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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.

## Manuals

You will need to purchase the manual from the CIT Copy Centre beside the Student Centre (in the same room as where you get your photos taken for your student card). Note that this is a cash-free area so you will need to put the appropriate amount of funds on your student card.

The Copy Centre will be open until 19:30 the first and second weeks of the term but from next week goes back to normal opening hours — closing at 16:30.

## Quiz 1 Results and Solutions

Below find the results. You are identified by the last four digits of your student number unless you are excelling. The marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes). One student didn’t write down their name. They should email me. Solutions and marking scheme here.

 S/N Quiz 1 R% Fahey 2.5 100 Buckley 2.5 100 Nolan 2.43 97.2 8332 2.37 94.8 8212 2.37 94.8 1486 2.24 89.6 8426 2.24 89.6 2128 2.24 89.6 9464 2.17 86.8 6645 2.17 86.8 9896 1.97 78.8 2237 1.71 68.4 3481 1.71 68.4 2942 1.71 68.4 3703 1.71 68.4 1321 1.45 58 7879 1.12 44.8 2073 0.99 39.6 9555 0.53 21.2 Not registered but did test. 0.53 21.2 2257 0.53 21.2 7209 0.39 15.6 8354 0.39 15.6 2070 0.33 13.2 3872 0.26 10.4 2092 0 0 1298 ? 4402 ? 8425 ?

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.

## Manuals

You will need to purchase the manual from the CIT Copy Centre beside the Student Centre (in the same room as where you get your photos taken for your student card). Note that this is a cash-free area so you will need to put the appropriate amount of funds on your student card.

The Copy Centre will be open until 19:30 the first and second weeks of the term.

## Week 1

In Week 1 we explained the kind of thing that we would be looking at in this module. We did a quick review of integral calculus.

## Week 2

In Week 2 we will look at $u$-substitutions. Then we will study Integration by Parts: this is the start of the new material (i.e. not MATH6019 material).

We will not have a Maple Lab until Week 3.

Arguably, the three central concepts in the theory of differential calculus are that of a function, that of a tangent and that of a limit. Here we introduce functions and tangents.

## Functions

When looking at differential calculus, two good ways to think about functions are via algebraic geometry and interdependent variables. Neither give the proper, abstract, definition of a function, but both give a nice way of thinking about them.

### Algebraic Geometry Approach

Let us set up the plane, $\Pi$. We choose a distinguished point called the origin and a distinguished direction which we call ‘positive $x$‘. Draw a line through the origin in the direction of positive $x$. This is the $x$-axis. Choose a unit distance for the $x$-direction.

Now, perpendicular to the $x$-axis, draw a line through the origin. This is the $y$-axis. By convention positive $y$ is anti-clockwise of positive $x$. Choose a unit distance for the $y$-direction.

This is the plane, $\Pi$:

Now points on the plane can be associated with a pair of numbers $(a,b)$. For example, the point a distance one along the positive $x$ and five along the negative $y$ can be denoted by the coordinates (1,-5):

Similarly, I can take a pair of numbers, say (-1,3), and this corresponds to a point on the plane.

This gives a duality:

points on the plane $\Leftrightarrow$ pairs of numbers

Now consider the completely algebraic objects

$x=3\,,\,\,y=2\,,\,\,y=x\,,\,\,y=x^2\,,\,\,y=x+2\,,\,\,x^2+y^2=1$.

We define the transfer function of a black box model as

$\displaystyle H(s)=\frac{Y(s)}{X(s)}$,

where $X(s)$ is the Laplace transform of the input and $Y(s)$ is the Laplace transform of the output. This yields:

$Y(s)=H(s)\cdot X(s)$.

$H(s)$ is a rational function and therefore has zeroes and roots (removable poles and roots are removed) and therefore so does $Y(s)$:

$\displaystyle Y(s)=K\cdot \frac{\prod_{j=1}^m(s-z_j)}{\prod_{i=1}^n(s-p_i)}$.

These $\{z_j\}$ are the zeroes of $Y(s)$ and the $\{p_i\}$ are the poles of $Y(s)$This means that we have (assuming $m) a partial fraction expansion:

$\displaystyle Y(s)=\sum_{i=1}^n\frac{C_i}{s-p_i}$.

