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 12

We finished the section on integrals related to the inverse trigonometric functions. We revisited work and then spoke about centroids of laminas and centres of gravity of volumes of revolution.

## Week 13

I will be available to any and all students at the following times and venues:

• Review Lecture Tuesday 15:00 B260
• Review Lecture Wednesday 11:00 B214
• Review Lecture Thursday 11:00 B260
• Review Tutorial Friday 11:00 B260
• Review Tutorial Friday 13:00 B245

The Review Lectures will be conducted as follows

1. Students can ask any question and I will answer it on the whiteboard. If we run out of questions
2. I will start going through the Winter 2013 paper (which was given out in class). If we finish this paper
3. I will help ye one to one.

The Review Tutorials will be conducted as follows

1. Students can ask any question and I will answer it on the whiteboard. If we run out of questions
2. I will help ye one to one. You can work on whatever you want but I will be recommending a centroids and centres of gravity.

For your study, I would strongly recommend that you attempt this paper before the review lectures. This is your best chance of seeing how you stand.

## Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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 12

In the tutorial ye tried to do a line integral but I don’t know how ye got on as I was showing ye your tests. We finished off double integrals and triple integrals and we spent one class looking at systems of differential equations.

## Week 13

I will be available at the following times and venues:

• Review Tutorial Tuesday 13:00 A123L (for those who are missing out on their Monday tutorial)
• Review Tutorial Tuesday 14:00 A123L (for the rest of ye)
• Review Lecture Wednesday 12:00 B228
• Review Lecture Wednesday 14:00 B212
• Review Lecture Thursday 10:00 B228

The Review Lectures will be conducted as follows

1. Students can ask any question and I will answer it on the whiteboard. If we run out of questions
2. I will start going through a combination of the sample and the Autumn 2013 paper (which were given out in class). If we finish this ‘paper’
3. I will help ye one to one.

The Review Tutorials will be conducted as follows

1. Students can ask any question and I will answer it on the whiteboard. If we run out of questions
2. I will help ye one to one. You can work on whatever you want but I will be recommending a triple integral.

For your study, I would strongly recommend that you attempt these papers before the review lectures. This is your best chance of seeing how you stand.

## Exam Format

Five questions do four:

Q. 1 — a question on each chapter.

Q. 2 — questions on chapter 1

Q. 3 — questions on chapter 2

Q. 4 — questions on chapter 3

Q. 5 — questions on chapter 4

As in this sample.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘glyn advanced modern engineering math’ you will see that the library have an E-Book resource.

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.

## Linear Algebra — Test Solutions

More comments on Linear Algebra next week but until then here are the solutions to the test.

## Week 12

We spoke about sampling in more detail and also introduced control charts.

## Week 13

We will hold a review class on Wednesday 7 May in the usual room. First off, the layout of your exam is the same as Autumn 2013: do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.

I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.

## Formulae to Learn?

I spoke about asking Dr. Michael Brennan about putting the confidence interval formula on the exam paper… I just realised this is moot as the exams have already gone for printing. Therefore there are two formulae you will need to know.

The first is the confidence interval for small samples (here I give three presentations):

$\bar{x}\pm t\frac{\sigma}{\sqrt{n}}$

$\bar{x}-t\frac{\sigma}{\sqrt{n}}\leq \mu\leq \bar{x}+t\frac{\sigma}{\sqrt{n}}$

$\left[\bar{x}-t\frac{\sigma}{\sqrt{n}},\bar{x}+t\frac{\sigma}{\sqrt{n}}\right]$

Here $\bar{x}$ is the sample mean, $n$ is the sample size, $t$ is a number of standard deviations, read from a table, depending on $n-1$ and the level of uncertainty. $\sigma$ is the population standard deviation which we approximate using the sample standard deviation, $\sigma\approx s$. Finally $\mu$ is the population mean, the number we are trying to get a handle on.

