$f\left(x,y\right)=x\mathrm{cos}\left(y\right)$
(a) Find $f\left(7,9\right)$ and $f\left(7.1,9.05\right)$ and calculate $\mathrm{\Delta }z$.
$f\left(7,9\right)=?$
$f\left(7.1,9.05\right)=?$
$\mathrm{\Delta }z=?$
(b) Use the total differential dz to approximate $\mathrm{\Delta }z$.
$dz=?$
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Foreckije

(a) Find $f\left(7,9\right)$ and $f\left(7.1,9.05\right)$ and calculate $\mathrm{\Delta }z$
$f\left(7,9\right)=7\mathrm{cos}\left(9\right)=-6.3779$
$f\left(7,9\right)=-6.3779$
$f\left(7.1,9.05\right)=7.1\mathrm{cos}\left(9.05\right)=-6.6072$
$f\left(7.1,9.05\right)=-6.6072$
$\mathrm{\Delta }z=f\left(7.1,9.05\right)-f\left(7,9\right)=-6.6072-\left(-6.3779\right)$
$\mathrm{\Delta }z=-0.2293$

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yotaniwc
b) ${f}_{x}=\mathrm{cos}\left(y\right)⇒{f}_{x}\left(7,9\right)=\mathrm{cos}9$
${f}_{y}=-x\mathrm{sin}\left(y\right)⇒{f}_{y}\left(7,9\right)=-7\mathrm{sin}\left(9\right)$
$dx=0.1$, $dy=0.05$
$dz={f}_{x}dx+{f}_{y}dy=\left(0.1\right)\mathrm{cos}9+0.05\left(-7\mathrm{sin}\left(9\right)\right)=-0.2354$