Copyright 2001
Department of Mathematics
University of Georgia
Athens, Georgia
Carol W. Penney
John Gosselin
Closest Point
In this project you will use the tools of Maple to solve a max-min problem. You will first set up the function you wish to minimize. You will then search for critical points and study the limiting behavior of your function for large values of
. You will then determine the minimum value of your function and give some justification as to why it is the minimum.
Project
Let
k
denote the number of letters in your last name. Your goal is to find the point
Q
on the graph of
that is closest to the point
P
= (1.65+0.1*k,2.0). The following animation illustrates the problem.
![[Maple Plot]](images/closest_point3.gif)
Express the distance from
P
to an arbitrary point on the graph of
as a function of
x
. Make a plot of your distance function and explain what it suggests about the point
Q
. What happens to the distance function as
x
gets very large in either direction? Use the tools you have learned in calculus to determine the coordinates of the point
Q
. What is the minimum distance? Use either a sign analysis of the first derivative or the second derivative test to show that the value of
x
you have found does indeed correspond to a local minimum of the distance function.
Extra Credit
Let P and Q denote the same points as in the project. The following animation is the same as the one above except that small segments of the tangent lines to the graph of f( x ) have been added.
![[Maple Plot]](images/closest_point5.gif)
Determine an equation of the line tangent to the graph of
at the point
Q.
Also determine an equation of the line passing through the points
P
and
Q
. Make a plot of f(
x
) and the two lines. You might want to add the option
scaling=constrained
to your plot to have the same scales on the horizontal and vertical axes. If
Q
is the closest point on the graph of f(
x
) to
P
, what relationship do you think should hold between the two lines? Can you verify this mathematically?
Could you have solved the original problem by using this condition? If you feel that you can, use this approach to find the point
Q
on the parabola
that is closest to the point (2,0). Show your work.
The Most Common Maple Commands
Academic Honesty Statement:
Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.
I understand that I may work with others if I give them credit in this statement. I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.
I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.
Signed: