Science:Math Exam Resources/Courses/MATH215/December 2011/Question 03 (b)
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) •
Question 03 (b) 

For the following ODE (b) Determine the sign of in the regions separated by the equilibrium points. 
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 

Recall that the derivative can only be zero (and hence change sign) at the fixed points. Therefore these points divide our space into regions where the derivative is either positive or negative. How many regions are there? 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. From part (a) we have that the equilibrium points are and therefore these points divide our derivative space into four regions where the derivative can be positive or negative, . We have that and we first note that is always positive (unless y = 2) and so the sign of the derivative will be determined by the first term only. For this term we get Therefore Note: We separate the domain of increase for the derivative at the point y=2 because strictly speaking the derivative is not increasing at this point. 