This quiz tests the work covered in Lecture 11 and corresponds to Section 2.5 of the
textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et
al.).

There is an applet at http://www.walter-fendt.de/m11e/deriv12.htm which allows
you to put in the formula for a function and then it draws the first and second
derivative functions.

Which of the following statements are correct? Exactly one option must be correct)

Choice (a) is incorrect

Try again, the function
is decreasing at $A\phantom{\rule{0.3em}{0ex}}.$

Choice (b) is correct!

The function is decreasing
and concave down at $A\phantom{\rule{0.3em}{0ex}},$
so $\frac{dy}{dx}<0\text{and}\frac{{d}^{2}y}{d{x}^{2}}0\text{at}A\phantom{\rule{0.3em}{0ex}}.$

Choice (c) is incorrect

Try again, the function
is decreasing at $A\phantom{\rule{0.3em}{0ex}}.$

Choice (d) is incorrect

Try again, the function
is concave down at $A\phantom{\rule{0.3em}{0ex}}.$

Let $P\left(t\right)$ be the number of
alien sightings at time $t\phantom{\rule{0.3em}{0ex}}.$
We are told that both ${P}^{\prime}\left(t\right)$
and ${P}^{\u2033}\left(t\right)$
are positive. Which of the statements below best reflect this? Exactly one option must be correct)

Choice (a) is correct!

The number of sightings are increasing so
${P}^{\prime}\left(t\right)>0$ and the rate of increase
is increasing so ${P}^{\u2033}\left(t\right)>0\phantom{\rule{0.3em}{0ex}}.$

Choice (b) is incorrect

Try again, since
${P}^{\u2033}\left(t\right)>0$ the
rate of sightings must be increasing.

Choice (c) is incorrect

Try again, since
${P}^{\prime}\left(t\right)>0$ the number
of sightings must be increasing.

Choice (d) is incorrect

Try
again, since ${P}^{\prime}\left(t\right)>0$
the number of sightings must be increasing.

Consider the graph of $y=f\left(x\right)$
below. At which point on the graph is ${f}^{\prime}\left(x\right)>0$
and ${f}^{\u2033}\left(x\right)<0\phantom{\rule{0.3em}{0ex}}.$
Exactly one option must be correct)

Choice (a) is incorrect

Try again, ${f}^{\prime}\left(x\right)<0$
and ${f}^{\u2033}\left(x\right)>0$
at $A\phantom{\rule{0.3em}{0ex}}.$

Choice (b) is incorrect

Try
again, ${f}^{\prime}\left(x\right)>0$
and ${f}^{\u2033}\left(x\right)>0$
at $B\phantom{\rule{0.3em}{0ex}}.$

Choice (c) is correct!

$f$ is increasing and
concave down at $C$
so ${f}^{\prime}\left(x\right)>0$
and ${f}^{\u2033}\left(x\right)<0$
at $C\phantom{\rule{0.3em}{0ex}}.$

Choice (d) is incorrect

Try
again, ${f}^{\prime}\left(x\right)<0$
and ${f}^{\u2033}\left(x\right)<0$
at $D\phantom{\rule{0.3em}{0ex}}.$

Choice (e) is incorrect

There is a correct answer. You need to find a point where
$f$ is
increasing and concave down.

A function $f$ is
decreasing for $x\ge 2$
and $f\left(2\right)=20\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}{f}^{\prime}\left(2\right)=-2$
and ${f}^{\u2033}\left(x\right)>0$
for $x\ge 2\phantom{\rule{0.3em}{0ex}}.$

Which of the following is a possible value for
$f\left(4\right)\phantom{\rule{0.3em}{0ex}}?$ Exactly one option
must be correct)

Choice (a) is incorrect

Try again, since $f$
is decreasing $f\left(4\right)<20\phantom{\rule{0.3em}{0ex}}.$

Choice (b) is incorrect

Try again,
since $f$ is
decreasing $f\left(4\right)<20\phantom{\rule{0.3em}{0ex}}.$

Choice (c) is correct!

Since
$f$ is
decreasing $f\left(4\right)<20$
and since ${f}^{\u2033}\left(x\right)>0$
the function cannot continue to decrease at the same rate as it was decreasing
at $x=2$ therefore
$f\left(4\right)>16\phantom{\rule{0.3em}{0ex}}.$

Choice (d) is incorrect

Try again,
since ${f}^{\u2033}\left(x\right)>0$
the function cannot continue to decrease at the same rate as it was decreasing
at $x=2$ therefore
$f\left(4\right)>16\phantom{\rule{0.3em}{0ex}}.$