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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Definitions:

Gestalt Grouping

Principles in psychology that describe how humans naturally organize visual elements into groups or unified wholes.

Proximity

The physical or psychological closeness between people, objects, or ideas, often affecting perceptions and relations.

Stimulus

An external event or factor that induces a response or reaction in a biological organism.

Gestalt Grouping

Principles in psychology that explain how humans naturally organize visual elements into groups or unified wholes when certain principles are applied, such as similarity or proximity.