OUESTION $2[15$ MARKS$]$

$($a$)$ Given the function $f(x,y,z)=(3x^2+3y^2)sin(4z).$

$($i$)$ Determine the domain and range of the function.

$($ii$)$ Calculate the partial derivative in higher dimension, $f_{ax}.$

$[5$ Marks$]$

$($b$)$ Given the function $f(x,y,z)=x^3{y}^2(5z+3).$

$($i$)$ Calculate the rate of change of the function in the direction of $v=($:$1,-2,2$:$)$ at the point $(1,2,-4)-\frac{48}{18},5x^3{y}^2+3x^3{y}^2$

$($ii$)$ Determine the directions $($in unit vectors$)$ where the function changes most rapidly at the point $(1,2,-4).$

$60,36\frac{F}{f1},96$

$[10$ Marks$]$