Have been emailed to you.

We continued looking at partial fractions and then the inverse Laplace Transform.

We had about 30 minutes of tutorial time on Thursday.

We miss out on Monday due to St Patrick’s Day.

We will have two tutorials on Wednesday (p.121, p.112, p.103, p.100, p.99) before starting to look at Differential Equations on Thursday.

You will then be able to begin Assignment 2 after Thursday.

Assignment 2 will have a hand-in time and date of 12:00 8 April: the Monday of Week 11. Assignment 2 is in the manual, P. 164. Once we get someway into the examples on p.123 you should be able to make a start.

In Week 9 we will finish looking at differential equations, and then have two possibly three tutorials.

We will have another tutorial on Monday. On Wednesday we will look at Systems of Differenial Equations and have a tutorial on Thursday.

We will look at double integrals and then have one or two tutorials.

We will look at triple integrals and then have one or two tutorials.

We will review the Summer 2018 paper.

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

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>As mentioned in previous weeks, I need to postpone the lecture of Tuesday 19 March, Week 8.

This will now take place the next night, Wednesday 20 March 2019.

Two students have indicated that they cannot attend this class: I will record as much of the class as possible but my camcorder usually doesn’t have the battery nor memory to record all 2.5 hours… but I’ll do my best.

As mentioned briefly in Week 6, for *most *students, Chapters 1 and 2 are easier and you will want to do well on them. Things are going to get a little harder for the rest of the semester and you will want to try and do homework (see below) regularly.

We did a quick revision of differentiation before looking at Parametric Differentiation and starting to look at Related Rates.

If you want to look back here are two videos:

We mentioned briefly the Reuleaux Triangle and it’s use in the “Harry Watt Square Drill Bit”. See here for an animation of how this works.

We will finish looking at Related Rates and then look at Implicit Differentiation.

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.

I recommend strongly that everyone completes P.102, Q.1.

After that you can look at:

- P.102, Q. 2
- P.112, Q. 1-5

If you want to do more again, look at P.113, Q.6-9

Test 2, worth 15% and based on Chapter 3, will probably take place Week 11, 9 April. It might have to be pushed out to after Easter if we don’t make good progress on Chapter 3.

…and marking scheme have been emailed to you.

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>Last year I didn’t have these until Week 8.

Last year I didn’t have these until Week 9.

We looked at finite differences.

In VBA we have VBA Assessment 1.

We will do a (written) Shooting Method example and start looking at partial differential equations by looking at Laplace’s Equation.

In VBA we have MCQ VI and will do the Boundary Value Problems lab.

The following is a proposed assessment schedule:

**Week 11**, 20% Second VBA Assessment, Based (roughly) on Weeks 6-9**Week 12,**40% Written Assessment(s), Based on Weeks 1-11

Study should consist of

- doing exercises from the notes
- completing VBA exercises

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

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.

Please ask questions in the lab about questions you have gotten wrong.

]]>

Last year the results were not made available until Week 8. I will do my best to have them to you comfortably before Week 8.

We had an Undetermined Coefficients Tutorial on Monday.

Tuesday, we started looking at “The Engineer’s Transform” — the Laplace Transform. We looked at the first shift theorem, and how the Laplace Transform interacts with differentiation. We started looking at partial fractions.

We had our Undetermined Coefficients Concept MCQ on Thursday. This showed up serious deficiencies in our ability to carry out differentiation. I may or may not come up with some interventional material.

We will continue looking at partial fractions and the inverse Laplace Transform. If we can finish Section 3.3 (doubtful) we will have tutorial time.

Assignment 2 will have a hand-in time and date of 12:00 8 April: the Monday of Week 11. Assignment 2 is in the manual, P. 164. Once we get someway into the examples on p.123 you should be able to make a start.

If you download Maple (see Student Resources), there is a *Maple Tutor* that is easy to use and will help you with Gaussian Elimination. Open up Maple and go to Tools -> Tutors -> Linear Algebra -> Gaussian Elimination.

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

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>As mentioned in previous weeks, I need to postpone the lecture of 19 March, Week 8.

This will now take place the next night, Wednesday 20 March 2019.

Two students have indicated that they cannot attend this class: I will record as much of the class as possible but my camcorder usually doesn’t have the battery nor memory to record all 2.5 hours… but I’ll do my best.

As mentioned briefly in Week 6, for *most *students, Chapters 1 and 2 are easier and you will want to do well on them. Things are going to get a little harder for the rest of the semester and you will want to try and do homework (see below) regularly.

We finished Chapter 2 by looking at Cramer’s Rule and then we did a Concept MCQ followed by tutorial time.

A video of a Cramer’s Rule Example

We will do a quick revision of differentiation. We will then look at Parametric Differentiation and Related Rates.

