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teaching:topics:calculus:introduction

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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 17:37] (current)
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. +\(\small0\,​\)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 speed 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  
 +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.
  
teaching/topics/calculus/introduction.1598671493.txt.gz · Last modified: 2020/08/29 13:24 by simon