On Monday we wrote down the Calculus of Limits. We stated that parts (iii) and (iv) have nasty proofs and left them on the webpage. We proved parts (i) and (ii) and showed in part (v) that if the limit exists, it must have that form. We showed that for any polynomial

, that

. Finally we proved the special limit:

In the tutorial we ran out of questions fairly early, so I threw the exercises up on the projector and allowed ye work away. Ye put up your hand if ye wanted assistence.

On Wednesday we considered infinite limits – when does a function tend to infinity and what does a function do as

tends to infinity. Also we gave the crucial definition of a continuous function.

Finally I stated that the Sample Test will be up before or on Wednesday 20 – the test is on 27/10/10. Question 1 will be from the exercises, Question 2 from past exam papers, and Question 3 will be about definitions.

**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.10 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

Q. 1\(ix), 2,3,4,5 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise2.pdf

*More exercise sheets*

Section 2 from Problems

*Past Exam Papers*

Q. 1(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

Q . 1(b), 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 1(b) from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/Autumn/MS2001A09.pdf

Q. 1(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 1(b), 2(a)(ii-iii), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 1(a) from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 2(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 2(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 1(b), 3(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 1(b)from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 1(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 1(b), 2(a) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 1(b) from http://booleweb.ucc.ie/ExamPapers/exams2002/Maths_Stds/ms2001.pdf

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

Q. 3 from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

*From the Class*

1. *Prove the following proposition:*

Suppose that are functions for which for all *. *If exists then so does and moreover they are equal.

*2. Investigate, for an odd natural number,*

*3. Investigate*

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October 16, 2010 at 11:47 am

Fiachra LandersWould it be possible for you to put up the solutions for some of the worksheets/sample tests? A lot of the class is struggling with the worksheets but if there were some worked out solutions, or even just the answers to some of the worksheets it would be a great help. Thank you

October 18, 2010 at 9:40 am

J.P. McCarthyTo be honest with you I don’t think I will.

Firstly, by 1.30 at last Tuesday’s tutorial we ran out of questions. That suggests one of three things to me (I say “you” here but I just mean any student in the class, rather than you Fiachra):

1. You didn’t attempt any exercises and so have no questions

2. You did exercises but you had no problem with them and so have no questions

3. You’re shy/ afraid/ unwilling to ask too many questions.

Now you told me that ye were having difficulty with the sheets. That suggests to me that ye did indeed attempt exercises, so 1 is out. Clearly 2. is out and all that leaves is 3. You have a resource in tutorials and ye plainly didn’t use it.

Come to tomorrow’s tutorial with a list of specific questions, and, time-willing, we can get through all of them.

As it stands you have worked examples presented in lectures and, when questions are asked, at tutorials. Also Wills’ notes have many worked examples.

Finally at present there are no solutions to past papers online and I am loathe to put some up because students will then learn off how to do the few questions that do regularly occur. I would much rather ye sit down with a pen, and paper (and indeed your notes), and figure out for yourself these solutions – it would be far, far more beneficial.

I understand that ye are in the business of passing exams, but I am in the business of of sending on students with a solid grounding in MS 2001 for MS 2002 and MS 3003.

Learning off examples is not the way to go – if you do the work and figure out the solutions (even find similar examples in your notes, or Wills’ notes), then ‘learning off’ the examples wouldn’t be too bad because you’ve at least gone through the logic in your head.

Now if two weeks down the line you are asking questions at tutorials and ye are still having difficulties, please feel free to contact me again.

Also I am very open to answering questions put in comments on the webpage.