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Values, change, and rates of change

Page history last edited by Ben Dreyfus 9 years, 1 month ago

Class content >Modeling with mathematics > Math recap

 

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, that can vary with time, position, or parameters;
  • Δq -- a change in that quantity;
  • dq/dt -- a rate of change of that quantity (derivative), here shown with respect to time.

 

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 my 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. This equation is not confusing if you have a really simple problem -- one with only one distance and only one time interval. But if you have a more realistic and interesting 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 Δs 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 most 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

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