For those doing Project Maths at Leaving Cert, an initial stumbling block will be on how to revise. The best way to revise mathematics is by doing exercises and a ready supply of these is traditionally found in past exam papers:
http://examinations.ie/index.php?l=en&mc=en&sc=ep&formAction=agree
However you might be under the impression that these past papers are useless with the new syllabus: not necessarily. For those doing Project Maths in 2011, the syllabus is to be found at:
Strand 5, “Functions”, doesn’t become examinable until 2012. However this topic includes differentiation and integration, which will certainly be examinable as per the old syllabus.
Essentially the past papers may be done partially. Paper 1, Q. 6-8 is definitely fair game. Figuring out which of the other questions on past papers are still relevant involves cross-checking the new syllabus with the old.
is increasing on 
; a corollary of which is the existence of positive 
th roots. We defined rational powers. Finally we introduced the idea of the curve; and showed a few examples on the projector.
differs from 
.
contains no invertible elements and is contained in a maximal ideal. I saw that there is a bijection between the set of characters of 
be functions that are differentiable at some 
. If 
, then 
is differentiable at 
with![\left(\frac{f}{g}\right)'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^2}](https://s0.wp.com/latex.php?latex=%5Cleft%28%5Cfrac%7Bf%7D%7Bg%7D%5Cright%29%27%28a%29%3D%5Cfrac%7Bf%27%28a%29g%28a%29-f%28a%29g%27%28a%29%7D%7B%5Bg%28a%29%5D%5E2%7D&bg=ffffff&fg=545454&s=0&c=20201002)
:
![=\frac{g(a)[f(a+h)-f(a)]-f(a)[g(a+h)-g(a)]}{g(a+h)g(a)}](https://s0.wp.com/latex.php?latex=%3D%5Cfrac%7Bg%28a%29%5Bf%28a%2Bh%29-f%28a%29%5D-f%28a%29%5Bg%28a%2Bh%29-g%28a%29%5D%7D%7Bg%28a%2Bh%29g%28a%29%7D&bg=ffffff&fg=545454&s=0&c=20201002)
![\Rightarrow \frac{q(a+h)-q(a)}{h}=\frac{g(a)\left[\frac{f(a+h)-f(a)}{h}\right]-f(a)\left[\frac{g(a+h)-g(a)}{h}\right]}{g(a+h)g(a)}](https://s0.wp.com/latex.php?latex=%5CRightarrow+%5Cfrac%7Bq%28a%2Bh%29-q%28a%29%7D%7Bh%7D%3D%5Cfrac%7Bg%28a%29%5Cleft%5B%5Cfrac%7Bf%28a%2Bh%29-f%28a%29%7D%7Bh%7D%5Cright%5D-f%28a%29%5Cleft%5B%5Cfrac%7Bg%28a%2Bh%29-g%28a%29%7D%7Bh%7D%5Cright%5D%7D%7Bg%28a%2Bh%29g%28a%29%7D&bg=ffffff&fg=545454&s=0&c=20201002)
on both sides:![q'(a)=\left(\frac{f}{g}\right)'(a)=\frac{g(a)f'(a)-f(a)g'(a)}{[g(a)]^2}](https://s0.wp.com/latex.php?latex=q%27%28a%29%3D%5Cleft%28%5Cfrac%7Bf%7D%7Bg%7D%5Cright%29%27%28a%29%3D%5Cfrac%7Bg%28a%29f%27%28a%29-f%28a%29g%27%28a%29%7D%7B%5Bg%28a%29%5D%5E2%7D&bg=ffffff&fg=545454&s=0&c=20201002)
. We presented and proved the sum, scalar, product and quotient rules of differentiation. The proof of the quotient rule is on this page. We did the derivative of 
for 
. As a corollary we showed that polynomials are differentiable everywhere. Finally we wrote down the Chain Rule.
, 
, 
, 
and 
(the last two proved non-rigorously). Finally we wrote down Rolle’s Theorem.
is separating if for all 
, 
such that 
). Also a number of results were derived that concerned the existence of functionals which were dominated on one set by another (e.g. 5.12). Finally some exercises on extreme points; for example every unit vector in a Hilbert space is an extreme point of the closed unit ball ![B_1^H[0]](https://s0.wp.com/latex.php?latex=B_1%5EH%5B0%5D&bg=ffffff&fg=545454&s=0&c=20201002)
.
. I saw a number of examples of function spaces… basically it was “An Introduction to Normed Algebras” and it is fairly straightforward with some very nice results such as the Gelfand-Mazur Theorem. I have done exercises 6.1-6.6 of 8.
J.P. McCarthy: Math Page 
Recent Comments