Values, change, and rates of change

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 use consistent symbols to express these differences:

• $q$ — a quantity, typically determined by a measurement.  (for example, the GPS location where you are reading this page)
• $\Delta q$— a change in that quantity.  (e.g., the 0.4 mile change in location you will need to get from one class to another)
• $\frac{dq}{dt}$ — a rate of change of that quantity (derivative), here shown with respect to time (for example, the speed with which you run to class after you realize that class is starting in 5 minutes).

In general, we'll use a "capital delta" (Δ) to indicate that we mean a change in the quantity that follows the delta symbol.

Be careful! In school science classes (especially physics), these distinctions are often suppressed in an attempt to make things look "simpler". 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. A better way to write that equation is $\Delta x = v\Delta t$.  This form highlights that the velocity $v$ connects a change in position to a change in time. 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 on the change in position (0.4 miles) and the time difference (5 min).

The evil equation may work if you have a really simple problem — one with only one distance and only one time interval.  (Things could go terribly wrong if you used e.g. GPS position or class start time to calculate a velocity.)

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 "$\Delta x = v \Delta t$" you can get very confused. The Δ symbols remind you to look at a particular change and this helps disentangle all the different quantities.]

Although these distinctions seem obvious, in our experience, a large number of errors, both calculational and conceptual, are caused by confusing these three concepts. This is actually quite natural and a result of the brain's penchant to make quick associations that I call one-step thinking. While "thinking fast"* is often useful and may be selected for by evolution, it can easily lead you astray in a science class! Even if you read this page carefully 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!

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 Follow-On pages.

* D. Kahnemann, Thinking Fast and Slow (Farrar, Strauus, & Giroux, 2011)

Joe Redish and Wolfgang Losert 9/3/12