Firstly; there will be no MS 2001 lecture on Monday 6 December at 3 p.m. Instead you will have an MS 2003 lecture at this time in WG G 08. The 12 p.m. MS 2003 lecture on Wednesday December 1 in WG G 08 will now be an MS 2001 lecture.   Indeed it will be the final MS 2001 lecture as Wednesday 8 December is a test day and the week after is review week. The morning lecture at 9 a.m. on Wednesday 1 December will still go ahead.
On Monday we finished off the section on Implicit Differentiation.  We did the example of a circle and emphasised the use of the chain and product rules in this area. We used implicit differentiation techniques to establish the power rule:
$\frac{d}{dx}x^n=nx^{n-1}$
for $n\in\mathbb{Q}$, $x\in(0,\infty)$, thus extending the rule we proved for integers.  We started a new chapter – Curve Sketching and Max/ Min Problems. We defined local maximum/ minimum and proved that if a function, continuous on a closed interval, takes an absolute max/ min at a point inside the interval, and is differentiable there, then the derivative must be zero.
In the tutorial we outlined Exercise Sheet 3, Q. 3(i)-(v) by stating where the functions were differentiable and what rule could be used to find the derivative where differentiable. We did Q. 4(i), Q. 10(i),(ii) and finally Q. 12. With a test in three weeks ye need to keep up the work on exercises.
On Wednesday we introduced critical points, the Closed Interval Method and the Second Derivative Test.
Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Use the Closed Interval Method to do Q. 8 (i), (iii) from

Other Exercise Sheets

Section 4 Q. 1-3 from Problems

Past Exam Papers

Those questions in bold are to be done using the Closed Interval Method. Those questions in italic request the critical points of a function $f:\mathbb{R}\rightarrow \mathbb{R}$ rather than $f:[a,b]\rightarrow \mathbb{R}$. In these questions the ‘endpoints’ $\pm\infty$ are not considered critical points.

Q. 5(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 4(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 3, 4(a), 5(b), 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 4(c), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

From the Class

1. Prove Proposition 5.1.1 in the case that of $x_1$ is an absolute minimum.