Consider the function

$$z = f(x,y) = x^{2}+5xy^{3}.$$
(a) Find $f(5,-3)$.
$z = f(5,-3) =$

(b) Find a function $g(x,y,z)$ whose level zero set is equal to the graph of $z = f(x,y)$ and such that the coefficient of $z$ in $g(x,y,z)$ is $1$.
The level set $g(x,y,z) =$ $= 0$ is the same as the graph of $z = f(x,y)$.

(c) Find the gradient of $g$. Write your answer as a row vector of the general form $\langle a, b, c \rangle$.
$\nabla g(x,y,z) =$

(d) Use $\nabla g$ to find a vector $\vec{n}$ perpendicular (or normal) to the graph of $z = f(x,y)$ at the point $\left(5,-3,-650\right)$. Write your answer as a row vector of the general form $\langle a, b, c \rangle$.
$\vec{n} =$

(e) Find an equation for the tangent plane to $z = f(x,y)$ at the point $\left(5,-3,-650\right)$. Enter your answer as an equation.