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This strategy is by no means optimal nor exhaustive. It is for students who are struggling with basic integration and anti-differentiation and need something to help them start calculating straightforward integrals and finding anti-derivatives.

TL;DR: The strategy to antidifferentiate a function $f$ that I present is as follows:

1. Direct
2. Manipulation
3. $u$-Substitution
4. Parts

Quadratics are ubiquitous in mathematics. For the purposes of this piece a quadratic is a real-valued function $q:\mathbb{R}\rightarrow \mathbb{R}$ of the form

$q(x)=ax^2+bx+c$,

where $a,\,b,\,c\in \mathbb{R}$ such that $a\neq 0$. There is a little bit more to be said — particularly about the differences between a quadratic and a quadratic function but for those this piece is addressed to (third level: non-maths; all second level), the distinction is unimportant.

## Geometry

The basic object we study is the square function, $s:\mathbb{R}\rightarrow \mathbb{R}$, $x\mapsto x^2$:

All quadratics look similar to $x^2$. If $a>0$ then the quadratic has this $\bigcup$ geometry. Otherwise it looks like $y=-x^2$ and has $\bigcap$ geometry

The geometry dictates that quadratics can have either zero, one or two real roots. A root of a function is an input $x$ such that $f(x)=0$. As the graph of a function is of the form $y=f(x)$, roots are such that $y=f(x)=0\Rightarrow y=0$, that is where the graph cuts the $x$-axis. With the geometry of quadratics they can cut the $x$-axis no times, once (like $s(x)=x^2$), or twice.

There are a number of ways of explaining why you cannot divide by zero. Here are my two favourites.

## Any Set of Numbers Collapses to a Single Number

How old are you? Zero years old.

How tall are you? Zero metres old.

How many teeth do you have? Zero.

How many Superbowls has Tom Brady won? Zero

Yep, if you allow division by zero you only end up with one number to measure everything with.

## Quiz 8 Results

Below find the results.  Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes). The QPP is your Quiz Percentage Points (a max of 25 at the moment). MPP is your Maple percentage points. GPP is your gross percentage points.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 R% QPP MPP GPP/25 1298 1.6 1.2 1.6 49.5 9.9 5.63 15.53 2327 1.4 1 2.2 72.1 14.42 7.50 21.92 2073 2.5 2.5 2.5 92.5 18.49 7.50 25.99 8332 2.5 2.5 0.8 72.0 14.39 7.50 21.89 9464 2.5 0 2.5 80.4 16.07 7.50 23.57 3872 1.4 0 0 26.8 5.36 5.63 10.99 1480 2.3 2.5 2.5 91.9 18.37 7.50 25.87 2237 1.4 1.6 0.9 39.0 7.79 7.50 15.29 2070 2.5 2.5 2.2 73.0 14.6 7.50 22.10 3481 2.3 2.5 2.1 69.3 13.85 7.50 21.35 3703 2.5 0.3 2.5 68.8 13.76 7.50 21.26 7879 2.4 2.5 2.2 72.1 14.42 7.50 21.92 2257 1.8 1.3 1.9 53.4 10.67 7.50 18.17 8425 2.4 1.3 1.5 50.2 10.03 7.50 17.53 9896 0 0 0 9.9 1.97 0.00 1.97 8212 2.5 1.5 2.5 92.2 18.44 7.50 25.94 1486 0 0 0 35.7 7.14 1.88 9.02 5484 2.5 2.5 0 76.7 15.33 7.50 22.83 1321 1.9 1.5 2 51.2 10.23 7.50 17.73 2128 1.5 1.6 0.6 67.1 13.42 7.50 20.92 2942 2.5 1.4 1 68.3 13.65 7.50 21.15 6645 2.5 2.5 1.9 95.4 19.07 7.50 26.57 8426 2.5 2.5 2.5 67.2 13.44 5.63 19.07 2092 1.6 1 0 14.0 2.8 5.63 8.43 8354 1.9 0.5 0 24.6 4.91 7.50 12.41

