= Solution
The first-order <multivariable Taylor expansion> is
$$
f(x,y)=e^{y_0}\bigl[x_0^2+2x_0(x-x_0)+x_0^2(y-y_0)\bigr]+O\bigl((x-x_0)^2+(y-y_0)^2\bigr).
$$
Taking $(x_0,y_0)=(2,0)$ gives the <linear approximation>
$$
\boxed{f(2.03,-0.01)\simeq4+4(0.03)+4(-0.01)=4.08.}
$$
To assess its <significant figures>, the quadratic <Taylor polynomial> supplies the correction $h^2+4hk+2k^2=-0.0001$ for $h=0.03$, $k=-0.01$. Direct evaluation gives $2.03^2e^{-0.01}=4.0798963599\ldots$. Thus the estimate is \b[correct to four significant figures]: both round to $4.080$, whereas to five significant figures the exact value is $4.0799$ and the approximation is $4.0800$. The unexpectedly small quadratic correction results partly from cancellation of its mixed and unmixed terms.
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