Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 22

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MATH100 A December 2023

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Other MATH100 A Exams

• December 2023 •

Question 22
Sketch the phase line for the differential equation ${\textstyle x'=(x+1)(x-4)}$ . Clearly indicate the equilibrium points, and indicate with arrows whether ${\textstyle x}$ is increasing or decreasing on either side of each equilibrium point.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall: if $x$ is at an equilibrium point, then the function $x$ is not changing with time. Given that the derivative is the rate of change, what is happening to the derivative at an equilibrium point?

Hint 2
How does the derivative relate to whether a function is increasing or decreasing?

Checking a solution serves two purposes: helping you if, after having used all the hints, 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.
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Solution
We know that if the function $x$ is at an equilibrium point, then it is not changing, which means the derivative of the function is zero. Since we are given the differential equation in factored form, we can read off the zeros easily: ${\textstyle x'=0}$ when ${\textstyle x=-1,4}$ ; these are the equilibrium points. To determine if the function is increasing or decreasing on either side of the equilibrium points, we look at whether ${\textstyle x'}$ is positive or negative. When ${\textstyle x<-1}$ , ${\textstyle x'}$ is positive. When ${\textstyle -1<x<4}$ , ${\textstyle x'}$ is negative. And when ${\textstyle x>4}$ , ${\textstyle x'}$ is positive. So, we have that the function is increasing for ${\textstyle x<-1}$ , decreasing for ${\textstyle -1<x<4}$ , and increasing for ${\textstyle x>4}$ , which means ${\textstyle x=-1}$ is an unstable equilibrium and ${\textstyle x=4}$ is a stable equilibrium. The phase line representing all of this is given below. phase line