We discussed the
syllabus
and course policies. This was followed by a short
quiz. Finally,
we discussed a bit of history. The study of area is one of
the oldest endeavors in math, going back at least to ancient
Egypt (where taxes were partially determined by the area of
the land one owned). We then discussed
Archimedes' discovery
that the volume of a sphere is 2/3 the volume of the smallest
cylinder containing that sphere, as well as his approach to
finding the area under an upside-down parabola. Then I recounted
the story of the discovery of
Archimedes Palimpsest and the recovery
of The Method,
Archimedes' text on a method for determining areas and volumes.
This course is primarily concerned with this problem as well. Incredibly,
it turns out that the study of lengths, areas, and volumes are intimately linked
to the study of many natural phenomena, from determining how hot a cup of coffee
is five minutes after being poured, to approximating the size of the
n^{th} prime number, to predicting the fluctuations of a given animal
population. More next time!

We discussed the quiz from last time. In particular, we saw that the symbolic answer using an integral sign, while techincally correct, is rather uninformative. We then discussed several different approximations you came up with to approximate the given area, including filling it with triangles, surrounding it with rectangles, and approaches between these two. We also learned more about each other by sharing `fun facts'.

We reviewed trigonometry. We started by giving a proper (unambiguous!)
definition of sine and cosine. Then we discussed a bunch of properties
of trig functions. We then turned to derivatives. After some discussion
I gave an intuitive description of the tangent line and the derivative,
involving zooming in on a given point. Then we turned that into a precise
formula. We finished by applying that formula to find the derivative of
the function x^{2} at 3.

We started by discussing why the most important aspect of the homework is not success, but the willingness to struggle. We also reiterated some points from the Help section of this webpage. Next, we took the quiz. After discussing one of the solutions (see the posted solution set), we talked about the two standard notations for the derivative of a function and what to do to take the derivative of the sum of functions (and why that works). We ended with stating the product rule and giving a heuristic explanation (not a proof, but an idea of why it might look the way it does) using areas of rectangles.