Each of the following was a modeling assignment for a second semester calculus course. The assigment is designed to require students to recognize common shapes and match them with known functions. The assignment is given after the topics using parametric form have been studied.
The assignment has two section. First the students must find a parametric model for the action described. This is often broken into multiple parts which they put together in a foreshadowing of vector techniques. The second is to use their model to find useful parameters and/or calculate derivatives. Note that the illustrations below are answer key versions: the yellow line tracing the curve is not provided for the students. They are required to use their own model. Note that the students have Mathematica available for graphing their model equations.
Traditionally textbooks have asked parametric model questions such as the cycloid and Witch of Agnesi by giving verbal descriptions. This does not match reality in which motion based problems are typically seen not read. Further my students seem almost incapable of transferring words into images. This skill is tackled through other assignments. This modeling project strips away the interpretation aspect allowing me to develop and assess solely their ability to match functions to motions.
Please also note, in an attempt to teach students that solutions are slowly and iteratively developed in life, projects involve revisions. This means that if a student's model has flaws, which would effect the second half of the project, they can submit, receive feedback, improve, and resubmit for substantial credit.
Guido is at the fair and decides to ride on a ferris wheel. Being an adventurous sort he begins swinging his gondola back and forth. Use the illustration to understand the description.
Guido did not read the Turnagain Arm tide tables and found himself stranded on a rock as the tide came back into the arm. A rescue helicopter was sent to pick him up. Unable to land, they let down a rope for Guido to grab. After rising the helicopter began moving forward. Guido, not this wisest person on earth, began swinging back and forth on the rope.
Find a model for the position of the end of the rope. The model should include as variables the forward speed of the helicopter, the length of the rope, and the speed of Guido's swinging (in terms of seconds per radian). Use the illustration to understand the description.
A helicoptor is flying in a straight line at