91Ó°ÊÓ

Use least squares to find the exponential curve \(y=B e^{A x}\) for the following tables of points. $$ \begin{array}{c|r} x & \multicolumn{1}{|c} {y} \\ \hline 0 & 1 \\ 1 & 2 \\ 2 & 5 \\ 3 & 10 \end{array} $$

Evaluate each iterated integral. \(\int_{3}^{5} \int_{0}^{y}(2 x-y) d x d y\)

Use Lagrange multipliers to maximize and minimize each function subject to the constraint. (The maximum and minimum values do exist.) $$ f(x, y)=x+2 y, \quad 2 x^{2}+y^{2}=72 $$

For each function, evaluate the stated partials. $$ f(x, y)=e^{x^{2}+y^{2}}, \text { find } f_{x}(0,1) \text { and } f_{y}(0,1) $$

For the given function and values, find: a. \(\Delta f\) b. \(d f\) $$ \begin{array}{l} f(x, y)=e^{x}+x y+\ln y \\ x=0, \quad \Delta x=d x=0.05 \\ y=1, \quad \Delta y=d y=0.01 \end{array} $$

For each function, evaluate the stated partials. $$ g(x, y)=(x y-1)^{5}, \text { find } g_{x}(1,0) \text { and } g_{y}(1,0) $$

Use Lagrange multipliers to maximize and minimize each function subject to the constraint. (The maximum and minimum values do exist.) $$ f(x, y)=12 x+30 y, \quad x^{2}+5 y^{2}=81 $$

For the given function and values, find: a. \(\Delta f\) b. \(d f\) $$ \begin{array}{l} f(x, y)=\ln \left(x^{2}+y^{2}\right), x=6, \quad \Delta x=d x=0.1, \quad y=8 \\\ \Delta y=d y=0.2 \end{array} $$

The price-earnings ratio of a stock is defined as \(R(P, E)=\frac{P}{E}\) where \(P\) is the price of a share of stock and \(E\) is its earnings. Find the price- earnings ratio of a stock that is selling for \(\$ 140\) with earnings of \(\$ 1.70 .\)

In a laboratory test the combined antibiotic effect of \(x\) milligrams of medicine \(\mathrm{A}\) and \(y\) milligrams of medicine \(\mathrm{B}\) is given by the function $$ f(x, y)=x y-2 x^{2}-y^{2}+110 x+60 y $$ (for \(0 \leq x \leq 55,0 \leq y \leq 60\) ). Find the amounts of the two medicines that maximize the antibiotic effect.