91Ó°ÊÓ

Can one have two closed sets \(A\) and \(B\) which are disjoint (and not empty) and such that dist \((A, B)=0\) ?

Sketch the level surfaces for the function \(f(x, y, z)=x^{2}+y^{2}-z^{2}\)

Show that the intersection \(\bigcap_{1}^{x} I_{n}\) of the nested sequence of intervals \(\left\\{I_{n}\right\\}\) is empty in the following cases: (a) \(I_{n}\) is the open interval \(0

Discuss the behavior of the sequence \(\left\\{a_{n}\right\\}\), where $$ a_{n}=n+1+1 / n+(-1)^{n} n $$

By constructing an example, show that the union of an infinite collection of closed sets does not have to be closed.

Solve for the points \(P\) and \(Q\) if $$ \begin{aligned} 2 P+3 Q &=(0,1,2) \\ P+2 Q &=(1,-1,3) \end{aligned} $$

Let \(\left\\{R_{n}\right\\}\) be a sequence of closed bounded rectangles in the plane, with \(R_{1} \supset R_{2} \supset R_{3} \supset \cdots ;\) describe \(R_{n}\) by $$ R_{n}=\left\\{\text { all }(x, y) \text { with } a_{n} \leq x \leq b_{n}, c_{n} \leq y \leq d_{n}\right\\} $$ Prove that \(\bigcap_{1}^{\alpha} R_{n} \neq \varnothing\)

Let \(F(x, y, z, t)=(x-t)^{2}+y^{2}+z^{2} .\) By interpreting this as the temperature at the point \((x, y, z)\) at time \(t\), see if you can get a feeling for the behavior of the function.

Find the equation of the hyperplane in 4 -space which goes through the point \(p_{0}=\) \((0,1,-2,3)\) perpendicular to the vector \(\mathrm{a}=(4,3,1,-2)\).

Solve for \(P\) and \(Q\) if $$ \begin{aligned} 3 P+Q &=(1,0,1,-4) \\ P-Q &=(2,1,2,3) \end{aligned} $$