Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 10 (b)
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Question 10 (b) 

The differential equation where , describes the rate of change of a population over time, subject to environmental constraints and the threat of extinction.
(b) Explain and justify in one or two sentences what happens to the population over time if the initial population satisfies . 
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! 
Hint 

Make a partition of interval and analyze the sign of on each interval. For example, if , then increases on the interval. 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. As mentioned in the hint, we distinguish three cases: Case 1: . In this case, . The population increases to approach . Case 2: . In this case, . The population remains as the constant . Case 3: . In this case, . The population decreases to approach . In conclusion, the population approaches over time, that is, Answer: over time. 