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## VBA Assessment 1 – Results

I would hope to have these with ye by the end of Week 8.

## Written Assessment 1 – Results

I would hope to have these with ye by the end of Week 8

## Week 7

In the first lecture you sat your first written assignment.

In the second lecture we looked more at boundary value problems (in particular the Shooting Method and Goal Seek). We started talking about Finite Differences.

In VBA we looked at Runge-Kutta methods.

## Written Assessment 1

Written Assessment 1 takes place Tuesday 13 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up. There are far more things I could examine.

Roughly, everything up to p. 57 is examinable. More specifically:

### Maclaurin/Taylor Series

Examples 1 & 2 on p. 17; Q. 1-2 on p.19

### ODEs in Engineering

p.22, Example, Q. 1-4

### Euler Method

p.29, Examples 1-4; p.38, Q. 1-6, 8-9; p.48, Q. 1, 5(a)

### Three Term Taylor Method

p.35, Example; p. 36, Examples 1-2; p.38, Q. 6-7, 10-14; p.48, Q. 3

### Heun’s Method

p.43, Examples 1-2; p.48, Q. 2, 4, 5(b)

### Second Order Differential Equations

p.52, Example; p.54, Example; p. 56, Q. 1-14 (some repetition here).

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of the assessment (both are provided on the assessment paper). More information in last week’s weekly summary.

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of the assessment (both are provided on the assessment paper). More information in last week’s weekly summary.

## Written Assessment 1

Written Assessment 1 takes place Tuesday 13 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up. There are far more things I could examine.

Roughly, everything up to p. 57 is examinable. More specifically:

### Maclaurin/Taylor Series

Examples 1 & 2 on p. 17; Q. 1-2 on p.19

### ODEs in Engineering

p.22, Example, Q. 1-4

### Euler Method

p.29, Examples 1-4; p.38, Q. 1-6, 8-9; p.48, Q. 1, 5(a)

### Three Term Taylor Method

p.35, Example; p. 36, Examples 1-2; p.38, Q. 6-7, 10-14; p.48, Q. 3

### Heun’s Method

p.43, Examples 1-2; p.48, Q. 2, 4, 5(b)

### Second Order Differential Equations

p.52, Example; p.54, Example; p. 56, Q. 1-14 (some repetition here).

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.

## Assessment

Considering our progress, I have decided to swap the positions of the first and second assessments. This time last year I had Section 1.2.3 completed but have (necessarily) slowed down this year. Last year Section 1.2.3 was tested both in the first (written) assessment and in the second (VBA) assessment.

It is more important that Section 1.2.3 is tested in the written component therefore the decision to switch the assessments.

Due to this change, the information in Sections 3.6 and 3.7 is now out of date.

The following is the proposed assessment schedule:

1. Week 6, 20% First VBA Assessment, More Info Below
2. Week 7, 20 % In-Class Written Test, More Info in Week 5
3. Week 11, 20% Second VBA Assessment, More Info in Week 9
4. Week 12, 40% Written Assessment(s), More Info in Week 10

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of 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 Euler’s Method): similar to Exercise 1 on p. 114 and also Exercise 1 on p.109 (except perhaps implementing Heun’s Method).

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. 105.

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 jumped forward and looked at Heun’s Method in the 09:00 class. We went back then and looked at the Three Term Taylor Method in the afternoon. We stated in the afternoon that Heun’s Method gives the same answer as the Three Term Taylor, and without the need for implicit differentiation.

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.

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

We finished our work on Taylor Series, and used it to analyse the errors when using the Euler Method.

In VBA we should all have been able to finish Lab 2 (certainly up to automatic selection, p.108). Those who did so started on a Newton Cooling program. Students who have not completed these two programs (damper and Newton Cooling), are advised to work on them outside class.

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.

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

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

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, big $\mathcal{O}$ notation, etc.

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

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.

## Week 10

We looked at finite differences for the Heat Equation.

In VBA we implemented same.

## Easter

You will receive regular correspondence from me over the next week or so such as sample VBA Assessment 2, sample 40% Written Assessment, sample solutions of both, better programs.

Some of you need to put in a serious effort to get yourselves up to scratch or you will fail this module. It is up to you to be organised and be keeping an eye on the email correspondence.

Feel free to email me any questions that you might have. If you put in the effort I will reciprocate.

## Week 9

We did a little revision of the earlier material.

In VBA we will look at finite difference methods for Laplace’s Equation.

## Week 10

We will look at finite differences for the Heat Equation.

In VBA we will implement same.