Applying the Inverse Laplace transform we have an input:

$y(t)=\sum_{i=1}^n C_ie^{p_it}$.

These $p_i$ are complex numbers and depending on their nature we get different behaviours.

A positive real pole is a pole $p\in\mathbb{R}$ such that $p>0$. This corresponds to an output $e^{pt}$ which tends to infinity as $t\rightarrow \infty$. This is divergent or unstable behaviour.

A negative real pole $p=-a$ with $a>0$ corresponds to an output $\displaystyle e^{pt}=e^{-at}=\frac{1}{e^{at}}\rightarrow 0$ as $t\rightarrow \infty$. This is convergent or stable behaviour.

A zero pole $p=0$ yields an output $e^{pt}=e^{0\cdot t}=e^0=1$ which is a constant output (which is considered stable).

A purely imaginary pole $p=k i$ is, via Euler Formula, corresponds to oscillatory behaviour:

$e^{pt}=e^{ikt}=\cos(kt)+i\sin(kt)$.

A genuinely complex pole $p=a+ik$ ($a>0$) with a positive real part (in the right-half plane) corresponds to the following output

$e^{pt}=e^{(a+ik)t}=e^{at+ikt}=e^{at}e^{ikt}=e^{at}(\cos(kt)+i\sin(kt))$.

While the $e^{ikt}$ component is oscillations, the $e^{at}$ goes to zero so we get behaviour that looks like:

Note that this behaviour is unstable.

When we have complex pole $p=-a+ik$ ($a>0$) with a strictly negative real part (a pole in the left-half plane), then we have $e^{pt}=e^{-at}(\cos(kt)+i\sin(kt))$. Note that we have oscillations but modulated by a decreasing $e^{-at}$. This is the motion of an underdamped harmonic oscillator:

This is inherently stable behaviour.

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.

## Continuous Assessment Summary

The results of the individual components of the CA are lower down — in particular Quiz 11.

Unless you won or came second in the league, you are identified by the last four digits of your student number. QPP is your QPP out of 20, MM are your Maple Marks out of 100 (Maple is worth 10% so say 80% gives you 8 Marks for your final grade). GPP is your Gross Percentage Points for CA (out of 30). PP is your Passing Percentage for the final paper — what you need on the final paper to pass. Finally FP is your First Percentage — what you need on the final paper to get a first: 70%.

 S/N QPP MM GPP PP FP Kelliher 20.0 93.2 29.3 15.3 58.1 Kiely 19.8 94.3 29.2 15.4 58.2 3281 19.9 89.8 28.9 15.9 58.7 5527 20.0 88.6 28.9 15.9 58.8 8416 19.7 84.1 28.1 17.0 59.8 8403 18.1 89.8 27.1 18.5 61.3 4198 18.5 85.2 27.0 18.5 61.4 6548 18.5 79.5 26.4 19.4 62.3 1864 17.2 86.0 25.8 20.3 63.1 8478 17.2 83.5 25.5 20.7 63.6 8556 17.5 76.1 25.1 21.3 64.2 7878 16.6 77.0 24.3 22.4 65.3 5546 16.1 77.3 23.8 23.1 66.0 2567 15.2 84.1 23.6 23.4 66.3 8603 15.7 77.3 23.4 23.7 66.6 1852 14.4 88.6 23.3 23.9 66.8 8455 14.4 78.4 22.2 25.4 68.3 2859 14.0 78.4 21.8 25.9 68.8 7950 10.4 63.4 16.7 33.2 76.1 4775 9.2 75.0 16.7 33.3 76.1 9464 8.6 60.0 14.6 36.2 79.1 7209 5.5 45.0 10.0 42.8 85.7 5553 1.3 45.0 5.8 48.9 91.7

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.

## Continuous Assessment

You are identified by the last four digits of your student number unless you are winning the league. The individual quiz marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes — now this includes missed quizzes — before I was doing the best non-zero but now I am including the zeros if a zero is in your best eight), MPP is your Maple Percentage Points for the biweekly lab, MT your mark on the Maple Test and MM your Maple Marks (as a percentage). GPP is your Gross Percentage Points (for best eight quizzes and Maple). Most of the columns are rounded but column 11, for quiz ten, is correct — as is GPP.

