If the x 1 x 2 x n are chosen to be the roots of the

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If the x 1 , x 2 , . . . , x n are chosen to be the roots of the Legendre polynomial of degree n show that the formula is exact for all polynomials of degree 2 n 1 or less. Conclude that the method of approximate integration given by is likely to be improved by this choice of x j . There is substantially more that can be said on Gaussian quadrature but (supposing the course to say no more) the rules of the game say that you are not expected to know more than the syllabus demands. The next example of an essay question is, in my opinion, rather more demanding. Q. 3. Assuming any results about exp and log that you need, define x α for x > 0 and α > 0 and establish the basic results concerning the function x mapsto→ x α . Here the problem is to decide which are the basic properties. Perhaps the reader should make her own list before looking at mine. 2
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Q. 3 . Set x α = exp( α log x ) . Stating carefully any results about exp and log that you need prove the following results. (Here x, y > 0 , α and β are real and n is strictly positive integer.) (i) ( xy ) α = x α y α . (ii) x ( α + β ) = x α x β . (iii) x αβ = ( x α ) β . (iv) dx α dx = αx α 1 . (v) x n = n bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright xx . . . x . Of course there are other ways to define x α and the examiner would accept these 2 . If I was marking Question 3, I think that I would give full marks to an essay which included a satisfactory treatment of (v) (it is essential that our new definition of x n should coincide with the old) together with three out of the four points (i), (ii), (iii) and (iv). Perhaps you disagree with my list (remember that I only ask for three out of the first four points). Do you disagree that points (i) to (v) are basic or do you feel that that I should have included more points? Some people might feel that I should have included the fact that x 0 = 1 (though this follows from (ii)) or shown explicitly that our new and old definitions of x α coincide when α is rational (although this follows from (v), I would certainly do this in any lecture course). Others might wish to include the fact that x α → ∞ as x → ∞ if α > 0. On the other hand, most people would agree that the Taylor series for (1 + x ) α is not a basic result in this context. There is room for disagreement but I hope that, after reflection, most mathematicians will agree that my list was reasonable (particularly if my marking scheme was flexible). My final example is a fully fledged essay. Q. 4. Write an essay on M¨obius maps. I shall use this as my typical example and I shall assume only the material contained in the first year course ‘Algebra and Geometry’. Question 4 is a full fledged essay because it involves organising material in a different way from the course (results on M¨obius maps are scattered throughout the lectures), there is too much material to cover in detail in a short essay so we must select which topics to treat and, even after making a choice of topics, it is not possible to give all the proofs in full so we must choose what to prove, what to sketch and what to state.
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