Let y = f ( x ) = ( <msubsup> x 1 2 </msubsup> +

Waldronjw 2022-07-15 Answered
Let y = f ( x ) = ( x 1 2 + 2 x 2 , x 1 x 2 3 x 1 )
Is the linear approximation just f ( y ) = f ( x ) + A ( y x ) whenever y is approximately near x?
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Answers (1)

Wade Atkinson
Answered 2022-07-16 Author has 12 answers
A differentiable function f : R 2 R 2 can be approximated by it's derivative in the sense that given a point, say, ( 1 , 1 ), you can write:
f ( 1 + δ x , 1 + δ y ) f ( 1 , 1 ) + D f ( 1 , 1 ) ( δ x , δ y )
where D f ( 1 , 1 ) is a matrix applied to the vector ( δ x , δ y ). So, yes, if y is near x (as ( ( 1 + δ x , 1 + δ y )) is near ( ( 1 , 1 ))), then you can approximate the difference f ( y ) f ( x ) by the derivative evaluated at x, applied to the difference y x. That is:
f ( y ) f ( x ) + D f ( x ) ( y x )
Compare this to the one dimensional case, where the tangent line to the graph of a differentiable function locally approximates the graph, in the sense that:
f ( y ) f ( x ) + f ( x ) ( y x )
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