91Ó°ÊÓ

Find an exponential regression curve for each data set. $$ \begin{array}{|r|r|r|r|r|} \hline x & 1 & 2 & 3 & 4 \\ \hline y & 8 & 25 & 72 & 225 \\ \hline \end{array} $$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=2 x^{2}+y^{2}-x y ; x+y=8 $$

Find an exponential regression curve for each data set. $$ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 4 & 6 & 8 \\ \hline y & 13 & 7 & 3.7 & 1 \\ \hline \end{array} $$

Evaluate. $$ \int_{0}^{2} \int_{0}^{x}\left(x+y^{2}\right) d y d x $$

The Haycock formula for approximating the surface area \(S,\) in square meters \(\left(\mathrm{m}^{2}\right),\) of a human is given by $$ S(h, w)=0.024265 h^{0.3964} w^{0.5378} $$ where \(h\) is the person's height in centimeters and \(w\) is the person's weight in kilograms. (Source: www.halls.md.) Use the Haycock approximation to estimate the surface area of a person whose height is \(165 \mathrm{~cm}\) and whose weight is \(80 \mathrm{~kg}\).

Find the four second-order partial derivatives. $$f(x, y)=x^{4} y^{3}-x^{2} y^{3}$$

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Minimum: \(f(x, y)=2 x^{2}+y^{2}+2 x y+3 x+2 y ;\) \(y^{2}=x+1\)

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Maximum: \(f(x, y, z, t)=x+y+z+t ;\) \(x^{2}+y^{2}+z^{2}+t^{2}=1\)

The Mosteller formula for approximating the surface area, \(S,\) in \(\mathrm{m}^{2},\) of a human is \(s=\frac{\sqrt{h w}}{60}\) where \(h\) is the person's height in centimeters and \(w\) is the person's weight in kilograms. (Source: www.halls.md.) a) Compute \(\frac{\partial S}{\partial h}\). b) Compute \(\frac{\partial S}{\partial w}\). c) The change in \(S\) due to a change in \(w\) when \(h\) is constant is approximately \(\Delta S \approx \frac{\partial S}{\partial w} \Delta w\) Use this formula to approximate the change in someone's surface area given that the person is \(170 \mathrm{~cm}\) tall, weighs \(80 \mathrm{~kg}\), and loses \(2 \mathrm{~kg}\).