## Quiz 7 Results

Below find the results.  Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes). The QPP is your Quiz Percentage Points (a max of 25 at the moment). MPP is your Maple percentage points. GPP is your gross percentage points.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 Q7 R% QPP MPP GPP/25 1298 1 0.8 1.6 1.2 47.4 8.3 5.63 13.93 2327 3 1.9 1.4 1 69.8 12.22 7.50 19.72 2073 3 2.5 2.5 2.5 91.4 15.99 7.50 23.49 8332 1 1.9 2.5 2.5 77.7 13.59 7.50 21.09 9464 2 2.1 2.5 0 77.5 13.57 7.50 21.07 3872 1 0 1.4 0 30.6 5.36 5.63 10.99 1480 2 2.5 2.3 2.5 90.7 15.87 7.50 23.37 2237 0 1.2 1.4 1.6 39.4 6.89 7.50 14.39 2070 2 2.5 2.5 2.5 70.9 12.4 7.50 19.90 3481 2 2.4 2.3 2.5 67.1 11.75 7.50 19.25 3703 1 2.3 2.5 0.3 64.3 11.26 7.50 18.76 7879 2 1.5 2.4 2.5 69.8 12.22 7.50 19.72 2257 1 1.6 1.8 1.3 50.1 8.77 7.50 16.27 8425 1 1.8 2.4 1.3 48.7 8.53 7.50 16.03 9896 0 0 0 0 11.3 1.97 0.00 1.97 8212 3 2.5 2.5 1.5 91.1 15.94 7.50 23.44 1486 0 2.5 0 0 40.8 7.14 1.88 9.02 5484 1 1.9 2.5 2.5 87.6 15.33 7.50 22.83 1321 1 1.2 1.9 1.5 47 8.23 7.50 15.73 2128 3 2.3 1.5 1.6 73.3 12.82 7.50 20.32 2942 2 1.8 2.5 1.4 72.3 12.65 7.50 20.15 6645 3 2.5 2.5 2.5 98.1 17.17 7.50 24.67 8426 0 1.2 2.5 2.5 62.5 10.94 5.63 16.57 2092 0 0 1.6 1 16 2.8 5.63 8.43 8354 0 1.7 1.9 0.5 28.1 4.91 7.50 12.41

## Quiz 6 Results

Below find the results.  Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes). The QPP is your Quiz Percentage Points (a max of 19.5 at the moment). MPP is your Maple percentage points. GPP is your gross percentage points.

 S/N Q1 Q2 Q3 Q4 Q5 Q6 R% QPP MPP GPP/19.5 1298 0.6 0.8 1.6 47.3 7.1 3 10.1 2327 2.5 1.9 1.4 74.8 11.2 4.5 15.72 2073 2.5 2.5 2.5 89.9 13.5 4.5 17.99 8332 0.6 1.9 2.5 73.9 11.1 4.5 15.59 9464 1.8 2.1 2.5 90.5 13.6 4.5 18.07 4402 0 0 0 0 0 0 0 3872 1.2 0 1.4 35.7 5.36 4.5 9.86 1480 1.5 2.5 2.3 89.1 13.4 4.5 17.87 2237 0 1.2 1.4 35.3 5.29 4.5 9.79 2070 2.3 2.5 2.5 66 9.9 4.5 14.4 3481 1.5 2.4 2.3 61.7 9.25 4.5 13.75 3703 1.4 2.3 2.5 73.1 11 4.5 15.46 7879 2.2 1.5 2.4 64.8 9.72 4.5 14.22 2257 1.2 1.6 1.8 49.8 7.47 4.5 11.97 8425 1 1.8 2.4 48.2 7.23 4.5 11.73 9896 0 0 0 13.1 1.97 0 1.97 8212 2.5 2.5 2.5 96.3 14.4 4.5 18.94 1486 0 2.5 0 47.6 7.14 1.5 8.64 5484 1 1.9 2.5 85.5 12.8 4.5 17.33 1321 1.02 1.2 1.9 44.9 6.73 4.5 11.23 2128 2.5 2.3 1.5 74.8 11.2 4.5 15.72 2942 1.8 1.8 2.5 75 11.3 4.5 15.75 6645 2.5 2.5 2.5 97.8 14.7 4.5 19.17 8426 0 1.2 2.5 56.3 8.44 3 11.44 2092 0.2 0 1.6 12 1.8 3 4.8 8354 0.2 1.7 1.9 29.4 4.41 4.5 8.91

## Quiz 5 Results

Below find the results.  Your best eight quizzes go to the 20% mark for quizzes. The R % column is your running percentage (for best eight quizzes). The QPP is your Quiz Percentage Points. MPP is your Maple percentage points. GPP is your gross percentage points.

