For the Instructor

"One thrust of calculus reform has been to place greater emphasis on problem solving and to present students with more realistic applications as well as open ended problems." [Calculus by Smith and Minton]  This project (a series of modules) was designed with this in mind and should be assigned only after students have mastered the first and second derivatives and their use in determining maximum and minimum functions values, intervals of increase and decrease, intervals of concavity, and points of inflection. Students must also be able to read, understand and modify Maple code as appropriate. The ultimate goal of this project is completion of Module F, which is a challenging, realistic, and somewhat open-ended optimization problem.

At Hollins, after completing Sections 3.1-3.5 (Linear Approximations, Newton's Method, Maximum and Minimum Values, Increasing and Decreasing Functions, Concavity) from Calculus by Smith and Minton, we consider the following "warm up" coaster problem, which is an extension of a problem from Section 3.3 of that text.

A section of roller coaster is in the shape of y = x5 - 4x3 - x + 10 for x between -2 and 2.

1. Find (exactly) the intervals of increase and decrease and explain their coaster meaning.
2. Find (exactly) all relative extrema and explain their coaster meaning.
3. Find (exactly) all points of inflection and explain their coaster meaning.
4. Find (exactly) the points of steepest ascent and descent on this coaster.
5. Find the angles of steepest ascent and descent.

We then ask questions such as "How could we design our own coaster?"; "What points are important points?"; "What type of safety criteria are needed?"; "How can we compare coasters?" and "Which coasters are best?".  We then proceed to the interactive coaster project.

All students should start with module A, as it introduces the basic ideas of the project -- use of functions to fit peak and valley points, points of steepest ascent/descent, angles of steepest ascent/descent, and our mathematical definition of thrill.  In this module, students can "watch" the fitting process on real-life coasters and then use the collected data (peak/valley coordinates and slope at steepest point) to calculate the thrill of portions of the real-life coasters.  (Students may struggle with which coaster tracks to use, but this is part of the fun.) Their collected data will be used in later modules, when students are asked to "do" the fitting process for themselves and compare answers.

You may choose which modules from B through E to assign. These modules include design and analysis of single-drop coasters using trigonometric (B) and polynomial (C) functions, as well as design and analysis of several-drop coasters using trigonometric (D) and polynomial (E) functions. In these modules, students model the paths of coasters by fitting these functions (in slope and height) to specified peak and valley points -- both those collected from module A and those created by students.  In each design, students must pay attention to stated safety criteria and calculate thrill.  These modules include questions and assignments for the students (indicated in red).

Note: If you don't have Maple installed, you can still see the functionality of our Maple worksheets in a "static" form that can be viewed in your browser window. Click on any of the links in the following table to see these "filled-in" worksheets. If you prefer to work in some other computer algebra system, you can use these as models for setting up your own worksheets.

 Browser-readable (HTML) exports from Maple worksheets Module B Module C Module D Module E Module F

As previously mentioned, the ultimate goal of this project is completion of Module F.  In this module, students must use what they have learned to design the "ultimate" coaster -- a coaster that satisfies all safety criteria and that has maximum thrill.  This is a challenging assignment that requires considerable ingenuity, creativity, and tweaking of the ideas and Maple code in previous modules. Students must decide what type(s) of function(s) to use and what types of slope conditions. For example, should they use fewer taller hills or more shorter hills?  The previous modules give students the necessary tools to experiment with different design strategies.  Some advanced Maple code is included but is certainly not necessary for success.

Finally, we mention that there are many ways to use these modules.  For example, portions of these modules may be used to introduce the ideas of intervals of increase and decrease, maximum and minimum values, intervals of concavity, and points of inflection, though that is not how they were intended.  Likewise, many portions of these modules may be extended.  For example, what is the coaster meaning of the third derivative? -- or what role could differential equations play in the design of coasters?

For the Instructor

For the Instructor

"One thrust of calculus reform has been to place greater emphasis on problem solving and to present students with more realistic applications as well as open ended problems." [Calculus by Smith and Minton]  This project (a series of modules) was designed with this in mind and should be assigned only after students have mastered the first and second derivatives and their use in determining maximum and minimum functions values, intervals of increase and decrease, intervals of concavity, and points of inflection. Students must also be able to read, understand and modify Maple code as appropriate. The ultimate goal of this project is completion of Module F, which is a challenging, realistic, and somewhat open-ended optimization problem.

At Hollins, after completing Sections 3.1-3.5 (Linear Approximations, Newton's Method, Maximum and Minimum Values, Increasing and Decreasing Functions, Concavity) from Calculus by Smith and Minton, we consider the following "warm up" coaster problem, which is an extension of a problem from Section 3.3 of that text.

A section of roller coaster is in the shape of y = x5 - 4x3 - x + 10 for x between -2 and 2.

1. Find (exactly) the intervals of increase and decrease and explain their coaster meaning.
2. Find (exactly) all relative extrema and explain their coaster meaning.
3. Find (exactly) all points of inflection and explain their coaster meaning.
4. Find (exactly) the points of steepest ascent and descent on this coaster.
5. Find the angles of steepest ascent and descent.

We then ask questions such as "How could we design our own coaster?"; "What points are important points?"; "What type of safety criteria are needed?"; "How can we compare coasters?" and "Which coasters are best?".  We then proceed to the interactive coaster project.

All students should start with module A, as it introduces the basic ideas of the project -- use of functions to fit peak and valley points, points of steepest ascent/descent, angles of steepest ascent/descent, and our mathematical definition of thrill.  In this module, students can "watch" the fitting process on real-life coasters and then use the collected data (peak/valley coordinates and slope at steepest point) to calculate the thrill of portions of the real-life coasters.  (Students may struggle with which coaster tracks to use, but this is part of the fun.) Their collected data will be used in later modules, when students are asked to "do" the fitting process for themselves and compare answers.

You may choose which modules from B through E to assign. These modules include design and analysis of single-drop coasters using trigonometric (B) and polynomial (C) functions, as well as design and analysis of several-drop coasters using trigonometric (D) and polynomial (E) functions. In these modules, students model the paths of coasters by fitting these functions (in slope and height) to specified peak and valley points -- both those collected from module A and those created by students.  In each design, students must pay attention to stated safety criteria and calculate thrill.  These modules include questions and assignments for the students (indicated in red).

Note: If you don't have Maple installed, you can still see the functionality of our Maple worksheets in a "static" form that can be viewed in your browser window. Click on any of the links in the following table to see these "filled-in" worksheets. If you prefer to work in some other computer algebra system, you can use these as models for setting up your own worksheets.

 Browser-readable (HTML) exports from Maple worksheets Module B Module C Module D Module E Module F

As previously mentioned, the ultimate goal of this project is completion of Module F.  In this module, students must use what they have learned to design the "ultimate" coaster -- a coaster that satisfies all safety criteria and that has maximum thrill.  This is a challenging assignment that requires considerable ingenuity, creativity, and tweaking of the ideas and Maple code in previous modules. Students must decide what type(s) of function(s) to use and what types of slope conditions. For example, should they use fewer taller hills or more shorter hills?  The previous modules give students the necessary tools to experiment with different design strategies.  Some advanced Maple code is included but is certainly not necessary for success.

Finally, we mention that there are many ways to use these modules.  For example, portions of these modules may be used to introduce the ideas of intervals of increase and decrease, maximum and minimum values, intervals of concavity, and points of inflection, though that is not how they were intended.  Likewise, many portions of these modules may be extended.  For example, what is the coaster meaning of the third derivative? -- or what role could differential equations play in the design of coasters?