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  • 03-08-2017
  • Mathematics
contestada

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (6, 6, 3) and use it to approximate the number 6.022 + 5.972 + 2.992 .

Respuesta :

LammettHash
LammettHash LammettHash
  • 12-08-2017
Near the point (6, 6, 3), we have the approximation

[tex]f(x,y,z)\approx f(6,6,3)+f_x(6,6,3)(x-6)+f_y(6,6,3)(y-6)+f_z(6,6,3)(z-3)[/tex]

where the subscripted [tex]f[/tex]'s refer to the corresponding partial derivatives.

[tex]f(x,y,z)=x^2+y^2+z^2[/tex]
[tex]\implies\begin{cases}f_x(x,y,z)=2x\\f_y(x,y,z)=2y\\f_z(x,y,z)=2z\end{cases}[/tex]

So

[tex]f(6.02,5.97,2.99)\approx80.8288[/tex]

Compare to more precise value,

[tex]f(6.02,5.97,2.99)\approx80.8214[/tex]
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