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FACULTY NOTES
The LTAs and Spinoffs are designed so that each professor can implement them in a way that is consistent with his/her teaching style and course objectives. This may range from using the materials as out-of-class projects with minimal in-class guidance to doing most of the work in class. The LTAs and Spinoffs are amenable to small group cooperative work and typically benefit from the use of some learning technology. Since the objective of the LTAs and Spinoffs is to support the specific academic goals you have set for your students, the Faculty Notes are not intended to be prescriptive. The purpose of the Faculty Notes is to provide information that assists you to take full advantage of the LTAs and Spinoffs. This includes suggestions for instruction as well as answers for the exercises.
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FACULTY NOTES
SPINOFF 16A
Modeling The Space Shuttle Landing: The Cubic Spline
This Spinoff is suitable for a Calculus I class. The students could do LTA 16 first, but the Spinoff has been written with an introduction that gives the students the background necessary from the LTA. The purpose of this Spinoff is to examine the exponential piece that models the transitional phase and see if the glide slopes on entry and exit for that phase match the corresponding linear pieces on either side. Once students compute the derivative and evaluate it at each endpoint, they will see that the glide slopes do not, in fact, match, and therefore the exponential piece causes the function to be non-differentiable at those two points. The next task, then, is to find a more suitable function. The function suggested in the Spinoff is a cubic spline, a third degree polynomial that is constructed to satisfy the two point conditions and the two derivative conditions. This results in a system of four equations in four unknowns with very unwieldy coefficients. The Spinoff briefly introduces students to matrix methods for solving a system of equations, but it is unrealistic to expect students to solve this particular system without access to a computer algebra system of some kind. Directions are given for using a TI-89™, a TI-92™, and Derive ™. Upon completion of the Spinoff, students could be asked to suggest other functions which might also work.
Solutions
and 2) Here is the final graph:
The glide slope during the first linear phase is arctan(0.36184) = 19.9°. The glide slope during the final linear phase is arctan(0.02701) = 1.5°. The glide slope is the arctangent of the slope of the line.
4) The derivative of the exponential piece is y ′ =0.016985734105 1.000534612( x )
NASA - AMATYC - NSF
- The polynomial model is
y = − 1.18515832066 10 ⋅ -8^ x^3^ + 0.000214958911022 x^2 − 0.862588999214 x +1127.
- The derivative is y ′ = − 3.55547496198 x^2 + 0.000429917822044 x −0. f ′ (2650) = .0270099999975, which agrees with the slope of the linear function to four decimal places. f ′ (7500) = .361840000002, which agrees with the slope of the linear to five decimal places. A graph of the new piecewise function is:
