## Assignment 1

Assignment 1 has a hand-in time and date of 12:00 Friday 1 March (Week 5). Submit in class or to A283.

Read the P.51 and P.52 instructions carefully. You will be submitting an Excel file, 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 first 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. I recommend that you start ASAP.

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.

Regarding Q. 1.3.5, Assignment 1, on P.62 of the manual. The intention with Q. 1.3.5 (b) really is for you to engage in some problem solving skills to come up with a clever way of implementing the Jacobi Method in Excel.
It should still be doable by hand but if it takes a large number of iterations to converge (to two significant figures), Excel is far more suitable.
It is possible that it could take a small number of iterations to converge to two significant figures (say two or three iterations) — which is no problem by hand — but potentially it could take more (at least six). I don’t really want people spending loads of time doing iterations by hand, so I will give 3/4 marks for part (b) if you do six iterations by hand. If you want to keep going – by hand – until convergence (to two significant figures) you can of course get the 4/4 marks – but you need to ask yourself is it worth your time to keep going for the sake of one mark (out of 60… out of 15% —- that is 0.25% of your final grade).
If it converges with fewer than six iterations then happy days for you, you can get 4/4.
If it doesn’t, you might be better off trying to come up with a way of doing the question in Excel if you really want all the marks.
You can still answer part (c) if you do six iterations and do not yet have convergence.

## Test 1

Test 1, worth 15%, takes place from 19:00 to 20:05 sharp, Tuesday 26 February in the usual lecture venue. There is a sample on P.45 of the notes to give you an idea of the length and layout only.

Almost everything in Chapter 1 is examinable. This means:

• P.23, Q.1-10
• P.32, Q.1-8 [Q.9 is too long and Q.10 is not examinable]
• P.39, Q.1-11

Additional practise questions may be found by looking at past 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. 41-44.

If you want questions answered you have two options:

• ask me questions via email, perhaps with a photo to show your work
• ask me questions via the comment function on this website

## Week 4

We had Concept MCQ about vectors and then we started looking at Chapter 2: Matrices. We did some examples of matrix arithmetic and looked at Matrix Inverses — “dividing” for Matrices. This will allow us to solve matrix equations. Here find a note that answers the question: why do we multiply matrices like we do?

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6 (5 & 8 March) in your usual lab time. You will not be allowed any resources – but the library of code (p.148) and these formulae will appear on the assessment:

The following is the proposed layout of the assessment:

### Q. 1: Numerical Solution of Initial Value Problem [80%]

Examples of initial value problems that might be arise include:

• Damping

$\displaystyle \frac{dv}{dt}=-\frac{\lambda}{m}v(t)$;           $v(0)=u$

• The motion of a free-falling body subject to quadratic drag:

$\displaystyle \frac{dv}{dt}=g-\frac{c}{m}v(t)^2$;           $v(0)=u$

• Newton Cooling

$\displaystyle \frac{d\theta}{dt}=-k\cdot (\theta(t)-\theta_R)$;           $\theta(0)=\theta_0$

• The charge on a capacitor

$\displaystyle \frac{dq}{dt}=\frac{E}{R}-\frac{1}{RC}q(t)$;           $q(0)=0$

Students have a choice of how to answer this problem:

• The full, 80 Marks are going for a VBA Heun’s Method implementation (like Lab 3).
• An Euler Method implementation (like Lab 2), gets a maximum of 60 Marks.

You will be asked to write a program that takes as input all the problem parameters, perhaps some initial conditions, a step-size, and a final time, and implements Heun’s Method (or possibly Euler’s Method): similar to Exercise 1 on p. 122 (except possibly implementing Heun’s Method) and also Exercise 1 on p.128 (except without the “conditional” derivative).

If you can write programs for each of the four initial value problems above you will be in absolutely great shape for this assessment.

### Q. 2: Using your Program [20%]

You will then be asked to use your program to answer a number of questions about your model. For example, assuming Heun’s Method is used, consider the initial value problem (3.7) on p. 119.

1. Given, $v_0=0.2$, $m=3$, $\lambda=1.5$, $h=0.01$, approximate $v(0.3)$.
2. Given, $v_0=0.4$, $m=30$, $\lambda=1.5$, $h=0.1$, investigate the behaviour of $v(t)$ for large $t$.
3. Given $v_0=0.2$, $m=0.1$, $\lambda=1.5$, $h=0.5$, $T=10$, run the Heun program. Comment on the behaviour of $v(t)$. Run the same program except with $h=0.05$. Comment on the behaviour of $v(t)$.
4. Given, $v_0=0$, $m=3$, $\lambda=1.5$, $h=0.1$, $T=2$, run the Heun program. Comment on the behaviour of $v(t)$.

