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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.**

## Test 1 Results

Have been emailed to you. I will send on solutions in time.

## Week 7

In Week 7 we finished off the first chapter and began the second chapter: on Integration.

## Week 8

In Week 8 we will start evaluating some straightforward integrals using the Fundamental Theorem of Calculus. Then we will look into anti-differentiation in more detail.

**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.**

## Test 1 Results

Pending…

## Week 6

In Week 6 we did some more max/min problems and even some applied examples.

## Week 7

In Week 7 we will finish off the first chapter and begin the second chapter: on Integration.

**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.**

## Test 1

The first 15% test will take place at 9** am Tuesday 18 October in B149**** **(Week 6). You can find a sample in the course notes, after the section on rates of change. It is a test that could arguably take 42 minutes but I’ll give ye from 09.05 — 10.00. You will be given a copy of these tables. Don’t worry I’ll scribble out the “UCC”!

Note that the format will be the same of this.

- Differentiation from First Principles
- Tangent Lines
- Differentiate by Rule
- Differentiate by Rule
- Differentiate by Rule
- Rates of Change/ Geometry of Graph
- Rate of Change/ Geometry of Graph

## Week 5

In Week 5 we spoke about applications of differentiation to rates of change and finding the maximum/minimum of a function.

## Week 6

In Week 6 we will apply what we learned to optimisation problems.

## Catch-Up Tutorial for BioEng 2-B

**11:00,** **Wednesday 12 October in B245**

## Test 1

The first 15% test will take place at 9** am Tuesday 18 October in B149**** **(Week 6). You can find a sample in the course notes, after the section on rates of change I think… actually I am not 100% where it is. It is a test that could arguably take 42 minutes but I’ll give ye from 09.05 — 10.00. You will be given a copy of these tables. Don’t worry I’ll scribble out the “UCC”!

Note that the format will be the same of this.

- Differentiation from First Principles
- Tangent Lines
- Differentiate by Rule
- Differentiate by Rule
- Differentiate by Rule
- Rates of Change
- Rate of Change/ Geometry of Graph,

but this is dependent on our progress in lectures.

## Week 4

In Week 4 we covered the Chain Rule in depth.

## Week 5

In Week 5 we will finish off our Chain Rule examples then talk about applications of differentiation to rates of change.

## Test 1

The first 15% test will take place at 9** am Tuesday 18 October in B149**** **(Week 6). You can find a sample in the course notes, after the section on rates of change I think… actually I am not 100% where it is. It is a test that could arguably take 42 minutes but I’ll give ye from 09.05 — 10.00. You will be given a copy of these tables. Don’t worry I’ll scribble out the “UCC”!

Note that the format will be the same of this.

- Differentiation from First Principles
- Tangent Lines
- Differentiate by Rule
- Differentiate by Rule
- Differentiate by Rule
- Rates of Change
- Rate of Change/ Geometry of Graph,

but this is dependent on our progress in lectures.

## Week 3

In Week 3 we covered everything in the notes up to the quotient rule.

## Week 4

In Week 4 we will talk about another way of combining functions — composition — and how to differentiate them — the Chain Rule.

## Week 2

In Week 2 we learned how to differentiate from first principles and also how to differentiate sums of differentiable functions.

## Week 3

In Week 3 we will continue talking about the duality between algebra & geometry and perhaps begin to talk about differentiating some more functions.

# Introduction

In Ireland at least, we first encounter fractions at age 6-8. At this age, because of our maturity, while we might be capable of some conceptional understanding, by and large we are doing things by rote and, for example, multiplying fractions is just something that we do without ever questioning *why* fractions multiply together like that. This piece is aimed at second and third level students who want to understand why the ‘calculus’ of fractions is like it is.

Mathematicians can in a rigorous way, write down what a fraction is… this piece is pitched somewhere in between these constructions — perhaps seen in an undergraduate mathematics degree — and the presentation of fractions presented in primary school. It is closer in spirit to a rigorous approach but makes no claims at *absolute *rigour (indeed it will make no attempt at rigour in places). The facts are real number axioms.

# Defining Fractions

We will define fractions in terms of *integers *and *multiplication*.

To get the integers we first define the *natural numbers*.

### Definition 1: Natural Numbers

The set of

natural numbersis the set of counting numbers,

together with the operations of

addition(+) andmultiplication.

## Manuals

The manuals are available in the Reprographic/Copy Centre. You will need these notes for next week.

## Week 1

In week one we spoke about the questions that the derivative was originally formulated to answer. Namely how do we find the tangent to a curve.

## Week 2

In Week 2 we will look at calculating some derivatives.

## Tutorials

Tutorials start properly *next* week:

- Monday 17:00 in B189
- Friday 09:00 in B185

As discussed on Tuesday, you must make at least ONE of these classes every week.

## Test 2 Results

Have been emailed to you.

## Week 12

We did some more examples on work. We covered a quick section on mean and root-mean-square values and then we spoke about differential equations.

## Week 13 – Revision

Tutorial times and venues as normal.

The lecture times have beeb given over to going over last year’s exam paper. When the paper is completed I will take questions from the class. If there are no questions I will help one-to-one.

## Test 2

I was hoping to have your results today however I got delayed with some other work so I am no unsure of when I will have them (and I won’t make another promise). Not too long hopefully.

## Week 11

We did some more examples on area. We had a quick look at volume and then we spoke about applications to work.

## Week 12

We will do some more examples on work. We will have a quick section on mean and root-mean-square values and then we will talk about differential equations.

## Week 13 – Revision

Tutorial times and venues as normal.

The lecture times will be given over to going over last year’s exam paper. When the paper is completed I will take questions from the class. If there are no questions I will help one-to-one.

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