Science:Math Exam Resources/Courses/MATH102/December 2017/Question 07 (b)
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Question 07 (b)
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.
(b) For what values of does the word fail to catch on and simply disappear from the language? Choose the best answer.
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
We are looking for the values of such that is always negative.
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According to the Hint, the word disappears if always hold true. Rewriting
we see that always hold true when , that is, when .