The second set of formulae are like confidence intervals for the control charts aka the inner and outer control limits (ICL & OCL). These should be considered versions of the above formula and I will display them as such… the $A_i$ depends on the level of uncertainty and the sample size, and plays the role of $\frac{t}{\sqrt{n}}$, while the mean range, $\bar{w}$, is a measure of the spread\deviation of the data, and plays the role of $\sigma$. The role of $\bar{x}$ is played by $\bar{\bar{x}}$ — the grand mean:

ICL: $\bar{\bar{x}}\pm A_{0.025}\bar{w}$,

OCL: $\bar{\bar{x}}\pm A_{0.001}\bar{w}$

ICL: $\bar{\bar{x}}- A_{0.025}\bar{w}\leq \bar{x}\leq \bar{\bar{x}}+ A_{0.025}\bar{w}$,

OCL: $\bar{\bar{x}}- A_{0.001}\bar{w}\leq \bar{x}\leq \bar{\bar{x}}+ A_{0.001}\bar{w}$

ICL: $[\bar{\bar{x}}- A_{0.025}\bar{w}, \bar{\bar{x}}+ A_{0.025}\bar{w}]$,

OCL: $[\bar{\bar{x}}- A_{0.001}\bar{w}, \bar{\bar{x}}+ A_{0.001}\bar{w}]$

Here $\bar{\bar{x}}$ is the grand mean — that is the mean of all the data — which may conveniently be calculated as the mean of the sample means. The $A_{i}$ are numbers read from a table, depending on the sample size $n$$\bar{w}$ is the mean range — the mean of the sample ranges. Finally $\bar{x}$ is a sample mean.

The smaller numbers are the lower limits and the larger numbers are the upper limits.

## Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

• P. 130 Q. 1-13
• P.139, Q. 1-7, particularly Q. 1

I wouldn’t necessarily advise looking at ALL of these questions: I would focus on the exam questions over the other questions.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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.

Results Pending…

## Week 11

We will have our test and then we will continue looking at these integrals related to the inverse trig functions… some of which will require us to complete the square. We might also revisit work.

## Week 12

We will finish the section on integrals related to the inverse trigonometric functions. We will revisit work and then talk about centroids of laminas and centres of gravity of volumes of revolution. One lecture might have to spill into Week 13 due to my illness on the Week 10 Tuesday afternoon.

## Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

## Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package MathematicaMathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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 11

In the tutorial we did some last minute revision for the test. We continued working with double integrals and their applications.

## Week 12

In the tutorial ye can try your hand at a line integral and a double integral. We will finish off double integrals and triple integrals and we will spend one class looking at systems of differential equations.

## Week 13

We will have five classes at the usual times and in the usual venues. We will start by going through the sample paper. I would urge you very strongly to try this paper out over the Easter break.

When this is finished I will take questions and after that I will help one-to-one.

## Exam Format

Five questions do four:

Q. 1 — a question on each chapter.

Q. 2 — questions on chapter 1

Q. 3 — questions on chapter 2

Q. 4 — questions on chapter 3

Q. 5 — questions on chapter 4

As in this sample.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

## Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package MathematicaMathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘glyn advanced modern engineering math’ you will see that the library have an E-Book resource.

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 11

We looked at the normal distribution in more detail and talked about sampling.

## Week 12

We will talk about sampling in more details and also introduce control charts.

## Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

• P. 125 Q. 1-21, particularly Q. 5-10

I wouldn’t necessarily advise looking at ALL of these questions: I would focus on the exam questions over the other questions.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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.

## Test 2

The second test will be on this Tuesday 8 April February at 11:00 in B214. Ye have the test that I gave last year in the notes.

## Week 10

We started the new chapter by looking at integration by parts and integrations related to the inverse trigonometric functions.

## Week 11

We will have our test and then we will continue looking at these integrals related to the inverse trig functions… some of which will require us to complete the square. We might also revisit work.

## Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

## Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package MathematicaMathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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.

## Test 2

The second test will be on this Wednesday 9 April February at 12:00 in B228. Ye can find a sample here.

It is vital that ye have a look at all the additional notes as well as perhaps section 3.3.2

## Week 10

In the tutorial we looked at doing full Laplace transform questions. On Thursday we did the sample test. On Wednesday we finished off  line integrals and started talking about double integrals.

