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teaching:topics:calculus:introduction [2020/08/29 13:24] simon [ideas to consider as you start Calculus] |
teaching:topics:calculus:introduction [2022/08/30 17:32] simon [Motion] |
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describe things we observe around us. | describe things we observe around us. | ||
- | Dealing properly with ideas like "infinitessimals" or "infinite" is difficult, we need to be quite careful | + | 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. |
- | 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 an introduction to the way we work with infinitessimals in Calculus. | + | |
- | Calculus is a powerful method when modelling things that have changing magnitudes | + | Calculus is a powerful method when considering things that have changing magnitudes |
which are measured in continuous rather than discrete units --- especially where | which are measured in continuous rather than discrete units --- especially where | ||
we can find functions that describe some relationship between these measurements. | 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, | 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, | 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 develop mathematical thinking and new mathematical ideas. | + | part of the way we develop mathematical thinking and new mathematical ideas --- including new ways to |
+ | model (that is describe, predict and discuss) the behavior of things we see around us. | ||
====Motion==== | ====Motion==== | ||
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metres are both measurements that are not restricted to whole numbers. Length and time (at least on any | 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 | 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 `%%10m//s%%` moves `10` metres every second. | + | object goes during each unit of time --- a sprinter running at \(\small10\,\)m/s moves \(\small10\) metres every second. |
Acceleration is a measure of how quickly speed is changing. We discovered that this last property is | 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 | much more regular, that there are very specific rules that natural objects seem to obey concerning | ||
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== == | == == | ||
We know how to find the average speed during a journey: | We know how to find the average speed during a journey: | ||
- | \(\quad\dfrac{\texttt{distance travelled}}{\texttt{time taken}},\quad\) | + | \(\quad\small\dfrac{\texttt{distance travelled}}{\texttt{time taken}},\quad\) |
but then how do we find the speed at some instant, when it might be continuously changing? In that instant | but then how do we find the speed at some instant, when it might be continuously changing? In that instant | ||
- | `0m` was travelled during `0s` but we cannot just do \(0\div0\), it will not give us a sensible answer. | + | `0`m was travelled during `0`s but we cannot just do \(\small0\div0\), 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 | 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 | graph we draw with position (that is distance from some fixed place) on the vertical axis and time (since | ||
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So **in this case** the slope, | So **in this case** the slope, | ||
- | \(\quad\dfrac{\tt rise}{\tt run},\quad\) is the speed of a moving object described by that line. | + | \(\ \small\dfrac{\tt rise}{\tt run},\quad\) is the //velocity// 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) like in the graph described we can use Real numbers, positive | ||
+ | or 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 | 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 | + | context is analagous to //velocity//. 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. | calculate this property of a curve? This will be our first exploration in calculus. | ||