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Values, change, and rates of change (2013)

Page history last edited by Joe Redish 4 years, 4 months ago

Class Content I >Modeling with mathematics > Math recap

2.2.5

 

Figuring out what is going to happen next and how to respond to changes are among the most important things any organism has to deal with in order to survive and flourish.  An extremely valuable part of science is making predictions, though it's not the whole story -- explanation and understanding are equally important characteristics of science.

 

As a result, when we quantify in science it is critically important to pay close attention to the differences between a quantity, a change in that quantity, and the rate of change of that quantity. In this class, we will use consistent symbols to express these differences:

 

  • q -- a quantity, typically determined by a measurement.  (e.g., the GPS location where you are reading this wiki page)
  • Δq -- a change in that quantity.  (e.g., the 1 mile change in location you will need to get to class)
  • dq/dt -- a rate of change of that quantity (derivative), here shown with respect to time.  (e.g., the speed with which you run to class after you realize that it starts in 5 minutes)

 

We will in general use the marker "capital delta" (Δ) to indicate that we mean a change in the quantity that follows the delta symbol.

 

Although these distinctions seem obvious, in our experience, a large number of errors, both calculational and conceptual, are caused by students confusing these three concepts. Even if you read this page at the beginning of the class, you are likely to make this error many times! Let each time you make this mistake help you raise your sensitivity to the important differences among them!

 

[One possible reason for this is that in school science classes (especially physics), these distinctions are often suppressed. You might, for example have seen the (evil) equation, "d = vt" (distance = velocity times time).  The equation is crossed out to remind you not to use it in that form. The correct equation, of course, is Δx = vΔt   which highlights that velocity v connects a change in position to a change in time.  Simply put, in our example above, the speed with which you run to class is not based on the GPS position of the classroom and the time the class starts, but of course the change in position (1 mile) and the time difference (5 min).

 

The evil equation is not confusing if you have a really simple problem -- one with only one distance and only one time interval.  (You saw how things could go terribly wrong if you used e.g. GPS position or class start time to calculate a velocity in the example above.)  Now, if you have a more complicated problem (for example, the tortoise and the hare), where you have three different time intervals, three different distances, and two different velocities, with the distances and time intervals having different starting points, writing "d=vt" instead of "Δx = v Δt" you can get very confused. The Δ symbols remind you to look at a particular change and this helps disentangle all the different quantities.]

 

The importance of changes and rates of change in science is one reason why calculus, where derivatives are studied, is considered as a pre- or co-requisite to many serious science classes. As with a lot of math, the way we think about and use derivatives -- and their inverse, integrals -- in our class may be conceptually different from the way you learned them in your math classes.  We discuss briefly what you need to know about them in the pages linked to below:

 

Follow-ons

 

Joe Redish 7/15/11

Wolfgang Losert 9/3/2012

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