A model for the population, $P$
(in millions), of a country is
$$\frac{dP}{dt}=0.698P-0.0087{P}^{2}$$
where $t$ is
measured in years from 2000.

According to this model, what is the maximum population that the country can support?

Give your answer in millions, to the nearest million.

According to this model, what is the maximum population that the country can support?

Give your answer in millions, to the nearest million.

*Correct!*

*Incorrect.*

*Please try again.*

Note that the population will not
increase if $\frac{dP}{dt}=0.$

A model for the population, $P$
(in millions), of a country is
$$\frac{dP}{dt}=0.698P-0.0087{P}^{2}$$
where $t$ is measured in
years from 2000 and $P\left(0\right)=12$.

According to this model, what is the maximum growth rate of the population?

Give your answer in millions per year, to the nearest million.

According to this model, what is the maximum growth rate of the population?

Give your answer in millions per year, to the nearest million.

*Correct!*

$\frac{dP}{dt}$ is a maximum
when $P\doteq 40$.

*Incorrect.*

*Please try again.*

Hint:
Draw a graph of $\frac{dP}{dt}$
against $P$; look
for the value of $P$
at which $\frac{dP}{dt}$
is a maximum.

Suppose that a population develops according to
$$\frac{dP}{dt}=0.05P-0.0001{P}^{2},$$
and that $P\left(0\right)=250$.

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

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

*Choice (a) is incorrect*

The population size increases initially, but will not continue to
increase if $\frac{dP}{dt}=0$.

*Choice (b) is correct!*

Initially $\frac{dP}{dt}>0$,
so $P$ is
increasing, but $\frac{dP}{dt}=0$
when $P=500$.

*Choice (c) is incorrect*

Initially $\frac{dP}{dt}>0$,
so $P$ is increasing.

*Choice (d) is incorrect*

Initially $\frac{dP}{dt}>0$,
so $P$ is
increasing.

Suppose that a population develops according to
$$\frac{dP}{dt}=0.05P-0.0001{P}^{2},$$
and that $P\left(0\right)=1000$.

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

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

*Choice (a) is incorrect*

Initially $\frac{dP}{dt}<0$,
so $P$ is
decreasing.

*Choice (b) is incorrect*

Initially
$\frac{dP}{dt}<0$, so
$P$ is decreasing.

*Choice (c) is correct!*

Initially $\frac{dP}{dt}<0$,
so $P$ is
decreasing, but $\frac{dP}{dt}=0$
when $P=500$.

*Choice (d) is incorrect*

The population size decreases initially, but will not continue to
decrease if $\frac{dP}{dt}=0$.

If $\frac{12}{\left(x+1\right)\left(x-5\right)}=\frac{a}{x+1}+\frac{b}{x-5}$, what is the
value of $b$?

*Correct!*

*Incorrect.*

*Please try again.*

Note
that $ax-5a+bx+b=12$. Now equate
coefficients of $x$,
and constant terms, on either side of this equation.

Using long division and partial fraction decomposition, determine which of the
following is equal to the rational fraction
$$\frac{{x}^{4}+3{x}^{3}+3{x}^{2}+3x+3}{{x}^{2}+3x+2}.$$
Exactly one option must be correct)

*Choice (a) is incorrect*

Remember that you can check your answer by putting all terms over a common denominator.

*Choice (b) is incorrect*

Remember
that you can check your answer by putting all terms over a common denominator.

*Choice (c) is correct!*

*Choice (d) is incorrect*

Remember that you can check your answer by putting all terms over a common
denominator.

Find the indefinite integral $\int \frac{dx}{{x}^{2}-4x+3}.$

(In each case, $C$ is an arbitrary constant.) Exactly one option must be correct)

(In each case, $C$ is an arbitrary constant.) Exactly one option must be correct)

*Choice (a) is incorrect*

Check by
differentiating.

*Choice (b) is incorrect*

Check
by differentiating.

*Choice (c) is incorrect*

Check by differentiating.

*Choice (d) is correct!*

Evaluate ${\int}_{-1}^{1}\frac{dx}{4-{x}^{2}}.$

Give your answer as an approximation correct to 3 decimal places.

Give your answer as an approximation correct to 3 decimal places.

*Correct!*

*Incorrect.*

*Please try again.*

Try writing
$\frac{1}{4-{x}^{2}}$ as
partial fractions.

Find the particular solution to the differential equation
$\frac{dy}{dx}=y+{y}^{2},$

given that $y\left(0\right)=2$. (Zero or more options can be correct)

For example, choice (a) should be True.

For example, choice (b) should be False.

For example, choice (c) should be False.

For example, choice (d) should be True.

given that $y\left(0\right)=2$. (Zero or more options can be correct)

*There is at least one mistake.*

For example, choice (a) should be True.

*There is at least one mistake.*

For example, choice (b) should be False.

*There is at least one mistake.*

For example, choice (c) should be False.

*There is at least one mistake.*

For example, choice (d) should be True.

*Correct!*

*True**False**False**True*

Find the general solution to the differential equation
$\frac{dP}{dt}=0.1P-0.004{P}^{2}.$

(In each case, $C$ is an arbitrary constant.) Exactly one option must be correct)

(In each case, $C$ is an arbitrary constant.) Exactly one option must be correct)

*Choice (a) is correct!*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*