Concepts of graphing functions

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We use the language of calculus to describe graphs of functions.

In this section, we review the graphical implications of limits, and the sign of the first and second derivative. You already know all this stuff: it is just important enough to hit it more than once, and put it all together.

Sketch the graph of a function $f$ which has the following properties:

$f(0)=0$
$\displaystyle \lim _{x \to 10^+} f(x) = +\infty$
$\displaystyle \lim _{x \to 10^-} f(x) = -\infty$
$f'(x)<0$ on $(-\infty ,0) \cup (6,10) \cup (10,14)$
$f'(x)>0$ on $(0,6) \cup (6,10) \cup (14,\infty )$
$f''(x)<0$ on $(4,10)$
$f''(x)>0$ on $(-\infty ,4) \cup (10,\infty )$

Try this on your own first, then either check with a friend or check the online version.

The first thing we will do is to plot the point $(0,0)$ and indicate the appropriate vertical asympote due to the limit conditions. We also mark all of the places where $f'$ or $f''$ change sign.

Now we classify the behaviour on each of the intervals:

On $(-\infty , 0)$ , $f$ is
and concave
On $(0, 4)$ , $f$ is
and concave
On $(4, 6)$ , $f$ is
and concave
On $(6, 10)$ , $f$ is
and concave
On $(10, 14)$ , $f$ is
and concave
On $(14, \infty )$ , $f$ is
and concave

Utilizing all of this information, we are forced to sketch something like the following:

Sketch the graph of a function $f$ which has the following properties:

$f(0)=1$
$f(6)=2$
$\displaystyle \lim _{x \to 6^+} f(x) = 3$
$\displaystyle \lim _{x \to 6^-} f(x) = 1$
$f'(x)<0$ on $(-\infty ,1)$
$f'(x)>0$ on $(1,6)$
$f'(x) = -2$ on $(6, \infty )$
$f''(x)<0$ on $(2.5,5)$
$f''(x)>0$ on $(-\infty ,2.5) \cup (5,6)$

Try this on your own first, then either check with a friend or check the online version.

The first thing we will do is to plot the points $(0,1)$ and $(6,2)$ , and the ‘‘holes” at $(6,3)$ and $(6,1)$ due to the limit conditions. We can immediately draw in what $f$ looks like on $(6,\infty )$ since it is linear with slope $2$ , and must connect to the hole at $(6,2)$ . We also mark all of the places where $f'$ or $f''$ change sign.

Now we classify the behaviour on each of the intervals:

On $(-\infty , 1)$ , $f$ is
and concave
On $(1, 2.5)$ , $f$ is
and concave
On $(2.5, 5)$ , $f$ is
and concave
On $(5, 6)$ , $f$ is
and concave

Utilizing all of this information, we are forced to draw something like the following:

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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