91Ó°ÊÓ

Find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of \(s\) and \(t\). $$ \begin{array}{ll} w=y^{3}-3 x^{2} y & s=0, \quad t=1 \\ x=e^{s}, \quad y=e^{t} & \end{array} $$

Describe the domain and range of the function. $$ f(x, y)=\sqrt{4-x^{2}-y^{2}} $$

A company manufactures two types of sneakers, running shoes and basketball shoes. The total revenue from \(x_{1}\) units of running shoes and \(x_{2}\) units of basketball shoes is \(R=-5 x_{1}^{2}-8 x_{2}^{2}-2 x_{1} x_{2}+42 x_{1}+102 x_{2}\), where \(x_{1}\) and \(x_{2}\) are in thousands of units. Find \(x_{1}\) and \(x_{2}\) so as to maximize the revenue.

Find an equation of the tangent plane to the surface at the given point.\(z=\sqrt{x^{2}+y^{2}}\) \((3,4,5)\)

Find the limit and discuss the continuity of the function. $$ \lim _{(x, y, z) \rightarrow(1,2,5)} \sqrt{x+y+z} $$

Find both first partial derivatives. $$ z=\frac{x^{2}}{2 y}+\frac{4 y^{2}}{x} $$

Find the directional derivative of the function at \(P\) in the direction of \(Q\). $$ f(x, y)=x^{2}+4 y^{2}, \quad P(3,1), Q(1,-1) $$

Use Lagrange multipliers to find the indicated extrema, assuming that \(x, y\), and \(z\) are positive. Minimize \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) Constraint: \(x+y+z=1\)

Find \(z=f(x, y)\) and use the total differential to approximate the quantity. $$ \sqrt{(5.05)^{2}+(3.1)^{2}}-\sqrt{5^{2}+3^{2}} $$

In Exercises \(17-20\), use a computer algebra system to graph the surface and locate any relative extrema and saddle points. \(z=\frac{-4 x}{x^{2}+y^{2}+1}\)