## Calculus with Applications (10th Edition)

$\sqrt 8.05^{2}+5.97^{2} \approx 10 + 0.022= 10.022$ Calculator gives $\sqrt 8.05^{2}+5.97^{2}\approx10.0221$ The error is approximately 0.0001
We are given: $\sqrt 8.05^{2}+5.97^{2}$ Let $f(x,y)=\sqrt x^{2}+y^{2}$ with $x=8, dx=0.05, y=6, dy=-0.03$ then use dz to approximate $\Delta z$ $dz=f_{x}(x,y).dx+d_{y}(x,y).dy$ $dz=(\frac{1}{2\sqrt x^{2}+y^{2}}.2x)dx+(\frac{1}{2\sqrt x^{2}+y^{2}}.2y)dy$ $dz=(\frac{x}{\sqrt x^{2}+y^{2}})dx+(\frac{y}{\sqrt x^{2}+y^{2}})dy$ $dz=\frac{4}{5}(0.05)+\frac{3}{5}(-0.03)$ $dz=0.022$ Thus, $\sqrt 8.05^{2}+5.97^{2} \approx 10 + 0.022= 10.022$ Calculator gives $\sqrt 8.05^{2}+5.97^{2}\approx10.0221$ The error is approximately 0.0001