Science:Math Exam Resources/Courses/MATH102/December 2017/Question 07 (a)
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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 where 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 represents... A. ...how appealing the word is when you first hear it (larger means more appealing). B. ...how easy the word is to remember once you start using it (larger means easier to remember). C. ...how easy the word is to forget once you start using it (larger means easier to forget). D. ...how annoyed people that don’t use the word get when they hear it (larger 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 , then is increasing. Otherwise, it is decreasing. 
Hint 2 

Observe that is a fraction, so that . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. According to the description, is the number of users of the word. Notice that, as a fraction, is valued between and (and is nonnegative in particular). For larger , 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: . 