x, y, and xy respectively at the point (-1, 2).
Given the function f(x, y) = 21 + x - 2y + 4x^2 + xy + 6y^2
∂f/∂x = 1 + 8x + y
∂f/∂y = -2 + x + 12y
∂^2f/∂x^2 = 8
∂^2f/∂y^2 = 12
∂^2f/∂x∂y = 1
So, the Taylor series of f(x,y) = 21 + x - 2y + 4x^2 + xy + 6y^2 about the point (-1, 2) is:
f(x,y) = 21 + (-1 - 1) + 2(-2) + (1/2!)[8(x + 1)^2 + 2(1)(x + 1)(y - 2) + 12(y - 2)^2] + ...
= 21 + (-2) - 4 + (1/2!)[8(x + 1)^2 + 2(x + 1)(y - 2) + 12(y - 2)^2] + ...
= 21 - 6 + (1/2!)[8(x^2 + 2x + 1) + 2(x)(y - 2) + 2(x + 1)(y - 2) + 12(y^2 - 4y + 4)] + ...
= 15 + (1/2!)[8(x^2 + 2x + 1) + 2(x)(y - 2) + 2(x + 1)(y - 2) + 12(y^2 - 4y + 4)] + ...
This is the Taylor series of the function f(x, y) = 21 + x - 2y + 4x ^ 2 + xy + 6y ^ 2 in maximum order about the point (- 1, 2)
Therefore,the Taylor series of the function f(x, y) = 21 + x - 2y + 4x ^ 2 + xy + 6y ^ 2 in maximum order about the point (- 1, 2) is 15 + (1/2!)[8(x^2 + 2x + 1) + 2(x)(y - 2) + 2(x + 1)(y - 2) + 12(y^2 - 4y + 4)] + ...