## Week 3

We then did some further study on the Euler Method. The global error with the Euler Method is $\mathcal{O}(h)$ and we need to reduce this by coming up with a better method or adjusting the Euler Method.

We looked at the Three Term Taylor Method as a better method. To employ the Three Term Taylor Method we need implicit differentiation, which means more pen-and-paper work.

We also looked at calculating a Maclaurin Series

In VBA we finished off the Euler Method Lab 2. Ideally everyone should have done up to p.125 (note in questions 2 and 3 that $u$ should be $0.2$.

Some students did or started Exercise 1 and/or Exercise 2 on p. 126.

## MCQ League

Unless you are excelling, you are identified by the last five digits of your student number.

Please ask questions in the lab about questions you have gotten wrong. Students in red appear to not have a good handle on the material and should consider putting in extra time outside class in doing exercises (in the manuals).

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.

## Assignment 1

Assignment 1 has a hand-in time and date of 11:00 Thursday 27 February (Week 5).

See the Week 2 Summary for more.

## Week 3

On Monday, we looked at applications to temperature distribution, where the Jacobi Method is used to find approximate solutions to a diagonally dominant linear system. We had about ten minutes of tutorial time.

To understand more what is going on watch the following video, which shows how the approximations to the solution iterate:

On Wednesday we had two tutorials on Chapter 1, including a second Concept MCQ.

On Thursday we started work on Chapter 2 — the method of undetermined coefficients for solving linear odes.

## Week 4

We will continue our work on Chapter 2 on Monday and Wednesday AM, then we will have two tutorials Wednesday PM and Thursday.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Test 1

Test 1, worth 15%, takes place from 19:00 to 20:05 sharp, Tuesday 25 February in the usual lecture venue. There is a sample on P.53 of the notes to give you an idea of the length and layout only.

Everything in Chapter 1 is examinable. Additional practise questions may be found by looking at past MATH6040 exam papers (usually vectors are Q. 1, sometimes Q. 2).

You will want to be familiar with all the concepts in the Vector Summary, P. 49-52.

If you want questions answered you have three options:

• hand me up written work next week, which I will correct, scan, and email back to you
• ask me questions via email, perhaps with a photo to show your work
• ask me questions via the comment function on this website

## Week 3

In Week 3 will found out how to calculate a vector product, and we looked at the applications of vectors to work and moments

We managed about 30 minutes of tutorial time.

## Week 2

In pre-recorded videos, we developed the Euler Method for approximating the solution of differential equations. As we will need Taylor Series to analyse the error in this approximation — and improve Euler’s Method — we started looking at that. We kind of rushed it, but we used it to analyse the Euler Method.

If you have not yet watched the video lectures please do so before Tuesday.

In VBA we started programming the Euler Method to solve the problem of a damper. We did MCQ 1. If you were missing you should do the two Excel (not VBA — you can do that next week) exercises on p.121-123.

## MCQ League

Unless you are excelling, you are identified by the last five digits of your student number.

Please ask questions in the lab about questions you have gotten wrong. Students in red appear to not have a good handle on the material and should consider putting in extra time outside class in doing exercises (in the manuals).

The standings after Week 2:

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.

## Assignment 1

Assignment 1 has a hand-in time and date of 11:00 Thursday 27 February (Week 5).

Read the P.54 and P.55 instructions carefully. You will be submitting an Excel file, to Canvas, and written work (including a print out of your Excel work).

Note in particular:

• Work submitted after the deadline will be assigned a mark of ZERO. Hand up whatever you have on time.
• Only Partial Pivoting has to be done using Excel.
• Note that if you are doing Gaussian Elimination by hand you must use exact fractions and square roots rather a decimal approximation.
• I advise that you do the questions out roughly because small mistakes are inevitable.

The files you need to complete this assignment have been emailed to you. If you don’t want to calculate your $c_i$ and $P$ they are calculated in MATH7021A1 – Student Data.

We have now covered enough in class for you to do all of the Assignment except for Problem F. After Monday we will have enough covered in class.

WARNING!

This gives a good opportunity for collaboration but remember collaboration does not mean one student solving the problem and everyone else copying that student’s work. I demand originality of presentation here and you should at least understand what you hand up. If you are unsure of what I mean by this please email me immediately as if I have students who have clearly copied the answer word-for-word from another student they will all be sharing the marks.

Start early so you have enough time to complete the assignment properly and get good learning from it.

THIS IS A LEARNING ACTIVITY NOT JUST A GRADED ACTIVITY. THE CHAPTER ONE EXAM QUESTION IS WORTH 24.5% OF YOUR FINAL GRADE WHILE THIS ASSIGNMENT IS WORTH JUST 15%. THINK ABOUT WHAT THIS MEANS.

