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.

## Test 2

The second test will take place 7 December 2011 at 09:00 in WGB G 05 (not the same room as the lecture).

Everything from Section 2.4 Continuity on Closed Intervals to Section 4.5 Asymptotes and Asymptotics (inclusive of both) is examinable.

Please find a Sample. Note that this is a new sample. I don’t want any of ye thinking just that because Test 1 was very similar to last year’s Test 1 and Sample that Test 2 will be very similar to last year’s Test 2 and sample. The sample is to show you the structure of the test. The paragraphs above and below describe what can come up. Hence have a cautious look at last year’s Sample, Test A and Test B (the latter two with solutions).

Question 1 will be taken from the exercise sheets, Question 2 from a past exam paper, and Question 3 will be on definitions. In a slight change from Test 1, Q.1 is now worth 5/12.5 or 40%, Q. 2 is worth 4/12.5 or 32% and Q. 3 is worth 3.5/12.5 or 28% (1 correct = 1 mark, 2 correct = 2 marks, 3 correct = 3 marks + 0.5 mark bonus).

For Q. 3 of the test, you need to know the following definitions and theorems: continuous on a closed interval, Intermediate Value Theorem, absolute maximum/ minimum on a closed interval, differentiable, Rolle’s Theorem, Mean Value Theorem, local maximim/ minimum, critical points, closed interval method, twice differentiable, concave up/ down, asymptotics, vertical asymptote. Q.3 is a harder question and the thinking behind this is that you can get 72% a bare first if you get all of Q.1 and Q.2 — but you will have to be even better than this to get a higher mark. To test your knowledge of the definitions I have devised some much harder exercises.

## This Week

In lectures, we covered sections 4.1 and 4.2.

In the tutorial we answered exercises Q. 11, 13(i) and 14(i) from Exercise Sheet 3.

## Problems

You need to do exercises – all of the following you should be able to attempt.

When asked to find the critical points of a function defined on the entire real line (rather than just on a closed interval $[a,b]$), the ‘endpoints’, $\pm\infty$ are not considered critical points.

Do as many as you can/ want in the following order of most beneficial:

### Wills’ Exercise Sheets

Q. 2 from Exercise Sheet 4.

More Exercise Sheets

Section 4; Q. 1,2 from Problems.

### Past Exam Papers

Nothing from Summer 2010.

Nothing from Autumn 2010.

Nothing from Summer 2009.

Nothing from Autumn 2009.

Nothing from Summer 2008.

Nothing from Autumn 2008.

Nothing from Summer 2007.

Q. 3(b) from Autumn 2007.

Nothing from Summer 2006.

Q. 4(a) Autumn 2006.

Q. 4(b) from Summer 2005.

Nothing from Autumn 2005.

Nothing from Summer 2004.

Q. 4(a) from Autumn 2004.

Q. 4(a) from Summer 2003.

Nothing from Autumn 2003.

Nothing from Summer 2002.

Nothing from Summer 2001.

Nothing   from Summer 2000.

### From the Class

1. Prove Proposition 4.1.2.
2. Prove that polynomials are infinitely differentiable.

## Supplementary Notes

The Weierstrass function — a function that is continuous but not differentiable at any point.