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MS 2001: Week 3

October 6, 2010 in MS 2001 | Leave a comment

We finished off the preceding week’s work by solving an inequality involving the absolute value function and a rational function. Also we did an example showing how to bound the absolute value of a rational function on an interval (using the triangle inequality and the reverse triangle inequality).

We began a new chapter: Limits & Continuity. We gave the \varepsilon–\delta definition of the limit of a function.  We did a few examples. We defined left- and right- hand limits and proved that a limit exists if and only if its left- and right-hand limits exist and are equal.

In one example, we showed a limit did not exist as the left- and right-hand limits did not agree. We then did two examples using the \varepsilon–\delta definition but noted we wouldn’t always have to do this due to the Calculus of Limits. We wrote down the five results and said we will prove the first and second, put the third and fourth on the webpage, and do a sketch of the fifth.

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.7, 8, 9 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

More exercise sheets

Q. 3, 4 from Problems

Past Exam Papers

Q. 1(a), 2 from  http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

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

Q. 1(a), 2 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/Autumn/MS2001A09.pdf

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

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

Q. 2 from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 1, 2(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 1, 2(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 1, 2(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 1, 2(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

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

Q. 1(a), 2 from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

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

Q. 1(a), 3 from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 1, 2(a) 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. 1(a), 2 from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

MS 2001: Week 2

September 30, 2010 in MS 2001 | Leave a comment

We described the graphs of f(x-a), f(ax), f(x)+a, af(x), for a\in\mathbb{R}, in terms of the the graph of f(x) (assuming we know everything about the graph of f(x)).

We defined the quadratic function and what a root of a function is. We derived the familiar x=-b\pm\dots equation for the roots of the quadratic, and explained how the b^2-4ac term determines the nature of the roots. We noted the inherent symmetry of the quadratic, and how if we know the graph of f(x)=x^2 we have a rough idea of the shape of a quadratic by looking at the sign of the squared term.

We defined a polynomial function – and noted that the quadratic function is an example of one. We wrote down the Factor Theorem (without proof – a proof is given on this webpage). We wrote down the Fundamental Theorem of Algebra (whose proof is beyond the scope of the course). We defined a rational function and described a method of solving inequalities involving rational functions (although we made an assumption about polynomials – namely that they are continuous – this assumption will be shown to be true in the next few weeks).

We defined the absolute value function and proved some of its properties. Finally we described the sets \{x\in\mathbb{R}:|x|<\delta\} and \{x\in\mathbb{R}:|x-a|<\delta\}.

Exercises

Q.4-6 and 7(i) from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

Q. 1-2 from Problems

From class:

Let x\in\mathbb{R}, x< 0. Prove that

\left|\frac{1}{x}\right|=\frac{1}{|x|}

MS 2001: Announcement 2 (Tutorial Clashes)

September 29, 2010 in MS 2001 | Leave a comment

If you have a clash with Tuesday’s tutorial – which starts next Tuesday (1-2, G 04) please  email me ASAP at jippo@campus.ie. At present four students have contacted me. If there are only four students I will accomodate them seperately and have a ready made room available in the Western Gateway Builiding.

If you come to me after not responding to repeated announcments, emails and announcements on the webpage I may not be as predisposed to helping. If there are only four students then this second tutorial will be open to them only.

If you do have a clash with Tuesday’s tutorial, please email me with an indication of which out of the following times are NOT free on your timetable:

Monday        2-3
Thursday      3-4
Friday            11-12

MS 2001: Week 1

September 27, 2010 in MS 2001 | Leave a comment

After introductions, we considered the type of questions that will be addressed in MS 2001. Some examples were:

  1. What is the slope of a function f(x)?
  2. How can we find the local maxima and minima of f(x)?
  3. Which two numbers add to 20 and have greatest product?
  4. If a farmer has 100 m of fencing and wants to fence a rectangular field beside a river; what dimensions confer the largest area?
  5. Which rectangle of a given area (permimeter) has the smallest perimeter (greatest area)?
  6. What is the distance from a point to a line?
  7. Sketch

\frac{2x^2+x+1}{x+1}

8.   What is \sqrt{5}? What are the roots of x^3-x+1

We stated that we knew how to answer these questions from MS 1001 but that we were going to develop the axiomatic and rigourous theory behind these answers. We discussed the axiomatic model. We proceed by writing down a set of axioms and deducing the conclusions from these.

As an example we wrote down the field axioms of the real numbers. From these we deduced that x\times 0=0 for all x\in\mathbb{R}. This was the first example of a result that we don’t have to assume: it follows from the axioms. We stated that the “rules” of algebra are the consequences of these axioms. We wouldn’t revisit this ground and the “rules” of algebra are assumed in this module.

We wrote down the axioms of inequality and derived some familiar properties of inequalities from these; including 1>0. We introduced some notation for intervals. Finally we defined what an even and an odd function is. We also defined what it means for a function to be increasing or decreasing. Finally we gave the example of constant function – a function that is even and both increasing and decreasing.

Exercises

Q.1-3 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

From the notes:

Let a\in\mathbb{R}, a\neq 0. Prove that if a>0  then 1/a>0.

MS 2001: Announcement 1

September 22, 2010 in MS 2001 | Leave a comment

For the student who was enquiring which modules he was to attend please note there are actually six maths studies modules. I am unaware of how many or which modules you registered for. Please see sit.ucc.ie for details or contact the registration office.

