91Ó°ÊÓ

The value of residential property for tax purposes is usually much lower than its actual market value. If \(v\) is the market value, the assessed value for real estate taxes might be only \(40 \%\) of \(v .\) Suppose that the property \(\operatorname{tax}, T,\) in a community is given by the function $$T=f(r, v, x)=\frac{r}{100}(.40 v-x)$$ where \(v\) is the market value of a property (in dollars), \(x\) is a homeowner's exemption (a number of dollars depending on the type of property), and \(r\) is the tax rate (stated in dollars per hundred dollars). (a) Determine the real estate tax on a property valued at \(\$ 200,000\) with a homeowner's exemption of \(\$ 5000,\) assuming a tax rate of \(\$ 2.50\) per hundred dollars of net assessed value. (b) Determine the tax duc if the tax rate increases by \(20 \%\) to \(\$ 3.00\) per hundred dollars of net assessed value. Assume the same property value and homeowner's exemption. Does the tax due also increase by \(20 \% ?\)

Let \(f(p, q)=1-p(1+q) .\) Find \(\frac{\partial f}{\partial q}\) and \(\frac{\partial f}{\partial p}.\)

Find the values of \(x, y,\) and \(z\) that minimize \(x y+x z-2 y z\) subject to the constraint \(x+y+z=2.\)

Both first partial derivatives of the function \(f(x, y)\) are zero at the given points. Use the second-derivative test to determine the nature of \(f(x, y)\) at each of these points. If the second derivative test is inconclusive, so state. $$f(x, y)=x^{4}-4 x y+y^{4},(0,0),(1,1),(-1,-1)$$

The productivity of a country is given by \(f(x, y)=300 x^{2 / 3} y^{1 / 3},\) where \(x\) and \(y\) are the amount of labor and capital. (a) Compute the marginal productivities of labor and capital when \(x=125\) and \(y=64.\) (b) Use part (a) to determine the approximate effect on the productivity of increasing capital from 64 to 66 units, while keeping labor fixed at 125 units. (c) What would be the approximate effect of decreasing labor from 125 to 124 units while keeping capital fixed at 64 units?

The volume \((V)\) of a certain amount of a gas is determined by the temperature ( \(T\) ) and the pressure ( \(P\) ) by the formula \(V=.08(T / P) .\) Calculate and interpret \(\frac{\partial V}{\partial P}\) and \(\frac{\partial V}{\partial T}\) when \(P=20, T=300\).

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of \(f(x, y)\) at each of these points. If the second-derivative test is inconclusive, so state. $$f(x, y)=-2 x^{2}+2 x y-y^{2}+4 x-6 y+5$$