Science talk:Math Exam Resources/Courses/MATH104/December 2012/Question 6 (c)

I feel rather weird about over and underestimate in this question. I don't know the syllabus of MATH104/184 in 2012, but I think we can talk about over or underestimate if we can show that concavity of the graph of y=y(x) doesn't change on [1,1.02]. I am not able to find any method to evaluate the sign of f(x) for all x between 1 and 1.02.

This is a valid point. When we made this question last year we intended "near-by" to mean that for points near (1,1) f(x) IS the Taylor polynomial. We then intend that 1.02 is one of these "near-by" points and so the second derivative is always -2 leading to the conclusion from the solution. Technically one should be more rigorous about the error bounds and ensure that 1.02 is close enough to be validated by the Taylor approximation but we decided that was too difficult. Anyway, a solution like what is currently here was a full-marks solution.

Thanks for this clarification Iain. I think that the wording expect, instead of show that, makes it clear enough that the question asks for the intuitive answer, rather than another complicated calculation.

I added some extra detail to the solution to expand the understanding of linear approximations.

I think that the original hint was just a short-form version of the eventual solution and so I changed them to be helpful without being overly revealing.