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Model for the time it takes an ant to build a tunnel. We employ the derivative and antiderivative to model the time it takes an ant to dig a tunnel of length x. This is a paper and pencil activity and does not need technology.

Data Fitting

A competition: Fitting the best parabola to the St. Louis Arch. This is a curve fitting contest: find the parabola that looks the most like the (catenary shaped) St. Louis arch. It could be given any time after the definite integral has been developed.
Batter Up: The Physics of Power in Baseball. This is an application for definite integrals. The problem uses data which leaves a good reason to use a numerical method such as the trapezoid rule.
Function Fitting to Descriptions of Curves. FUNCFIT is a fun classroom activity to familiarize students with the shapes of various functions. Groups write non-technical geometric descriptions of graphs and exchange them with other groups. Then students try to recreate graphs from the descriptions. The exercise is appropriate for pre-calculus or beginning calculus students.
A Golf ball's Flight With and Without Lift. The classical projectile (golf ball in this case) motion problem is contrasted with a more realistic one accounting for lift forces due to the spin on the ball. The problem is still written for students new to projectile motion. Rather than having to understand a more complicated model, data is presented and the students are expected to use computer software to fit an appropriate function to the data.
Population Rank Modelling. Fitting a function to the population vs. rank data for major cities in a region and conjecturing (1) how this function changes over time and (2) how plots for different societies, e.g. rural, industrial, compare.
Protozoa - How fast can they grow? How far can they reach? Fitting the population data to the logistic growth model using differenced data on the differential equation or actual data on the solved equation. Several approaches are suggested and comparison is of interest.
Data analysis through mathematical model of data collected from a "rock" experiment. An experiment in data collection and analysis is described in which collective groups of subjects are analyzed as to their abilities to differentiate masses of rocks. A reasonable sigmoidal function is needed to sit the data and comparisons between groups of subjects leads to fun discussion as to which group is better.
Modeling World Class Sprinters in 100 Meter Dash. The Hill-Keller differential equation model for runners is used to ascertain parameters for two world-class runners by fitting the solution to the differential equation to world record runs with split times. An issue is whether or not the model distinguishes the runner's abilities, or at least the runner's approach to the sprint.
Sound Bite length for Presidential Election - Predictions. Over time the average length of uninterrupted speech offered by a Presidential candidate on the evening news has decreased. Where is the average length of these sound bites headed? We take data from a New York Times article and see if politicians are doomed to a "Yup!" "Nope!" soundbite in the future and if so when?
Building a Path with Steps and Pavement. This exercise uses the notions of arc length and slope to determine the cost of building a path up a slope that has steps when it's steep and is paved otherwise.
The Tide Is Turning. Based on Coast Guard data of water levels in port over a 24 hour period we are asked to offer a mathematical model to "match" the data and to make intermediate predictions on height based on that model. Estimating with trigonometric functions proves to be very successful.
Tape thickness and speed. It's easy to estimate how thick a piece of paper is by measuring a ream of 500 sheets. This (fairly easy) problem deals with estimating the tape thickness on a video cassette, and then asks for a function which represents the tape driver speed. The problem is accessible to precalculus students, but accuracy and modeling issues also make it an appropriate application of polar integration!

Difference Equations

A Simple Dynamic System: modeled with differences. This problem is intended for late in a calculus sequence as an introduction to modeling with discrete dynamical systems and matrix algebra.
Feigenbaum Diagrams and Extensions for Cantor Sets. The goal of this problem set is to familiarize students with bifurcations of maps. The problems require knowledge of the derivative, and are appropriate for a course in chaos, discrete math, or calculus (given about a week to spare.) Students are motivated through the beauty of various types of bifurcation diagrams.
Optimization of solutions to discrete dynamical systems. Comparing various investment plans in which we pay a broker's fee for finding higher interest rates using discrete dynamical systems.

Differential Equations

Comparing two models for a dissolving capsule. Simple differential equation models for dissolution of spherical material in solution are compared.
Parachute - the Full Experiences. A simple model of a sky diver's trip to the ground can yield reasonable position and velocity functions but by furthering considering the acceleration, the models require modification. This problem explores this issue and is appropriate for students who are prepared to set up and solve (at least with a computer) first order ODEs.
Polution Concentrations in Industry Lake. A model for lake with pollutants is studied (1) first assuming instantaneous mixing and then (20 with diffusion in a two layered lake. Model results are compared for which is more realistic and why.
Resonance in a Simple Dynamic System. A study of a n oscillator in which we attempt to ascertain the maximum frequency response, where frequency response is the amplitude of the output signal as a function of the frequency of the input signal to a second order differential equation modeling a spring-dashpot system.
Issues involved in Torricelli Model of tank leaking fluid. Several slit configurations in the side of a cylindrical can are considered to develop specific "leaking" patterns using Torricelli's Law.
Modeling an Oil Slick Growth. Can we use differenced data taken at unknown starting times to ascertain the size of a growing oil slick? Either difference or differential equation models will permit discovery if student are willing to plot and do some differencing.