 S/N Q1 Q2 Q3 Q4 Q5 R% QPP MPP GPP O’Sullivan, K. 2.17 2.5 2.5 2.5 2.5 97.4 12.2 3 15.2 Murphy 2.37 2.5 2.07 2.5 2.5 95.5 11.9 3 14.9 Connolly 2.17 2.5 2.5 1.8 2.1 88.6 11.1 3 14.1 Fahey 2.5 2.5 2.07 1.5 2.5 88.6 11.1 3 14.1 2073 0.99 2.5 2.5 2.5 2.5 87.9 11 3 14 5484 2.43 2.5 2.5 1 1.9 82.6 10.3 3 13.3 2327 2.5 1.72 1.2 2.5 1.9 78.6 9.82 3 12.8 2128 2.24 0.4 2.28 2.5 2.3 77.8 9.72 3 12.7 2942 1.71 1.7 1.74 1.8 1.8 70 8.75 3 11.8 8332 2.37 1.22 2.5 0.6 1.9 68.7 8.59 3 11.6 3703 1.71 1.2 1.85 1.4 2.3 67.7 8.46 3 11.5 2070 0.33 0.2 2.07 2.3 2.5 59.2 7.4 3 10.4 7879 1.12 2.5 0 2.2 1.5 58.6 7.32 3 10.3 1486 2.24 2.4 0 0 2.5 57.1 7.14 1.5 8.64 3481 1.71 0.3 1.04 1.5 2.4 55.6 6.95 3 9.95 8426 2.24 2.5 0 0 1.2 47.5 5.94 1.5 7.44 2257 0.53 0.6 1.74 1.2 1.6 45.4 5.67 3 8.67 1298 0.46 1.9 1.74 0.6 0.8 44 5.5 1.5 7 1321 1.45 0.4 0.76 1.02 1.2 38.6 4.83 3 7.83 8425 0 0.4 1.63 1 1.8 38.6 4.83 3 7.83 3872 0.26 0 2.5 1.2 0 31.7 3.96 3 6.96 2237 1.71 0 0.98 0 1.2 31.1 3.89 3 6.89 8354 0.39 0 0.22 0.2 1.7 20.1 2.51 3 5.51 9896 1.97 0 0 0 0 15.8 1.97 0 1.97 7209 0.39 0 0 0 0 3.12 0.39 0 0.39 2092 0 0 0 0.2 0 1.6 0.2 1.5 1.7 4402 0 0 0 0 0 0 0 0 0

## Quiz 4 Results

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). The QPP is your Quiz Percentage Points. MPP is your Maple percentage points. GPP is your gross percentage points. One student didn’t write down their name for Quiz 4. They should email me.

 S/N Q1 Q2 Q3 Q4 R% QPP MPP GPP O’Sullivan, K. 2.17 2.5 2.5 2.5 96.7 9.67 3 12.7 Murphy 2.37 2.5 2.07 2.5 94.4 9.44 3 12.4 Connolly 2.17 2.5 2.5 1.8 89.7 8.97 3 12 1480 2.5 2.5 2.07 1.5 85.7 8.57 3 11.6 2073 0.99 2.5 2.5 2.5 84.9 8.49 3 11.5 5484 2.43 2.5 2.5 1 84.3 8.43 3 11.4 2327 2.5 1.72 1.2 2.5 79.2 7.92 3 10.9 2942 1.71 1.7 1.74 1.8 69.5 6.95 3 9.95 8332 2.37 1.22 2.5 0.6 66.9 6.69 3 9.69 7879 1.12 2.5 0 2.2 58.2 5.82 3 8.82 2128 2.24 0.4 2.28 0 49.2 4.92 3 7.92 2070 0.33 0.2 2.07 2.3 49 4.9 3 7.9 3703 1.71 1.2 1.85 0 47.6 4.76 3 7.76 8426 2.24 2.5 0 0 47.4 4.74 1.5 6.24 1298 0.46 1.9 1.74 0.6 47 4.7 1.5 6.2 1486 2.24 2.4 0 0 46.4 4.64 1.5 6.14 3481 1.71 0.3 1.04 1.5 45.5 4.55 3 7.55 2257 0.53 0.6 1.74 1.2 40.7 4.07 3 7.07 3872 0.26 0 2.5 1.2 39.6 3.96 3 6.96 1321 1.45 0.4 0.76 1.02 36.3 3.63 3 6.63 8425 0 0.4 1.63 1 30.3 3.03 3 6.03 2237 1.71 0 0.98 0 26.9 2.69 3 5.69 9896 1.97 0 0 0 19.7 1.97 0 1.97 8354 0.39 0 0.22 0.2 8.1 0.81 3 3.81 9555 0.53 0 0 0 5.3 0.53 0 0.53 7209 0.39 0 0 0 3.9 0.39 0 0.39 2092 0 0 0 0.2 2 0.2 1.5 1.7 4402 0 0 0 0 0 0 0 0

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.

## Quiz 3 Results

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). The QPP is your Quiz Percentage Points. MPP is your Maple percentage points. GPP is your gross percentage points. One student didn’t write down their name for Quiz 3. They should email me.

