The SketchResponse tool is an extensible, gradeable sketching interface.
Bringing engineering themes, problems, and projects into the Calculus classroom.
One of the main goals in calculus is to deepen students' intuition for functions and their graphs. To support students learning to sketch qualitative graphs of functions, the SketchInput Tool was designed. The sketch input tool consists of a student-facing web application and a collection of server-side processing and grading code.
To find out more about how we designed the sketchresponse tool and see some preliminary relsults, check out the papers!
Try the sketchinput tool. Test the grading functionality by solving the problem and hitting the check button.
Observe that the student-facing portion provides a set of (customizable) semantically-meaningful drawing tools.
Sketch the function \( \ \displaystyle f(x)=\frac{2x^2}{x^2-1} .\)
Modeling a zipline is a rich problem that involves many aspects of calculus, differential equations, and mechanics. There are many ziplines of different lengths, different maximum velocities, appropriate for different rider body masses. There are two limiting models. The simplest model assumes an elliptic path, that is you assume that the mass of the cable is zero, and all of the mass is coming from the rider (reasonable for short ziplines and heavy riders). The other limiting case assumes a catenary path by supposing that all of the mass comes from the cable, and the rider is essentially massless (reasonable model for the longest ziplines in the world). Most ziplines, however, fall somewhere in the middle, where the mass of the rider and the mass of the cable are of the same order of magnitude.
Using an energy argument and assuming no energy loss, how does a small change in the total vertical drop experienced during a zipline ride change the maximum possible velocity of the rider?
Assuming that a zipline rider follows a frictionless, elliptic path, describe the point on the path where the rider experiences the maximum velocity.
Assuming that the mass of the zipline cable is much greater than the mass of the rider, what is the shape of the path traversed by the rider? In other words, what is the shape of the hanging cable?
See the mathlet visualization and interactive demonstration, created by MIT student Peter Kleinhenz, of the quasistatic position of a person on a zipline as a function of the arc length traversed, the total vertical drop, and the ratio of rider mass to cable mass.
Comments or questions?
