The Quadratic Taylor Approximations Of $f ( x , y )$

question 49

Essay

The quadratic Taylor approximations of $f ( x , y )$ at the points $( - 1 - 1 ) , ( 1,1 )$ , (2,0)and (0,2)are given respectively by: $$\begin{array} { l } Q _ { 1 } ( x , y ) = - 3 + 3 ( x + 1 ) ^ { 2 } + 4 ( y + 1 ) ^ { 2 } \\Q _ { 2 } ( x , y ) = 1 - 3 ( x - 1 ) ^ { 2 } + 4 ( y - 1 ) ^ { 2 } \\Q _ { 3 } ( x , y ) = - 2 - 9 ( x - 2 ) - 6 ( x - 2 ) ^ { 2 } - 2 y ^ { 2 } \\Q _ { 4 } ( x , y ) = 8 + 3 x + 24 ( y - 2 ) + 22 ( y - 2 ) ^ { 2 }\end{array}$$ Find an approximate value of $f ( 0.2,1.9 )$ .

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