About Calculus Projects

The following projects were made possible through the support of the Class Alumni Fund Grant and the Class of 1960 Fellowship.

SketchResponse Tool

The SketchResponse tool is an extensible, gradeable sketching interface.

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Zipline Design

Bringing engineering themes, problems, and projects into the Calculus classroom.

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SketchResponse Tool

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.

Read about it

To find out more about how we designed the sketchresponse tool and see some preliminary relsults, check out the papers!

Making the tool

Preliminary results

Engage with it

Try the sketchinput tool. Test the grading functionality by solving the problem and hitting the check button.

Try the demo

Download it

This tool is available as an open source tool on github.

Github link


Observe that the student-facing portion provides a set of (customizable) semantically-meaningful drawing tools.

Sketch the function \( \ \displaystyle f(x)=\frac{2x^2}{x^2-1} .\)

Zipline Design

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.

Sample Calculus Problems

Linear approximation

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?

Mathlet Visualization

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?