First thing on Monday we tried to rearrange the Monday 6 December lecture. We failed miserably but someone had the bright idea to swap with MS 2003. I have emailed the lecturer and she agrees in principle so we should have a definite plan soon. The second test is also slated for Wednesday 8 December.
Anyway, we stated and proved Rolle’s Theorem. We stated and proved the Mean Value Theorem and verified the Mean Value Theorem for a quadratic.
In the tutorial nobody asked any questions, so I threw the exercises up on the projector and allowed ye work away. Ye put up your hand if ye wanted assistance. In the end I did questions 5 (iii),(iv) & 6 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf.
On Wednesday we proved the intuitively true theorem that the sign of a derivative determines whether the function is increasing or decreasing. We used this theorem to prove that $x^n$ is increasing on $(0,\infty)$; a corollary of which is the existence of positive $n$th roots. We defined rational powers. Finally we introduced the idea of the curve; and showed a few examples on the projector.

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

Past Exam Papers

From the Class

1. Prove Proposition 4.2.1 for the case that the minimum, $m$ differs from $f(a)$.

2. Drawings can be deceptive! Draw a function that is continuous on a closed interval but not differentiable at any point in the interval. What does your drawing suggest? Now see http://en.wikipedia.org/wiki/Weierstrass_function

3. Prove Proposition 4.2.3 (iii)