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\Cb{Rationale for Individual Worksheets}
\noindent \Bf{Code:}
\begin{description}
\item[\Star]  indicates that the worksheet contains some important problems---these will have stars next to them on the worksheet itself.
\item[\Star \Star]  indicates that the worksheet contains generally important material.
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\noindent \Bf{Usage:}   The worksheets may be used as either supplements to the lecture or as homework or as in-class cooperative lab sheets.  It is important that the students do the ``Suggested Problems'' on the worksheet before tackling the problems on the worksheet.  In ``cooperative lab'' mode, this can be enforced by assigning these (or similar routine problems) in the previous class meeting and then occasionally giving 3 minute quizzes on one or two of these problems at the beginning of the class meeting.


\noindent \Bf{Caution:}  It is certainly not expected that every problem on every worksheet will be treated by every  instructor---not even the principal author will do that!   This is the first trial of these worksheets and your judgement should be exercised.  The \Star\ ratings of the individual worksheets are the opinion of a few of the Calculus team and may serve as a guide to minimum usages.  The principal author has deliberately included some things that are not of the highest priority in an engineering curriculum since these materials will be available nationwide through the internet anonymous ftp site at the Center for Wave Phenomena.

\subsection*{Chapter 1-Four worksheets}
\begin{description}
\item[WS1\Star] The first two (from the text) problems involve use of two equations (i.e. more than one) and use of parameters (instead of specific numbers).  These are easy problems from the text that set the right tone for the course.  The next two problems again involve symbols rather than numbers and cover easy, but important, facts about absolute values.  The remainder of the worksheet studies inequalities with an emphasis on use of general parameters, but the approach and/or content may not be to every instructors' liking.

\item[WS2\Star \Star] Functions defined in various ways besides formulas--the definition in terms of graphs is important for building intuition and the definition in terms of tabled numbers is important because it mirrors field acquisition of data.  This worksheet also introduces our first \Mma\ statement: \Tt{Plot}.

\item[WS3\Star \Star] Techniques for solving word problems; use of parameters; St. Olaf material.  \Mma\ \Tt{Table} command.  A fairly tough problem (the `asteroid problem') is included---``students should be encouraged to take some time with at least some problems''.  This problem will be revisited several times (including the next worksheet).  

\item[WS4\Star] This one uses the asteroid problem to illustrate equation solving using \Tt{Plot} and \Tt{Table} in a tough situation (transcendental equation with many roots). It contains an \Bf{Honor} problem (\# 4) posed as a generalization of (\# 2) which is from the text.  It also (\# 7) introduces the important notions of \Em{secant} line and \Em{tangent} line encouraging the student to bring out common misconceptions about the latter, prior to getting enmeshed in techniques for computing slopes and derivatives.
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\subsection*{Chapter 2-Four worksheets}
\begin{description}
\item[WS1\Star] Reinforces the word problem techniques introduced in Chapter 1.  Affords students the opportunity to discover some important slope facts for themselves before getting enmeshed in technical statements of theorems.  Exercise in getting information about a function defined only by a graph (motivated by St. Olaf material).  Has \Bf{Honor} problem connected with the asteroid  problem.

\item[WS2\Star \Star] Use of \Mma\ to interactively explore limits.  Other useful limit techniques.  Functions and rules in \Mma.  First compound \Mma\ statement introduced to make a strong tool for exploring limits numerically.

\item[WS3\Star] Discusses function composition and decomposition (needed skill for using chain rule effectively).  Advanced problem on \Em{infinitely} iterated function composition.  Limit and continuity concepts tested.

\item[WS4] First problem is application of the Intermediate Value Theorem.  The rest of the worksheet is on the method of bisection (a rival to the graphical and tabular approaches to equation solving introduced earlier).
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\subsection*{Chapter 3-Ten worksheets}
\begin{description}
\item[WS1\Star]  Graphical techniques for approximating the derivative as a \Em{function}.     Series of ``which is the function and which is the derivative'' problems. Intuitive peek at maximum/minimum problems.   New \Mma\ technique: plotting two functions on one graph.

\item[WS2\Star]  The book does a ``ball'' problem with initial velocity 96 (example 4).  After extensive calculations, it turns out that the final velocity is -96.  The authors make no comment about this remarkable coincidence.  We return to the theme of using parameters to prove that $v_0$ always equals $v_f$ and use this more general form of the answers to check that their  dimensions are correct.   To reinforce a more scientific attitude to such problems, an additional problem discusses the inverse square law effect.  Other, simpler, word problems reinforce the use of parameters and/or the scientific approach towards assessing the reasonableness of proposed models.  Finally, there is a problem involving a function defined by \Em{data} instead of by formula.

\item[WS3\Star]  Problem on inverse square law---this material will be referred to repeatedly throughout the year and is not treated at all in the text.  New \Mma\ material:
\Tt{D, Simplify, f'[x]}.  The theme is to use these tools to \Em{check} calculations done by hand---\Em{not} to use \Mma\ to replace drilling these fundamental tools (power rule, product rule, quotient rule).

\item[WS4\Star]  Applied problems that call for using several equations.  New \Mma\ material: \Tt{Expand, Factor}.  Again, the theme is to use these tools to \Em{check} calculations done by hand---\Em{not} to use \Mma\ to replace drilling the fundamental chain rule (but the last problem is one that would be abysmal to do by hand).

\item[WS5\Star]  Problem to check that the role of inverse functions in calculus is understood.  New \Mma\ material: \Tt{Together}.  Problem to make sure scope of the generalized power is understood.  Problem that asks students to generalize from specific cases.

\item[WS6]  Optional worksheet testing whether the maximum value property and its application to the closed interval extremum problem is well understood by asking students to produce examples of various situations.

\item[WS7\Star \Star] Word problem techniques including the text's ``5 step process'', ``plausibility'', and ``dimensional'' checks.  Extra information on how to do the latter.  Careful selection of max/min problems with usual emphasis on using parameters to get general solutions.

\item[WS8\Star]  An important illustration of doing the \Em{general} problem---here, a final question is posed that simply can't be tackled without keeping the parameters in the answer.  More word problems and some trig differentiations with ``shape''.

\item[WS9\Star] Distinguish differentiating identities from differentiating equations.  Use \Mma\ to compute implicit derivatives---first use of \Tt{Solve} (non-numeric context).  Use of \Mma\ in graphing implicit functions.  Students ask to generalize word problems and make use of their generalized solutions. 

\item[WS10]  Note: this worksheet is superceded at Mines by  a lab project.  Manipulation of algebraic equations.  Newton's method using \Mma.  Introduction to \Tt{Solve, NSolve} and \Tt{FindRoot}.  Application to the ``asteroid problem''.


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