Science:Math Exam Resources/Courses/MATH102/December 2017/Question 07 (a)

MATH102 December 2017

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q10 (a)  •  Q10 (b)  •  Q10 (c)  •  Q10 (d)  •  Q11 (a)  •  Q11 (b)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q12 (d)  •  Q13 (a)  •  Q13 (b)  •  Q13 (c)  •  Q13 (d)  •  Q14 (a)  •  Q14 (b)  •  Q15 (a)  •  Q15 (b)  •  Q16  •  

   Other MATH102 Exams

  •  December 2014  •  December 2011  •  December 2012  •  December 2013  •  December 2016  •  December 2015  •  

Question 07 (a)

New words are constantly being introduced into a language. The usage of one particular new word is thought to grow according to a logistic equation $$u'=2u(1-u)-au$$ where ${\textstyle u}$ is the fraction of the population using the word. The usage increases at a rate proportional to the product of the fraction of people who use the word and the fraction that don’t use it and decreases at a rate proportional to the number of users. (a) The parameter ${\textstyle a}$ represents... A. ...how appealing the word is when you first hear it (larger ${\textstyle a}$ means more appealing). B. ...how easy the word is to remember once you start using it (larger ${\textstyle a}$ means easier to remember). C. ...how easy the word is to forget once you start using it (larger ${\textstyle a}$ means easier to forget). D. ...how annoyed people that don’t use the word get when they hear it (larger ${\textstyle a}$ means more annoyed).

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
If $u'\geq 0$, then $u$ is increasing. Otherwise, it is decreasing.

Hint 2
Observe that $u$ is a fraction, so that $0\leq u\leq 1$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. According to the description, ${\textstyle au}$ is the number of users of the word. Notice that, as a fraction, ${\textstyle u}$ is valued between ${\textstyle 0}$ and ${\textstyle 1}$ (and is non-negative in particular). For larger ${\textstyle a}$, the derivative is smaller. This describes how easy the word is not used anymore by the users, or how easy the word is to forget once you start using it. Answer: ${\textstyle {\color {blue}C}}$.