## Week 2

We finished the Gaussian Elimination examples on Monday — then had some tutorial time Wednesday AM — and then Wednesday PM we will began to look at applications of linear systems to traffic and pipe flow.

We started Thursday with some Gaussian Elimination Concept MCQs.

Then we finished the section on pipe flow and had 10 minutes tutorial time.

## Week 3

On Monday, we will look at applications to temperature distribution, where the Jacobi Method is used to find approximate solutions to a diagonally dominant linear system.

Wednesday am we will finish looking at Chapter 1, and possibly have another Concept MCQ. Wednesday pm we will have more tutorial time.

On Thursday we will start work on Chapter 2 — the method of undetermined coefficients for solving linear odes.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Week 2

In Week 2 we finished talking about the scalar product before doing some tutorial work on the p.29 exercises.

We started talking about the vector product, but we did not show how to calculate it.

## Homework/Study

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises via email, or even better on this webpage.

I would invite you to complete the exercises we started in class:

• P.29, Q.1-11

In addition, you can hand up your work in class and I will have it correct for the week after. You may need the formula that the area of a triangle is equal to $\displaystyle \frac12 ab\sin C$.

## Week 3

In Week 3 we will find out how to calculate a vector product, and we will look at the applications of vectors to work and moments

We might manage a little tutorial time.

## Test 1

If we finish the Vectors chapter in Week 3 the test will be in Week 5 (this is very likely): otherwise we will push this out to Week 6. Official notice will be given in Week 3 (or Week 4 if necessary). There is a sample test in the notes.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

If you are a little worried about your maths this semester, perhaps after the Diagnostic Test or in general, I would just like to remind you about the Academic Learning Centre. Next week, some students will receive emails detailing areas of maths that they should brush up on. The timetable is here: there is some availability after 17:00 on Mondays and Tuesdays.

## Student Resources

We tell four tales of De Morgan.

In each case we have something that looks like AND, something that looks like OR, and something that looks like NOT.

## Sets

### The Collection of Objects

Consider a universe of discourse/universal set/ambient set  $U$. When talking about people this might be the collection of all people. When talking about natural numbers this might be the set $\mathbb{N}$. When talking about real numbers this might be the set $\mathbb{R}$. When talking about curves it might be the set of subsets of the plane, $\mathcal{P}(\mathbb{R}\times \mathbb{R})$, etc.

The collection of objects in this case is the set of subsets of $U$, denoted $\mathcal{P}(U)$.

Suppose, for the purposes of illustration, that

$U=\{1,2,3,4,5\}$.

Consider the subsets $A=\{2,3,5\}$, and $B=\{1,3,5\}$ .

in the obvious way.

### AND

Note that two objects are contained both in $A$ AND in $B$. We call the set of such objects the intersection of $A$ AND $B$, $A\cap B$:

$A\cap B=\{3,5\}$.

We can represent the ambient set $U$, as well as the sets $A$ and $B$ — and the fact that they intersect — using a Venn Diagram:

We can demonstrate for a general $A$ and $B$ ‘where’ the intersection is:

## Week 1

In Week 1, by briefly looking at a number of examples (many of which we have seen before), we had a review of some central ideas from approximation theory such as approximation, measurement error, accuracy & precision, iteration, convergence, meshing, error, etc.

We started looking at where ordinary differential equations come into Engineering.

In VBA we had a quick review lab, focussing on plotting data, command buttons, message boxes, input boxes, If-statements and do-loops.

If you have not completed Lab 1 (p.116), I recommend that you do at least up to the first Do-Loop exercise.

## MCQ League

To add a bit of interest to the Ungraded Concept MCQs, I will keep a league table. There are cash prizes (€20, €10, €5) for first, second, and third at the end of the league.

Unless you are excelling, you are identified by the last five digits of your student number.

The standings after Week 1:

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 started the first chapter on Linear Algebra. Essentially, for us, simultaneous equations. We looked at Gaussian Elimination including Partial Pivoting, which is required in the presence of rounding. We looked at three examples, and had a tutorial on Thursday.

## Week 2

We will finish the Gaussian Elimination examples on Monday — then have some tutorial time Wednesday AM — and then Wednesday PM we will begin to look at applications of linear systems to traffic and pipe flow.

## Assignment 1

Assignment 1 has a hand-in date of  Thursday 27 February (Week 5). More information next week once the class list has settled down.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Strike!

There is 24 hour strike this Tuesday, and so we will have our Week 2 class on Wednesday 5 February, 19:00-22:00, in the same room, B188.

## Diagnostic Test

If you haven’t already, you are invited to take the following ‘Diagnostic Test’:

click here

This ‘Test’ does not go towards your grade, but allows me to give you some feedback on where you are in terms of material you have seen before that will be used in this module.

## Manuals

The manuals are available in the Copy Centre and must be purchased before next Wednesday’s class.