Science:Math Exam Resources/Courses/MATH110/December 2012/Question 04

MATH110 December 2012

  •  Q1 (a)  •  Q1 (b)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q2 (e)  •  Q2 (f)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q3 (d)  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q10  •  

   Other MATH110 Exams

  •  December 2011  •  December 2010  •  April 2012  •  April 2011  •  December 2012  •  April 2013  •  December 2013  •  April 2014  •  December 2014  •  December 2015  •  April 2017  •  April 2016  •  December 2016  •  December 2017  •  April 2018  •  

Question 04
Explain why there exists a number $c$ in the interval $(1,2)$ satisfying the equation $\displaystyle -x^{3}+x^{2}+2x=\ln(x)+1.$ Justify your answer, citing appropriate theorems.

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
Try rearranging the equation for zero on one side, which means you now need to show that at some point this equation equals zero. This should remind you of a particular theorem...

Hint 2
Use the Intermediate Value Theorem.

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. 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. As suggested by the hint, we will solve for zero on one side of the equation, giving: $-x^{3}+x^{2}+2x-\ln(x)-1=0$ We now need to show that there is a number c in the interval (1,2) that makes the left side equal to zero. This should remind you of the Intermediate Value Theorem (IVT). The statement of the IVT says If f(x) is continuous on [a,b] and for some number L, f(a) < L < f(b), then there exists c in (a,b) such that f(c) = L." In our scenario, the left side of the equation is f(x), a = 1, b=2, and L = 0'. In order to use the IVT, we need to check that the function is continuous on the interval [1,2]. As each term of the function is continuous on the interval, the whole function is also continuous on the interval. Now we compute f(1) and f(2). $f(1)=-1+1+2-\ln(1)-1=1>0$ $f(2)=-8+4+4-\ln(2)-1<0$ Because f(1) < 0 < f(2), we can now use the IVT to conclude that there exists a number c in the interval (1,2) such that f(c) = 0.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Intermediate value theorem, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag