Homework 2: Graphing

$\newenvironment {prompt}{}{} \newcommand {\DeclareMathOperator }[2]{\@OldDeclareMathOperator {##1}{##2}\immediate \write \myfile {\unexpanded {\DeclareMathOperator }{\unexpanded {##1}}{\unexpanded {##2}}}} \newcommand {\ungraded }[0]{} \newcommand {\reserved@a }[2]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{} \newcommand {\vnameref }[1]{\unskip ~\nameref {##1}\@vpageref [\unskip ]{##1}} \newcommand {\ref }[0]{\@ifstar \@refstar \T@ref } \newcommand {\pageref }[0]{\@ifstar \@pagerefstar \T@pageref } \newcommand {\nameref }[0]{\@ifstar \@namerefstar \T@nameref } \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{}$

Online Problems

Consider the function $f:\mathbb {R}^2\rightarrow \mathbb {R}$ given by $f(x,y) = x^2+4y^2 -2.$ What is the domain of $f$ ?

$\mathbb {R}$ $\mathbb {R}\setminus \{0\}$ $[0,\infty )$ $(0,\infty )$ $\mathbb {R}^2$ $\mathbb {R}^2\setminus \{(0,0)\}$

What is the range of $f$ ? $\textrm {Range }f = \answer {[-2,\infty )}$

Is $f$ onto?

yes no

We would like to restrict the codomain of the function $f$ so that it becomes onto. We’ll describe our new codomain as the set of numbers $a$ in $\mathbb {R}$ such that some condition holds. Which condition gives us the largest possible codomain such that $f$ is onto?

$a\in \mathbb {R}$ $a \geq 0$ $a > 0$ $a\neq 0$ $a = 0$ $a \geq 2$ $a > 2$ $a\neq 2$ $a = 2$ $a \geq -2$ $a > -2$ $a\neq -2$ $a = -2$

Is $f$ one-to-one?

yes no

We would like to restrict the domain of the function $f$ , so that it becomes one-to-one. We’ll describe our new domain as the set of points $(x,y)$ in $\mathbb {R}^2$ such that some condition(s) hold. Which condition(s) give us the largest possible domain such that $f$ is one-to-one?

$x\neq 0$ $x\geq 0$ $x > 0$ $y\neq 0$ $y\geq 0$ $y > 0$

Let $f:\mathbb {R}^3\rightarrow \mathbb {R}^3$ be the function defined by $f(\vec {x}) = 3\vec {x} + \textbf {i} - 2\textbf {j}.$

Find the component functions of $f$ in terms of $x$ , $y$ , and $z$ .

$\begin{align*} f_1(x,y,z) &= \answer{3x+1}\\ f_2(x,y,z) &= \answer{3y-2}\\ f_3(x,y,z) &= \answer{3z} \end{align*}$

Consider the linear function $f:\mathbb {R}^3\rightarrow \mathbb {R}^2$ given by $f(\vec {x}) = A\vec {x}$ , where $A = \left (\begin {array}{ccc} 1&5&2\\ -2&0&1 \end {array}\right ),$ and $x = \left (\begin {array}{c}x_1\\x_2\\x_3\end {array}\right )$ .

(a): Determine the component functions of $f$ in terms of $x_1$ , $x_2$ , and $x_3$ . $\begin{align*} f_1(x_1,x_2,x_3) &= \answer{x_1 + 5x_2 + 2x_3}\\ f_2(x_1,x_2,x_3) &= \answer{-2x_1 + x_3} \end{align*}$
(b): Is $f$ one-to-one?
Yes No
(c): Is $f$ onto?
Yes No

Consider the function $f(x,y) = -xy.$ What is the shape of the level curve at height $0$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

What is the shape of the level curve at height $1$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

What is the shape of the level curve at height $-1$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

What is the shape of the level curve at height $2$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

Which of the following is the graph of $f$ ?

Consider the function $f(x,y) = |x|.$ What is the shape of the level curve at height $0$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

What is the shape of the level curve at height $1$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

What is the shape of the level curve at height $-1$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

What is the shape of the level curve at height $2$ of $f$ ?

Empty A single line Two intersecting lines Two parallel lines Circle Ellipse Parabola Hyperbola

Which of the following is the graph of $f$ ?

Which of the following is the graph of the ellipsoid $\frac {x^2}{9}+ y^2 + \frac {z^2}{4} = 1?$

Is there a function $f(x,y)$ such that the graph of $f$ is the ellipsoid above?

Yes No

Why is this impossible?

It wouldn’t be one-to-one. It wouldn’t be onto. There would be multiple inputs with the same output. A single input would need to have two outputs.

Classify the quadric surface defined by the equation $x^2 + 4y^2 + z^2 + 8y = 0.$

Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Elliptic Cone Hyperboloid of One Sheet Hyperboloid of Two Sheets

It is centered at the point $\answer {(0,-1,0)}$ .

Which of the following is the graph of the quadric surface given above?

Classify the quadric surface defined by the equation $2x^2 + 2y^2 - 8y - z + 4 = 0$

Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Elliptic Cone Hyperboloid of One Sheet Hyperboloid of Two Sheets

It is centered at the point $\answer {(0,2,-4)}$ .

Which of the following is the graph of the quadric surface given above?

Written Problems

Consider the function $f(x,y,z) = \frac {4}{\sqrt {9-x^2-y^2-z^2}}.$

(a): What is the domain of $f$ ? Describe this domain as a region in $\mathbb {R}^3$ .
(b): What is the range of $f$ ?

Consider the function $f(x) = x^2 + y^2 - 4.$

(a): Draw at least five level curves of $f$ .
(b): Use these level curves to sketch the graph of $f$ .

Draw the graph of the surface in $\mathbb {R}^3$ determined by the equation $x = y^2/4 - z^2/9.$ Use level curves and/or sections to justify why your drawing is correct.

Professional Problem

(a): Suppose we have a surface such that the $x$ -sections at $x=C$ are always $z=y^2$ . Draw and describe this surface.
(b): Suppose we have a surface such that the level sets at $z=C$ are always $x^2+y^2=1$ . Draw and describe this surface.
(c): Suppose we have a surface such that the level sets at $z=C$ are always given by $g(x,y)=1$ , for some function $g$ of $x$ and $y$ . Describe this surface, and draw the surface for some “generic” function $g$ .

Images were generated using CalcPlot3D.