Calculus Concepts Discussion

This discussion reinforces your understanding of the ideas underlying derivatives and integrals from section 2.1 of our textbook.

Background

If you live in the Rochester or Monroe County area, you might remember that we had a tornado on July 29 (it was small and didn’t do any damage beyond toppling some trees). If you live anywhere in the eastern half of New York state, you had tornado warnings (unrelated, as far as I know) over the next few days. At least in the case of Monroe County’s tornado, television weather reporters spent about half an hour making more or less continuous announcements of where the tornado was, where it was heading, and how fast.

Discussion

Before class time (1:30 PM) on Thursday (Sept. 3), post one or more responses (i.e., thoughts, questions of your own, answers, responses to other posts, etc.) to one or more of the following questions.

Speed from Position

The following statements about measuring the position and speed of a tornado or storm are at least approximately right, though I’m not a weather expert so I might be missing something: Weather radar will tell a weather reporter the tornado’s exact position, as a direction and distance from the radar antenna. If it’s Doppler radar (which pretty much all weather radar is these days), it can also measure speed towards or away from the radar antenna — but if the tornado is moving at an angle to the antenna, i.e., neither exactly towards it nor exactly away from it, that measurable speed is only part of the tornado’s total speed. Assuming that these statements are true, how could you estimate a tornado’s total speed (and direction) from radar data?

The tornado’s speed most likely changes with time. How would you make your estimated speed more accurately estimate the tornado’s speed at a precise instant in time (i.e., the instantaneous speed at that time)?

Position from Speed

Now imagine a weird alternate universe in which weather radar can exactly measure speeds and directions, but not positions. Thanks to an on-the-ground weather spotter network, alternate universe you knows a tornado’s position at one moment in time, but after that moment you only have speed and direction measurements. How could you estimate the tornado’s changing position from the initial position and these speed and direction measurements? (Hint: If you know the tornado’s speed at some moment, can you estimate how far it moves in a short time interval starting at that moment?)

How could you make your position estimates more accurate?

Putting It Together

The syllabus defined calculus as “the mathematics of change.” Is there any sense in which either or both of the above scenarios involve “mathematics of change”? If so, how do they involve it?

Section 2.1 in the textbook introduces the ideas of derivatives and integrals. What, if anything, do those ideas have to do with the above scenarios?