HM/MA 004, Assignment 8, due Friday, 1 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.)
- Defending the calculus philosophically. (12 points)
Take a couple of paragraphs to consider Newton's discussion
of the nature of "first and last ratios" (Reader,
section 12.B4, pp. 393--394). What are they? Do you buy
Newton's argument that it is possible to know an "ultimate
ratio" before it gets to nothing? Why or why not?
- Defending the calculus mathematically. (30 points)
- a. (8 points)
An anti-calculus Dutch
mathematician named Bernard Nieuwentijdt complained that
Leibniz's method of differentials was not really general,
because it could not find the differential of an expression
like z=y^x where x and y
both are variable quantities.
Based on what you know of the workings of Leibniz's calculus,
was he right? Can you find such a differential? If so, what
is it, and if not, why not?
- b. (14 points)
Leibniz tried a new
trick suggested by his colleague Johann Bernoulli to solve
Nieuwentijdt's challenge problem in part (a): he started out by
taking logarithms of both sides of the equation z=y^x..
Using this hint and the algorithms of Leibniz's calculus (as well as
the differential-calculus rule that for any variable
quantity w, d(log w) = 1/w dw), find
the desired differential d(y^x).
- c. (8 points)
When the exponent x is taken to
be some constant a, what does your expression for
the desired differential d(y^x) turn into?
- Do the two forms of the calculus solve
each other's problems? (12 points)
Can you apply Newton's ``o-method''
for finding fluxions (from Reader, sections 12.A1 and 12.A5,
pp. 381--382, 385) to get the fluxion or differential of w
in Leibniz's minimization problem on the
equation w = h(l^(1/2)) + r(m^(1/2)) (from Reader,
section 13.A3, pp. 432--433)? How, or why not?
- Should we even be doing calculus? (12 points)
Take a couple of paragraphs to explain the gist of Newton's
argument about geometry and algebra (from Reader, section 12.D3,
pp. 412--413). Is he, like Kepler, renouncing the use of
"Algebraick Expressions" in mathematics? If so, why, and if
not, what does he mean?