teaching:topics:calculus:introduction
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teaching:topics:calculus:introduction [2021/01/13 13:30] simon [ideas to consider as you start Calculus] |
teaching:topics:calculus:introduction [2022/08/30 17:37] simon [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 \(10\,\)m/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 | ||
| - | `0`m was travelled during `0`s but we cannot just do \(0\div0\), it will not give us a sensible answer. | + | \(\small0\,\)m was travelled during \(\small0\,\)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 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// |
| - | 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. | and the `\ tt{run}\ \ ` is the //time taken// for that to happen. | ||
| So **in this case** the slope, | So **in this case** the slope, | ||
| - | \(\quad\dfrac{\tt rise}{\tt run},\quad\) is the //velocity// of a moving object described by that line. | + | \(\ \small\dfrac{\tt rise}{\tt run},\ \) 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 | Speed is a number --- when we want to consider direction as well we speak of **velocity**, which is a | ||
teaching/topics/calculus/introduction.txt · Last modified: 2022/08/30 17:37 by simon
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