Problem 39
Suppose that the temperature \(T\) on the circular plate \(\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}\) is given by \(T=2 x^{2}+y^{2}-y .\) Find the hottest and coldest spots on the plate.
Problem 39
Find the domain of each function. (a) \(f(w, x, y, z)=\frac{1}{\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}}\) (b) \(g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\exp \left(-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}\right)\) (c) \(h\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sqrt{1-\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)}\)
Problem 40
If \(f(x, y, z)=\left(x^{3}+y^{2}+z\right)^{4}\), find each of the following: (a) \(f_{x}(x, y, z)\) (b) \(f_{y}(0,1,1)\) (c) \(f_{z z}(x, y, z)\)
Problem 40
Sketch (as best you can) the graph of the monkey saddle \(z=x\left(x^{2}-3 y^{2}\right) .\) Begin by noting where \(z=0\)
Problem 40
Let \(f\), a function of \(n\) variables, be continuous on an open set \(D\), and suppose that \(P_{0}\) is in \(D\) with \(f\left(P_{0}\right)>0 .\) Prove that there is a \(\delta>0\) such that \(f(P)>0\) in a neighborhood of \(P_{0}\) with radius \(\delta\).
Problem 41
If \(f(x, y, z)=e^{-x y z}-\ln \left(x y-z^{2}\right)\), find \(f_{x}(x, y, z)\)
Problem 41
Find the shape of the triangle of largest area that can be inscribed in a circle of radius \(r\). Hint: Let \(\alpha, \beta\), and \(\gamma\) be the central angles that subtend the three sides of the triangle. Show that the area of the triangle is \(\frac{1}{2} r^{2}[\sin \alpha+\sin \beta-\sin (\alpha+\beta)]\). Maximize.
Problem 41
The French Railroad Suppose that Paris is located at the origin of the \(x y\) -plane. Rail lines emanate from Paris along all rays, and these are the only rail lines. Determine the set of discontinuities of the following functions. (a) \(f(x, y)\) is the distance from \((x, y)\) to \((1,0)\) on the French railroad. (b) \(g(u, v, x, y)\) is the distance from \((u, v)\) to \((x, y)\) on the French railroad.
Problem 42
Let \((a, b, c)\) be a fixed point in the first octant. Find the plane through this point that cuts off from the first octant the tetrahedron of minimum volume, and determine the resulting volume.
Problem 42
Identify the graph of \(f(x, y)=x^{2}-x+3 y^{2}+\) \(12 y-13\), state where it attains its minimum value, and find this minimum value.
