Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 11 (d)

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MATH100 B December 2024

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q10  •  Q11 (a)  •  Q11 (b)  •  Q11 (c)  •  Q11 (d)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q13  •  Q14  •  

   Other MATH100 B Exams

  •  December 2023  •  December 2024  •  

Question 11 (d)
In this question, we’ll investigate solutions to the differential equation $$\frac{dy}{dt}=(y-1)(3-y).$$ Is your numerical approximation from (c) using Euler’s method larger or smaller than $y(1)$? You may use your previous work, but you must explain your 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!

Hint
From your sketch in part (a), does the curve look concave up or concave down? What does this imply about whether Euler's method will yield an over approximation or an under approximation?

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
From the sketch, we see $y(x)$ is concave down. That means its tangent lines lie above the curve, so linear approximations produce over approximations. Hence, the first step of Euler's method produces an over approximation. Additionally, $y(x)$ is increasing, so an initial over approximation plus a linear approximation will produce an over approximation in the second step as well.

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