MS2001: Week 11 & 12

December 6, 2011 in MS 2001 | 6 comments

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.

These Weeks

In lectures, we finished off the notes.

Review Week

I will hold a tutorial on Tuesday at the same time and place. I should have the results of Test 2 by then.

Appraisal

Thank you very much for your work on the appraisal. Sorry about all the hairstyles.

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PhD Progression Report

December 2, 2011 in Research, Research Updates | Leave a comment

The following runs a thread through what I’ve looked at over the past year: Progression Report.

MS2001: Week 10

November 23, 2011 in MS 2001 | 2 comments

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

See here for full details.

This Week

In lectures, we covered sections 4.3 to Summary 4.5.5 inclusive.

In the tutorial we answered Q. 9 (ii) from Exercise Sheet 1, Q. 10 (iii), 16 from Exercise Sheet 3 and Q.2 from Exercise Sheet 4.

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ST 1021: Question about Expectation

November 19, 2011 in General, Grinds | Leave a comment

Consider the following question. X is supposed to represent the sale price of a hotel room, while Y represents the cost price. Therefore the profit is given by X-Y. I am going to use the term expected average as opposed to the more standard expected value or expectation. 

There is a problem with the interpretation and I wouldn’t treat this particular exercise with much importance.

Suppose that X and Y are independent random variables with distributions

 

Find the expected average of the profit on a single room. Find the expected average of the profit on 1,000 rooms. Find the probability that the profit on 1,000 rooms is less than 20,000.

Solution : The expected average of a variable is given by:

\mathbb{E}[X]=\sum_ix_i\mathbb{P}[X=x_i]=\sum_ix_ip_i.

Now expected average is linear:

\mathbb{E}[X+\lambda Y]=\sum_i(x_i+\lambda y_i)p_i=\sum_{i}x_ip_i+\lambda\sum_i y_i p_i

=\mathbb{E}[X]+\lambda\mathbb{E}[Y].

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Project Maths: A More Geometric Approach to De Moivre’s Theorem and Complex Roots

November 19, 2011 in General, Grinds, Leaving Cert, MA 1008, MS 3011 | 3 comments

Introduction

This is just a short note to provide an alternative way of proving and using De Moivre’s Theorem. It is inspired by the fact that the geometric multiplication of complex numbers appeared on the Leaving Cert Project Maths paper (even though it isn’t on the syllabus — lol). It assumes familiarity with the basic properties of the complex numbers.

Complex Numbers

Arguably, the complex numbers arose as a way to find the roots of all polynomial functions. A polynomial function is a function that is a sum of powers of x. For example, q(x)=x^2-x-6 is a polynomial. The highest non-zero power of a polynomial is called it’s degree. Ordinarily at LC level we consider polynomials where the multiples of x — the coefficients — are real numbers, but a lot of the theory holds when the coefficients are complex numbers (note that the Conjugate Root Theorem only holds when the coefficients are real). Here we won’t say anything about the coefficients and just call them numbers.

Definition

Let a_n,\,a_{n-1},\,\dots,\,a_1,\,a_0 be numbers such that a_n\neq 0. Then

p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,

is a polynomial of degree n.

In many instances, the first thing we want to know about a polynomial is what are its roots. The roots of a polynomial are the inputs x such that the output p(x)=0.

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MS2001: Week 9

November 17, 2011 in MS 2001 | Leave a comment

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).

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MS 2001: Module Materials Appraisal Feedback

November 14, 2011 in MS 2001, MS 2002 | Leave a comment

Just a chance for me to give some feedback on some of your most common… feedback. Of course your opinions are correct – they are your opinions. Here are my opinions on some of your opinions.

Normally the failing of teaching evaluation, is that ye never see the fruits of your criticisms. However, as I am currently drafting the MS 2002 notes ye have had a chance to improve your lot in this module at least.

A lot of people felt that through a combination of not leaving ye enough time to fill in notes, talking to fast or just going to fast, that ye didn’t have enough time to digest explanations. At least for the rest of this module we can address that.

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MS2001: Week 8

November 9, 2011 in MS 2001 | Leave a comment

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.

MS2001: Week 7

November 2, 2011 in MS 2001 | Leave a comment

I HAVE TAKEN CARE OF THE DUPLICATE STUDENT NUMBER FOUR DIGIT ENDINGS — ALL OF THESE STUDENTS HAVE RECEIVED AN EMAIL SO YOU DON’T HAVE TO CHECK FOR A DUPLICATE. I HAVE EDITED THE MARK DISPLAY TO TAKE THIS INTO ACCOUNT (the duplicates are last five digits)

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

On Wednesday, we covered from (but not including) Corollary 3.1.6 to Rolle’s Theorem (although we only started the proof).

In the tutorial we answered exercises Q. 1, 3 & 8(a) from Exercise Sheet 3.

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. 3, 5, 7, 8 (b), 9 & 10 from Exercise Sheet 3.

More Exercise Sheets

Q. 3 from from Problems.

Past Exam Papers

Q. 1(c) & 4(b) from Summer 2010.

Q. 1(c) & 4 from Autumn 2010.

Q. 1(c) & 4 from Summer 2009.

Q. 1(c) & 4 from Autumn 2009.

Q. 4(a) from Summer 2008.

Q. 1(c) & 4(a) from Autumn 2008.

Q. 1(c) & 3 from Summer 2007.

Q. 1(c) from Autumn 2007.

Q. 3(b), 4(b) & 5(a) from Summer 2006.

Q. 3(b) & 4(b) Autumn 2006.

Q. 4(a) from Summer 2005.

Q. 4 & 5(a) from Autumn 2005.

Q. 4 from Summer 2004.

Q. 4(b) from Autumn 2004.

Q. 4(b) from Summer 2003.

Q. 4 from Autumn 2003.

Q. 4 & 5(a) from Summer 2002.

Q. 1(b), 5(b) & 6(b) from Summer 2001.

Q. 1(b),  from Summer 2000.

From the Class

Nothing here.

Supplementary Notes

The proof of the chain rule and the “chain rule by rule” here.

Proof of the Cauchy-Schwarz Inequality for Euclidean Spaces

November 2, 2011 in General, Grinds | Leave a comment

Theorem: Cauchy-Schwarz Inequality

Let a_1,a_2,\dots,a_n and b_1,b_2,\dots,b_n be sequences of real numbers. Then we have

\left|\sum_{i=1}^na_ib_i\right|\leq\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum_{i=1}^nb_i^2}.

Proof : Consider the following quadratic function f:\mathbb{R}\rightarrow\mathbb{R}:

f(x)=\sum_{i=1}^n(a_ix+b_i)^2.

Note at this point that f(x)\geq0 for all x\in\mathbb{R}.

f(x)=\sum_{i=1}^n(a_i^2x^2+2a_ib_ix+b_i^2)

=\left(\sum_{i=1}^na_i^2\right)x^2+\left(2\sum_{i=1}^na_ib_i\right)x+\sum_{i=1}^nb_i^2.

That is f is a \bigcup or `+x^2‘ positive quadratic so has one or no roots. That means the roots are real and repeated or complex so that we have b^2-4ac\leq 0 where f(x)=ax^2+bx+c:

\left(2\sum_{i=1}^na_ib_i\right)^2-4\left(\sum_{i=1}^na_i^2\right)\left(\sum_{i=1}^nb_i^2\right)\leq0

\Rightarrow \left(\sum_{i=1}^na_ib_i\right)^2\leq \left(\sum_{i=1}^na_i^2\right)\left(\sum_{i=1}^nb_i^2\right)

Now take square roots (remembering \sqrt{x^2}=|x|.) \bullet