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Test Results

First of all results are down the bottom. You are identified by the last four digits of your student number (or five if these four digits are shared by another student). The scores are itemized as you can see. At the bottom there are some average scores.

If you would like to see your paper or have it discussed please email me.

Students with no score were either absent in which case they score zero — or certified absent in which case their marks carry forward to the summer exam.

Solutions and Remarks

Solution are found here.

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.

## [EDIT] Definition Questions

If you really want to get good at your definition questions check out these (tough) exercises..

## Next Week

We will have an additional tutorial instead of a lecture on Monday 24.

## This Week

On Monday, we covered from (but not including) Definition 3.1.1 to (and including) Corollary 3.1.6. On Wednesday we went through Questions 1 & 2 from 2010/11’s Test 1 A and we did Q. 3 from the sample test.

In the tutorial we answered exercises Q. 1 (vii), 3 (b), 4(ii) & 8(iii) from Exercise Sheet 2.

## 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. 1, 2, 3(i) & (iii), 4, 8(a) & (b) from Exercise Sheet 3.

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 1

Firstly, some good news. I am going at a different pace to last year so in fact we have covered everything we need to do for Test 1 already. All of Chapter 1 and Sections 2.1, 2.2 and 2.3 of Chapter 2 are all that will be examinable in Test 1. This means that Section 2.4 Continuity on Closed Intervals and Chapter 4 Differentiability are not  examinable.

Please find the Sample, Test 1 A and Test 1 B (second 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. Q.1 is worth 4/12.5 or 32%, Q. 2 is worth 5/12.5 or 40% 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: even, odd, increasing, decreasing, quadratic, roots, polynomial, rational function, absolute value, limit, one-sided limit, continuous at a point, continuous, composition. 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.

The first in-class test will take place on 26 October 2011. Any material presented in class, up to and including 19 October is examinable. The test is worth 12.5% of your continuous assesment mark for MS 2001. A sample test (without solutions) — as well as last years test (with solutions) shall be posted here on 12 October 2011. Although if you are willing to look through the “MS 2001: Continuous Assessment” category you will find these easily enough.

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.

We covered from Proposition 1.4.3 to Proposition 2.1.5 inclusive. Here are the proofs of the product and quotient  rules for the calculus of limits.

## 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:

### More exercise sheets

Q. 4 from Problems

### Past Exam Papers

Q . 1(a), 2(a) fromhttp://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 1(a), 2(a)(i), (b) fromhttp://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 1(a), 2(a), 3(b) fromhttp://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 1(a), 2(a), 3 fromhttp://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 1(a), 2(a), 3 fromhttp://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

### Tutorial Quiz Questions

These are not necessarily of exam standard but are more an exercise to help your understanding. The quiz we never did this week as ye actually did some exercises and asked questions.

We covered from Proposition 1.1.4 about the inequality relation — up to but not including Proposition 1.4.3.

I will make a decision about tutorial clashes on or about Monday.

Exercises

Q. 1,2 and 4 – 7 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

Q. 2 – 3 from Problems

Stephen Wills: Supplementary Notes.

This week’s exercises

Q.3 from Exercise Sheet 1.

The lecture notes are available from An Scoláire, College Road (across from the main College Road entrance to UCC). They are for sale at €12 — just ask for the notes for MS2001.

http://www.goldenpages.ie/an-scolaire-cork-city/2/

Questions 1 (ii) — (vii) from here.

With all of these questions the first thing we want to do is plug in the value; i.e. if we are evaluating $\lim_{x\rightarrow a}f(x)$ we should first try $f(a)$. If we find that $f(a)$ is undefined (e.g. division by zero), we will usually have to ‘factor out the bad stuff’. The reason this works is because of the following theorem:

### Theorem

Suppose that $f(x)=g(x)$ except perhaps at $x=a$. Then

$\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}g(x)$.

Take for example

$\lim_{x\rightarrow 0}\frac{x^2}{x}$.

Now if $f(x)=x^2/x$, then $f(0)=0/0$ — which is undefined. However we can cancel an $x$ above and below… well what we really do is as follows:

$\frac{x^2}{x}=x\times\underbrace{\frac{x}{x}}_{=1}=x$.

However this is only true in the case where $x\neq 0$; i.e. we have that $x^2/x=x$ for all $x\neq 0$ — and use the above theorem to say that

$\lim_{x\rightarrow 0}\frac{x^2}{x}=\lim_{x\rightarrow 0}x=0$.

Determine whether the following functions are defined and where they are continuous.

$f(x)=\frac{x+\sin \pi x}{x^2+3x+2}$$g(x)=\frac{x^2}{x+1}$.

### Solution

As $\sin \pi x$ is defined everywhere, $f(x)$ is defined as long as the denominator, $x^2+3x+2\neq0$. Now

$x^2+3x+2=x^2+2x+x+2=x(x+2)+1(x+2)=(x+1)(x+2)$,

therefore $x^2+3x+2\neq0\Leftrightarrow x\neq -1,-2$. So $f(x)$ is defined for all $x\in\mathbb{R}$ except $x=-1,-2$. As $x+\sin\pi x$ is continuous (as the sum of a polynomial and a sine) and $x^2+3x+2$ is continuous (as a polynomial), $f(x)$ is continuous as long as $x^2+3x+2\neq 0\Leftrightarrow x\neq-1,-2$.

Similarly $g(x)$ is defined for, and continuous at, all $x+1\neq0\Leftrightarrow x\neq -1$.