The Maple Test went very poorly… I have sent ye on worked solutions — please see the Remarks therein.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP MT MM GPP Kelliher 3 3 3 3 2 3 3 3 3 2.5 100 20.0 7.5 1.8 93.2 29.3 Kiely 2 3 2 3 3 3 2 3 2 2.4 98 19.7 7.5 1.9 94.3 29.1 5527 2 3 3 3 3 3 3 3 3 2.4 100 20.0 7.5 1.4 88.6 28.9 3281 2 3 3 3 2 3 3 3 2 2.4 99 19.8 7.5 1.5 89.8 28.8 8416 2 1 2 3 3 1 2 3 3 2.5 97 19.4 7.5 0.9 84.1 27.8 8403 2 1 2 3 1 2 2 3 2 1.8 89 17.8 7.5 1.5 89.8 26.8 6548 2 1 2 3 2 2 2 3 3 2.4 91 18.2 7.5 0.5 79.5 26.1 4198 0 1 2 3 2 3 3 3 2 1.1 86 17.2 7.5 1.0 85.2 25.7 8478 2 2 2 2 0 3 2 2 2 1.8 85 17.0 7.5 0.9 83.5 25.3 1864 1 2 3 2 1 1 2 3 1 2 82 16.4 7.5 1.1 86.0 25.0 7878 2 2 2 2 2 2 2 2 2 0.6 83 16.6 7.5 0.2 77.0 24.3 8556 1 1 2 2 0 2 3 3 1 2.1 81 16.3 7.5 0.1 76.1 23.9 2567 2 2 2 1 2 2 2 2 2 1.1 76 15.2 7.5 0.9 84.1 23.6 8603 1 2 2 2 0 2 3 2 2 0.8 78 15.7 7.5 0.2 77.3 23.4 1852 1 1 2 2 2 2 1 1 2 2.3 72 14.3 7.5 1.4 88.6 23.2 5546 0 0 1 1 2 2 3 2 3 2.2 73 14.6 7.5 0.2 77.3 22.3 8455 0 1 1 1 2 2 2 1 2 1.5 64 12.8 7.5 0.3 78.4 20.6 2859 2 1 0 1 2 2 2 3 0 0.5 61 12.1 7.5 0.3 78.4 19.9 7950 0 0 1 0 2 1 3 3 0 1.3 51 10.1 6 0.3 63.4 16.4 4775 1 0 1 0 0 1 2 2 1 0.4 44 8.7 7.5 0.0 75.0 16.2 9464 1 1 2 1 1 1 2 0 0 0 43 8.6 6 0.0 60.0 14.6 7209 2 1 2 1 0 0 0 0 0 0 28 5.5 4.5 0.0 45.0 10.0 5553 0 1 0 0 0 0 0 0 0 0 7 1.3 4.5 0.0 45.0 5.8

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.

## Continuous Assessment

You are identified by the last four digits of your student number unless you are winning the league. The individual quiz marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes — now this includes missed quizzes — before I was doing the best non-zero but now I am including the zeros if a zero is in your best eight), MPP is your Maple Percentage Points and the GPP is your Gross Percentage Points (for best eight quizzes and Maple). Most of the columns are rounded but column nine, for quiz eight, is correct.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP GPP Kelliher 3 3 3 3 2 3 3 3 2.5 100 20.0 7.5 27.5 Casey 2 3 3 3 3 3 3 3 2.5 100 20.0 7.5 27.5 3281 2 3 3 3 2 3 3 3 2.4 99 19.8 7.5 27.3 8335 2 3 2 3 3 3 2 3 2.3 98 19.6 7.5 27.1 8416 2 1 2 3 3 1 2 3 2.5 91 18.1 7.5 25.6 8403 2 1 2 3 1 2 2 3 2.4 87 17.4 7.5 24.9 6548 2 1 2 3 2 2 2 3 2.5 87 17.3 7.5 24.8 4198 0 1 2 3 2 3 3 3 2.3 86 17.2 7.5 24.7 8478 2 2 2 2 0 3 2 2 2 84 16.9 7.5 24.4 7878 2 2 2 2 2 2 2 2 2.2 83 16.6 7.5 24.1 1864 1 2 3 2 1 1 2 3 1.15 78 15.6 7.5 23.1 8603 1 2 2 2 0 2 3 2 2.3 77 15.5 7.5 23.0 2567 2 2 2 1 2 2 2 2 1.9 76 15.2 7.5 22.7 8556 1 1 2 2 0 2 3 3 1.2 74 14.8 7.5 22.3 1852 1 1 2 2 2 2 1 1 2.1 65 12.9 7.5 20.4 5546 0 0 1 1 2 2 3 2 2.5 62 12.4 7.5 19.9 8455 0 1 1 1 2 2 2 1 2.2 60 12.0 7.5 19.5 2859 2 1 0 1 2 2 2 3 0 58 11.6 7.5 19.1 7950 0 0 1 0 2 1 3 3 0 44 8.8 6 14.8 9464 1 1 2 1 1 1 2 0 0 43 8.6 6 14.6 4775 1 0 1 0 0 1 2 2 1.4 43 8.6 7.5 16.1 7209 2 1 2 1 0 0 0 0 0 28 5.5 4.5 10.0 5553 0 1 0 0 0 0 0 0 0 7 1.3 4.5 5.8

