### Rationale for workshop problems, fall 2006

Why the problems in workshop #2?

1. and 2. and 3. All of these problems are simple examples of geometric/physical modeling situations. Such situations will arise repeatedly in Math 151. Successful modeling will strengthen skills which most Math 151 students need in other courses and will need professionally.
1. This exploration using a "fairy tale" is intended to show students that limits and function values can be distinct.
Why the problems in workshop #1?
1. Students entering a calculus course should have verbal, algebraic, geometric and logical skills. Neglecting deficiencies in these areas makes the course much more difficult. These problems are an attempt to detect such deficiencies. The writeups should be evidence that students can communicate technical material clearly and carefully.
2. Students should be able to handle graphical information. They should be able to transform the graphs in response to simple algebraic queries, and also answer simple algebra questions about functions defined graphically.
3. Polynomials are among the simplest building blocks of functions in calculus and in many other subjects. This problem seeks to determine if students can create polynomials with prescribed roots, degrees, and positivity.
4. Students should be able to "model" and understand simple physical and geometric problems using the tools of calculus. Here's a rather simple problem, which additionally requests that students use a calculator to understand some of the model.
5. Absolute value is a mystery to many students. In this course and elsewhere absolute value is used to compare quantities. Again, algebraic and logical skill is needed to complete this problem and to be sure that the results agree with the graphical information. This interaction is important. Piecewise linear functions occur in the tax code (as mentioned during the lecture) and also are important enough so that massive amounts of computer time are devoted to maximizing and minimizing such functions (a major portion of industrial engineering and operations research).