I am emailing a link of this to everyone on the class list every Wednesday morning. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.

## This Week

In lectures, we covered from Rolle’s Theorem to the end of the chapter on differentiation.

In the tutorial we answered exercises Q. 2, 5, 6 & 7(b) from Exercise Sheet 3. I had been very confused by Q. 7(b). We showed that $f(x)=x+\sin x$ had a horizontal tangent at $x=\pi$ but I couldn’t see how it could have a unique max or min given that $f(0)=0$ and $f(2\pi)=2\pi$ — but as we shall see in the next ten days, while $f'(x)=0$ is necessary for a (differentiable) max or min, it is not sufficient. That is

(differentiable) maximum or minimum $\Rightarrow$ $f'(x)=0$

but

$f'(x)=0\not\Rightarrow$ (differentiable) maximum or minimum

necessarily. There is a third possibility, that of a saddle point that is neither a local maximum nor minimum. This is the case with $f(x)=x+\sin x$ as this plot shows (by the way that Wolfram Alpha is an unbelievable piece of kit — have a play around with it).

## Test 2

Just giving fair warning about test 2 — it will be held on December 7. More details next week.

## 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

Q. 6, 11, 12, 13, 14, 16 & 17 from Exercise Sheet 3.

Q. 1 from Exercise Sheet 4.

More Exercise Sheets

Nothing from Problems.

### Past Exam Papers

Q. 4(a) from Summer 2010.

Q. 3 from Autumn 2010.

Q. 3(b) from Summer 2009.

Q. 3 from Autumn 2009.

Q. 3(b) & 4(b) from Summer 2008.

Q. 4 from Autumn 2008.

Q. 4 from Summer 2007.

Q. 4(a) from Autumn 2007.

Q. 4(a) & 5(b) from Summer 2006.

Q. 5(a) & 6(a) Autumn 2006.

Q. 5(a) from Summer 2005.

Nothing from Autumn 2005.

Q. 5(a) & 6(a) from Summer 2004.

Q. 3(a), 5(a) & 6(a) from Autumn 2004.

Q. 5(a) & 6(a) from Summer 2003.

Q. 5(a) & 6(a) from Autumn 2003.

Nothing from Summer 2002.

Q. 4(c), 5(a) & 6(a) from Summer 2001.

Q. 4(c), 5 & 6(a)   from Summer 2000.

### From the Class

1. Prove Rolle’s Theorem in the case where $f(a)\neq m$.
2. Prove that the function $F(x)$ defined in the proof of the Mean Value Theorem satisfies $F(a)=F(b)$.
3. Prove Proposition 3.2.3 (iii)
4. Prove that $2012^n>2011$ for $n\geq2$$n\in\mathbb{N}$.

## Supplementary Notes

A list of implicitly defined curves.