If you want to look ahead here are two videos:

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. Here are good exercises for Matrices. Feel free to try questions in the exercises that are not listed here (p.66, p.73, p.84, or are in exam papers, see below):

- P.63, Q.1-4
- p.79, Q.1-2
- p.95, Q.3-5

Test 2, worth 15% and based on Chapter 3, will probably take place Week 11, 9 April. It might have to be pushed out to after Easter if we don’t make good progress on Chapter 3.

…and marking scheme have been emailed to you.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

]]>VBA Assessment 1 is taking place this week, Week 6.

Tuesday 14:20-16:00 will run 14:20-16:10

Friday 09:05-10:45 will run 09:05-10:55

More information in the Week 4 weekly summary.

In the Week 5 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.

Written Assessment 1 takes place Tuesday 12 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 – and replaces Section 1.6.1 of the manual.

However there are far more things I could examine.

Roughly, everything up to but not including Runge Kutta Methods (p.64). Some examples of questions I could ask include:

p.13, Examples 1-4; p.15, Q.1-4

Example, p. 15; p.34 Example

Examples 1 & 2 on p. 24; Q. 1 on p.27

p.29, Examples 1-4; p. 38, Q.1-5, 8-9

p. 35, Examples 1-2; p.39, Q.7, 10-14

p.38, Q. 6; p. 42, Examples 1-2l p. 47, Q.4-5

p.50, Example. p.51, Example. p.55, Q. 1-3, 5-14

We looked at boundary value problems (in particular the Shooting Method and Goal Seek).

In VBA we have VBA Assessment 1.

We will do a (written) Shooting Method example and then look at finite differences.

In VBA we have MCQ V and do the Runge Kutta lab.

The following is a proposed assessment schedule:

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

Study should consist of

- doing exercises from the notes
- completing VBA exercises

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.

Please ask questions in the lab about questions you have gotten wrong.

]]>

*Consider the following:*

*A rectangular block moves across a stationary horizontal surface with acceleration (the question had m/s but the m/s is repeated as m/s and so includes the unit).*

**There is a serious problem with this question and that is that the asymmetry in the problem means that there is an ambiguity: is the block moving left to right or right to left? I am going to assume the block moves from left to right. One would hope not to see such ambiguity in an official exam paper.**

*Two particles of mass placed on the block, are connected by a taut inextensible string. A second string passes over a light, smooth, fixed pulley to a third particle of mass which presses against the block as shown in the diagram.*

*If contact between the particles and the block is smooth, find the magnitude and direction of the resultant forces acting on the particles.*

Note firstly that there are *two *accelerations at play. The acceleration of the block relative to the horizontal surface, , and the accelerations of the particles relative to the block, say :

We draw all the forces (*I lazily didn’t add arrows to the force vectors*):

We know that the normal forces for the particles on top of the block because their vertical acceleration is zero and so the sum of the forces in that direction must be zero, and as the down forces are equal for both, necessarily the up forces must be equal too.

The accelerations of the particles on top are (it would be if the block was moving right-to-left), while the acceleration of the particle is in the horizontal direction, and in the vertical direction. Thus we can form four equations via Newton’s Second Law :

(A)

(B)

(C)

(D)

So we have . If we add (A)+(B)+(D) we get

.

We can thus find using (A):

.

Let us impose the following axis on the system:

Therefore the magnitude of the force on the first particle is and the direction is in the -direction. As the acceleration and mass of the second particle is the same as the first, by Newton’s second law, the resultant force must be acting on the second particle. Therefore (in order that .

Now turning our attention to the particle. The force in the negative direction is

,

and so the resultant force on this particle is :

Call this vector . We find the magnitude using Pythagoras Theorem:

.

The angle of the resultant force is, below the positive -axis:

.

*If contact between the particles and the block is rough, for what same value of the coefficient of friction will the particles remain at rest relative to the block?*

Let the coefficient of friction be given by . We have the same acceleration (labeling) but now additional friction forces:

The normal forces are the same, and so the frictions are , , and . This gives three equations:

If the particles are at rest relative to the block :

Add these together:

.

*A bucket with mass and a block with mass are hung on a pulley system, as shown.*

*The pulley with the mass effectively has a mass of .*

*Find the magnitude of the acceleration with which the bucket and the block are moving.*

First, the accelerations. We will assume that the moves ‘down’ (if it doesn’t its acceleration will come out as negative).

If the mass moves down then the mass-pulley system goes ‘up’. Let be the acceleration that the mass-pulley system has.

**Problem: **Convince yourself that the mass has acceleration . We draw the forces:

Now we write down two equations via Newton’s Second Law (the masses are ‘known’).

For the mass-pulley system:

, (A)

and for the other mass, the bucket:

Multiply both sides by two:

. (B)

Add equations (A) + (B):

.

This is the acceleration of the block. The acceleration of the block is twice this:

.

*Find the magnitude of the tension force by which the rope is stressed.*

i.e. find the tension. From Newton’s Second Law for the bucket, we have

**Exercise**

Show that this simplifies to

]]>

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

Work that is handed in late will be assigned a mark of ZERO so hand in what you have one time.

More information in last week’s Weekly Summary.