Exponential Models

Variable Frequencies. A string vibrating at a constant frequency can be modeled by A sin(2 Pi w t) where w is the frequency of vibration. However, if the frequency changes over time, then w is not the frequency. This set of exercises explores how to find the true frequency given w as well as how to find w given the desired variable frequency.
Musical Exponents: Modeling in music with exponentials. Here's a fun classroom exercise to introduce exponential functions as an application to music.
Music in Mathematica. This lab exercise is designed for students with access to Mathematica. The problem builds on OCTAVE (also at this WWW site) by using exponential functions to write music.


Archimedes Method to Approximate Pi. This analytic geometry problem is well described by its title. It fits well early in a calculus course.
How to cut a circular cake in three equal pieces. Determine several regular ways to cut a circular cake in thirds.
The Volume of a Tetrahedron. By folding a triangle, create a tetrahedron and find its volume. This is a short yet challenging problem to enhance 3D visualization. While it could be solved with calculus, a nicer solution relies only on skills learned in HS geometry.


Area Fitting. We seek to determine parameters in functions so that integrated areas will meet required specifications.
Computing the area of a complicated region -- An Integration Lab. This lab is devoted to measuring the area of a region using several techniques (both numerical and analytical) and then to contrasting the results.
How Much Does a Beam Bend? This problem shows an application of calculus to the analysis of one of the simplest load carrying structures: the beam. In particular, you will be finding the amount of deflection (bend or sag) in a beam carrying a distributed load. The calculus skill you will be applying is integration.
How to cut a hemispherical loaf of French bread so that each of 8 guests gets the same amount of crust. The claim is that by slicing parallel sections of equal altitudes from a sphere we get identical surface areas of these sections. The problem is posed in terms of equity of crust distribution for French bread.
We determine the volume of a circular bore in a hollow cylinder. Boring a small cylindrical hole into another solid cylinder to determine a set center of mass and volume is the interest here.
Inertial Navigation: Determining position and velocity with accelerometer. An inertial guidance system breaks down, only giving acceleration information. Where are we? Where are we heading?
Moonlight in Vermont - Don't let the moon phase you - at least without plotting it!! Here's a simple question to ask, but a hard one to answer! How much of the lit moon is visible? One solution is made easier with spherical coordinates.
Painting a Hemispherical Dome. Sphere is designed as a big homework problem using spherical coordinates (among other things!) The problem revolves -- no pun intended -- around issues of painting a hemispherical dome.

Inverse Problems

Determining the mass distribution of a bar from observed forces using physical principles. Determine the gravitational force at a point from a nearby mass and determine the underlying mass distribution given force information.


