Science:Math Exam Resources/Courses/MATH100 A/December 2024/Question 10 (a)

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MATH100 A 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 (a)  •  Q10 (b)  •  Q11 (a)  •  Q11 (b)  •  Q11 (c)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q13  •  Q14  •  

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Question 10 (a)
We would like to find a rational approximation of the cube root of 2, which we will think of as a solution to the equation $x^3-2=0$. Suppose we want to start at an integer. Find consecutive integers m, m+1 that bracket the root (i.e. one is smaller than the root, one is larger). Which one is likely closer, and why?

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
To bracket a root of $f(x)$, we need two consecutive integers where the function changes sign. Start by evaluating $f(x)=x^3-2$ at small integer values.

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

The first thing we can observe is that the function $f(x)=x^3-2$ is increasing. It's easy to check that $x=0$ returns $f(x)=-2<0$. Plugging in, $x=1$, we get $1-2=-1<0$. So no sign change between 0 and 1. Next, plugging $x=2$, the function gives $f(x)=8-2=6>0$. Since going from 1 to 2, the function changes sign, we know that the exact zero of $f(x)$ should be somewhere in the interval $[1,2]$. In order to determine which integer the root is closest to, we can check the sign of derivative, or alternatively plug in directly $f(1.5)=1.5^2-2>0$, so the root is likely closer to 1.

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