On Parallel and Perpendicular Lines

June 5, 2018 in A1 Leaving Cert Maths, General, Grinds, Leaving Cert Maths | Leave a comment

The purpose of this post is to briefly discuss parallelism and perpendicularity of lines in both a geometric and algebraic setting.

Lines

What is a line? In Euclidean Geometry we usually don’t define a line and instead call it a primitive object (the properties of lines are then determined by the axioms which refer to them). If instead points and line segments – defined by pairs of points P,Q – [PQ] are taken as the primitive objects, the following might define lines:

Geometric Definition Candidate

A line, \ell, is a set of points with the property that for each pair of points in the line, P,Q\in \ell,

[PQ]\cap \ell=[PQ].

In terms of a picture this just says that when you have a line, that if you take two points in the line (the language in comes from set theory), that the line segment is a subset of the line:

line

Exercise:

Why is this objectively not a good definition of a line.

Once we move into Cartesian\Coordinate Geometry we can perhaps do a similar trick. We can use line segments, and their lengths to define slope, (slope = rise over run) and then define a line as follows:

Algebraic Definition Candidate

A line, \ell, is a set of points such that for all pairs of distinct points P,Q\in\ell, the slope is a constant.

This means that if you take two pairs of distinct points in a line \ell, and then calculate the slopes between them, you get the same answer, and therefore it makes sense to talk about the slope of a line, m.

line2

This definition, however, has exactly the same problem as the previous. The definition we use isn’t too important but I do want to use a definition that considers the line a set of points.

The Equation of a Line

We can use such a definition to derive the equation of a line ‘formula’ for a line of slope m containing a point (x_1,y_1).

Suppose first of all that we have an x\text{-}y axis and a point P(x_1,y_1) in the line. What does it take for a second point Q(x,y) to be in the line?

line4

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MATH7021: Spring 2018, Week 12

April 30, 2018 in MATH7021 | 4 comments

Student Feedback

If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.

 

Week 12

On Monday we finished the module by looking at triple integrals.

The Wednesday 09:00 lecture will be a tutorial. In this class and your usual tutorial we will look at the P.182, P. 192, P. 163, & P.116 exercises. If these are completed you will be recommended to revise either by trying Chapter 1 & 2 exercises or perhaps by looking at the Summer 2017 paper.

As next Monday is a bank holiday, we will begin the Summer 2017 Paper (in your notes) revision on Thursday.

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MATH7021: Spring 2018, Week 11

April 26, 2018 in MATH7021 | Leave a comment

Student Feedback

If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.

Assignment 2

Assignment 2 has been corrected and your results emailed to you.

Week 11

We spent three lectures looking at double integrals, in particular their application to second moments of area. We set of integration over a cylinder by looking at polar coordinates.

In the Wednesday tutorial we worked on the p. 163 (primarily) and p. 182 exercises.

Week 12

On Monday we will finish the module by looking at triple integrals.

We will therefore have three tutorials where we will look at the P.182, P. 192, P. 163, & P.116 exercises. If these are completed you will be recommended to revise either by trying Chapter 1 & 2 exercises or perhaps by looking at the Summer 2017 paper.

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MATH6038: Spring 2018, Week 11

April 26, 2018 in MATH6038 | 2 comments

Student Feedback

If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.

Week 11

We looked at the normal distribution.

In Maple looked at Binomial and Poisson random variables.

Week 12 – Maple Test

The Maple Test should take no more than one hour but I am giving ye extra time. For various reasons, I have decided to schedule next week’s class as:

  • 19:05 – 20:25: Sampling and Control Charts
  • 20:25 – 20:45: Break
  • 20:45 – 22:00: Maple Test

The Maple Test will be open book. You have a sample Maple Test (this is also in the notes) with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2 (Lab 4).

We will speak about sampling in more detail and also introduce control charts.

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Some Leaving Cert Applied Maths Notes

April 23, 2018 in Leaving Cert Applied Maths | Leave a comment

For those planning on focusing on questions one to five and ten:

Applied Maths some Notes

MATH6038: Spring 2018, Week 10

April 19, 2018 in MATH6038 | Leave a comment

Week 10

We did a lot of probability — reliability block diagrams, the binomial distribution, the Poisson distribution.

Those who missed the class I recorded some of it here.

Ironically I recorded the same lectures in 2016 as you can see here.

 

Week 11 – Maple Night

We will look at the normal distribution and talk about sampling.

In Maple we will look at Binomial and Poisson random variables.

Week 12 – Maple Test

We will speak about sampling in more detail and also introduce control charts.

The Maple Test will be open book. You have a sample Maple Test with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2.

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MATH7021: Spring 2018, Week 10

April 19, 2018 in MATH7021 | 6 comments

Assignment 2

Assignment 2 has a hand-in date of 17:00 23 April: the Monday of Week 11. Assignment 2 is in the manual, P. 149..

Week 10

The Monday lecture was another tutorial on the P.116 exercises. In the Wednesday lectures we worked on systems of differential equations. In the Thursday lecture we worked on systems of differential equations exercises.

In the Wednesday tutorial we continued with the full Laplace Transform questions.

Week 11

In the Monday and Wednesday lectures we will make a start on the final chapter by looking at double integrals.

In the Wednesday tutorial we will work on the p. 163 and p. 182 exercises. This work will continue on Thursday.

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MATH7016: Spring 2018, Week 10

April 17, 2018 in MATH7016 | Leave a comment

Week 10

We looked at finite differences for the Heat Equation. This completes the examinable written material.

In VBA we implemented same.

Week 11 — 2nd 20% VBA Assessment

In the 09:00 class we will have a revision session, geared towards the 20% VBA Assessment 2. This will look at the 20% VBA Assessment 2 Tutorial Sheet It might therefore be a good idea to go through this before next week.

In the 12:00 class we will have a revision session, geared towards the 40% Written Assessment 2. This will look at the 40% Written Assessment 2 Tutorial Sheet It might therefore be a good idea to go through this before next week.

Formulae will be provided in the VBA 2 Assessment.

To understand how your student numbers generate constants (see below) see this VBA Test 2 from last year (do not read this as a sample – it included e.g. the Heat Equation which you will not be examined on (in VBA) and the Laplace’s Equation might be slightly simpler than what ye will have).

See last week’s Weekly Summary for the Format of the VBA Assessment 2.

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MATH7021: Spring 2018, Week 9

April 12, 2018 in MATH7021 | Leave a comment

Assignment 2

Assignment 2 has a hand-in date of 17:00 23 April: the Monday of Week 11. Assignment 2 is in the manual, P. 149.

Assignment 1

If you were absent today and want to view your submission next week please email me.

Week 9

We said a few things about damped harmonic oscillators on Monday and the rest of the week was spent working on the p.116 exercises.

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MATH6038: Spring 2018, Week 9

April 12, 2018 in MATH6038 | Leave a comment

Week 9

We made a good start on probability, talking about random variables, independence, mutual exclusivity, conditional probability, and tree diagrams.

You can read about the child paradox here.

Week 10

We will have a lot of probability to do — reliability block diagrams, the binomial distribution, the Poisson distribution.

You have a sample Maple Test with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2.

We do not have enough probability done to have a Maple Lab until…

Week 11 – Maple Night

We will look at the normal distribution and talk about sampling.

Week 12 – Maple Test

We will speak about sampling in more detail and also introduce control charts.

The Maple Test will be open book and you will have already received a sample test with solutions.

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