56bc61…: “Up to the short tree”

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How You Can (And Should) Get More Practice!

Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for you.

Theoretically Easier Difficulty Problem

You don’t need to factor or graph the function

The answer is only asking for the maximum possible relative extrema, not how many local extrema the function actually has.

Consider the function $f(x) = \sage {p1fDisp}$ . What are the maximum number of relative extrema that $f(x)$ could have? $\answer [id=c]{\sage {p1ans1}}$ .

Theoretically Medium Difficulty Problem

You don’t need to factor or graph the function

The answer is only asking for the maximum possible relative extrema, not how many local extrema the function actually has.

Consider the function $f(x) = \sage {p2fDisp}$ . What are the maximum number of relative extrema that $f(x)$ could have? $\answer [id=c]{\sage {p2ans1}}$ .

You don’t need to factor or graph the function

The answer is only asking for the minimum possible relative extrema, not how many local extrema the function actually has.

What is the minimum number relative extrema that $f(x)$ could possibly have? $\answer [id=b]{\sage {p2ans2}}$

Remember that local/relative extrema must come in pairs

Since you know the minimum number of possible extrema from the previous problem, and you know they must come in pairs, add an even number to the minimum (as long as you stay below the maximum!)

Enter any number that could be a valid number of possible local extrema for $f(x)$ .

$\answer [id=a]{\sage {p2ans1}}$

Theoretically Harder Difficulty Problem

You don’t need to factor or graph the function

The answer is only asking for the maximum possible relative extrema, not how many local extrema the function actually has.

Consider the function $f(x) = (\sage {p3f1}) + (\sage {p3f2}) + (\sage {p3f3}) + (\sage {p3f4})$ . What are the maximum number of relative extrema that $f(x)$ could have? $\answer [id=cc]{\sage {p3ans1}}$ .

You don’t need to factor or graph the function

The answer is only asking for the minimum possible relative extrema, not how many local extrema the function actually has.

What is the minimum number relative extrema that $f(x)$ could possibly have? $\answer [id=b]{\sage {p3ans2}}$

Remember that local/relative extrema must come in pairs

Since you know the minimum number of possible extrema from the previous problem, and you know they must come in pairs, add an even number to the minimum (as long as you stay below the maximum!)

Enter any number that could be a valid number of possible local extrema for $f(x)$ .

$\answer [id=a]{\sage {p3ans1}}$

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	$\sqrt{\blue{[?]}}$
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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	$\nu$
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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	$\Omega$

How You Can (And Should) Get More Practice!

Theoretically Easier Difficulty Problem

Theoretically Medium Difficulty Problem

Theoretically Harder Difficulty Problem

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