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

E3 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…CNT Nanocomposite 3-Point Bend Results

0

5

10

15

20

25

0 0.02 0.04 0.06 0.08 0.1 0.12

Displacement (inches)

Lo

ad

(lb

s)

Neat Epoxy

0.015 wt. % as-received

0.15 wt. % as received

0.15 wt. % oxidized

Engineering Inspires

Bring E3 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 E3 to the Classroom - Core Elements

Functions changing with respect to a variableDistance-Velocity-Acceleration → vectors changing with

respect to timeShear-Moment-Displacement → vectors changing with respect

to lengthOther 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 E3 to the Classroom – Lead-in

Bringing E3 to the Classroom – Day 1 Discuss nanotubes,

nanocomposites, nanotechnology, E3 research

Review physics concepts

Introduce differential equations

Bringing E3 to the Classroom – Day 2 Discuss Modulus,

Moment of Inertia, and their effect

2 more differential equations

Student activity

Bringing E3 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 SyllabusAnalysis 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 E3 RET Program

BFFs