HM/MA 004, Assignment 11, due Friday, 22 November 2002, 10:00 AM
(Note on collaboration: If you do the homework in pairs
or groups, please list the name(s) of your collaborator(s)
on what you each turn in.)
- Analysis with infinite series. (15 points)
In Assignment 7, we found an algebraic justification for
Leibniz's "quotient rule" for finding the differential of
the quotient of two variables. Euler justifies this same
rule in a different way, as follows: Given two functions of x,
namely p and q,
their values at x+dx will be p+dp and q+dq. Euler's
first step in finding the differential of (p/q) is to
state that ((p+dp)/(q+dq)) =
(p+dp)((1/q) - (dq/q^2)).
- i. Justify this step by means of expansion into
infinite series, as we saw Euler do in his derivation
of a series for the exponent.
- ii. Starting from the above result of the first step,
explain how the standard quotient rule for
d(p/q) can be derived from it.
- Ideas for final paper. (20 points)
Write two separate brief "project proposals" (about a paragraph each)
for your final paper due 18 December. This
is described in the syllabus as a 13--15 page research paper;
comparable projects of other kinds, however, can also be
proposed for approval. As before, you should propose two
separate topics, and note which one you prefer. Suggested
sample topics fall into the category of renowned problems
solved---or controversies caused---by calculus; they include
the quadrature of the circle, L'Hopital's rule, the
brachistochrone/isochrone problem, the curve of constant
subtangents, the philosophical debates on infinites and
infinitesimals, the disputes over convergence of series, the
role of rigor, applications of calculus to astronomy,
pedagogical problems with calculus, curvature of curves, etc.
Don't hesitate to look for ideas in the remaining part of the
syllabus that we haven't covered yet; you don't have to know
a lot about your proposed topics already, but you should form
a vague sense of what they're about, how they illustrate the
historical importance of the calculus, and why they interest you.
- Overview of the calculus---another approach. (5 points)
Write a limerick on some person or issue connected with the calculus in
the eighteenth century. (Yes, a limerick, why not?)