91Ó°ÊÓ

Find the values of \(x\) and \(y\) that minimize \(2 x^{2}+x y+y^{2}-y\) subject to the constraint \(x+y=0.\)

Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for each of the following functions. $$f(x, y)=x e^{x^{2} y^{2}}$$

Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for each of the following functions. $$f(x, y)=\ln (x y)$$

Let \(f(x, y)=10 x^{2 / 5} y^{3 / 5} .\) Show that \(f(3 a, 3 b)=3 f(a, b)\).

Find the values of \(x\) and \(y\) that minimize \(2 x^{2}-2 x y+y^{2}-2 x+1\) subject to the constraint \(x-y=3.\)

Let \(R\) be the rectangle consisting of all points \((x, y)\) such that \(0 \leq x \leq 2,2 \leq y \leq 3 .\) Calculate the following double integrals. Interpret each as a volume. $$\iint_{R}\left(x y+y^{2}\right) d x d y$$

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. $$f(x, y)=-8 y^{3}+4 x y+4 x^{2}+9 y^{2}$$

The present value of \(A\) dollars to be paid \(t\) years in the future (assuming a \(5 \%\) continuous interest rate) is \(P(A, t)=A e^{-0.05 t} .\) Find and interpret \(P(100,13.8)\).

Let \(R\) be the rectangle consisting of all points \((x, y)\) such that \(0 \leq x \leq 2,2 \leq y \leq 3 .\) Calculate the following double integrals. Interpret each as a volume. $$\iint_{R} e^{-x-y} d x d y$$

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. $$f(x, y)=2 x^{3}+2 x^{2} y-y^{2}+y$$