Comments about workshop problems

Problem #1 in Workshop 5 (the afternoon of October 5, 2007)
This problem studies how the height of liquid which is filling several containers of different shapes changes. Here is part of a student message to me about the problem:

...just by looking at the graph, how do you tell whether the graph is differentiable or not??
My reply included the following:
I've tried to address that specifically with several examples in class. Also various answers to this question are discussed in sections 3.1 and 3.2 of the text and the problems for these sections. Do YOU know ANY specific examples of functions which are NOT differentiable? I hope that the answer is yes. Examples could include

  1. f(x)=0 if x is NOT 0 and 1 if x=0.
  2. f(x)=0 if x<=0 and 1 if x>0.
  3. f(x)=-x if x<0 and x if x>=0 (this is |x|, which is really also a piecewise defined function).
Please quickly sketch a graph of each of these functions, and examine their behavior near 0. You should be thinking about average rates of change (slopes of secant lines) and instantaneous rates of change (slopes of tangent lines, if there are any good candidates for tangent lines).

I think that the real difficulty of analyzing this workshop problem is for students to sketch eligible curves. By this I mean the following: there is a little bit of quantitative data given (numbers: 10, 30, 5). So from this a student should be able to figure out what the starting height is, what time is needed to fill the containers, and what the ending height is.

What's really needed is the shape of each of the the height curves between the start and the end, accompanied by some explanation of why "you" (or the anonymous "student") believes that the curves have these shapes. I joked with several students after today's lecture that they could build containers and measure the heights as the containers are filled. There are no measurements given on the varying widths of the containers, so only qualitative aspects of the curves can be noted. But it is exactly these qualitative aspects which I would like you to address (continuity and differentiability).

It takes some effort is needed to see how the "theory" involved in these words interact with the more realistic setting of this problem. That's why I assigned the problem.

I hope this message is helpful.

Maintained by and last modified 10/5/2007.