 S/N Q1 Q2 Q3 R% QPP MPP GPP Nolan 2.43 2.5 2.5 99.1 7.43 1.5 8.93 Connolly 2.17 2.5 2.5 95.6 7.17 1.5 8.67 O’Sullivan, K. 2.17 2.5 2.5 95.6 7.17 1.5 8.67 1480 2.5 2.5 2.07 94.3 7.07 1.5 8.57 1486 2.24 2.4 92.8 4.64 1.5 6.14 8212 2.37 2.5 2.07 92.5 6.94 1.5 8.44 8332 2.37 1.22 2.5 81.2 6.09 1.5 7.59 2073 0.99 2.5 2.5 79.9 5.99 1.5 7.49 2327 2.5 1.72 1.2 72.3 5.42 1.5 6.92 2942 1.71 1.7 1.74 68.7 5.15 1.5 6.65 2128 2.24 0.4 2.28 65.6 4.92 1.5 6.42 3703 1.71 1.2 1.85 63.5 4.76 1.5 6.26 8426 2.24 2.5 0 63.2 4.74 1.5 6.24 1298 0.46 1.9 1.74 54.7 4.1 0 4.1 7879 1.12 2.5 0 48.3 3.62 1.5 5.12 3481 1.71 0.3 1.04 40.7 3.05 1.5 4.55 2257 0.53 0.6 1.74 38.3 2.87 1.5 4.37 3872 0.26 0 2.5 36.8 2.76 1.5 4.26 2237 1.71 0 0.98 35.9 2.69 1.5 4.19 1321 1.45 0.4 0.76 34.8 2.61 1.5 4.11 2070 0.33 0.2 2.07 34.7 2.6 1.5 4.1 9896 1.97 0 0 26.3 1.97 0 1.97 8354 0.39 0 0.22 8.13 0.61 1.5 2.11 8425 0 0.4 8 0.4 1.5 1.9 9555 0.53 0 0 7.07 0.53 0 0.53 9951 0.53 0 0 7.07 0.53 0 0.53 7209 0.39 0 0 5.2 0.39 0 0.39 4402 0 0 0 0 0 0 0 2092 0 0 0 0 0 0 0

## Maple Labs

The Maple Lab takes place in C128.

The 15 individuals who responded to the poll and wanted to start at 18:00 (below) are guaranteed a spot in the early sitting. If there is space for more they can be accommodated, otherwise they have to start at 19:15 and do Maple at 20:55. Note there is a smaller class for the later Maple. If you are struggling with Maple you could consider switching to the later class to get more attention.

The following is the proposed schedule for Weeks 3, 5, 7, 8, 10, 11:

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

Wednesdays 18:00-19:05 – Maple Lab in room C128

Wednesdays 19:15-19:30 – Weekly Quiz in C212

Wednesdays 19:30-20.50 – Theory class in room C128

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

Wednesdays 19:15-19:30 – Weekly Quiz in C128

Wednesdays 19:30-20:50 – Theory class in room C212

Wednesdays 20.55-22:00 – Maple Lab in room C128

In weeks where there is no Maple Lab, the quiz will take place from 19:00-19:15 sharp.

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.

## Quiz 2 Results

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). The QPP is your Quiz Percentage Points. MPP is your Maple percentage points. GPP is your gross percentage points. One student didn’t write down their name for Quiz 1. They should email me.

 S/N Q1 Q2 R% QPP MPP GPP Fahey 2.5 2.5 100 5 1.5 6.5 Nolan 2.43 2.5 98.6 4.93 1.5 6.43 B Murphy 2.37 2.5 97.4 4.87 1.5 6.37 8426 2.24 2.5 94.8 4.74 1.5 6.24 9464 2.17 2.5 93.4 4.67 1.5 6.17 6645 2.17 2.5 93.4 4.67 1.5 6.17 1486 2.24 2.4 92.8 4.64 1.5 6.14 2327 2.5 1.72 84.4 4.22 1.5 5.72 7879 1.12 2.5 72.4 3.62 1.5 5.12 8332 2.37 1.22 71.8 3.59 1.5 5.09 2073 0.99 2.5 69.8 3.49 1.5 4.99 2942 1.71 1.7 68.2 3.41 1.5 4.91 3703 1.71 1.2 58.2 2.91 1.5 4.41 2128 2.24 0.4 52.8 2.64 1.5 4.14 3481 1.71 0.3 40.2 2.01 1.5 3.51 9896 1.97 0 39.4 1.97 0 1.97 1298 0 1.9 38 1.9 0 1.9 1321 1.45 0.4 37 1.85 1.5 3.35 2237 1.71 0 34.2 1.71 1.5 3.21 2257 0.53 0.6 22.6 1.13 1.5 2.63 9555 0.53 0 10.6 0.53 0 0.53 9951 0.53 0 10.6 0.53 0 0.53 2070 0.33 0.2 10.6 0.53 1.5 2.03 8425 0 0.4 8 0.4 1.5 1.9 7209 0.39 0 7.8 0.39 0 0.39 8354 0.39 0 7.8 0.39 1.5 1.89 3872 0.26 0 5.2 0.26 1.5 1.76 4402 0 0 0 0 0 0 2092 0 0 0 0 0 0