Homework 2: Graphing

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 $f$ ?

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

Is $f$ onto?

Is $f$ one-to-one?

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?

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?
(c): Is $f$ onto?

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

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

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

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

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 $f$ ?

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

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

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

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?

Why is this impossible?

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

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 $2x^2 + 2y^2 - 8y - z + 4 = 0$

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

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

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.

(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 $g$ .

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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