SUNY Geneseo Department of Mathematics
Wednesday, January 23
Math 221 03
Spring 2019
Prof. Doug Baldwin
(No Previous Lecture)
To Math 221, Calculus 1.
I’m Doug Baldwin.
A story about Galileo: towards the end of his life, he got interested in mathematical descriptions of motion — how distance moved relates to time spent moving, acceleration, etc. In particular, he wanted to measure the distance-vs-time relationship for falling bodies. He couldn’t just drop a ball and time its fall (clocks of the time weren’t accurate enough), so he cleverly rolled balls down inclined planes, figuring that they were still “falling,” just more slowly than if dropped free. He also had some clever way of counting time, although each account I’ve seen gives a different clever way — he used his own pulse, or he used the beats of a song, or.... In any case, the time and distance measurements he saw were similar to these, although these are made up and conveniently vague about the units (since we’re vague about Galileo’s time units anyhow).
| Time | Distance | Speed |
|---|---|---|
| 1 | 1 | D/T = 1 |
| 2 | 4 | D/T = 2 |
| 3 | 9 | |
| 4 | 16 |
Galileo’s cool discovery: distance traveled is proportional to time squared.
What else can you say from this data? Anything about speed?
You can calculate a speed over each time interval by dividing the distance traveled by the time elapsed (the 3rd column above). But that gives you an average speed. Unlike the distance-vs-time formula, we didn’t get a formula you can plug a time into and get the instantaneous speed the ball is moving at that time. Is it possible to have such a formula? Maybe you can use smaller and small time intervals in calculating an average in order to come closer to instantaneous speed...?
It turns out that it wasn’t until calculus was invented (around 50 years later) that there were ways to answer these questions and get a full mathematical understanding of how speed, distance, and acceleration are related. This course will start with a very similar development, and then move on....
Here are some different ways in which you can be said to “understand” something, based on something called “Bloom’s taxonomy.” Which do you think will receive the most emphasis in this course? (The stars below represent your ideas and mine after discussion, more stars meaning more emphasis on that level.)
What are some things that would help you understand at that level?
Meet Madison Rodgers, our supplemental instruction leader.
Details of how this course will work.
What are some key things you want to know about that?
Read the syllabus, and see if you can find answers to the things you want to know. Come to class with any questions you have about it.