Category Archive

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 1

The first 15% test will take place at 5 pm this Thursday 24 October in B228 (Week 6). You can find a sample here. It is a test that could arguably take 40 minutes but I’ll give ye from 17:05 — 18:00. You will be given a copy of these tables. Don’t worry I’ll scribble out the “UCC”!

Note that the format will be the same of the sample:

1. Maclaurin Series OR Taylor Series (Taylor Series formula given)
2. Error Analysis
3. Taylor Series of Several Variables (formula given)
4. Euler\Three Term Taylor Method (Three Term Taylor Formula given)
5. Homogenous Second Order Ordinary Linear Differential Equations

Note that I like to make the sample harder than the test and the Q. 1 on the sample is a bit of a dinger alright!

Course Notes

At the moment they look like this.

Week 5

In Week 5 we study of second order ordinary linear differential equations.

Week 6

In Week 6 we will do two more examples of non-homogenous ordinary second order linear differential equations with boundary/initial conditions. Then we will move onto beam analysis.

Read the rest of this entry »

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 1

The first 15% test will take place at 5 pm Thursday 24 October in B228 (Week 6). You can find a sample here. I will give ye a copy of this sample on Monday. It is a test that could arguably take 40 minutes but I’ll give ye from 17:05 — 18:00. You will be given a copy of these tables. Don’t worry I’ll scribble out the “UCC”!

Note that the format will be the same of the sample:

1. Maclaurin Series OR Taylor Series (Taylor Series formula given)
2. Error Analysis
3. Taylor Series of Several Variables (formula given)
4. Euler\Three Term Taylor Method (Three Term Taylor Formula given)
5. Homogenous Second Order Ordinary Linear Differential Equations

Course Notes

At the moment they look like this.

Week 4

In Week 4 we looked at the Euler and Three Term Taylor Method for finding approximate solutions to first order ordinary differential equations. We started our work on homogenous second order ordinary linear differential equations.

Week 5

In Week 5 we will continue our study of second order ordinary linear differential equations.

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. You could also win a tablet device if you enter a competition that they are running.

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 asking can Euler’s Method ever be exact?

Maple & 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!

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“

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.

Course Notes

Last week they looked like this.

Week 3

In Week 3 we looked at Error Analysis and two-variable Taylor Series. We also began our study of differential equations.

Week 4

In Week 4 we will continue our study of ordinary first order differential equations and develop numerical techniques (the Euler and Three Term Taylor Methods) for approximating solutions to ODEs that we can’t solve exactly.

Test 1

Test 1 will be on the Thursday of Week 6. Expect a sample in Week 5.

Tutorials

Ted should has a division of the class such that one group this week is Thursday 15:00 while the other is 16:00. Both groups in B228.

Read the rest of this entry »

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.

Course Notes

At the moment they look like this.

Week 2

In Week 2 we reviewed partial differentiation.

Tutorials

Tutorials start properly this week. Ted should have a division of the class such that one group this week is Thursday 15:00 while the other is 16:00. Both groups in B228.

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. You could also win a tablet device if you enter a competition that they are running.

Maple & 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!

Week 3

In Week 4 we will have a look at Error Analysis and two-variable Taylor Series.

Test 1

Test 1 will be on the Thursday of Week 6. Expect a sample in Week 5.

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 attempting to explain why .

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“

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 one we showed that differential equations can arise in engineering. We discussed how some differential equations do not submit easily to analysis and that sometimes we would have to find approximate solutions. We then proceeds to review of calculus. Finally we answered the question of how calculators work by developing a theory of power series.

Tutorials

Tutorials start properly this week. Ted should have a division of the class such that one group this week is Thursday 16:00 while the other is 17:00.

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. You could also win a tablet device if you enter a competition that they are running.

Week 2

In Week 2 we will review partial differentiation and have a look at two-variable Taylor Series.

Test 1

Test 1 will be on the Thursday of Week 6. Expect a sample in Week 5.

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 addressing the strange conclusion that the Maclaurin Series of !

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“

Test 2 on Thursday 19 April at 8.25 p.m

Please find a sample test here.

Note that the format will be the same as this.

1. Write down the bending moment for three loadings.
2. Exam Standard Beam I
3. Exam Standard Beam II
4. Binomial Distribution
5. Poisson Distribution

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We developed our applications of step and impulse functions to beam equations. Relevant notes p.22 – 50

In the tutorial we worked on the sample test.

Next Week

We have a tutorial on Thursday. Please take this opportunity to nail down the differential equations. On the final exam there will be both a full question on beam equations and a full question on second order linear differential equations. Hence if you are comfortable with the material that is examinable in the test, feel free to move onto some of the exercises on beam equations.

Test Date

Tuesday 13 March at 6.45 p.m.

Timetable Changes

We are now going to schedule ourselves as follows:

Week 6: – & Tutorial

Week 7: Lecture/Test & –

Week 8: Lecture & Lecture

Week 9: Tutorial & Lecture

Week 10: Tutorial & Lecture

Week 11: – & Lecture

Week 12: Tutorial & Lecture

Sample Test Answers

Here I give you links to how I checked answers quickly using Wolfram Alpha. It takes Mathematica code (they are the same company) and will probably decipher your own stab at code also. Note that Wolfram Alpha gives us a lot more information than we need but that is the beauty of the thing really.

Question 1 — note there is a small typo here it should be rather than just 16.

Question 2 — it’s not evaluating the constants using the boundary conditions for some reason… the answer is

Question 3

Question 4 — the boundary conditions yield .

Question 5 — what is here and how do we write ?

Question 6 — again we need to realise that the notation here is different to ours. Applying the boundary conditions we get .

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We introduced step and impulse functions to model loads on a beam and we wrote down the equations relating the loading, the shearing force, the bending moment and the deflection. Relevant notes p.23 to p.43.

In the tutorial we worked on the exercises on second order linear differential equations in these notes.

Next Week

We have a tutorial on Tuesday. In this we will try to shore up the second order differential equations and the second order differential equations involving step and impulse functions. Remember the integration of impulse and step functions is given by (where each of the arrows indicate an integration)

, and then

.

Also each integration generates an addition constant .

In next weeks lectures we will continue our work on the beam equations.

Read the rest of this entry »

Consider this notice for the test on Tuesday 13 March at 6.45 p.m (just under three weeks away) (note that there is still a small chance that this tell will be held on Thursday 8 March at 8.15 p.m.).

Please find a sample test here

Note that the format will be the same as this.

1. Homogeneous Second Order Linear
2. Homogeneous Second Order Linear with boundary/initial conditions
3. Non-Homogeneous Second Order Linear
4. Non-Homogeneous Second Order Linear with boundary/initial conditions
5. Second Order Separable with Step and Impulse Functions
6. Second Order Separable with Step and Impulse Functions with boundary/initial conditions

Q. 5 and 6 will be covered Thursday night. Note that for questions 1-4 the roots will not be complex — although they could be fractions or surds (i.e. ye might need the formula).

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 jippo@campus.ie and I will add you to the mailing list.

Week 2

In week 2 we began to look at homogenous second order linear differential equations (with constant coefficients). These are differential equations of the form

,               (*)

where  are real constants and the solution is . We showed that to solve these equations we look at the auxiliary equation

.

Fact 1

In general, if the roots of of this equation are  and  then the solution to  (*) is given by

where and  are real constants.

Read the rest of this entry »