## MATH1003 Quizzes

Quiz 8: The Logistic Function; Partial Fractions
Question 1 Questions
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?

Correct!
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?

Correct!
$\frac{dP}{dt}$ is a maximum when $P\doteq 40$.
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)
 a) As $t\to \infty$, the population size increases without bound. b) As $t\to \infty$, the population size tends to 500. c) As $t\to \infty$, the population size decreases to zero. d) As $t\to \infty$, the population size remains at 250.

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)
 a) As $t\to \infty$, the population size remains at 1000. b) As $t\to \infty$, the population size increases without bound. c) As $t\to \infty$, the population size tends to 500. d) As $t\to \infty$, the population size decreases to zero.

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!
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)
 a) ${x}^{2}+\frac{1}{x+1}-\frac{1}{x+2}$ b) ${x}^{2}-\frac{1}{x+1}+\frac{1}{x+2}$ c) ${x}^{2}+1+\frac{1}{x+1}-\frac{1}{x+2}$ d) ${x}^{2}+1-\frac{1}{x+1}+\frac{1}{x+2}$

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)
 a) $ln|{x}^{2}-4x+3|+C$ b) $\frac{ln|{x}^{2}-4x+3|}{2x-4}+C$ c) $\frac{1}{2}ln\left|\frac{x-1}{x-3}\right|+C$ d) $\frac{1}{2}ln\left|\frac{x-3}{x-1}\right|+C$

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

Correct!
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)
 a) $\frac{2{e}^{x}}{3-2{e}^{x}}$ b) $\frac{2}{3{e}^{x}-2}$ c) $\frac{2{e}^{-x}}{3-2{e}^{-x}}$ d) $\frac{2}{3{e}^{-x}-2}$

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!
1. True
2. False
3. False
4. 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)
 a) $P=\frac{25}{1+C{e}^{-0.1t}}$ b) $P=\frac{25}{1+{e}^{-0.1t}}+C$ c) $P=\frac{0.1{e}^{t}}{1+0.004{e}^{t}}+C$ d) $P=\frac{C{e}^{t}}{1+C{e}^{t}}$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect