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Quiz 8: The Logistic Function; Partial Fractions

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
A model for the population, P (in millions), of a country is
dP-= 0.698P - 0.0087P2 dt
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.

 

Your answer is correct
Not correct. You may try again.
Note that the population will not increase if dP- dt = 0.
 

Question 2

 
 
A model for the population, P (in millions), of a country is
dP-= 0.698P - 0.0087P2 dt
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.

 

Your answer is correct
dP- dt is a maximum when P 40.
Not correct. You may try again.
Hint: Draw a graph of dP- dt against P; look for the value of P at which dP --- dt is a maximum.
 

Question 3

 
 
Suppose that a population develops according to
dP -dt = 0.05P - 0.0001P 2,
and that P(0) = 250.
Which one of the following statements is 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.

 

Not correct. Choice (a) is false.
The population size increases initially, but will not continue to increase if dP dt- = 0.
Your answer is correct.
Initially dP- dt > 0, so P is increasing, but dP- dt = 0 when P = 500.
Not correct. Choice (c) is false.
Initially dP- dt > 0, so P is increasing.
Not correct. Choice (d) is false.
Initially dP- dt > 0, so P is increasing.
 

Question 4

 
 
Suppose that a population develops according to
dP- = 0.05P - 0.0001P 2, dt
and that P(0) = 1000.
Which one of the following statements is 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.

 

Not correct. Choice (a) is false.
Initially dP- dt < 0, so P is decreasing.
Not correct. Choice (b) is false.
Initially dP --- dt < 0, so P is decreasing.
Your answer is correct.
Initially dP -dt < 0, so P is decreasing, but dP dt- = 0 when P = 500.
Not correct. Choice (d) is false.
The population size decreases initially, but will not continue to decrease if dP- dt = 0.
 

Question 5

 
 
If -----12----- (x+ 1)(x- 5) = --a-- x +1 + --b-- x - 5, what is the value of b?

 

Your answer is correct
Not correct. You may try again.
Note that ax - 5a + bx + b = 12. Now equate coefficients of x, and constant terms, on either side of this equation.
 

Question 6

 
 
Using long division and partial fraction decomposition, determine which of the following is equal to the rational fraction
4 3 2 x--+-3x2+-3x-+-3x-+3-. x + 3x+ 2
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

 

Not correct. Choice (a) is false.
Remember that you can check your answer by putting all terms over a common denominator.
Not correct. Choice (b) is false.
Remember that you can check your answer by putting all terms over a common denominator.
Your answer is correct.
Not correct. Choice (d) is false.
Remember that you can check your answer by putting all terms over a common denominator.
 

Question 7

 
 
Find the indefinite integral ∫ ----dx----. x2 - 4x+ 3
(In each case, C is an arbitrary constant.)
a) 2 ln ∣x - 4x + 3∣+ C   b) 2 ln∣x---4x+-3∣+ C 2x- 4
c) ∣ ∣ 1ln∣∣x--1∣∣+ C 2 ∣x- 3∣   d) ∣ ∣ 1ln ∣∣x--3∣∣+ C 2 ∣x- 1∣

 

Not correct. Choice (a) is false.
Check by differentiating.
Not correct. Choice (b) is false.
Check by differentiating.
Not correct. Choice (c) is false.
Check by differentiating.
Your answer is correct.
 

Question 8

 
 
Evaluate ∫ 1 dx 4---x2. -1
Give your answer as an approximation correct to 3 decimal places.

 

Your answer is correct
Not correct. You may try again.
Try writing ---1-- 4 - x2 as partial fractions.
 

Question 9

 
 
Find the particular solution to the differential equation dy- dx = y + y2,
given that y(0) = 2.
a) --2ex-- 3 - 2ex   b) ---2--- 3ex - 2
c) -2e-x--- 3- 2e-x   d) ---2---- 3e-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.
Your answers are correct
  1. True.
  2. False.
  3. False.
  4. True.
 

Question 10

 
 
Find the general solution to the differential equation dP- dt = 0.1P - 0.004P2.
(In each case, C is an arbitrary constant.)
a) P = ----25---- 1+ Ce- 0.1t   b) P = ---25--- 1 + e-0.1t + C
c) P = t --0.1e---- 1+ 0.004et + C   d) P = t -Ce---- 1+ Cet

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.