Category Archive

You are currently browsing the category archive for the ‘MATH6055’ category.

How does a Curve have an Equation?

November 1, 2017 in A1 Leaving Cert Maths, General, MATH6015, MATH6040, MATH6055 | 8 comments

In school, we learn how a line has an equation… and a circle has an equation… what does this mean?

The short answer is

points (x_0,y_0) on curve \longleftrightarrow solutions (x_0,y_0) of equation

however this note explains all of this from first principles, with a particular emphasis on the set-theoretic fundamentals.

Set Theory

A set is a collection of objects. The objects of a set are referred to as the elements or members and if we can list the elements we include them in curly-brackets. For example, call by S the set of whole numbers (strictly) between two and nine. This set is denoted by

S=\{3,4,5,6,7,8\}.

We indicate that an object x is an element of a set X by writing x\in X, said, x in X or x is an element of X. We use the symbol \not\in to indicate non-membership. For example, 2\not\in S.

Elements are not duplicated and the order doesn’t matter. For example:

\{x,x,y\}=\{x,y\}=\{y,x\}.

Read the rest of this entry »

MATH6055: Winter 2017, Week 7

October 25, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Catch-Up Material

If you haven’t watched these at this stage you really should:

Week 7

Disrupted by the Friday off, we did two lectures on Functions, looking at the definition of a function. We looked at what it means for a function to be 1-1, onto, and bijective. We looked at the composition of two functions as well as the concept of an inverse function.

Week 8

In Week 8, we lose another lecture with the bank holiday, but we should have an opportunity to better understand the function concepts after looking at arrow diagrams and the graph of a function.

Assessment 1

Results have been emailed to you. I will send ye on a copy in a week or two.

Assessment 2

Will be held in Week 11.

Study

Some students need to do extra work outside tutorials. Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

 

 

MATH6055: Winter 2017, Week 6

October 18, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 6

The storm interrupted us, costing us two lectures. You must watch the following videos to catch up: Inclusion-Exclusion Principle and the properties of relations. We have/had our test this Friday. A sample has been handed out.

Week 7

In Week 7 w will start the chapter on Functions.

Assessment 1

Test 1 will be held at 09:00, Friday 20 October in Week 6. Everything up to but not including relations is examinable. I have emailed ye a sample, a hard copy of which I gave out in tutorials.

Study

We are probably all going to have to put in some extra study before the test. Please try and find some extra time to try exercises, particularly with any new or harder material such as logs, cartesian products, and power sets. Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

 

 

MATH6055: Winter 2017, Week 5

October 13, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 5

In Week 5 we looked more at sets and explored Cartesian Products We introduced relations. Then on Friday we looked harder at elements, subsets, power sets and the empty set.

Week 6

In Week 6 we will look at the Inclusion-Exclusion Principle and the properties of relations. We should start the chapter on Functions and then have our Test 1 on Friday.

Assessment 1

Test 1 will be held at 09:00, Friday 20 October in Week 6. Everything up to but not including relations is examinable. I have emailed ye a sample, a hard copy of which I will provide you with next week.

Study

We are probably all going to have to put in some extra study before the test. Please try and find some extra time to try exercises, particularly with any new or harder material such as logs, cartesian products, and power sets. We all have only one hour of tutorials before the test and finding two  hours to do exercises will make a big difference. Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

 

 

MATH6055: Winter 2017, Week 4

October 6, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 4

In Week 4 we finished looking at logarithms and started the second chapter on Sets and Relations. I tried and failed to tell you about the barber paradox.

Week 5

In Week 5 we will look more at sets and set identities, and explore Cartesian Products and perhaps introduce relations.

Assessment 1

Test 1 will be held at 09:00, Friday 20 October in Week 6. Note this is different to your assessment schedule provided by the head of department, Tim Horgan. Expect a sample this time next week. Only material from Weeks 1-5 will be examined.

Study

We are probably all going to have to put in some extra study before the test. Please try and find some extra time to try exercises, particularly with any new or harder material such as logs. We all have two hours of tutorials before the test and finding two extra hours to do exercises will make a big difference. Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

 

 

MATH6055: Winter 2017, Week 3

September 29, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 3

In Week 3 we finished looking at quadratic equations and exponents, and started looking at logarithms.

Week 4

In Week 4 we will finish looking at logarithms and start the second chapter on Sets and Relations.

Assessment 1

Test 1 will be held at 09:00, Friday 20 October in Week 6. Note this is different to your assessment schedule provided by the head of department, Tim Horgan. Expect a sample this time next week. Only material from Weeks 1-5 will be examined.

Study

We are probably all going to have to put in some extra study before the test. Please try and find some extra time to try exercises, particularly with any new or harder material such as logs. We all have two hours of tutorials before the test and finding two extra hours to do exercises will make a big difference. Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

 

 

MATH6055: Winter 2017, Week 2

September 22, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 2

We finished looking at algebra and ‘easy’ equations. We started looking then at quadratic equations.

Week 3

In Week 3 we will finish looking at quadratic equations and look at exponents and logarithms.

 

Assessment 1

I said last week that Assessment 1 will be on  Friday 13 October (Week 5). I am not so sure now: it might be pushed into Week 6. Expect a sample, an proper notice, at least a week beforehand.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

 

 

Quadratics: A First Look

September 20, 2017 in A1 Leaving Cert Maths, General, Grinds, MATH6000, MATH6014, MATH6015, MATH6037, MATH6040, MATH6055, MATH7021 | 1 comment

Quadratics are ubiquitous in mathematics. For the purposes of this piece a quadratic is a real-valued function q:\mathbb{R}\rightarrow \mathbb{R} of the form

q(x)=ax^2+bx+c,

where a,\,b,\,c\in \mathbb{R} such that a\neq 0. There is a little bit more to be said — particularly about the differences between a quadratic and a quadratic function but for those this piece is addressed to (third level: non-maths; all second level), the distinction is unimportant.

Geometry

The basic object we study is the square function, s:\mathbb{R}\rightarrow \mathbb{R}, x\mapsto x^2:

graph1

All quadratics look similar to x^2. If a>0 then the quadratic has this \bigcup geometry. Otherwise it looks like y=-x^2 and has \bigcap geometry

The geometry dictates that quadratics can have either zero, one or two real roots. A root of a function is an input x such that f(x)=0. As the graph of a function is of the form y=f(x), roots are such that y=f(x)=0\Rightarrow y=0, that is where the graph cuts the x-axis. With the geometry of quadratics they can cut the x-axis no times, once (like s(x)=x^2), or twice.

Read the rest of this entry »

MATH6055: Winter 2017, Week 1

September 15, 2017 in MATH6055 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 1

We studied number and began to take a look at basic algebra.

Week 2

In Week 2 we will look further into algebra and look at equations.

Tutorials

Tutorials start properly in Week 2.

  • COMP1C-X: Tuesday at 15:00 in B241L
  • COMP1C-Y: Wednesday at 12:00 in B225

Read the rest of this entry »

Why Can’t I Divide by Zero?

September 13, 2017 in A1 Leaving Cert Maths, General, Grinds, MATH6000, MATH6015, MATH6037, MATH6038, MATH6040, MATH6055, MATH7016, MATH7019, MATH7021 | 3 comments

There are a number of ways of explaining why you cannot divide by zero. Here are my two favourites.

Any Set of Numbers Collapses to a Single Number

How old are you? Zero years old.

How tall are you? Zero metres old.

How many teeth do you have? Zero.

How many Superbowls has Tom Brady won? Zero

Yep, if you allow division by zero you only end up with one number to measure everything with.

Read the rest of this entry »