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# The University of Minnesota Calculus Initiative

The Geometry Center is assisting in the development of interactive technology-based modules for the engineering calculus sequence. These modules emphasize geometric concepts of calculus while examining applications of mathematics to the physical and life sciences.
Rainbow Lab
How are rainbows formed? Why do they only occur when the sun is behind the observer? If the sun is low on the horizon, at what angle in the sky should we expect to see a rainbow? This lab helps to answer these and other questions by examining a mathematical model of light passing through a water droplet.

Numerical Integration Lab
The fundamental theorem of calculus tells us that if we know the rate of change of some quantity, then adding up (or integrating) the rate of change over some interval will give the total change in that quantity over the same interval. But often scientists do not know a formula for a function, but can only experimentally know the value of the function at discrete times. Is it possible to "integrate" this discrete data? If so, how?

Beams, Bending, and Boundary Conditions Lab
Beams hold up the roof over your head and give support to the walls surrounding you. Beams are literally all around us. Engineers use beams to support and strengthen structures ranging from silos to bridges to towering skyscrapers. In this lab, we will explore the mathematics associated with the static deformation of beams. We will begin with the geometric concepts of centroids and moments of inertia, and then learn how different methods of supporting a beam contribute to the beam's ability to support loads.

Modeling Population Growth
Populations grow according to the number of individuals that are capable of reproduction. At the same time, their growth is limited according to scarcity of land or food, or the presence of external forces such as predators. In this module, we examine simple differential equations that model populations. We also introduce and explore powerful techniques for the geometric analysis of differential equations: phase space, equilibria, and stability.

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