Experiments for Teaching Calculus

1
Experiments for Teaching Calculus Authors G.V.
Loganathan (CEE) William Greenberg
(MATH) Lorraine Holub (MATH) Craig Moore
(CEE) CEE Department of Civil and
Environmental Engineering MATH Department of
Mathematics
2

• What is ESP?
• Emerging Scholars Program
• ESP is the students who are at risk in the
  scientific/engineering calculus courses.
• Students are placed in the program based on their
  Mathematics Readiness Score
• Mathematics Readiness Score is derived from a
  regression model, using the following criteria
• High school GPA
• Math SAT score
• Whether the student has had calculus in high
  school or not

3
Fall 1998
4
MATH 1205 Syllabus Topics Derivative as a
Limit Derivative as a Slope Beam and Weir
Experiments Derivative as Rate of Change Beam
and Weir Experiments Derivative of Composite
Functions Beam, Weir, and R/C Circuit
Experiments Higher Order Derivatives Beam
Experiment Maxima and Minima Beam Experiment
5

• Beam Experiment
• Students measure the following parameters and
  variables
• Length between supports
• Length from supports to weights
• Diameter of the rod
• Deflection, y for certain x values
• Students are given
• Moment of Inertia
• Modulus of Elasticity

6
(No Transcript)
7
Beam Experiment (continued) Radius of
Curvature The students differentiate the first
derivative to obtain the second derivative
as The second derivative is very nearly equal to
the reciprocal of the radius of the arc formed by
the deflected beam between its supports.
8
Beam Experiment (continued) Approximation of the
Deflection The students will use the calculated
value (dv/dx) and the measured value ?x to
predict the deflection at a certain point. The
students can compare this result to the measured
deflection.
9
Beam Experiment (continued) Questions and
Additional Problems Students answer thought
provoking questions One example involves a diving
board. Maximum Deflection (Maxima and
Minima) Students must check endpoints!
10
Weir Experiment Students Measure the following
parameters and variables Length and Width of the
rectangular tank (compute Area (A)) Width of the
weir Height of the crest of the weir (P) Depth
(h) with respect to time (t) Equilibrium Depth
(Heq)
11

• Weir Experiment (continued)
• Compute the Inflow, I and the average inflow,
  Iavg based on the rate of change
• Compute Q from the equation above
• Calibration of the Weir Discharge Coefficient (C)
  based on equilibrium.
• For dh/dt0, we have I(t)Q(t)

12
Weir Experiment (continued) Compute the
theoretical Outflow (Qtheor) for the sharp
crested rectangular weir using Cmeasured Students
compare and analyze the results
13
Highway bridge
Structural Lab experiment - Metal rod with two
pin supports
14
What we learned by the experience Experiments
should illustrate basic concepts (freshmen need
to understand the lab) Students should be able
to visualize the real world applications Experime
nts should illustrate engineering ideas Hands-on
experiments are excellent learning tools
15

• Added benefits of doing these labs
• Enhances the learning process by discussion
• Failure models
• Provide students with real world situations and
  disasters
• Material Properties
• Discuss brittleness
• Modulus of Elasticity
• Theoretical Concepts
• Moment of Inertia (Different cross-sections)
• Centroid
• Error Concepts
• How accurate can we measure physical items?

16
Conclusion The students learned to translate
observations into mathematical models. (Not just
word problems into math models.)
Observations
Problem Description
Identify Variables and Constants
Identify Process Rules
Solve the Mathematical Model
Interpret the Results