Tom Lehrer earned Bachelors' and Masters' degrees in Mathematics from Harvard and has taught math at Harvard, MIT, Wellesley, and the University of California at Santa Cruz. What he became most famous for, however, is writing some of the funniest satirical songs in the past 50 years. "Poisoning Pigeons in the Park," "Pollution," "Fight Fiercely, Harvard" ... all of these served as models for satirists from Mark Russell to the Capitol Steps. If you can find CDs of his work, I highly recommend them, although a lot of the songs are quite dated.

Lehrer sometimes wrote about science and math -- a lot of chemistry teachers still try and perform his version of "The Elements," a rapid-fire recitation of the periodic table done to the tune of Gilbert and Sullivan's "I Am the Very Model of a Modern Major General."

Here is a piece of video from 1997 in which Lehrer digs out some old songs for the 80th birthday celebration of a former professor of his. This is a song about the definition of the derivative:

This week has been brutal for finding time to blog for the class. I hope next week will be better.

We had noticed that the expression



kept appearing in various examples as a solution. We gave this expression its own notation, , and called it the derivative of f at a.

Computationally, this limit (if it exists) takes a lot of algebra to compute -- sometimes it can be quite difficult to do so. Conceptually, we can interpret the derivative at a point as either the slope of the tangent to the curve at a, or the instantaneous rate of change of f at a. For instance if f represents a position function for an object, the derivative at a would give the value of the object's instantaneous velocity there.

We also noted that the derivative of f at a results generally in a function of a, so we can think of the derivative as a function in its own right:



At long last, Friday was the day that our calculus "veterans" have waited for ... the revealing of the Power Rule.

We first noted that any function that is differentiable at a point a is also continuous at that point. However, the converse is not true: the function is continuous at 0 but not differentiable there. A function will not be differentiable at a particular point if there is a sharp corner there (as in the previous example), a discontinuity of any kind, or a vertical tangent line (e.g. at x = 0).

We took a moment to examine alternative notations for the derivative that our book uses frequently.

We then began developing formulas for derivatives of common functions. Using the definition of the derivative, we immediately saw that the derivative of a constant function is zero. Using our knowledge of linear functions, we saw that .

We then looked at several examples of derivatives of power functions. With the help of Pascal's Triangle and some slick factoring, we found that we could state the



The population rejoiced!

Let's see ... this week on Monday, we began exploring Mathematica and on Friday, we took our fourth quiz.

We'll figure out some other derivative rules on Monday. See you then.

The solutions to Exam #1 have been posted and are available from the Test and Quiz Solutions page.

Both sections may report to the Cabot Computer Lab today to begin working on the introductory labs, available from the Computer Labs link above.

There are still some who haven't taken Exam #1 for various reasons. They are taking it Monday and you will get your exams back Wednesday.

I will post the solutions on Monday.