Consider the function

$$f(x,y) = \left( \begin{array}{c} 5x^{2}+xy+2y^{3} \\ 3x+4y+\sin\!\left(xy\right) \end{array} \right).$$
(a) Find a general expression for the Jacobian matrix $\displaystyle Jf = \frac{\partial(f_1, f_2)}{\partial(x,y)}.$

$Jf =$

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

(b) Evaluate $Jf$ at the point $\boldsymbol{\vec{a}} = \langle 4, 0 \rangle.$

$Jf(\boldsymbol{\vec{a}}) =$

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