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Unformatted text preview: Thoughts on the Essay Question T. W. Korner November 5, 2008 Small print The opinions expressed in this note are the authors own. Even the best advice (and there is no reason to suppose that the advice here is the best advice) does not apply to all people and in all circumstances 1 . The advice given is intended for students taking 1A, 1B and, particularly, Part II of the Cambridge mathematics tripos. It does not apply to Part III which is a very different course with different objectives. I should very much appreciate being told of any corrections or possible improvements. This document is written in L A T E X2e and stored in the file labelled ~twk/FTP/Excess.tex on emu in (I hope) read permitted form. It also available via my web home page. My e-mail address is twk@dpmms . 1 What is an essay question? What students dislike most about essay questions is that they have to make choices for themselves about what to include and how to treat it. Since this dislike is often shared by examiners (giving students genuine choices makes marking much more difficult) many essay questions are simply dis- guised bookwork. Consider the following essay question. Q. 1. Write an essay on the Steinitz exchange lemma and its use in estab- lishing the notion of dimension. A little thought shows that it could be rewritten as follows. Q. 1 . Define the terms spanning set, linearly independent set and basis. State and prove the Steinitz exchange lemma. Show that any vector space V with a finite spanning set has a basis. Use the Steinitz exchange lemma to show that all bases of V have the same number of elements and so we can define the dimension of V . 1 Two days wrong! sighed the Hatter I told you butter wouldnt suit the works! he added, looking angrily at the March Hare. It was the best butter, the March Hare meekly replied. Yes, but some crumbs must have got in as well, the Hatter grumbled: you shouldnt have put it in with the breadknife. [Alice in Wonderland] 1 Now consider a possible essay question on numerical analysis. Q. 2. Suppose that we wish to find integraldisplay 1 1 f ( x ) d x but that it is expensive to obtain values of f . Discuss the use of orthogonal polynomials in evaluating the integral. In principle, this is a more open question than Question 1 but in practice it is limited by the fact that the essay must be on a subject that you have covered. If the only method of integration discussed in the course is Gaussian quadrature then the question can only be on Gaussian quadrature and can be restated as follows. Q. 2 . Explain the method of Gaussian quadrature. A little reflection converts this into a bookwork question....
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