Functions of two variables

Find the natural domain of $f(x,y)=\sqrt{x-y^2}$ and state whether it is an open, closed, bounded set.

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Find the natural domain of $f(x,y)=\ln (x^2+y^2-9)$ and state whether it is an open, closed, bounded set.

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Find the natural domain of $f(x,y)=\frac{\sqrt{4-x^2-y^2}}{x-y}$ and state whether it is an open, closed, bounded set.

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Find the level curves of $f(x,y)=x^2-y$ and the level curve that passes through the point $(2,1)$.

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Compute the partial derivatives of $f(x,y)=x^3-4x^2{y}^2+2y$.

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Compute the gradient of $f(x,y)=x\ln y+e^{x-y}$, defined for $y>0$, at the point $(1,1)$.

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Compute the second derivatives of $f(x,y)=x^2{e}^y$, verify that the mixed derivatives coincide and write the Hessian matrix at the point $(1,0)$.

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Write the equation of the tangent plane to the graph of $f(x,y)=x^{3}y-2y^2$ at the point with coordinates $x=1$, $y=1$.

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A firm has production function $Q(K,L)=10K^{\frac{1}{2}}{L}^{\frac{1}{2}}$ and employs $K=100$ and $L=400$. Using the differential, estimate the production when capital rises to $102$ and labor falls to $396$.

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Let $f(x,y)=x{y}^2$ and the curve $x(t)=t^2$, $y(t)=3-t$. Compute $g^{\prime }(1)$, where $g(t)=f(x(t),y(t))$.

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Prove that the limit $\underset{(x,y)\to (0,0)}{\lim }\frac{x^2-y^2}{x^2+y^2}$ does not exist.

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Given the production function $Q(K,L)=5K^{\frac{2}{5}}{L}^{\frac{3}{5}}$, compute the production and the marginal productivities for $K=32$ and $L=32$.

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A consumer has utility function $U(x,y)=x{y}^2$, with $x>0$ and $y>0$. Write the indifference curve through the bundle $(4,2)$ and compute the marginal rate of substitution $\frac{U_x}{U_y}$ at that point.

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Determine the type of returns to scale of the production function $Q(K,L)=3K^{\frac{1}{2}}{L}^{\frac{3}{4}}$ and compute the number by which production is multiplied when capital and labor double.

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