Consider the function $f(x,y)=4x^2+y^2+1.$

$($a$)$ Identify the domain and range of this function.

$($b$)$ Sketch some level curves of this function in the $xy-$plane.

$($c$)$ Use these level curves to sketch a graph of $z=f(x,y)$ on an $xyz$ coordinate system.

$($d$)$ Calculate $f_x(1,1)$ and $f_y(1,1).$ What do these numbers mean in terms of your graph?