$(8pts)$ Let $f(x,y)be$ a smooth function. The wettos gradf$(0,0),grad{f}_x(0,0),$ and $grad{f}_y(0,0)$ are

represented $by\frac{?}{\text{b}\text{a}\text{r}}(OA)\frac{,}{b}ar(OB),$ and $\frac{?}{\text{b}\text{a}\text{r}}(OC)$ respectively, $in$ the figure below.

$(a)$ Compute the unexumua unavasuve $\frac{?}{?}-$

$(b)Is$ there a direction $ifin$ which the rate $of$ change $offat(0,0)is-\sqrt[\phantom{2}]{5}?$

Circle your answer:

Yes

$No$

$Ifso,$ slowtch $ifin$ the figure above. $If$ not, explain why.

$(c)Iff(0,0)=2,$ estimate $f(0\mathrm{.}01,-0\mathrm{.}02)$ using the linear approdmation.

$(d)$ Determine the second$-$order partial derivatives $f_{}(0,0),f_w(0,0),$ and $f_{xy}(0,0).$