91Ó°ÊÓ

Extrema on a circle of intersection \(\quad\) Find the extreme values of the function \(f(x, y, z)=x y+z^{2}\) on the circle in which the plane \(y-x=0\) intersects the sphere \(x^{2}+y^{2}+z^{2}=4.\)

Extreme values on a helix Suppose that the partial derivatives of a function \(f(x, y, z)\) at points on the helix \(x=\cos t, y=\sin t\) \(z=t\) are $$ f_{x}=\cos t, \quad f_{y}=\sin t, \quad f_{z}=t^{2}+t-2 $$ At what points on the curve, if any, can \(f\) take on extreme values?

Suppose that \(T\) is to be found from the formula \(T=x\left(e^{y}+e^{-y}\right),\) where \(x\) and \(y\) are found to be 2 and \(\ln 2\) with maximum possible errors of \(|d x|=0.1\) and \(|d y|=0.02 .\) Estimate the maximum possible error in the computed value of \(T\)

You plan to calculate the area of a long, thin rectangle from measurements of its length and width. Which dimension should you measure more carefully? Give reasons for your answer.

Find an equation for and sketch the graph of the level curve of the function \(f(x, y)\) that passes through the given point. $$f(x, y)=\frac{2 y-x}{x+y+1}, \quad(-1,1)$$

Find three numbers whose sum is 9 and whose sum of squares is a minimum.

Find the dimensions of the rectangular box of maximum volume that can be inscribed inside the sphere \(x^{2}+y^{2}+z^{2}=4\).

Among all closed rectangular boxes of volume \(27 \mathrm{cm}^{3},\) what is the smallest surface area?

Define \(f(0,0)\) in a way that extends \(f\) to be continuous at the origin. $$f(x, y)=\frac{3 x^{2} y}{x^{2}+y^{2}}$$

Express \(v_{x}\) in terms of \(u\) and \(y\) if the equations \(x=v \ln u\) and \(y=u \ln v\) define \(u\) and \(v\) as functions of the independent variables \(x\) and \(y,\) and if \(v_{x}\) exists. (Hint: Differentiate both equations with respect to \(x\) and solve for \(v_{x}\) by eliminating \(u_{x}\).)