Average Velocity of a Bicyclist. This project has students gather (sample data is provided too) and analyze time-distance data from a bicycle ride to estimate the velocity by differencing the data over several time intervals.
Designing a Fair Ballpark. Some ball parks favor right hand hitters over left (and possibly vice versa.) This problem is concerned with the how to build the outfield fence (height) so that the ball park isn't biased for any directions. This is an involved problem accessible to students of projectile motion.
Studying the Path of Points on a Bifold Door as it Opens. We seek parametric equation (kinematics) description of points along a two panel folding door as well as attendant velocity and acceleration information to model physical motion.
Motion of a Panel. We seek to determine the position of the center of mass of a plate whose motion is governed by a constant velocity motion applied through a complex mechanism. The mathematics involves identifying the plane as complex coordinate system and using complex numbers to locate points on plane of motion.
Shooting a Cannonball Over a Wall. We present problems related to shooting a cannonball over a mountain to hit a target with a minimum velocity. Two dimensional projectile motion is appropriate.
Distance and Velocity Graphs from Car Ride Data. Here's a fun group project for differential calculus students. Students go on a drive, record speed and distance, and then analyze their data.
Catapult - Up and Away. We maximize the range of a projectile by backing up an incline in the opposite direction of the range to give some initial lift, but what is the best lift.
Cornering at Full Speed. Curvature is a topic often left for vector calculus, but this curvature problem was designed for calculus students who have just learned about position, velocity, and acceleration.
How deep is a well -- using sound of falling body striking the bottom of the well? We drop a small stone in a deep well. Given the time elapsed from release until we hear the splash determine the depth of the well.
Kind of a Drag. Kinematic descriptions of motion for a point on the dragging end of a stick which is hinged to a point on a rolling disk are developed and used to determine velocity and acceleration quantities.
Specialized Film Making. A camera tracks an object undergoing sinusoidal motion. We develop equations for the tracking camera and ascertain maximum angular acceleration and maximum focal length acceleration.
Flight Paths. Flight paths is a project comprised of 3 flight scenarios: crop dusting, a movie stunt pilot, and a passenger plane landing. Among other things, students model flight paths by choosing functions that satisfy necessary constraints, and then reflect on safety and/or comfort issues based on the g's that the plane pulls.
Torricelli's Fluid Law: A Dry Lab Approach. Dry lab data from an experiment in fluid flow using Torricelli's Law form the basis of this modeling activity.
Hit the Target. This projectile motion problem deals with the relationship between launch angle and starting height when attempting to hit a target.
Home Run Derby. In this projectile (base ball) motion problem, we seek to find out whether a player makes a home run given the dimensions of the stadium field as well as the ball's initial velocity and launch angle. This could be used as a class room activity or homework problem.
Off the Wall - Just a Game. We attempt to determine just how we should launch a rubber ball so that it bounces of the sides of a deep vertical box and finally goes through a slit near the bottom of the box in order to win a prize. We use our laws of falling bodies many times.
Clear the Hill, Hit the Target. We seek a minimal initial velocity on a projectile in order to clear a hill and hit a target behind the hill.
Pipes and Valves - Using a disk to cover or plug a pipe flow. How fast does the flow rate change as a valve is closed? Building on this question, PIPEPLUG (or even parts of it) could be used as a term project for multi-variate calculus.
Rendezvous for aircraft refueling. This is an open-ended problem for differential calculus (using notions of position and velocity) in which students determine the flight position of a refueling plane.
Setting up the Cutoff for Baseball Relay. This uses a simple differential equation to model the relay between an outfielder and an infielder throwing the ball home.
Relative Motion of Rod and Yoke. We seek to render parametric equations which describe the tip of a rod sitting astride a moving, circular-headed, piston, given the rules of motion of the piston.
Me and My Shadow - the former on the ground the latter on a dome. We seek to characterize the motion of my shadow on a dome from a light source on the ground behind me as I walk toward the dome.
At The Pass - Can Rose-Hulman's Solar Car Pass The University of Michigan's Solar Car in the allotted time? Given certain physical and legal restrictions can one car pass another in a given strip?
Designing a Fair Home Run Stadium. For a given playing field outline how high must the outfield fence be at each point in order to make a homerun equally likely in all fair directions?
Exploring times that a particle projected upward passes through the same point -- once going up and once coming down. First find out maximum height H a given projectile will go. Then test the time T(a) (0 < a < 1) it takes between when the projectile passes distance aH going up and then coming down. Develop T(a) as a function of a.


Tracking an object on a circular arc and optimizing an angular velociy and angular acceleration. We track an object moving along a circular path with a camera, ascertaining various quantities including angular acceleration of the camera's turning motion. Makes heavy use of parametric equation descriptors of position.
Inscribe a circle of maximum area in a region bounded above by a function and below by the x-axis. We seek to determine the maximum area circle which can be inscribed several selected regions using geometric optimization strategies.
Determine the process for manufacturing the most profitable enclosed box given associated costs and revenue parameters. We seek to determine first the maximum volume box cut from a fixed planar region and then the minimum cost box cut from comparable region.
Maximize volume of a box made from cutting corners off flat sheet and folding along resulting edges. We consider an extension of the problem of constructing a box by cutting four equal squares from a rectangle and folding to one in which the cutouts can be more general quadrilateral introducing another variable.
Maximizing the Volume of a Rectangular Box. We seek to determine the maximum volume (in terms of standard soda can volumes) of a box cut from a fixed planar region.
Put out the Camp Fire. This Oldie is a nice introductory calculus problem: find the shortest path from your car to a camp fire if you have to stop at the river bank to get water on the way. Next, find the shortest travel time (and corresponding route) possible given that you can walk faster without water than with it.
Cans With a Cone - Forming a Cone of Maximum Volume from a Flat Disk. The goal is to produce a cone of maximum volume by cutting a sector from a fixed circle and connecting the two radial edges.
Optimal Volume Paper Cone from Wire Paper Mold. From a fixed length of wire make a "Pac-Man" shaped perimeter (circle with sector cut out) so that if we fill the interior with paper mesh, cut out the paper, and fold to make a cone by connecting the two radial edges, we have a maximim volume cone.
Investigating Extreme Values of Function Combinations. Sometimes functions don't always look the way we expect them to. This short classroom exploration helps students understand the effects of adding functions together and parameter dependence.
Using Sound to Determine Subsurface Geological Structure. We determine the flow of soundwaves in an underground sonic boom through various media and we attempt to determine the shape of the underlying region given timings on soundwave propagation.
Getting Malled - Determining the geographical region for which travel to a mall is one hour or less given a road configuration. Determine the one hour driving neighborhood about a shopping mall when a high speed highway is put in place. Optimization of a function of one variable will help.
Maximum Area Rectangle in Function Boundary. Determination of maximal inscribed rectangle in a region which does not permit functional representation of an area function as an explicit function of one variable.
Determining the largest circle which we can inscribed in a geometric region. Determine the largest circle which can be inscribed in the region bounded by y = cos(x), x = 0, y = 0.
Maximum Horizontal Bounce Distance of Ball Bouncing On Inclined Ramp. Drop a ball on an inclined ramp from a fixed height. Of all the angles you can incline the ramp which permits the ball to bounce OUT the farthest.
The Road Less Paved. Here's a nice minimization problem to get your students thinking: Pave the smallest amount of connecting roads as possible between four homes located at the vertices of a one square mile field. Differential calculus of a single variable and an open-mind are prerequisites.
ROCK OUT! Best place to sit during battle of the bands! This is a straight forward homework assignment for students of max/min problems of functions of two variables. Here, we seek to minimize noise as a function of your position relative to (noisy) bands.
Curve Fitting and Optimization applied to HydroElectric Power. This realistic problem deals with controlling the flow rate through turbines to optimize power. It is meant to be done with Lagrange multipliers.

