Answer
$f\left( {4.1,7.9} \right) = \frac{{4.1}}{{7.9}} \simeq 0.51875$
Using a calculator:
$\frac{{4.1}}{{7.9}} \simeq 0.518987$
Work Step by Step
Let $f\left( {x,y} \right) = \frac{x}{y}$. So, the partial derivatives are
${f_x} = \frac{1}{y}$, ${\ \ \ }$ ${f_y} = - \frac{x}{{{y^2}}}$
We can consider $\frac{{4.1}}{{7.9}}$ as a value of $f\left( {x,y} \right) = \frac{x}{y}$. Thus,
$f\left( {4.1,7.9} \right) = \frac{{4.1}}{{7.9}}$
Using the linear approximation, Eq. (3) we have
$f\left( {a + h,b + k} \right) \approx f\left( {a,b} \right) + {f_x}\left( {a,b} \right)h + {f_y}\left( {a,b} \right)k$
For $\left( {a,b} \right) = \left( {4,8} \right)$ and $\left( {h,k} \right) = \left( {0.1, - 0.1} \right)$, we get
$f\left( {4.1,7.9} \right) \approx f\left( {4,8} \right) + {f_x}\left( {4,8} \right)\cdot0.1 + {f_y}\left( {4,8} \right)\cdot\left( { - 0.1} \right)$
$f\left( {4.1,7.9} \right) \simeq \frac{1}{2} + \frac{1}{8}\cdot0.1 + \frac{1}{{16}}\cdot0.1$
$f\left( {4.1,7.9} \right) = \frac{{4.1}}{{7.9}} \simeq 0.51875$
Using a calculator, we get $\frac{{4.1}}{{7.9}} \simeq 0.518987$
