91Ó°ÊÓ

Allow inequality constraints, optional but good. Minimize \(f=x^{2}+y^{2}\) subject to the inequality constraint \(x+y \geqslant 4\). Now the minimum is at ___________ and the multiplier is \(\lambda=\) __________ and \(f_{\text {min }}=\) ________. A small change to \(4+d k\) changes \(f_{\min }\) by what multiple of \(d k ?\)

The hours in a day are limited by \(x+y=24\). Write \(x^{2} y\) as \(x^{2}(24-x)\) and maximize to find the optimal number of hours to stay awake.

Calculate all eight third derivatives \(f_{x x x}, f_{x x y}, \ldots\) of \(f=\) \(x^{3} y^{3} .\) How many are different?

Around the point (1,-2) the temperature \(T=e^{-x^{2}-y^{2}}\) has \(\Delta T _\approx \quad \Delta x+\) In what direction u does it get hot fastest?

Near \(x=16\) draw the level curve \(x^{2} y=2048\) and the line \(x+y=24\). Show that the curve is convex and the line is tangent.

Allow inequality constraints, optional but good. A wire \(40^{\prime \prime}\) long is used to enclose one or two squares (side \(x\) and side \(y\) ). Maximize the total area \(x^{2}+y^{2}\) subject to \(x \geqslant 0, y \geqslant 0,4 x+4 y=40\).

If \(\sin z=x+y\) find \((\partial z / \partial x)\), in two ways: (1) Write \(z=\sin ^{-1}(x+y)\) and compute its derivative. (2) Take \(x\) derivatives of \(\sin z=x+y .\) Verify that these answers, explicit and implicit, are equal.

Choose \(g(y)\) so that \(f(x, y)=e^{x x} g(y)\) satisfies the equation. \(f_{x}=7 f_{y}\)

Choose \(g(y)\) so that \(f(x, y)=e^{x x} g(y)\) satisfies the equation. \(f_{y}=f_{x x}\)

Draw a contour map of the top of your shoe.