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In school, we learn how a line has an equation… and a circle has an equation… what does this mean?

points $(x_0,y_0)$ on curve $\longleftrightarrow$ solutions $(x_0,y_0)$ of equation

however this note explains all of this from first principles, with a particular emphasis on the set-theoretic fundamentals.

## Set Theory

set is a collection of objects. The objects of a set are referred to as the elements or members and if we can list the elements we include them in curly-brackets. For example, call by $S$ the set of whole numbers (strictly) between two and nine. This set is denoted by

$S=\{3,4,5,6,7,8\}$.

We indicate that an object $x$ is an element of a set $X$ by writing $x\in X$, said, $x$ in $X$ or $x$ is an element of $X$. We use the symbol $\not\in$ to indicate non-membership. For example, $2\not\in S$.

Elements are not duplicated and the order doesn’t matter. For example:

$\{x,x,y\}=\{x,y\}=\{y,x\}$.

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.

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

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

## Continuous Assessment Marks

I will send you an email either today or Monday regarding your continuous assessment marks.

## Week 12

We finished the quick section on mean and root-mean-square values and then we spoke about differential equations.

## Week 13 – Revision

• Tuesday 09:00 in B149
• Thursday 09:00 in B180
• Thursday 11:00 in B188

Sadly you have lost out on a tutorial and an extra revision class due to the test you have on Week 13.

The review will be given over to going over last year’s paper (at the back of your notes). If the paper is completed I will take questions from the class. If there are no questions I will help one-to-one.

## Test 2 Results.

Probably on Monday

## Week 11

We had a quick section on volume and then we spoke about applications to work. We started looking at the mean value of a function.

## Week 12

We will finish the 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 paper (at the back of your notes). 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

The second 15% test will take place at 9 am Tuesday 22 November in B149 (Week 11). A slight change in format to last year: this year the format will be as per this sample (although there will be space to write your answers into).

You will be given from 09.05 — 10.00. You will be given a copy of these tables.

## Week 10

We finished substitution and started partial fractions.

## Week 11

We will have a quick section on volume and then we will talk about applications to work.

## Test 2

The second 15% test will take place at 9 am Tuesday 22 November in B149 (Week 11). A slight change in format to last year: this year the format will be as per this sample (although there will be space to write your answers into).

You will be given from 09.05 — 10.00. You will be given a copy of these tables.

## Week 9

We worked on antidifferentiation by substitution.

## Week 10

We will continue to work on substitution and maybe start partial fractions.

## Test 1 Results

Have been emailed to you. I will send on solutions tomorrow.

## Test 2

The second 15% test will take place at 9 am Tuesday 22 November in B149 (Week 11). You can find a sample in the course notes.  A slight change in format to last year: this year the format will be as per this sample (although there will be space to write your answers into).

You will be given from 09.05 — 10.00. You will be given a copy of these tables.

## Week 8

We started evaluating some straightforward integrals using the Fundamental Theorem of Calculus.

## Week 9

We will start looking into antidifferentiation by substitution.

## Catch-up Tutorial for BioEng 2B

Due to the Bank Holiday, 2B missed out on a tutorial. We will have a replacement class, Wednesday 9 November at 11:00, in B165.

## Continuous Assessment

As can be seen here in the Module Descriptor, there will be two 15% tests: one in Week 6 and one in Week 11. I hope to give you two week’s notice of each and there are sample tests in the notes.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Student Resources

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.