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

Any students who missed a Maple lab are invited to do a double lab.

Here we do the Question 1 from the Quiz.

### 1.

In measuring a quantity $u$, the formula

$\displaystyle u=\frac{8x}{4x+y}$,

was used. Estimate a range of values of $u$ where the values of $x$ and $y$ were measured to be one and four with maximum errors of $0.04$ and $0.08$ respectively.

Solution: We think that

$\displaystyle u=u(1,4)=\frac{8(1)}{4(1)+4}=\frac{8}{8}=1$.

We estimate the error in this calculation of $u$ due to the errors in the measurements using the differential, $du'$:

$\displaystyle\Delta u\approx du'=\left|\frac{\partial u}{\partial x}\right|\Delta x+\left|\frac{\partial u}{\partial y}\right|\Delta y$.

Hence we need to calculate the partial derivatives of $u$. As written,

$\displaystyle u(x,y)=\frac{8x}{4x+y}$,

is a fraction as so you need the Quotient Rule to differentiate it:

$\displaystyle\frac{\partial u}{\partial x}=\frac{(4x+y)(8)-8x(4)}{(4x+y)^2}=\frac{32x+8y-32x}{(4x+y)^2}$

$\displaystyle =\frac{8y}{(4x+y)^2}$.

We want to evaluate this at the measurement $(x,y)=(1,4)$:

$\displaystyle\left.\frac{\partial u}{\partial x}\right|_{(x,y)=(1,4)}=\frac{8(4)}{(4(1)+4)}^2=\frac{1}{2}=0.5$.

However, when you are differentiating with respect to $y$, note that the top is a constant…therefore we can write

$u(x,y)=8x(4x+y)^{-1}$

and differentiate as follows using the Chain Rule:

$\displaystyle\frac{\partial u}{\partial y}=8x(-(4x+y)^{-2}(0+1))=-8x(4x+y)^{-2}$.

We want to evaluate this at the measurement:

$\left.\frac{\partial u}{\partial y}\right|_{(x,y)=(1,4)}=-8(1)(4(1)+4)^{-2}=-\frac{1}{8}=-0.125$.

Therefore we have

$\displaystyle \Delta u\approx `du'=|0.5|(0.04)+|-0.125|(0.08)$

$=(0.5)(0.04)+(+0.125)(0.08)=0.03$.

Now we present our answer and it is good practise to match the precision (# decimal places) of the calculation with that of the error:

$u=1.00\pm0.03$.

Remark: You might note that you could have originally written

$u(x,y)=8x(4x+y)^{-1}$

and used the product rule for differentiating with respect to $x$. It is a matter of taste which way you prefer to do it. This is a general situation. Suppose you have a function that is a fraction

$\displaystyle f(x)=\frac{u(x)}{v(x)}$.

Now you can use the Quotient Rule or else write

$f(x)=u(x)[v(x)]^{-1}$

and use the Product Rule.

## Quiz 7 Question Bank

The question bank for Quiz 7 (in Week ‘6’ — April 1) is as follows:

• P.75 Q. 5*, 6*, 9**
• P.84 Q. 1, 2, 4 (b), 6 (a)***, 6 (c)****

*apply the Newton-Raphson method three times — unless $x_2$ and $x_1$ agree to three decimal places (in which case you can stop).

**apply the Newton-Raphson method three times.

***approximate the first integral only

****approximate the third integral only

There is a summary of formulae on P.87. The Newton-Raphson, Trapezoidal and Simpson Rules are on the Sheet Tables. The $y_i$ of the sheet tables are the $f(x_i)$ of the formulae on P.87. You will have to learn the Midpoint Rule however.

There is no value in writing down the final answers alone — you will receive marks for full and correct solutions — but nothing for final answers without justification or skipping important steps. Please don’t learn off model solutions — you need to understand the material not just on a superficial level to do well later on. Quiz 7 runs from 19:15 to 19:30 sharp on Wednesday 1 April as it is a ‘Maple’ night.

The word on the Academic Learning Centre is that although the evening session perhaps might have been made exclusive to evening students, the fact of the matter is that they are not.

My departmental head suggested that if a group of ye want to get an improvement in your ALC experience, that ye should email questions to catherine.palmer@cit.ie in advance of the session. Dr Palmer said that this will allow her to more easily help ye.

## Maple Labs

We next have a Maple Lab Wednesday 1 April.

Group 1 – Starts at 18:00 and Finishes at 20:50:

• Wednesdays 18:00-19:05 – Maple Lab in room C219
• Wednesdays 19:15-19:30 – Weekly Quiz in C212
• Wednesdays 19:30-20.50 – Theory class in room C212

Group 2 – Starts at 19:15 and Finishes at 22:00:

• Wednesdays 19:15-19:30 – Weekly Quiz in C212
• Wednesdays 19:30-20:50 – Theory class in room C212
• Wednesdays 20.55-22:00 – Maple Lab in room C219

As you can see here you can download a student copy of Maple. Some students said that they were unable to open the file I sent in their Maple 17 — this is strange as I am actually using Maple 16!

## Week 8

In Week 8 we finished off looking at the Newton Raphson Method and learned how to numerically approximate integrals.

## Week ‘6’

In Week ‘6’ we start the final chapter on Laplace Transforms.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage. Anyone can give me exercises they have done and I will correct them. I also advise that you visit the Academic Learning Centre.

## Continuous Assessment

The Continuous Assessment is broken into Weekly Quizzes (20%) and Maple (10%). There will be eleven weekly quizzes and your eight best results will count (so 2.5% per quiz from eight quizzes). You will receive an email (i.e. this one) on Thursday/Friday detailing the examinable exercises. Maple consists of five labs and a Maple Test in the sixth lab. Satisfactory participation in labs gives you 1.5% and the Maple Test is worth 2.5%. More on this in the coming days.