Answer
$f\left( {2,3,1} \right) \simeq 8.7$
Work Step by Step
We are given:
$f\left( {2,3} \right) = 8$, ${\ \ }$ ${f_x}\left( {2,3} \right) = 5$, ${\ \ }$ ${f_y}\left( {2,3} \right) = 7$
Let $\left( {a,b} \right) = \left( {2,3} \right)$. Write $x = a + \Delta x$ and $y = b + \Delta y$.
By equation (3) we have the linear approximation:
$f\left( {a + \Delta x,b + \Delta y} \right) \approx f\left( {a,b} \right) + {f_x}\left( {a,b} \right)\Delta x + {f_y}\left( {a,b} \right)\Delta y$
For $\Delta x = 0$ and $\Delta y = 0.1$ we obtain the desired estimate:
$f\left( {2,3,1} \right) \approx f\left( {2,3} \right) + {f_x}\left( {2,3} \right)\cdot0 + {f_y}\left( {2,3} \right)\cdot0.1$
$f\left( {2,3,1} \right) \simeq 8 + 7\cdot0.1$
$f\left( {2,3,1} \right) \simeq 8.7$