## Week 4

We finished off a Three Term Taylor Method example and spoke again about Heun’s Method.

We also introduced second order differential equations and saw how to attack them numerically. In particular we looked at a real pendulum.

In VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

## Week 5

In the morning class we will finish looking at second order differential equations.

In the afternoon we will begin a quick study of Runge-Kutta Methods.

In VBA we have MCQ III and look at Lab 4, on Second Order Differential Equations.

## Assessment

The following is a proposed assessment schedule:

1. Week 6, 20% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 20 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 20% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 40% Written Assessment(s), Based on Weeks 1-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Ungraded Concept MCQ League Table

To add a bit of interest to the Ungraded Concept MCQs, I will keep a league table.

Unless you are excelling, you are identified by the last five digits of your student number. AW is the number of attendance warnings received.

## Assignment 1

Assignment 1 has a hand-in time and date of 12:00 Friday 1 March (Week 5).

Read the P.51 and P.52 instructions carefully. You will be submitting an Excel file, 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 first 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. I recommend that you start ASAP.

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 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 also had about ten minutes of tutorial time with this.

On Wednesday we showed the following video, which shows how the approximations to the solution iterate:

## Test 1

Test 1, worth 15%, takes place from 19:00 to 20:05 sharp, Tuesday 26 February in the usual lecture venue. There is a sample on P.45 of the notes to give you an idea of the length and layout only.

Almost everything in Chapter 1 is examinable. This means:

• P.23, Q.1-10
• P.32, Q.1-8 [Q.9 is too long and Q.10 is not examinable]
• P.39, Q.1-11

Additional practise questions may be found by looking at past 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. 41-44.

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

We start working with the cross product and looked at the applications of vectors to work and moments.

We had no tutorial time but have some active learning time with a Concept MCQ: Vector or Scalar?

## Week 3

We backtracked a little and found the Maclaurin Series

$\displaystyle \ln(\sec x)\approx \frac{1}{2}x^2+\frac{1}{12}x^4$.

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 could avoid implicit differentiation by looking at Huen’s Method, which is an adjustment of Euler’s Method in that it uses lines.

In VBA we finished off the Euler Method Lab 2 and looked at P. 122, Exercise 1. The first group also started P. 123, Exercise 2, but the later groups instead used the time for some theory revision.

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 12:00  Friday 1 March (Week 5).

Read the P.51 and P.52 instructions carefully. You will be submitting an Excel file, 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 rst 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 Q. 1.3.5. 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 started Monday with some Gaussian Elimination Concept MCQs. We probably should have waited until we finished the Gaussian Elimination examples, which we did straight afterwards.

Wednesday am we had some tutorial time — and then Wednesday pm began to look at applications of linear systems to traffic and pipe flow.

On Thursday we finished the section on pipe flow and had a little more 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 hopefully have another Concept MCQ. Wednesday pm we will have more tutorial time (and do the Concept MCQ if necessary).

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.

## Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

## Student Resources

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.

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here: there is some availability after 16:00 on Mondays and Tuesdays.

## Week 2

We finished the dot product examples.

Then we had extensive tutorial time in which we worked on:

• P.23, Q.1-10
• P.45, Sample Test, Q. 1, 2, 4
• P. 203, Winter 2018, Q. 1 (a), (b)

Then we started talking about the cross product.

## Homework Exercises

If you do any of the suggested exercises you can give them to me for correction. Please feel free to ask me questions about the exercises via email or even better on this webpage.

• P.23, Q.1-10
• P.45, Sample Test, Q. 1, 2, 4
• P. 203, Winter 2018, Q. 1 (a), (b)
• I also handed out an older exam paper: you might be able to do some of Q. 1 (a).

## Week 3

We will start working with the cross product and begin to look at the applications of vectors to work and moments. There will be no tutorial time.

## Test 1

If we finish the Vectors chapter the test will be in Week 5: 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.

## CIT Mathematics Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

## Student Resources

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 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.

In VBA we started programming the Euler Method to solve the problem of a damper. We did MCQ 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.

## Timetable

The Geotech Tutorial Groups are determined by whether or not you are taking MATH7021.

If you are taking MATH7021 your Geotech tutorial is Wednesday 12:00 and you attend the 14:00 MATH7021 class in A243L.

## 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 were a little disrupted by snow on Wednesday, effectively costing us 0.5 – 1 hours.

## Week 2

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

## Assignment 1 – Warning

Assignment 1 has a hand-in date of  Friday 1 March (Week 5). More information next week.

## Study

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

## CIT Mathematics Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.