Answer
$\Delta f = f\left( {3.53,8.98} \right) - f\left( {3.5,9} \right) \simeq 0.08$
Work Step by Step
We have $\nabla {f_{\left( {3.5,9} \right)}} = \left( {2, - 1} \right)$.
Let $P = \left( {a,b} \right) = \left( {3.5,9} \right)$ and $\Delta {\bf{v}} = \left( {h,k} \right) = \left( {0.03, - 0.02} \right)$.
Write:
$\Delta f = f\left( {3.53,8.98} \right) - f\left( {3.5,9} \right)$
Using Eq. (7) from Exercise 56, we get
$\Delta f \simeq \nabla {f_p}\cdot\Delta {\bf{v}}$
$\Delta f = f\left( {3.53,8.98} \right) - f\left( {3.5,9} \right) \simeq \nabla {f_{\left( {3.5,9} \right)}}\cdot\Delta {\bf{v}}$
$\Delta f = f\left( {3.53,8.98} \right) - f\left( {3.5,9} \right) \simeq \left( {2, - 1} \right)\cdot\left( {0.03, - 0.02} \right) = 0.08$
So, $\Delta f = f\left( {3.53,8.98} \right) - f\left( {3.5,9} \right) \simeq 0.08$.
