Answer
Using linear approximation:
$\sqrt {{{7.1}^2} + {{4.9}^2} + 69.5} \simeq 11.9958$
Using a calculator:
$\sqrt {{{7.1}^2} + {{4.9}^2} + 69.5} \simeq 11.9967$.
Work Step by Step
We have $f\left( {x,y,z} \right) = \sqrt {{x^2} + {y^2} + z} $.
The partial derivatives are
${f_x} = \frac{x}{{\sqrt {{x^2} + {y^2} + z} }}$, ${\ \ }$ ${f_y} = \frac{y}{{\sqrt {{x^2} + {y^2} + z} }}$, ${\ \ }$ ${f_z} = \frac{1}{{2\sqrt {{x^2} + {y^2} + z} }}$
Let $x=a+h$, $y=b+k$ and $z=c+m$. The linear approximation for three variables is given by
(1) ${\ \ \ \ }$ $f\left( {a + h,b + k,c + m} \right) \approx f\left( {a,b,c} \right)$
$ + {f_x}\left( {a,b,c} \right)h + {f_y}\left( {a,b,c} \right)k + {f_z}\left( {a,b,c} \right)m$
Write $\left( {a,b,c} \right) = \left( {7,5,70} \right)$ and $\left( {h,k,m} \right) = \left( {0.1, - 0.1, - 0.5} \right)$.
So,
$f\left( {7,5,70} \right) = 12$
${f_x}\left( {7,5,70} \right) = \frac{7}{{12}}$
${f_y}\left( {7,5,70} \right) = \frac{5}{{12}}$
${f_z}\left( {7,5,70} \right) = \frac{1}{{24}}$
Using equation (1) we get
$f\left( {7.1,4.9,69.5} \right)$
$ = \sqrt {{{7.1}^2} + {{4.9}^2} + 69.5} $
$ \simeq f\left( {7,5,70} \right) + {f_x}\left( {7,5,70} \right)\cdot0.1 + {f_y}\left( {7,5,70} \right)\cdot\left( { - 0.1} \right) + {f_z}\left( {7,5,70} \right)\cdot\left( { - 0.5} \right)$
$ \simeq 12 + \frac{7}{{12}}\cdot0.1 + \frac{5}{{12}}\cdot\left( { - 0.1} \right) + \frac{1}{{24}}\cdot\left( { - 0.5} \right)$
$ \simeq 11.9958$
Using a calculator, we obtain $\sqrt {{{7.1}^2} + {{4.9}^2} + 69.5} \simeq 11.9967$.
