Consider the function

$$f(x,y,z) = \left( \begin{array}{c} 5e^{xyz^{2}}+4y \\ 2x^{2}+xz \end{array} \right).$$
(a) Use the Jacobian $Jf(\boldsymbol{\vec{a}})$ to construct a linear (or affine) approximation $L(\boldsymbol{\vec{x}})$ to $f$ near the point $\boldsymbol{\vec{a}} = \left( \begin{array}{r} -1 \\ 0 \\ 1 \end{array} \right).$

$L(\boldsymbol{\vec{x}}) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

(b) Find the values of the function $f$ and the linear approximation $L$ at the nearby point $\boldsymbol{\vec{b}} = \left( \begin{array}{r} -1.02 \\ 0.01 \\ 0.98 \end{array} \right).$

$f(\boldsymbol{\vec{b}}) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

$L(\boldsymbol{\vec{b}}) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$