# f ( x , y ) = <mrow class="MJX-TeXAtom-ORD"> &#x221A; </mrow> ( 7 + 2

$f\left(x,y\right)=\surd \left(7+2xy\right)$Find the linear approximation at $\left(3,-1\right)$
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Schetterai
make a change of variable $x=3+h,y=-1+k.$. then
$z={\left(7+2\left(3+h\right)\left(-1+k\right)\right)}^{1/2}={\left(1-2h+6k+\cdots \right)}^{1/2}=1+\frac{1}{2}\left(-2h+6k\right)+\cdots =1-h+3k+\cdots$
the planar approximation of $z$ at $\left(3,-1\right)$ is $z=1-\left(x-3\right)+3\left(y+1\right).$
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Devin Anderson
Use
$L\left(x\right)-f\left(3,-1\right)=-\left(x-3\right)+3\left(y-\left(-1\right)\right)=3-x+3y+3.$