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.

## Continuous Assessment

You are identified by the last four digits of your student number unless you are winning the league. The individual quiz marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes), MPP is your Maple Percentage Points and the GPP is your Gross Percentage Points (for best eight quizzes and Maple). Most of the columns are rounded but column nine, for quiz eight, is correct.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP GPP Kelliher 2.5 2.5 2.5 2.5 2.3 2.5 2.5 2.5 99 19.8 7.5 27.3 8335 2.375 2.5 2.4 2.5 2.5 2.5 2.3 2.5 98 19.6 7.5 27.1 3281 2.4 2.5 2.5 2.5 2.1 2.5 2.5 2.5 98 19.5 7.5 27.0 5527 1.9 2.5 2.5 2.5 2.5 2.5 2.5 2.5 97 19.4 7.5 26.9 4198 0 1.2 2 2.5 1.7 2.5 2.5 2.5 85 14.9 7.5 22.4 8478 1.9 1.72 2.15 2.4 0 2.5 2 2.2 85 14.9 7.5 22.4 8416 2.3 1 2.2 2.5 2.5 1.2 2.4 2.5 83 16.6 7.5 24.1 8403 1.85 1.4 2.45 2.5 1.3 2.4 1.9 2.5 82 16.3 7.5 23.8 7878 2.15 2.2 2.25 1.9 2.1 1.5 1.9 1.9 80 15.9 7.5 23.4 6548 1.55 1 2.25 2.5 1.6 2.3 2.1 2.5 79 15.8 7.5 23.3 1864 1.3 2.4 2.5 2.1 1.24 1.1 2.4 2.5 78 15.5 7.5 23.0 8556 1.35 0.6 2.4 2.2 0 2 2.5 2.5 77 13.6 7.5 21.1 8603 0.6 1.8 1.85 2.4 0 2.4 2.5 1.6 75 13.2 7.5 20.7 2567 1.55 1.5 2.05 1.28 1.8 2.1 2.1 2.2 73 14.6 7.5 22.1 7950 0 0 0.8 0 1.8 1.2 2.5 2.5 70 8.8 6 14.8 2859 2.4 1 0 0.5 1.6 1.7 1.9 2.5 66 11.6 7.5 19.1 5546 0 0 1.1 1 1.9 1.5 2.5 1.9 66 9.9 7.5 17.4 1852 0.7 1.2 1.6 1.5 2.1 2.2 0.9 1.3 58 11.5 7.5 19.0 8455 0.2 0.9 1.05 0.7 1.9 2.2 2 1 50 10.0 7.5 17.5 9464 1.2 0.9 1.5 0.9 1.24 0.8 2.1 0 49 8.6 6 14.6 7209 1.9 1.2 1.75 0.5 0.16 0 0 0 44 5.5 4.5 10.0 4775 0.5 0.3 1 0 0.4 1.2 1.9 1.9 41 7.2 7.5 14.7 5553 0.4 0.8 0 0 0.1 0 0 0 17 1.3 4.5 5.8

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.

