Expand f(x, y) =x²+3y²-9x-9y+26 in Taylor's series upto second order about the point (2,2)
Answers
lets put (2,2) in the given cubic function :
2^2 + 2x2^2 - 9x2 - 9x2 + 26
or, (4 + 8) + 26 - 18
or, 12 + 8
or, 20. Answer.
The expansion of function in Taylor's series up to second order about the point (2,2) is the function it self that is x²+3y² -9x-9y+26.
Given that,
The function is f(x,y) = x²+3y²-9x-9y+26
We have to find the expansion of function in Taylor's series up to second order about the point (2,2).
We know that,
The Taylor's series expansion for a function up to second order is
F(x,y) = f(x₀,y₀)+ ()of(x₀,y₀) (x-x₀) + ()of(x₀,y₀) (y-y₀) + ()of(x₀,y₀) (x-x₀)² + ()of(x₀,y₀) (y-y₀)² + ()of(x₀,y₀) (x-x₀)(y-y₀)+......
So,
(x₀,y₀) = (2,2)
f(x₀,y₀) = 4+3(4)-9(2)-9(2)+26=6
= 2x-9
()of(x₀,y₀) = -5
= 6y-9
()of(x₀,y₀) = 3
= 2
()of(x₀,y₀) =2
= 6
()of(x₀,y₀) =6
()of(x₀,y₀) =0
So,
F(x,y) = 6 - 5(x-2) + 3(y-2) + ×2(x-2)² + ×6(y-2)²+0
F(x,y) = 6 - 5x + 10 + 3y - 6 + x² - 4x + 4 + 3(y²-4y+4)
F(x,y) = -5x + 3y + x² - 4x + 3y² - 12y + (6+10-6+4+12)
F(x,y) = x²+3y² -9x-9y+26
Therefore, the expansion of function in Taylor's series up to second order about the point (2,2) is the function it self that is x²+3y² -9x-9y+26.
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