teaching:topics:calculus:introduction [Simonweb wikinotes]

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=====ideas to consider as you start Calculus=====
The calculus you are learning is called **Infinitesimal Calculus**. Yes --- there //are// others! Mathematics
is a vast field and we only just touch the surface in school. Mostly 17th century maths, with occasional
more modern refinements and notation.

**Calculus** is concerned with //differentiation// and //integration//.

**Differentiation** is about finding the ratio between two vanishingly small quantities,

**Integration** is about adding up an infinite number of vanishingly narrow slices of something.

**Infinitessimals** is the name for these vanishingly tiny things 

They are a way to work with the slopes of curves and the areas under them. These properties are often very
useful when the curve is describing some real-world system, some pattern we observe around us. We will be
finding functions describing these values. We will be making mathematical models that can be used to
describe things we observe around us.

Dealing properly with ideas like <html><span id="infinitessimals">"infinitessimals"</span></html> or "infinite" is difficult, we need to be quite careful how we do it. They do not behave like numbers, so we cannot just treat them like a new kind of number. Look at [[teaching:topics:calculus:sequences#Sums of Sequences]] and [[teaching:topics:calculus:limits#Limits]] for a discussion about working with infinitessimals. Historically following these ideas is what led to Calculus. That path was very contraversial. It took a few generations of dedicated effort by many mathematicians to get there. Since then mathematicians have worked to make it more general and more rigorous, it has many very important applications.

Calculus is a powerful method when modelling things that have changing magnitudes
which are measured in continuous rather than discrete units --- especially where
we can find functions that describe some relationship between these measurements.
We use it in practical applications to help understand and predict behaviours of systems in Physics,
Biology, Meteorlogy, business, Economics, computing, society, most science and much analysis when trying to develop policy and manage organisations and society. It is also a very important part of pure mathematics,
part of the way we develop mathematical thinking and new mathematical ideas. 

====Motion====
We start exploring calculus (just like they did in the 17th century, while
inventing it) by trying to describe the way objects move. Seconds and
metres are both measurements that are not restricted to whole numbers. Length and time (at least on any
scale we are concerned with here!!) can take any real-number value. Speed is a number saying how far an
object goes during each unit of time --- a sprinter running at \(10\,\)m/s moves `10` metres every second.
Acceleration is a measure of how quickly speed is changing. We discovered that this last property is
much more regular, that there are very specific rules that natural objects seem to obey concerning
acceleration. 
== ==
We know how to find the average speed during a journey:
\(\quad\dfrac{\texttt{distance travelled}}{\texttt{time taken}},\quad\)

but then how do we find the speed at some instant, when it might be continuously changing? In that instant
`0`m was travelled during `0`s but we cannot just do \(0\div0\), it will not give us a sensible answer.
Instead we could consider the **rate of change of position** (over time!), or we could consider a
graph we draw with position (that is distance from some fixed place) on the vertical axis and time (since
some fixed start time) on the horizontal axis. Then we see that speed is the same as the //slope// of a
line on this graph --- the `\ tt{rise}\ \ ` is the same as //change in position//, which is the //distance travelled//
and the `\ tt{run}\ \ ` is the //time taken// for that to happen.

So **in this case** the slope,
\(\quad\dfrac{\tt rise}{\tt run},\quad\) is the speed of a moving object described by that line.

Speed is a number --- when we want to consider direction as well we speak of velocity, which is a vector. With only one dimension (say height) we can use Real numbers, positive and negative depending on direction,
instead of a vector.

So we have a //model// for the motion of an object presented as a graph, and we see that //slope// in this
context is analagous to //speed//. We can easily calculate the slope of a straight line, but how do we
calculate this property of a curve? This will be our first exploration in calculus.

There are [[notation#three notations for Calculus]], they come from quite different mathematical perspectives. You
should become familiar with each of them, they are used in different contexts. The
[[notation#why three|story]] behind these very different ways of writing derivatives might help you
get a better idea of the different ways we work with them. 

===Before we begin===
lets look at a closely related question about motion that remained unanswered for
2000 years ...\\
<wrap right>**=>[[teaching:topics:calculus:sequences#sums of sequences|a paradox, and sums of sequences]]**</wrap>\\ \\

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