91Ó°ÊÓ

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

Define the total differential of a function of two variables.

Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. [Hint: In Exercise 23, minimize \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+3 y=-1 .]\) $$ \begin{array}{ll} \underline{\text { Curve }} & \underline{\text {Point}} \\ \text { Circle: }(x-4)^{2}+y^{2}=4 \quad (0,10) \end{array} $$

Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. [Hint: In Exercise 23, minimize \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+3 y=-1 .]\) $$ \begin{array}{ll} \underline{\text { Surface }} & \underline{\text {Point}} \\ \text { Plane: } x+y+z=1& \quad(2,1,1) \end{array} $$

The volume of the red right circular cylinder in the figure is \(V=\pi r^{2} h .\) The possible errors in the radius and the height are \(\Delta r\) and \(\Delta h\), respectively. Find \(d V\) and identify the solids in the figure whose volumes are given by the terms of \(d V\). What solid represents the difference between \(\Delta V\) and \(d V ?\)

The formula for wind chill \(C\) (in degrees Fahrenheit) is given by $$ C=35.74+0.6215 T-35.75 v^{0.16}+0.4275 T \mathrm{}^{0.16} $$ where \(v\) is the wind speed in miles per hour and \(T\) is the temperature in degrees Fahrenheit. The wind speed is \(23 \pm 3\) miles per hour and the temperature is \(8^{\circ} \pm 1^{\circ} .\) Use \(d C\) to estimate the maximum possible propagated error and relative error in calculating the wind chill.

Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. Use the graphing utility to plot the points and graph the regression line. $$ (0,6),(4,3),(5,0),(8,-4),(10,-5) $$

Find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point.\(x y z=10, \quad(1,2,5)\)

Use Lagrange multipliers to find the dimensions of a right circular cylinder with volume \(V_{0}\) cubic units and minimum surface area.

Use Lagrange multipliers to find the dimensions of a rectangular box of maximum volume that can be inscribed (with edges parallel to the coordinate axes) in the ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\)