Partial Derivatives

Discover the Material (Total) Derivative! How does a cyclist perceive temperature changes as she pedals between two cities of different and changing temperatures? This classroom or homework exercise offers a fairly realistic and understandable application of the chain rule for partial derivatives.

Parameter Estimation

Reaction chemistry, parameter estimation, and optimization. After modeling a chemical reaction A->B->C with a set of linear, first-order differential equations we estimate kinetic parameters through a number of methods and determine the run time for the reaction which will optimize a financial return on the reaction.


What's the Beat? This exercise deals with the relationship between a beat's frequency and the frequency of the lower tone. (A beat (a wa-wa tone) is a superposition of two tones vibrating at close frequencies.) This makes for a good trigonometry review -- no calculus is required for this problem.
Design of Cams. Cam provides an excellent application to motivate students to study the relationships of position, velocity, acceleration, and jerk. The problem is designed as a project accessible to students who are familiar with the derivatives and anti-derivatives. In essence, the problem asks students to determine the "lift" function that satisfies certain properties, and then to note properties of the function and reflect on their solution.
Kinematic Analysis of a Reciprocating Engine. With the given realistic data, the main goal is to determine the piston's position with respect to time as well as the connecting rod's angle from the piston axis with respect to time. This problem is designed to be completed within a single class period once students are familiar with the concepts of position, velocity and acceleration.
The Railroad Track Problem. When a section of track buckles, the effects can be quite dramatic. Three models for such a scenario are considered. This can be used at the beginning of a calculus course.
Analysis of a Simple Structure. In this multivariate calculus application, we determine the position of a truss joint by minimizing potential energy. Physics preliminaries are provided.
Analyzing projected light on a flat screen from a point source of light inside an elliptical cylinder filled with water. We attempt to ascertain how fast the light cast by a centrally located ball of light sitting in the middle of an elliptical tank of water is passing along a flat panel which is tangent to the cylinder. We use some physics and the fact that light travels at different speeds in different media.


Bombs, Baseball, e, and the Poisson Distribution. A number of different data are considered as being random, i.e. satisfying the hypotheses of the Poisson Process. These include V2 rocket hits in London in World War II to no-hitters per season in baseball.


Mathematically describe a flat piece of sheet metal which, when folded about a cylinder of radius 1 unit, will produce a desired curved panel. We seek the mathematical description (equations) which describes the cut figure in a flat sheet of metal which when bent around a cylinder gives a prescribed shape.
Spelling out LOVE with functions. Here's a quick challenge for students learning how to plot parametrically: write a word with parametric curves.
What we can see on one hill from a nearby hill. We ask readers to place their eye on one mountain and (1) describe what they can see, (2) how much surface area they can see, and (3) nearest point - all relative to a nearby mountain.
Designing a Perfume Bottle. A number of attempts are given to make an aesthetically pleasing perfume bottle using integration using both numerical estimates and exact methods.
Point of Maximum Viewing Angle. From a given point Q what point P on a nearby surface gives the greatest angle of inclination as we view P from Q?