# Use suitable linear approximation to find the approximate values for given functions at the points i

Use suitable linear approximation to find the approximate values for given functions at the points indicated:
$f\left(x,y\right)=x{e}^{y+{x}^{2}}$
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bailaretzy33
Let $L\left(x,y\right)$=$f\left({x}_{0},{y}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0}\right)\left(y-{y}_{0}\right)$. Then $L\left(x,y\right)\approx f\left(x,y\right)$. Consider $\left({x}_{0},{y}_{0}\right)=\left(2,-4\right)$. Then,
$L\left(x,y\right)=2+9\left(x-2\right)+2\left(y+4\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}f\left(2.05,-3.92\right)\approx L\left(2.05,-3.92\right)=2.61$
Notice, from a calculator, $f\left(2.05,-3.92\right)=2.7192$
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Diya Bass
Denoting the partial derivatives by ${f}_{x}^{\prime }$ and ${f}_{y}^{\prime }$, the formula is:
$f\left({x}_{0}+h,{y}_{0}+k\right)=f\left({x}_{0},{y}_{0}\right)+{f}_{x}^{\prime }\left({x}_{0},{y}_{0}\right)h+{f}_{y}^{\prime }\left({x}_{0},{y}_{0}\right)k+o\left(‖\left(h,k\right)‖\right).$