How do you find the linear approximation of f ( x , y ) = 53 - 9 x 2 - y 2 at (2,-1)?

Question

How do you find the linear approximation of             f              (        x        ,        y        )              =                  53        -        9                  x          2                -                  y          2                     at (2,-1)?

Kody Larsen · Accepted Answer

The linear approximation to a function f of two variables (at a point) is the equation of the tangent plane to the surface (at that point).The equation of that tangent plane depends on the slope in each direction; the partial derivatives                     f        x                   and                     f        y                  . If the surface is      z    =          f              (        x        ,        y        )            , then the tangent plane at             (              x        0            ,              y        0            )       is      z    =          f              (                  x          0                ,                  y          0                )              +                  f        x                    (                  x          0                ,                  y          0                )                    (      x      -              x        0            )        +                  f        y                    (                  x          0                ,                  y          0                )                    (      y      -              y        0            )      You need to take the partials of your             f              (        x        ,        y        )            , evaluate them at             (      2      ,      -      1      )      , and simplify the equation.Hint: If you get stuck, look for previous examples of derivatives of a function to a power. Use the Chain Rule to do the job.*

How do you find the linear approximation of f(x,y) = sqrt(53-9x^2-y^2) at (2,-1)?

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