Worksheets developed by Warren Code with Sujatha Ramdorai for Math 104 and Math 184. (Page last updated December 14, 2012)
A note about use: Some of these could be printed and copied and directly and reused, others could be recreated by hand in a short time and adjusted as necessary. Having perfect digital versions is not the idea; rather, these are intended as a basis for creating worksheets tailored to particular class situations. Designing and coming up with ideas and examples is the longer part; producing a worksheet by hand can be done relatively quickly and still be or acceptable quality for classroom use. Indeed, handwritten items can be more suitable in many cases as they given a clearer model of expert work (what graph sketches look like, what notation is used and how, etc.).
These worksheets can be handed in and checked for progress by a grading TA. Note that the work is reduced if students are required to work in groups of 2 or 3.
Highly recommended: Try to keep any worksheet or portion that will be handed in on a single sheet - this is much better to deal with logistically and the sheets can be photocopied for the grader and still handed back to students the very next class with solutions posted online.
Final note: It is important for students to get a chance to work on these without knowing they will get the answer very soon (which can short-circuit their effort and lead to them just waiting for the answer all the time). You can also promote their working by circulating all around the room. Also, it will usually take them longer to do the work than you might expect; remember that in many cases this material is still fairly fresh, and they will not be as quick as on an exam.
Tips for success in the course and explanation of the purpose of activities. The purpose and value of things like in-class work and group discussion must be stressed early and repeated during the course to keep students on track.
Intended for the start of course "review" about functions and proprties like being 1-1. Gets students to remember some of these ideas and to see the terminology if they are not familiar with it. Can be run as a structured class time worksheet: have students work on a part or two at a time, then go over it with the class or check up and discuss using a clicker question (the solutions file includes a couple of clicker questions used).
Available since 2012W, this is useful to give students in the first week or two of classes. Along with the accompanying course notes, it introduces the key business terms for the course. This sample problem also has a fair bit of the analysis that will later be applied in the less linear cases. See also RCP Mini-review
Intended as further follow-up on the business terminology and basic objects like the demand curve (still linear here).
Intended to give students an extended example of (and motivation for) a piecewise function; students entering Calc 1 are quite unfamiliar with such functions and their finer points.
Not used in 2012W Math 104 (replaced by diagnostic quiz), but if enough time is available this is another example of piecewise linear functions (with sensible real-world appliction) and makes for a good discussion of the different between the tax owed (increasing, continuous function) and the tax rate (series of horizontal line segments) as a lead-up to the derivative as rate of change. Also useful for comparison with the Paint Store (if available).
Permits discussion of how to "prove" continuity or discontinuity at a point. Can also add to the task by defining continuity on an interval and having students distinguish between that and point continuity for the given graphed functions. The back (Q3) is a common exam-style question with a piecewise-defined function.
The language here needs a bit of exaplining; the idea is to poke at the hypotheses of the IVT and how they are connected to the picture.
Q1 emphasizes the choices of variable names in the definition of the derivative; x is given a very different role in the two standard versions (and a is used to emphasize that here in the bottom left). Naturally goes with the picture explanation of what is going on. The difference between use of parameter a and a specific input value like 1 is the point of the bottom right.
Q2 gives the picture for a nondifferentiable point (often not shown as an example) which also matches up with a similar webwork problem.
Q3 is just a way of checking how many rules students are already familiar with, before introducing them in the course (most student have had calculus before, and mechanical use of several derivative rules is often known well).
Connecting the picture of the tangent line to the derivative value, and determining the equation of the tangent line. Few students with a previous calculus course remember this connection by the time they get to Math 104. The second page has some practice drawing derivatives of functions with given graphs (definitely takes more time to get to in class).
Most students in Math 104 are very uncomfortable working with parameters in expressions. This worksheet is intended to refine the idea for them and have them work with some examples including derivative expressions that involve parameters (the distinction between variable and parameter is sometimes clearer when a derivative is involved). With enough time allocated, it also gives some practice at parameters in piecewise-defined functions (common on exams and webwork).
Has been used in M184, not M104, as students need more practice with derivative rules.
Provided to the entire course as home practice; the textbook does not support these types of business problems, so more are needed. This is a simpler set than the "Extra" Business Problems set.
Provided to the entire course as home practice; the textbook does not support these types of business problems, so more are needed. Some of these are quite challenging. Mark MacLean has created a sequence of pencasts for Q5:
Part 5a is at http://www.livescribe.com/cgi-bin/WebObjects/LDApp.woa/wa/MLSOverviewPage?sid=62Jhb1q6JH94,
Part 5b is at http://www.livescribe.com/cgi-bin/WebObjects/LDApp.woa/wa/MLSOverviewPage?sid=ZRlzKRSdzC0n,
Part 5c is at http://www.livescribe.com/cgi-bin/WebObjects/LDApp.woa/wa/MLSOverviewPage?sid=qpLzccVl7BMJ
The above files are applicable to material before the first midterm. This note was distributed as the back of another sheet to remind students of the importance of the activities, and also as a partial response to their questions generated through a short, online survey to check in with them in the week before their first midterm. Students very much appreciate when you check in like this and if you can change any small thing in response they are generally grateful.
Intended to mimic recent exam problems on elasticity (there are not very many practice problems overall).
A structured worksheet to lead through the class (for example, do not keep them too long from giving them the regular compound interest formula). This process will probably go slowly enough that almost nobody will get to the rabbit problem. This is also an opportunity - via the Loan Shark problem - to introduce e as the limit of compound interest of 100% over one year as the compounding frequency goes to infinity; the most common approches for the exploration activity are to plug in some numbers for large n (like compounding every second) and some student try (and are not able) to solve for the n where the interest formula will be equal to $3000.
The functions here could be assigned in class and this file can be posted for students after class. There is not very much practice in the current textbook on this topic, and webwork does not provide full solutions.
First developed in 2011W with David Kohler, these are a sequence of activities using structured in-class worksheets.
Introduction to the technical meanings of critical point and absolute maximum/minimum, along with what the graphs look like. Also the idea of a local max or min when the function has a discontinuity, and implications of the Extreme Value Theorem.
Problems taken from the Math 184 workshops. Includes optimization and a few related rates problems (better not to tell the students this at first). Intention is that students do the problem setup and not the full computation in class, as the translation from words to a diagram and symbols is often the biggest challenge. The diagrams may be included as hints on the back of the sheet - these are often the hardest part of these problems for the students.
The derivative chart worksheet template needs a chart to be drawn in for students to work with (the solutions give one example). The other two files are slides of related clicker questions.
First developed in 2011W with David Kohler, these are a sequence of activities using structured in-class worksheets that incoporate pre-class work assigned using Vista/Blackboard/Connect (as indicated on the homework sheets). The homeworks take most students 30-45 minutes to complete; participation points for completion were awarded to boost completion rates. The quiz has been used at the end of the Friday class to check in on student uptake of the concepts.
Original worksheet was hand-drawn. This worksheet emphasizes some key points about Tayor polynomials: some are degenerate (like T_2(x) for sin(x)), the higher degree is tighter to the actual function value than the lower degree, and how the coefficients are related to derivatives (and the graph of the function). It also gets students to focus on the form of the higher-order polynomials as including the lower-order terms as well.