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 five, for quiz four, is correct.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 R % QPP MPP GPP Kelliher 3 3 3 2.5 100 10.0 3 13.0 3281 2 3 3 2.5 99 9.9 3 12.9 8335 2 3 2 2.5 98 9.8 3 12.8 5527 2 3 3 2.5 94 9.4 3 12.4 7878 2 2 2 1.9 85 8.5 3 11.5 1864 1 2 3 2.1 83 8.3 3 11.3 8403 2 1 2 2.5 82 8.2 3 11.2 8478 2 2 2 2.4 82 8.2 3 11.2 8416 2 1 2 2.5 80 8.0 3 11.0 4198 0 1 2 2.5 76 5.7 3 8.7 6548 2 1 2 2.5 73 7.3 3 10.3 8603 1 2 2 2.4 67 6.7 3 9.7 8556 1 1 2 2.2 66 6.6 3 9.6 2567 2 2 2 1.28 64 6.4 3 9.4 7209 2 1 2 0.5 54 5.4 3 8.4 2859 2 1 0 0.5 52 3.9 3 6.9 1852 1 1 2 1.5 50 5.0 3 8.0 9464 1 1 2 0.9 45 4.5 3 7.5 5546 0 0 1 1 42 2.1 3 5.1 7950 0 0 1 0 32 0.8 1.5 2.3 8455 0 1 1 0.7 29 2.9 3 5.9 5553 0 1 0 0 24 1.2 3 4.2 4775 1 0 1 0 24 1.8 3 4.8

The solutions to three of the quiz questions may be found in the notes. The only other question is done here: Use partial fractions to decompose $\displaystyle \frac{11x-4}{x^2+2x-8}$. I give two worked solutions with explanations. Standard Method The first thing we must do is factorise the ‘bottom’ (denominator), $x^2+2x-8$. We multiply $(1)(-8)=-8$ and look at the factors of eight which are $\{(1,8),(2,4)\}$ and we can use $+4$ and $-2$ to rewrite $2x$: $\displaystyle x^2+2x-8=x^2+4x-2x-8=x(x+4)-2(x+4)=(x+4)(x-2).$

Now to each factor in the bottom we can associate a term in the partial fraction expansion. Here we have distinct linear factors, which is Rule II: $\displaystyle \frac{11x-4}{(x+4)(x-2)}\overset{!}{=}\frac{A}{x+4}+\frac{B}{x-2}$ $\displaystyle =\frac{A}{x+4}\cdot\frac{x-2}{x-2}+\frac{B}{x-2}\cdot \frac{x+4}{x+4}=\frac{A(x-2)+B(x+4)}{(x-2)(x+4)}$.

Therefore, for the partial fraction expansion we need $\displaystyle 11x-4=A(x-2)+B(x+4)$ for all values of $\displaystyle x$ (the bottoms are the same).

If they are equal for all values of $\displaystyle x$ we can input values of $\displaystyle x$ into both of them to generate equations in $\displaystyle A$ and $\displaystyle B$. For example, $\displaystyle x=2$: $\displaystyle 11(2)-4=A(0)+B(2+4)$ $\displaystyle \Rightarrow 18=6B$ $\displaystyle \Rightarrow B=3$,

and $\displaystyle x=-4$, $\displaystyle 11(-4)-4=A(-4-2)+B(0)$ $\displaystyle \Rightarrow -48=-6A$ $\displaystyle \Rightarrow A=8$, $\displaystyle \frac{11x-4}{x^2+2x-8}=\frac{8}{x+4}+\frac{3}{x-2}$.

Cover Up Method

Because there is no $\displaystyle x^2$ in the factorisation I can use the Cover Up Method. $\displaystyle \frac{11x-4}{(x+4)(x-2)}=\frac{\frac{11(-4)-4}{-4-2}}{x+4}+\frac{\frac{11(2)-4}{2+4}}{x-2}$ $\displaystyle =\frac{8}{x+4}+\frac{3}{x-2}$.

## Quiz 5 Question Bank

The question bank for Quiz 5 (in Week 7) is as follows:

• P. 58, Q. 1(iii) & (iv)*, 3, 4, 5** & 6

* there is a typo in the solution. The second term should be $8x^3yz$.

** there is a typo in the answer to Q.5(iii) – it should be $\displaystyle 2ye^{2xy}$ rather than $\displaystyle ye^{xy}$. The final answers will not be given on the quiz paper and neither is there any 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 5 runs from 19:15 to 19:30 sharp on Wednesday 18 March.

I am waiting for an update on the issue of day students taking over the evening session.

## Week 6

No classes next week but we will have a Maple night on April 1 in the normal rooms.

## Maple Labs

We have a Maple Lab Wednesday 18 March. 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 5

In Week 5 we finished off Partial Differentiation and used it to look at Error Analysis.

## Week 7

In Week 7 we will finish off Error Analysis and look at some numerical methods of solving equations and evaluating integrals. In Maple, we will look at Partial Differentiation and Error Analysis.

## 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.