91Ó°ÊÓ

For the functions \(f(x, y)\) : (a) sketch the cross-sections of the graph \(z=\) \(f(x, y)\) with the coordinate planes, (b) sketch several level curves of \(f,\) labeling each with the corresponding value of \(c,\) and (c) sketch the graph of \(f\). $$ f(x, y)=|x|+|y| $$

In Exercises \(2.5-2.9,\) sketch the level sets corresponding to the indicated values of \(c\) for the given function \(f(x, y, z)\) of three variables. Make a separate sketch for each individual level set. $$ f(x, y, z)=x^{2}+y^{2}+z^{2}, c=0,1,2 $$

Are the planes \(x-y+z=8\) and \(2 x+y-z=-1\) perpendicular? Are they parallel?

Let \(U\) be the set of points \((x, y)\) in \(\mathbb{R}^{2}\) that satisfy the given conditions. Sketch \(U,\) and determine whether it is an open set. Your arguments should be at a level of rigor comparable to those given in the text. $$ y>x $$

Provide the missing proofs of parts of Proposition 3.21 about properties of continuous functions. Here, \(U\) is an open set in \(\mathbb{R}^{n}, f, g: U \rightarrow \mathbb{R}\) are real-valued functions defined on \(U,\) and \(\mathbf{a}\) is a point of \(U\). If \(f\) and \(g\) are continuous at a, prove that their product \(f g\) is continuous at a. (Hint: Use an "add zero" trick and the previous exercise to show that there is a \(\delta_{1}>0\) such that, if \(\mathbf{x} \in B\left(\mathbf{a}, \delta_{1}\right),\) then \(\left.|f(\mathbf{x}) g(\mathbf{x})-f(\mathbf{a}) g(\mathbf{a})| \leq(|f(\mathbf{a})|+1)|g(\mathbf{x})-g(\mathbf{a})|+(|g(\mathbf{a})|+1)|f(\mathbf{x})-f(\mathbf{a})| .\right)\)

Sketch the level sets corresponding to the indicated values of \(c\) for the given function \(f(x, y, z)\) of three variables. Make a separate sketch for each individual level set. $$ f(x, y, z)=x^{2}+y^{2}, c=0,1,2 $$

Are the planes \(x-y+z=8\) and \(2 x-2 y+2 z=-1\) perpendicular? Are they parallel?

For the functions \(f(x, y)\) : (a) sketch the cross-sections of the graph \(z=\) \(f(x, y)\) with the coordinate planes, (b) sketch several level curves of \(f,\) labeling each with the corresponding value of \(c,\) and (c) sketch the graph of \(f\). $$ f(x, y)=x y $$

Sketch the level sets corresponding to the indicated values of \(c\) for the given function \(f(x, y, z)\) of three variables. Make a separate sketch for each individual level set. $$ f(x, y, z)=x^{2}+y^{2}-z, c=-1,0,1 $$

Let \(U\) be the set of points \((x, y)\) in \(\mathbb{R}^{2}\) that satisfy the given conditions. Sketch \(U,\) and determine whether it is an open set. Your arguments should be at a level of rigor comparable to those given in the text. $$ y \geq x $$