$[7\mathrm{.}51/10$ Points$]q,$ SCALCET$99\mathrm{.}4\mathrm{.}003[7\mathrm{.}51/10$ Points$]q,$ SCALCET$99\mathrm{.}4\mathrm{.}003$

Suppose that a population develops according to the logistic equation

$\frac{dP}{dt}=0\mathrm{.}01P-0\mathrm{.}0001P^2,$

where $t$ is measured in weeks.

$($a$)$ What is the carrying capacity?

$M=100$

What is the value of $k?$

$k=0\mathrm{.}01$

$($b$)$ A direction field for this equation is shown in the figure below.

Where are the slopes close to $0?($Enter your answers as a comma$-$separated list.$)$

$p=0,100$

Where are the slopes largest? $($Enter your answers as a comma$-$separated list.$)$

Which solutions are increasing? $($Enter your answer using interval notation.$)$

$P_{0}in(0,100)$

Which solutions are decreasing? $($Enter your answer using interval notation.$)$

$P_{0}in(100,)$

$($c$)$ Use the direction field to sketch solutions for initial populations of $20,40,60,80,120,$ and $140.$

What do these solutions have in common? How do they differ?

All of the solutions approach $P=100$ as $t$ increases. Also all the solutions are increasing. The solutions differ since some have an inflection point and some don't.

All of the solutions approach $P=100$ as $t$ increases. Also all the solutions are decreasing. The solutions differ since some have an inflection point and some don't.

Which solutions have inflection points? At what population levels do they occur?

The solutions that have $P_0=20$ and $P_0=40$ have inflection points at $P=50.$

The solutions that have $P_0=120$ and $P_0=140$ have inflection points at $P=100.$

The solutions that have $P_0=20,P_0=40,P_0=60,$ and $P_0=80$ have inflection points at $P=50.$

None of the solutions have inflectio

Suppose that a population develops according to the logistic equation

$\frac{dP}{dt}=0\mathrm{.}01P-0\mathrm{.}0001P^2,$

where $t$ is measured in weeks.

$($a$)$ What is the carrying capacity?

$M=100$

What is the value of $k?$

$k=0\mathrm{.}01$

$($b$)$ A direction field for this equation is shown in the figure below.

Where are the slopes close to $0?($Enter your answers as a comma$-$separated list.$)$

$p=0,100$

Where are the slopes largest? $($Enter your answers as a comma$-$separated list.$)$

Which solutions are increasing? $($Enter your answer using interval notation.$)$

$P_{0}in(0,100)$

Which solutions are decreasing? $($Enter your answer using interval notation.$)$

$P_{0}in(100,)$

$($c$)$ Use the direction field to sketch solutions for initial populations of $20,40,60,80,120,$ and $140.$

What do these solutions have in common? How do they differ?

All of the solutions approach $P=100$ as $t$ increases. Also all the solutions are increasing. The solutions differ since some have an inflection point and some don't.

All of the solutions approach $P=100$ as $t$ increases. Also all the solutions are decreasing. The solutions differ since some have an inflection point and some don't.

Which solutions have inflection points? At what population levels do they occur?

The solutions that have $P_0=20$ and $P_0=40$ have inflection points at $P=50.$

The solutions that have $P_0=120$ and $P_0=140$ have inflection points at $P=100.$

The solutions that have $P_0=20,P_0=40,P_0=60,$ and $P_0=80$ have inflection points at $P=50.$

None of the solutions have inflection points.

All of the solutions have inflection points at $P=100.$

$$

$($d$)$ What are the equilibrium solutions? $($Enter your answers as a comma$-$separated list.$)$

$P=0,100$

How are the other solutions related to these solutions?

The $,$ solutions move away from $P=0,$ and all nonzero solutions $,,$ approach $P=100$