Engineering in a Calculus Classroom Department of Aerospace Engineering, TAMU Advisors: Dr. Dimitris Lagoudas, Dr. Daniel Davis, Patrick Klein Jeff Cowley, Lesley Weitz Mike Vogel, Mathematics Teacher KIPP: Houston High School

Carbon Nanotubes Cylindrical carbon lattice Can be single- or multi-walled Strong and lightweight High thermal and electrical conductivity

Nanocomposites Composite of nanotube and matrix material Research explores how their properties can improve the properties of the matrix material Multifunctionality

E 3 Research – How do Nanocomposites Compare? Experiments on nanocomposite beam, thermal conductivity, electrical conductivity How do composite properties compare? Mathematical models calculating shear, displacement, etc.

Applications of Nanocomposites and Nanotubes Sports - Bicycles, golf equipment, hockey sticks, baseball bats Aerospace Prosthetics

Epoxy Beam 0.15% weight High surface area to volume

Beam-bending in the lab

Sample Results…

Engineering Inspires Bring E 3 experience to the classroom Connect engineering “See” and relate Calculus to physical world

Terms & Concepts: loads (point, distributed), shear, moments, modulus, Moment of Inertia, displacement Mathematics: link Calculus to physical phenomenon other than motion Hands-on exploration: Beam-bending Bringing E 3 to the Classroom - Core Elements

Functions changing with respect to a variable Distance-Velocity-Acceleration → vectors changing with respect to time Shear-Moment-Displacement → vectors changing with respect to length Other examples? Introduce beam concepts as part of introducing physics concepts Integration of “rate” functions as displacement Introduce solving differential equations General vs. Particular solutions Bringing E 3 to the Classroom – Lead-in

Bringing E 3 to the Classroom – Day 1 Discuss nanotubes, nanocomposites, nanotechnology, E 3 research Review physics concepts Introduce differential equations

Bringing E 3 to the Classroom – Day 2 Discuss Modulus, Moment of Inertia, and their effect 2 more differential equations Student activity

Bringing E 3 to the Classroom – Day 3, Group Project CAS – Maple, TI-nspire Student activity – test students’ modulus calculations Introduce Group Project

Group Projects Derive equations for simply-supported beam, distributed load, intermediate load on a cantilever, etc. Class presentations (time permitting), including beam demonstration Discuss applications to their beam project

Objectives – Solve Differential Equations

Objectives – Graph Connections

AP Calculus Syllabus Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function. Derivative as a function Corresponding characteristics of graphs of ƒ and ƒ’ Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’ Applications of derivatives Optimization, both absolute (global) and relative (local) extrema Modeling rates of change, including related rates problems Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration Interpretations and properties of definite integrals Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and accumulated change from a rate of change. Applications of antidifferentiation Finding specific antiderivatives using initial conditions, including applications to motion along a line

Pre/Post Test

Acknowledgements Dr. Dimitris Lagoudas and Dr. Daniel Davis Patrick Klein Jeff Cowley and Lesley Weitz NSF E 3 RET Program BFFs