ASSIGNMENT 6
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete justifications. You will be graded on the mathematical, logical and grammatical coherence and
elegance of your solutions. Your solutions must be typed, with your name and student number at the top of
the first page. If your solutions are on multiple pages, the pages must be stapled together.
Your written assignment must be handed in before the start of your recitation on Friday, October
24. The online assignment will close at 9:00 a.m. on Friday, October 24.
1. Find a quartic polynomial p(x) agreeing with the function f (x) = ex and its first four derivatives at x = 0.
2. Note that 12
that rr = ss .
1/2
=
1 1/4
.
4
Prove that there are infinitely many pairs of distinct numbers r and s such
3. This question refers to the 2010 paper “The rate of growth in scientific publication and the decline in
coverage provided by Science Citation Index” [1]. A copy of the paper is available on the course webpage.
In the second row of Table 2, the authors consider the graph of the function log10 (n(t)) on the interval
[1997, 2006] where n(t) is the number of publications in year t recorded in Compendex, an online index of
scientific engineering literature. In particular, in the second entry, the authors give the slope of the line
fit to the graph of this function; and in the fourth entry, the doubling time for n(t).
(The notation “log10 ” is used as follows: we say that log10 (a) = b if and only if a = 10b .)
(a) Find an expression for n(t) using the slope of 0.055 given by the authors. Be careful to define all of
your terms.
(b) The authors claim that the doubling time for n(t) is 5.5 years. Is that claim correct? Justify your
answer.
(c) Suppose the slope of 0.055 holds even if we extend the interval to the present day. In that case, what
is the current rate of growth of publications on Compendex?
[1] Larsen, P.O. and von Ins, M. “The rate of growth in scientific publication and the decline in coverage provided by
Science Citation Index.” Scientometrics 84 (2010), 575-603.