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\noindent {\bf Problem statement} Suppose $R(x,y) = v(x+y^2)$ where $v$ is a four times differentiable
function of {\it one} variable. Suppose you know also that:

$$v(0)=\alpha\,;\ 
v'(0)=\beta\,;\ 
v^{(2)}(0)=\gamma\,;\ 
v^{(3)}(0)=\delta\,;\ 
v^{(4)}(0)=\varepsilon\,.$$

\noindent Compute the following seven quantities in terms of $\alpha$,
$\beta$, $\gamma$, $\delta$, and $\varepsilon$:

\vskip -.1in

$$R(0,0)\,;\ 
{{\partial R}\over {\partial x}}(0,0)\,;\
{{\partial R}\over {\partial y}}(0,0)\,;\
{{\partial^{2} R}\over {\partial x^{2}}}(0,0)\,;\
{{\partial^{2} R}\over {\partial y^{2}}}(0,0)\,;\
{{\partial^{2} R}\over {\partial x \partial y}}(0,0)\,;\
{{\partial^{4} R}\over {\partial x^{2} \partial y^{2}}}(0,0)\,.$$











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