Answer
$f\left( { - 2.1,3.1} \right) \simeq 4.2$
Work Step by Step
We are given the equation of the tangent plane to $z = f\left( {x,y} \right){\rm{at}}\left( { - 2,3,4} \right)$:
$4x + 2y + z = 2$
We can write the equation as $z = L\left( {x,y} \right) = - 4x - 2y + 2$.
By definition of differentiability, the estimate of $f\left( { - 2.1,3.1} \right)$ is $L\left( { - 2.1,3.1} \right)$ for point near to $\left( {x,y} \right) = \left( {-2,3} \right)$. Thus,
$f\left( { - 2.1,3.1} \right) \approx L\left( { - 2.1,3.1} \right) = - 4\cdot\left( { - 2.1} \right) - 2\cdot3.1 + 2$
$f\left( { - 2.1,3.1} \right) \simeq 4.2$
