Science:Math Exam Resources/Courses/MATH102/December 2014/Question A 03

MATH102 December 2014

  •  QA 1  •  QA 2  •  QA 3  •  QA 4  •  QA 5  •  QA 6  •  QA 7  •  QA 8  •  QB 1  •  QB 2  •  QB 3  •  QB 4  •  QB 5  •  QB 6  •  QB 7  •  QC 1  •  QC 2(a)  •  QC 2(b)  •  QC 2(c)  •  QC 2(d)  •  QC 3  •  

   Other MATH102 Exams

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

Question A 03
Consider a differential equation $dy/dt=f(y)$. Shown in A-D is the phase line (state space) diagram ($f(y)$ versus $y$). Which of the following is the correct pairing of these sketches with the sketch of a solution $y(t)$ to the differential equation? Diagram A1, B4, C2, D3 A2, B4, C1, D3 A2, B4, C3, D1 A4, B2, C3, D1 A4, B2, C1, D3

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
If the derivative of a function is positive/negative in some interval or maximized/minimized at certain point, what doest it mean?

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. 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. We will pair each graph of $y'$ with the graph of corresponding function $y$ by the sign and the magnitude of $y'$. First look at diagram (A). It shows the $y'$ is always positive in the domain of interest, which means function $y$ must be a increasing function. Hence we know the solution sketch corresponding to (A) might be (2) or (4). Next we notice $y'$ has a local minimum in the mediate $y$ value which means the increasing rate of $y$ achieves its lowest value when the value of $y$ is about in the middle, which indicates (2) is the correct choice. Similarly, the plot of $y'$ in (B) indicates $y$ is an increasing function and its increasing rate first increases before the middle point and then decreases again. The only correct pairing is (B)(4). Next, from $y'$ in (C) we know $y$ is a decreasing function. Its decreasing rate first increases as the magnitude (absolute value) of $y'$ reaches its maximum in the middle and then decreases. Only (1) matches it. Finally, from (D) we get $y$ is a decreasing function but its decreasing rate slows down in the middle, which matches (3). So the correct answer is (b).

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Exact differential equation, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag