MATH7019: Week 3

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.

Repeat Students — particularly EXAM ONLY

There have been some changes made to MATH7019

– Second Order Linear Ordinary Differential Equations have been moved to MATH7021 and are no longer studied in MATH7019

– The chapter on Curve Fitting from MATH7021 has been introduced into MATH7019 however forward difference methods have been dropped altogether. The correlation coefficient has been added to this chapter. This material is being done first and will be completed by or in Week 3.

– Cantilevers have been added to the section on beam equations

– Regarding the chapter on Further Calculus, reviews of calculus topics have been spread out throughout the module

– No change to the chapter on Statistics

To find out the exact syllabus please consult the module descriptor.

Manuals

The manuals are priced at €14 and are available in the Reprographic Centre.

Week 3

In Week 3 we finished talking about Least Squares Curve Fitting and started the second chapter on Differential Equations.

Week 4

In Week 4 we will continue with second chapter on Differential Equations — with a particular emphasis on Beam Equations.

Quick Test: Academic Learning Centre

I would urge anyone having any problems with material that isn’t being addressed in the tutorials to use the Academic Learning Centre. As you can see the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

Based on the results of the Quick Test, I have already advised some of ye to attend the Academic Learning Centre.

Assessment 1

Assessment 1 will have a hand-in date of October 17. Expect to see the assignment early in Week 3.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

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. For example this question about a particular differential equation.

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

1. Log into Blackboard Learn

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 on the action link button next to the course 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, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

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

Additional Notes: E-Books

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.

Calculators

Please note the following taken from the CIT code of conduct for CIT examination candidates:

“Where a pocket calculator is used it must be silent, self-powered and non-programmable.

It may not be passed from one candidate to another. Instructions for its use may not be
brought into the Examination Hall.
The term ‘programmable’ includes any calculator that is capable of storing a sequence of
keystrokes that can be retrieved after the calculator is turned off or powers itself off. Note that the
capacity to recall, edit and replay previously executed calculations does not render a calculator
programmable, provided that this replay memory is automatically cleared when the calculator is
powered off. Also, the facility to store numbers in one or more memory locations does not render
a calculator programmable.
Calculators with any of the following mathematical features are prohibited:
• Graph plotting
• Equation solving
• Symbolic algebraic manipulation
• Numerical integration
• Numerical differentiation
• Matrix calculations
Calculators with any of the following features are prohibited
• Data Banks
• Dictionaries
• Language translators
• Text retrieval
• Capability of remote communication“

2 comments

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October 9, 2014 at 7:27 am

Student

Hi JP,

I’ve completed the problem one of the assignment and just wanted to clarify something before moving on.

Does the curve

$S=a\cdot t+\frac{b}{t}$

give normal equations:

$\sum St=\sum a+\sum \frac{b}{t^2}\Rightarrow \sum St=12a+b\sum\frac{1}{t^2}$

and

$\sum \frac{S}{t}=\sum at^2+\sum b\Rightarrow \sum \frac{S}{t}=a\sum t^2+12b$

Is this correct?

Could you also please let me know which questions will be covered in class Friday as I won’t be able to attend due to a wedding.

Thanks in advance.

Regards.

October 9, 2014 at 7:46 am

J.P. McCarthy

We will focus on the positives for now.

It is true that $\sum a=12a$ and $\sum b=12b$ .

Also you have correctly inferred that

$\displaystyle\sum \frac{b}{t^2}=b\sum\frac{1}{t^2}$ .

However the normal equations are incorrect.

Rather than give you the normal equations, I will derive a more general formula and see if you can figure out things from there. Of course you just need to get from your curve $S=at+b/t$ to the normal equations — you don’t have to go through all this every time — this is just showing you why the normal equations take the form that they do.

O.K., starting with a set of data $\{(t_i,S_i):i=1,\dots, n\}$ , suppose that we want to best-fit a curve of the form

$S(t)=a\cdot \theta(t)+b\cdot \lambda(t),$

in the least squares sense, where $\theta=\theta(t)$ and $\lambda=\lambda(t)$ are functions of $t$ .

Hence we wish to minimise the sum of the squared deviations of the data from the curve. The deviations are given by

$\delta_i=|S_i-(a\theta(t_i)+b\lambda(t_i))|$
$\Rightarrow \delta_i^2=|S_i-a\theta(t_i)-b\lambda(t_i)|^2=(S_i-a\theta(t_i)-b\lambda(t_i))^2$ ,

so we want to minimise the Sum of the Squared Deviations as a function of $a$ and $b$ :

$SSD(a,b)=\sum_{i=1}^n\delta_i^2=\sum_{i=1}^n(S_i-a\theta(t_i)-b\lambda(t_i))^2$ .

To find the minimum of a function of two variables we can find its minimums with respect to $a$ and with respect to $b$ separately (as was explained in lectures). To minimise with respect to $a$ we find where $\displaystyle \frac{\partial SSD}{\partial a}=0$ and to minimise with respect to $b$ we find where $\displaystyle\frac{\partial SSD}{\partial b}=0$ . Using the Sum and Chain Rules we find:

$\displaystyle\frac{\partial SSD}{\partial a}=\sum_{i=1}^n2(S_i-a\theta(t_i)-b\lambda(t_i))(-\theta(t_i))$ .

We want to find where this is equal to zero so we want

$\sum_{i=1}^n2(S_i-a\theta(t_i)-b\lambda(t_i))(-\theta(t_i))=0$
$\Rightarrow \sum_{i=1}^n (-S_i\theta(t_i))+\sum_{i=1}^n a\theta(t_i)^2+\sum_{i=1}^nb\lambda(t_i)\theta(t_i)=0$
$\Rightarrow \sum_{i=1}^n S_i\theta(t_i)=a\sum_{i=1}^n\theta(t_i)^2+b\sum \lambda(t_i)\theta(t_i)$ ,

which we usually just write as

$\sum S\theta(t)=a\sum \theta(t)^2+b\sum \lambda(t)\theta(t)$ .

Similarly we can show that to minimise $SSD$ with respect to $b$ we require

$\sum S\lambda(t)=a\sum\theta(t)\lambda(t)+b\sum\lambda(t)^2$ .

In conclusion, if $\theta=\theta(t)$ and $\lambda=\lambda(t)$ , and we want to find the best-fit curve, in the least squares sense, of the form

$S=a\theta+b\lambda$ ,

we solve the normal equations:

$\sum S\theta=a\sum \theta^2+b\sum\lambda\theta$ ,
$\sum S\lambda=a\sum\theta\lambda+b\sum\lambda^2$ .

Now if you can’t see how your normal equations are incorrect, and can’t find the correct normal equations, then get back to me.

In Friday’s class, apart from lectures, I would be trying to get the class to do P.66 Q. 1 and 4-8 and maybe P. 69 Q. 1-4.

Regards,
J.P.

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MATH7019: Week 3

Repeat Students — particularly EXAM ONLY

Manuals

Week 3

Week 4

Quick Test: Academic Learning Centre

Assessment 1

Study

Math.Stack Exchange

Maple Online & Wolfram Alpha

Additional Notes: E-Books

Calculators

Recent Comments

Categories

J.P. McCarthy Maths

2 comments

Leave a Reply Cancel reply

MATH7019: Week 3

Repeat Students — particularly EXAM ONLY

Manuals

Week 3

Week 4

Quick Test: Academic Learning Centre

Assessment 1

Study

Math.Stack Exchange

Maple Online & Wolfram Alpha

Additional Notes: E-Books

Calculators

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Categories

J.P. McCarthy Maths

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