Project #1 -- Landscape Architecture
Assignment
Assume you are working for a landscape architecture firm which has been hired
by a large company that wishes to redesign the entry way to their headquarters.
The centerpiece of the new entry will be a garden, bordered by a smooth wooden
frame. The frame must pass through a collection of one hundred specified points.
It will be designed in 10 stages, where each stage will produce a segment of the
frame that passes through ten of the one hundred points.
Your task is to design the first of the 10 segments by finding a ninth degree
polynomial whose graph passes through the ten points:
(0,3),(1,3),(2,-3),(3,5),(4,2),(-1,2),(-2,0),(-3,0),(-4,-1),(-5,1)
and prepare a report for your supervisor delineating your solution.
Mathematics Needed
You will need an understanding of the idea of a basis for a vector space, as well as
an understanding of the
Lagrange Interpolation Formula (explained on pages 47-49 of your text).
The Report
The report to your supervisor should contain (at least) the following sections:
Introduction and Summary : This section should give a brief account
of the subject matter discussed in your report. Someone should be able to
read this first section of your work and understand what the problem is
and your response to it. You should also provide a brief outline of
your method; include only the major ideas, and describe them in generality.
Method : This is the heart of the report, and the most
important part of this assignment. In this section you should not only
clearly describe what steps you took to solve the problem, you should also strive
to explain to the reader why the steps you took were the right ones to
take. In other words, offer a concise explanation of what you understand to be the
ideas behind the calculations. You may want to give a brief and very clear
explanation of the
Lagrange Interpolation Formula, where it came from, and how it was used to solve the
problem.
Please do not delve into the gory detail of your computations. It's unnecessary to
list every step of every process, and doing so is deathly boring for your
reader. It is sufficient to say what you did and why, show the key intermediate
steps of the calculation, and then show what
resulted from your calculation.
Conclusion : This section will probably be very short. It should
clearly state the polynomial you found (whose graph passes though the ten
given points), and provide a picture of the graph of that polynomial (copied
off of your graphing calculator? from computer lab? using calculus I
techniques?), and give a satisfying concluding comment to your report.
Points
This project will be worth 20 points towards your final grade for the course. The point
breakdown is
- Mathematical Methodology: Your
calculations, approach and procedure should be clear, complete and correct.
-- 10 points
- Clarity of your explanations: It is very
important to be able to communicate your solution clearly to non-experts - they may
be the ones making decisions based off your findings! -- 4 points
- Good Grammar: Grammar, spelling, general professionality of completed
document (the spell check
is just the beginning....) -- 3 points
- Mathematical Notation: Use correct
mathematical notation in the correct context -- 3 points
- Extra Credit Any imaginative ideas,
background stories, mathematical/artistic additions which extend the topic
in some (tasteful) way. -- 3 points
Project adapted from the files of Carl V. Lutzer