Try out the following calculations in Maple. Focus on precisely what information
is taken from the graph and how Maple is finding the roots of the equation.
> with(plots):
> eqn:=sin(x^2-1)^3 -2*x*cos(sin(x)+cos(x^2/5));
> plot(eqn);
> SLN:=solve(eqn);
> evalf(SLN);
> allvalues(SLN);
> x1:=fsolve(eqn,x=-2*Pi..-7*Pi/4);
> x2:=fsolve(eqn,x=-7*Pi/4..-Pi);
> x3:=fsolve(eqn,x=-Pi/2..0);
> x4:=fsolve(eqn,x=0..Pi/2);
> x5:=fsolve(eqn,x=Pi/2..Pi);
> x6:=fsolve(eqn,x=Pi..5*Pi/4);
> x7:=fsolve(eqn,x=5*Pi/4..2*Pi);
> display(plot(eqn),pointplot([[x1,0],[x2,0],[x3,0],[x4,0],[x5,0],[x6,0],[x7,0]]));
For this assignment we are going to use the method of Lagrange multipliers
to solve a problem. The first one is a warmup. The second will require
you to be somewhat smart about how to find the solutions (look at the
graph and use a guess to help Maple find the answer).
1. Assume $0 \leq a,b,c,d$ and $c\geq a$ and $d\geq b$.
Inscribe a rectangle with vertices $(0,0), (a,0), (0,b)$ and $(a,b)$
inside a triangle with vertices $(0,0), (c,0)$, $(0,d)$. Find the values
of $c$ and $d$ which minimize the area of the triangle (as a formula
in terms of $a$ and $b$).
Using the method of Lagrange multipliers, you should be able to solve this
by hand, but I would like you set up the equations, fill in the explanation
of how you use Maple to find the solution to this problem. How do you
know that your solution is in fact a minimum? What can you do to convince
the reader that your answer is correct?
2. Say that the Death Star is traveling towards the planet Alderaan
(outlined at (1+0.1*cos(t), 1+0.1*sin(t))) traveling along the trajectory
(3*(p-0.5)/(1+(p-0.5)^3), 3*(p-0.5)^2/(1+(p-0.5)^3))
for $0 \leq p \leq 20$. Plot the planet and the trajectory of the Death
Star. Find the values of $p$ and $t$ which minimize
the distance from a point on the curve to the planet's surface
subject to the constraint that the from the planet to the Death Star is
perpendicular to the trajectory. These are the points where the
Death Star should fire on the planet. Find the points and the distance to
from the Death Star to the planet and label and plot these on your
display. Do they by eye seem to be what you would expect?
Why? Label
your equations and variables. Make sure that you explain clearly
what the equations you are solving represent and what the solution
Maple finds represents (is it a minimum or a maximum? are you sure? why?).
You should open up a new worksheet and start from scratch. You will have to save
your work in a file and upload that file on to the course
moodle. Your
solution should be a sequence of commands where it is easy to change the input
string and after you execute the sequence of commands you should have the
correct output string. Please add documentation to your worksheet to explain how it
works. Just a few sentences is sufficient, but imagine that you were opening up the
worksheet for the first time and wanted to know what it did. You will be marked down
if what you write is not clear and coherent.
You should finish your assignment by Tuesday, December 4 by 11:59pm.
Assignments submitted after this date will be assessed a penalty of 10% per day.