## Week 11

In the tutorial ye can look at slightly harder full Laplace transform questions that require the first shift theorem. We will continue working with double integrals and their applications.

## Week 12

In the tutorial ye can try your hand at a line integral and a double integral. We will finish off double integrals and triple integrals and we will spend one class looking at systems of differential equations.

## Week 13

We will have five classes at the usual times and in the usual venues. We will start by going through the sample paper. I would urge you very strongly to try this paper out over the Easter break.

When this is finished I will take questions and after that I will help one-to-one.

## Exam Format

Five questions do four:

Q. 1 — a question on each chapter.

Q. 2 — questions on chapter 1

Q. 3 — questions on chapter 2

Q. 4 — questions on chapter 3

Q. 5 — questions on chapter 4

As in this sample.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

## Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package MathematicaMathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘glyn advanced modern engineering math’ you will see that the library have an E-Book resource.

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.

## Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on. Please complete your Maple catch up before 30 April.

## Summer 2011 Q. 2 (b)

This question came up last night however I said that there was an issue with interpretation but that I felt there was only one reasonable interpretation: that the question required a mix of the Poisson Distribution and the Binomial Distribution, and I have answered the question under that assumption below.

I stated however that you will not get a question like this in your exam because your exam will prompt you to use the Binomial, Poisson or indeed Normal distribution.

I have no idea where I got this question from… it wasn’t on any MATH6038 paper… a mystery!

A production department has 35 milling machines. The number of breakdowns on each machine averages 0.06 per week. Determine the probabilities of having

1. one machine breaking down in a week

2. less than three machines breaking down in a week

Solution:

1. With an average rate of occurrence, $\lambda=0.06$ given, assume that the number of times a machine breaks down in a week, $X$, has a Poisson distribution, $X\sim \text{Poi}[0.06]$. Therefore to calculate the probability that a machine breaks down during the week we look at the probability of the machine breaking down one or more times:

$\mathbb{P}[X\geq 1]=1-\mathbb{P}[\text{not-}(X=0)]=1-e^{-0.06}\approx 0.05824$.

2. Now we are looking at 35 ‘trials’ where success is defined as a machine breaking down so that we have $p=1-e^{-0.06}$ — and $1-p=1-(1-e^{-0.06})=e^{-0.06}$ (note that using 0.05824 introduces rounding error. Not the end of the world in an exam but perhaps in real life!).

Let $Y$ be the number of successes from the $n=35$ trials so that $Y$ has a binomial distribution; $Y\sim \text{Bi}[25,1-e^{-0.06}]$. We are looking at the probability of less than three machines breaking down in a week;

$\mathbb{P}[Y<3]=\mathbb{P}[(Y=0)\cup(Y=1)\cup(Y=2)]$

$=\mathbb{P}[Y=0]+\mathbb{P}[Y=1]+\mathbb{P}[Y=2]$

$={35\choose 0}(1-e^{-0.06})^0(e^{-0.06})^{35}+{35\choose 1}(1-e^{-0.06})^1(e^{-0.06})^{34}+$

${35\choose 2}(1-e^{-0.06})^2(e^{-0.06})^{33}$

$0.12246+0.26503+0.2786=0.4154$.

## Week 10

We looked at reliability block diagrams, the binomial distribution, the Poisson distribution and we started talking about the normal distribution.

## Week 11

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

## Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

• P. 108, Q. 1-7
• P. 112 Q. 1-17
• P.116, Q. 1-15

I wouldn’t necessarily advise looking at ALL of these questions: I would focus on the exam questions over the other questions.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

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 9

We finished up doing partial differentiation in the abstract and applied it to error analysis. Finally we looked at one big application of the Chain Rule: Related Rates.

## Week 10

We will start the new chapter by looking at integration by parts.

## Test 2

The second test will be on in Week 11. That is Tuesday 8 April February at 11:00 in B214. Ye have the test that I gave last year in the notes.

## Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

## Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package MathematicaMathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!