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MATH1003 Quizzes

Quiz 8: The Logistic Function; Partial Fractions
Question 1 Questions
A model for the population, P (in millions), of a country is dP dt = 0.698P 0.0087P2 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.

Correct!
Incorrect. Please try again.
Note that the population will not increase if dP dt = 0.
A model for the population, P (in millions), of a country is dP dt = 0.698P 0.0087P2 where t is measured in years from 2000 and P(0) = 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.

Correct!
dP dt is a maximum when P40.
Incorrect. Please try again.
Hint: Draw a graph of dP dt against P; look for the value of P at which dP dt is a maximum.
Suppose that a population develops according to dP dt = 0.05P 0.0001P2, and that P(0) = 250.
Which one of the following statements is correct? Exactly one option must be correct)
a)
As t , the population size increases without bound.
b)
As t , the population size tends to 500.
c)
As t , the population size decreases to zero.
d)
As t , the population size remains at 250.

Choice (a) is incorrect
The population size increases initially, but will not continue to increase if dP dt = 0.
Choice (b) is correct!
Initially dP dt > 0, so P is increasing, but dP dt = 0 when P = 500.
Choice (c) is incorrect
Initially dP dt > 0, so P is increasing.
Choice (d) is incorrect
Initially dP dt > 0, so P is increasing.
Suppose that a population develops according to dP dt = 0.05P 0.0001P2, and that P(0) = 1000.
Which one of the following statements is correct? Exactly one option must be correct)
a)
As t , the population size remains at 1000.
b)
As t , the population size increases without bound.
c)
As t , the population size tends to 500.
d)
As t , the population size decreases to zero.

Choice (a) is incorrect
Initially dP dt < 0, so P is decreasing.
Choice (b) is incorrect
Initially dP dt < 0, so P is decreasing.
Choice (c) is correct!
Initially dP dt < 0, so P is decreasing, but dP dt = 0 when P = 500.
Choice (d) is incorrect
The population size decreases initially, but will not continue to decrease if dP dt = 0.
If 12 (x+1)(x5) = a x+1 + b x5, 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 x4 + 3x3 + 3x2 + 3x + 3 x2 + 3x + 2 . Exactly one option must be correct)
a)
x2 + 1 x+1 1 x+2
b)
x2 1 x+1 + 1 x+2
c)
x2 + 1 + 1 x+1 1 x+2
d)
x2 + 1 1 x+1 + 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 dx x2 4x + 3.
(In each case, C is an arbitrary constant.) Exactly one option must be correct)
a)
ln|x2 4x + 3| + C
b)
ln|x2 4x + 3| 2x 4 + C
c)
1 2 ln x 1 x 3 + C
d)
1 2 ln x 3 x 1 + 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 11 dx 4 x2.
Give your answer as an approximation correct to 3 decimal places.

Correct!
Incorrect. Please try again.
Try writing 1 4x2 as partial fractions.
Find the particular solution to the differential equation dy dx = y + y2,
given that y(0) = 2. (Zero or more options can be correct)
a)
2ex 32ex
b)
2 3ex2
c)
2ex 32ex
d)
2 3ex2

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 dP dt = 0.1P 0.004P2.
(In each case, C is an arbitrary constant.) Exactly one option must be correct)
a)
P = 25 1+Ce0.1t
b)
P = 25 1+e0.1t + C
c)
P = 0.1et 1+0.004et + C
d)
P = Cet 1+Cet

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