Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 23

MATH100 A December 2023

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9  •  Q10  •  Q11  •  Q12  •  Q13  •  Q14  •  Q15  •  Q16  •  Q17  •  Q18  •  Q19  •  Q20  •  Q21  •  Q22  •  Q23  •  Q24  •  Q25  •  Q26  •  Q27(a)  •  Q27(b)  •  Q27(c)  •  Q28(a)  •  Q28(b)  •  Q29(a)  •  Q29(b)  •  Q30  •  

   Other MATH100 A Exams

  •  December 2023  •  

Question 23
Which of the following is the slope field for the differential equation ${\frac {\mathrm {d} y}{\mathrm {d} x}}=x+y$? Select all that apply. Slope field (i) Slope field (ii) Slope field (iii) Slope field (iv)

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
Try finding the coordinates of the points $(x,y)$ where ${\frac {\mathrm {d} y}{\mathrm {d} x}}=0$. Then think about the points where ${\frac {\mathrm {d} y}{\mathrm {d} x}}>0$.

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. According to the differential equation, ${\frac {\mathrm {d} y}{\mathrm {d} x}}=0$ if and only if $x+y=0$, which occurs precisely along the line $y=-x$. This rules out slope fields (i) and (iii), which do not have horizontal slopes along the line $\{y=-x\}$ in the fourth quadrant. To choose between slope fields (ii) and (iv), note that the slopes should be positive at the points $(x,y)$ for which $x+y>0$. These are the points that lie above the line $\{y=-x\}$. We see that slope field (iv) has lines of negative slope above the line $y=-x$ in the fourth quadrant. Thus, the only possible slope field for the differential equation is (ii).