At the moment I am aware of three Irish students and two English students who have a clash with this module. Please email me ASAP if you have a timetable clash (even if it is the same clash as one of these).

For those students studying GA 2014 there is an unfortunate clash with Monday’s lecture. As it stands, for those taking Maths Studies, MS 2001 is a compulsory rather than an elective module. In contrast GA 2014 is an elective module (for some students at least the choice is 2/6 for Irish). Typically the onus is on the student to select elective modules which do not clash with their compulsory modules. I consulted with the Maths Department in order to find an alternate solution (i.e. a different lecture time for Monday’s MS 2001), however none was forthcoming. As it stands you will have to change your Irish module selection in order to avoid a clash with MS 2001. Feel free to approach the Irish department, however it may be equally difficult for them to change the lecture time. I’m sorry we couldn’t solve this problem fully but as you can appreciate my hands were tied.

For those students studying the EN module which clashes with the Tuesday tutorial please hang tight while we come to some arrangement.

MS 2001: Problems

September 20, 2010 in MS 2001 | 1 comment

Some additional problems for you (largely taken from Wills notes):

Problems

How to define ‘Greater Than’

September 20, 2010 in General, MS 2001 | Leave a comment

How can a statement like “5 is greater than 4” be quantified? Or is it just obvious? If we know anything about mathematics we know that there is no way we can assume something as obvious, there must be an axiomatical contruct that puts a rigorous meaning on “5 is greater than 4“.

The first attempt would be to say that 5=4+1 so 5 is “1 more” than 4 so must be bigger. This translates to 5-4=1: “5 is greater than 4 because 5-4 is positive“. Careful! 4=5+(-1) so 4-5=-1: “4-5 is negative“. But what does positive and negative mean? Easy? Positive is greater than zero… At this point a stronger construct is needed:

Definition: Call a set P\subset \mathbb{R} positive if for all a,b\in P

  1. a+b\in P
  2. ab\in P
  3. Given x\in \mathbb{R} either x\in P, -x\in P or x=0

If we think carefully, this definition concurs exactly with that of the naive notion of positive. So we can say that “5 is greater than 4 because 5-4 is positive.”

Definition: Given a,b\in\mathbb{R},  a is said to be greater than b, a>b, if a-b is positive.

MS 2001: General Information

September 20, 2010 in MS 2001 | Leave a comment

Lecturer: Mr. J.P. McCarthy

Office: Mathematics Research, Western Gateway Building
Meetings by appointment via email only.

Email: jippo@campus.ie

Read the rest of this entry »

An inductive proof of the Factor Theorem

September 8, 2010 in Leaving Cert, MS 2001 | 2 comments

At Leaving Cert you are only required to prove the Factor Theorem for cubics. This is a more general proof.

The Factor Theorem is an important theorem in the factorisation of polynomials. When (x-k) is a factor of a polynomial p(x) then p(x)=(x-k)q(x) for some polynomial q(x) and clearly k is a root. In fact the converse is also true. Most proofs rely on the division algorithm and the remainder theorem; here a proof using strong induction on the degree of the polynomial is used. See Hungerford, T.W., (1997), Abstract Algebra: An Introduction. Brooks-Cole: U.S.A for the standard proof (this reference also describes strong induction).

Factor Theorem
k\in\mathbb{C} is a root of a polynomial p(x) if and only if (x-k) is a factor: p(x)=(x-k)q(x) where q(x) is a polynomial.

Proof:
Suppose deg p(x)=1. Then p(x)=ax+b and p(k)=ak+b=0\Rightarrow k=-b/a. Clearly p(x)=(x-(-b/a))a and q(x)=a is a polynomial (of degree 0).

Suppose for all polynomials of degree less than or equal to n-1, that k a root implies (x-k) a factor.  Let p(x)=\sum_ia_ix^i be a polynomial of degree n and assume k\in\mathbb{C} a root. Now

p(x)-p(k)=\sum_{i=0}^na_ix^i-\sum_{i=0}^na_ik^i

\underset{p(k)=0}{\Rightarrow} p(x)=\sum_{i=1}^n a_i(x^i-k^i)

Now each x^i-k^i for i=1,\dots,n-1 is a polynomial of degree less than or equal to n-1, with a root x=k, hence (x-k) is a factor of each. Thence x^i-k^i=(x-k)q_i(x) where q_i(x) is a polynomial:

p(x)=\sum_{i=1}^{n-1}a_i(x-k)q_i(x)+a_n(x^n-k^n)

Now x^n-k^n=(x-k)x^{n-1}+kx^{n-1}-k^n but kx^{n-1}-k^n is a polynomial of degree n-1 with root k and hence (x-k) is a factor:

x^n-k^n=(x-k)x^{n-1}+(x-k)h(x)

Hence

p(x)=(x-k)\lbrack x^{n-1}+h(x)+\sum_{i=1}^{n-1}a_iq_i(x)\rbrack

 

Hence (x-k) a factor \bullet

MS 2001: Useful Links

July 10, 2010 in MS 2001 | Leave a comment

Stephen Wills MS 2001 homepage: http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/MS2001.html

Notes from there: http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/MS2001_notes.pdf

Past Exam Papers: http://booleweb.ucc.ie/ExamPapers/maths_studies.html