Science:Math Exam Resources/Courses/MATH102/December 2012/Question A 01

MATH102 December 2012

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Question A 01
The functions f(x) = x2 and g(x) = x3 are equal at x = 0 and at x = 1. Between x = 0 and x = 1, for what value of x are their graphs furthest apart? (a) x = 1/2 (b) x = 1/3 (c) x = 2/3 (d) x = 1/4 (e) x = 3/4

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
The question is asking about the distance between two graphs. Try sketching a picture of the two graphs (it doesn't have to be 100% accurate for this technique to work!), and identifying the distance between the two at a point between 0 and 1. Now find a formula for that distance.

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

As stated in the hint, sketching a picture of the two graphs between 0 and 1 is a good first step. Identify the distance between the two graphs and write a formula for it. The distance between the two graphs at a point x can be given by $D(x)=x^{2}-x^{3}$ Now this is simply a max/min problem (see the keyword "furthest apart" in the question): we are looking for the maximum of $D(x)$. First we find critical points by differentiating, setting equal to zero, and solving for x: $D'(x)=2x-3x^{2}$ $0=x(2-3x)$ So we have critical points x = 0 and x = 2/3. We know x = 0 isn't the correct answer as we already know that the graphs are equal there. Thus, the point of furthest distance must be x = 2/3 or (c).

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