engineering in a calculus classroom department of aerospace engineering, tamu advisors: dr. dimitris...
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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