Question $1[25$ Marks$]$

a$)$ Find the total differential $dz$ for the function $z=f(x,y)=x^2{y}^2-3xy+x.$ If $x$ changes from $2$ to $1\mathrm{.}97$ and $y$ changes from $3$ to $3\mathrm{.}02,$ compare the values of $dz$ and $z.$

$(8)$

b$)$ If $f(x,y)$ has a directional derivative of $8$ in the direction of vec$(V{)}_1=($:$3,-4$:$)$ and a directional derivative of $1$ in the direction of vec$(V{)}_2=($:$12,5$:$)$ at the point $(-1,2),$ what is the directional derivative for $f(x,y)$ at point $(-1,2)$ in the direction of vec$(V{)}_3=($:$3,-5$:$)?$ Find the maximum rate of change for $f(x,y)$ at point $(-1,2)$ and the direction in which it occurs.

c$)$ Find the volume of the solid that lies below the surface $z=2x+y$ and above the $x-y$ plane, bounded by lines $x=0,y=1,$ and $x=\sqrt[\phantom{2}]{y}$ in first quadrant of the $x-y$ plane.