One final warning: do not give your work to others to copy. If there is a lack of originality of presentation I will be dividing marks between those who copy each other and the person who did the original work will be penalised along with those who copy them.

We finished our work on Chapter 2 — the method of Undetermined Coefficients — Wednesday PM. The Thursday class was (will be as I write) a tutorial.

On Monday we will have a tutorial on Undetermined Coefficients.

On Wednesday AM we will have a Concept MCQ, and then crack into Chapter 3, by looking at “The Engineer’s Transform” — the Laplace Transform.

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

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

]]>As mentioned in previous weeks, I need to postpone the lecture of 19 March, Week 8.

Three possibilities to catch up are:

- the next night, Wednesday 20 March 2019
- the Tuesday before Easter, Tuesday 16 April 2019
- the Wednesday after Easter, Wednesday 24 April 2019

Please fill in this Doodle poll by selecting all the days that you *can *attend.

Hopefully we can find a day that suits everyone but if people cannot make a day that is otherwise popular I will record the lecture.

I hope to have these out to you within 24 hours.

Test 2, worth 15% and based on Chapter 3, will probably take place Week 11, 9 April (or possibly Week 12, 30 April if we don’t go for a class Wednesday 20 March).

We had our test and then we talked about linear systems, and determinants.

We had no tutorial time.

We will finish Chapter 2 by looking at Cramer’s Rule. When we finish talking about Cramer’s Rule we will do a quick revision of differentiation, hopefully including some tutorial time.

VBA Assessment 1 will take place in Week 6, (5 & 8 March) in your usual lab time.

Tuesday 10:05-11:45 will run 10:05 to 11:55

Tuesday 14:20-16:00 will run 14:20-16:10

Friday 09:05-10:45 will run 09:05-10:55

More information in last week’s weekly summary.

In the Week 5 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.

Written Assessment 1 takes place Tuesday 12 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 – and replaces Section 1.6.1 of the manual.

However there are far more things I could examine.

Roughly, everything up to but not including Runge Kutta Methods (p.64). Some examples of questions I could ask include:

p.13, Examples 1-4; p.15, Q.1-4

Example, p. 15; p.34 Example

Examples 1 & 2 on p. 24; Q. 1 on p.27

p.29, Examples 1-4; p. 38, Q.1-5, 8-9

p. 35, Examples 1-2; p.39, Q.7, 10-14

p.38, Q. 6; p. 42, Examples 1-2l p. 47, Q.4-5

p.50, Example. p.51, Example. p.55, Q. 1-3, 5-14

Given an initial value problem (IVP):

,

I want you to understand the following:

First of all:

- The Two Term Taylor Method is the Euler Method and uses lines to approximate .
- If the solution of the IVP is a line, the error in the Euler Method aka the Two Term Taylor Method is zero. Note that the the second derivative of a line is zero.
- The global error for Euler’s Method and so Euler’s Method aka the Two Term Taylor Method is said to a
*first order method*(because the error is ). - The Three Term Taylor Method uses parabolas, degree two polynomials, to approximate .
- If the solution of the IVP is a parabola, the error in the Three Term Taylor Method is zero. Note that the
*third*derivative of a parabola is zero. - The global error for the Three Term Taylor Method is and so the Three Term Taylor Method is said to be a
*second order method.* - If is dependent on , then the Three Term Taylor Method requires implicit differentiation.
- With further implicit differentiation, we have the -Term Taylor Method which uses a degree polynomials to approximate .
- If the solution to the IVP is an a degree polynomial, the error in the -Term Taylor Method is zero. Note that the -st derivative of a degree polynomial is zero.
- The global error for the -Term Taylor Method is and so the -Term Taylor Method is said to be a
*-th order method.*

The Euler Method is *the *first order Runge-Kutta Method.

Now what we want for a second order Runge-Kutta Method is that:

- Like the Three Term Taylor Method, it has zero error when the solution is a parabola.
- Like the Three Term Taylor Method, the global error is .

It does this by taking a weighted average of two slopes: the slope at , and , the slope at any other point between and .

Two such schemes are presented in the notes.

Now what we want for an -th order Runge-Kutta is that:

- Like the -Term Taylor Method, is has zero error when the solution is a degree polynomial.
- Like the -Term Taylor Method, the global error is .

It does this by taking a weighted average of slopes: , where the slope at , and the other are slopes at points between and .

See p.108-109 for other Runge-Kutta Theory exercises.

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

In the afternoon, we will looked of Runge-Kutta Methods. A summary is given above and you will want to understand all of this for Written Assessment 2.

In VBA we had MCQ IV and looked at Lab 4, on Second Order Differential Equations.

We will look at boundary value problems (in particular the Shooting Method and Goal Seek).

In VBA we have MCQ V and VBA Assessment 1.

The following is a proposed assessment schedule:

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

Study should consist of

- doing exercises from the notes
- completing VBA exercises

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.

Please ask questions in the lab about questions you have gotten wrong.

]]>