Freely rising air bubbles in water sometimes assume the shape of a spherical cap, a shape also known as the "big bubble". Is it possible to find some objective function involving a combination of a bubble's attributes for which the big bubble is the optimal shape? Following the basic idea of the definite integral, we define a bubble's surface as…

This article presents an application of standard undergraduate ODE techniques to a modern engineering problem, that of using a tuned mass damper to control the vibration of a skyscraper. This material can be used in any ODE course in which the students have been familiarized with basic spring-mass models, resonance, and linear systems of ODEs.…

Mathematical modeling occupies an unusual space in the undergraduate mathematics curriculum: typically an "advanced" course, it nonetheless has little to do with formal proof, the usual hallmark of advanced mathematics. Mathematics departments are thus forced to decide what role they want the modeling course to play, both as a component of the…

This project served as a capstone event for the United States Military Academy sophomore Calculus II course. This multi-disciplinary problem-solving exercise motivated the link between math and biology and many other fields of study. The seven-lesson block of instruction was developed to show students how mathematics play a role in every…

This article is a project that takes students through the process of forming a mathematical model of bicycle dynamics. Beginning with basic ideas from Newtonian mechanics (forces and torques), students use techniques from calculus and differential equations to develop the equations of rotational motion for a bicycle-rider system as it tips from…

This project was designed to be used in a freshman calculus class whose students had already been introduced to logistic functions and basic data modeling techniques. It need not be limited to such an audience, however; it has also been implemented in a topics in mathematics class for college upperclassmen. Originally intended to be presented in…

As part of an integrative learning experience at the end of a sophomore Calculus II course at the United States Military Academy, this project served as a multidisciplinary problem-solving exercise that explored the connections among mathematics, biology, and other fields of study. During a seven-lesson block of instruction, this module was…

This project was used in a mathematical modeling and introduction to differential equations course for first-year college students. The students worked in two-person groups and were given three weeks to complete the project. Students were given this project three weeks into the course, after basic first order linear differential equation and…

This article presents material that has been used as a classroom activity in a calculus-based probability and statistics course. The application was used in the first few lessons of this course. Students had three previous semesters of math, including calculus (single and multivariable), differential equations, and a course in mathematical…

We detail a student project that incorporates the story of the 1918 Influenza Pandemic, which, between the summer of 1918 and the spring of 1919, infected one-fifth to one-half of the world's population and killed more U.S. soldiers fighting in World War I than the war itself. The project uses Susceptible, Infected, Recovered, and Dead (SIRD)…

This paper describes details of development of the general birth and death process from which we can extract the Poisson process as a special case. This general process is appropriate for a number of courses and units in courses and can enrich the study of mathematics for students as it touches and uses a diverse set of mathematical topics, e.g.,…

The material presented here was used for a semester-long capstone project for a first semester freshman course entitled Mathematical Modeling and Introduction to Calculus. The goals for the students in this work were twofold: first, enable the students to gain insight into an actual problem that affects millions of people in the United States and…

This project was developed as an interdisciplinary application of the optimization of a single-variable function. It was used in a freshman-level single-variable calculus course. After the first month of the course, students had been exposed to the concepts of the derivative as a rate of change, average and instantaneous velocities, derivatives of…

A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…

This project is one of four that were issued to first semester sophomore undergraduates at the United States Military Academy as part of an integrated learning experience at the end of their Calculus II course work. This project was used during a short, seven lesson block of instruction that was intended to capitalize on their recent academic…