## Continuous Assessment

You are identified by the last four digits of your student number unless you are winning the league. The individual quiz marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes), MPP is your Maple Percentage Points and the GPP is your Gross Percentage Points (for best eight quizzes and Maple). Most of the columns are rounded but column eight, for quiz seven, is correct.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP GPP Kelliher 3 3 3 3 2 3 2.5 99 17.3 6 23.3 8335 2 3 2 3 3 3 2.3 98 17.1 6 23.1 3281 2 3 3 3 2 3 2.5 97 17.0 6 23.0 5527 2 3 3 3 3 3 2.5 97 16.9 6 22.9 8478 2 2 2 2 0 3 2 84 12.7 6 18.7 4198 0 1 2 3 2 3 2.5 83 12.4 6 18.4 8416 2 1 2 3 3 1 2.4 81 14.1 6 20.1 7878 2 2 2 2 2 2 1.9 80 14.0 6 20.0 8403 2 1 2 3 1 2 1.9 79 13.8 6 19.8 8603 1 2 2 2 0 2 2.5 77 11.6 6 17.6 6548 2 1 2 3 2 2 2.1 76 13.3 6 19.3 1864 1 2 3 2 1 1 2.4 75 13.0 6 19.0 8556 1 1 2 2 0 2 2.5 74 11.1 6 17.1 2567 2 2 2 1 2 2 2.1 71 12.4 6 18.4 5546 0 0 1 1 2 2 2.5 64 8.0 6 14.0 7950 0 0 1 0 2 1 2.5 63 6.3 4.5 10.8 2859 2 1 0 1 2 2 1.9 61 9.1 6 15.1 1852 1 1 2 2 2 2 0.9 58 10.2 6 16.2 8455 0 1 1 1 2 2 2 51 9.0 6 15.0 9464 1 1 2 1 1 1 2.1 49 8.6 6 14.6 7209 2 1 2 1 0 0 0 44 5.5 4.5 10.0 4775 1 0 1 0 0 1 1.9 35 5.3 6 11.3 5553 0 1 0 0 0 0 0 17 1.3 4.5 5.8

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.

## Note: Simpson’s Rule Example, p.5 (b), Second Integral

Yes, the answer in the notes WAS wrong. The answer should be close to 0.5870 (as we found on the board.)

## Continuous Assessment

You are identified by the last four digits of your student number unless you are winning the league. The individual quiz marks are out of 2.5 percentage points. Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes), MPP is your Maple Percentage Points and the GPP is your Gross Percentage Points (for best eight quizzes and Maple). Most of the columns are rounded but column seven, for quiz six, is correct.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP GPP Kelliher 3 3 3 3 2 2.5 99 14.8 4.5 19.3 8335 2 3 2 3 3 2.5 99 14.8 4.5 19.3 3281 2 3 3 3 2 2.5 97 14.5 4.5 19.0 5527 2 3 3 3 3 2.5 96 14.4 4.5 18.9 8478 2 2 2 2 0 2.5 85 10.7 4.5 15.2 7878 2 2 2 2 2 1.5 81 12.1 4.5 16.6 8403 2 1 2 3 1 2.4 79 11.9 4.5 16.4 4198 0 1 2 3 2 2.5 79 9.9 4.5 14.4 8416 2 1 2 3 3 1.2 78 11.7 4.5 16.2 6548 2 1 2 3 2 2.3 75 11.2 4.5 15.7 8603 1 2 2 2 0 2.4 72 9.1 3 12.1 1864 1 2 3 2 1 1.1 71 10.6 4.5 15.1 2567 2 2 2 1 2 2.1 69 10.3 4.5 14.8 8556 1 1 2 2 0 2 68 8.6 4.5 13.1 1852 1 1 2 2 2 2.2 62 9.3 4.5 13.8 2859 2 1 0 1 2 1.7 58 7.2 4.5 11.7 5546 0 0 1 1 2 1.5 55 5.5 4.5 10.0 7950 0 0 1 0 2 1.2 51 3.8 3 6.8 8455 0 1 1 1 2 2.2 46 7.0 4.5 11.5 7209 2 1 2 1 0 0 44 5.5 4.5 10.0 9464 1 1 2 1 1 0.8 44 6.5 4.5 11.0 4775 1 0 1 0 0 1.2 27 3.4 4.5 7.9 5553 0 1 0 0 0 0 17 1.3 4.5 5.8