From: Stephen Dalton
Date: November 30, 2009 9:03:10 PM EST
To: James Propp 
Subject: Comments on 92.305 Assignment #9

- 5.2.1: In order to pull a constant out of the limit in this problem, it was necessary to cite the differentiability of f(x).  The differentiability of f(x) is what guarantees the existence of the limit of the difference quotient - and remember that only when the limit exists are we allowed to bring a constant multiple out of the limit!

- 5.2.2: Many students wrote $f(g \circ h)(x)$ to denote the function $f$ multiplied by the function $g \circ h$.  But the notation $f(g \ circ h)$ actually means "$f$ evaluated at $g \circ h$," and the notation "$f(g \circ h)(x)$" has ambiguous meaning.  Just remember that when a letter used to denote a function comes with an open parentheses right after it, that means we are evaluating the function at whatever comes inside the parentheses.  So in order to denote "the function $f$ multiplied by the function $g \circ h$," we would want to write either $f \times (g \circ h)$, or perhaps $f(x)(g \circ h)(x)$, if the point at which we are evaluating the function is important.

That's all!

-Steve