MTH 2001: Project 2 Instructions • Each group must choose one problem to do, using material from chapter 14 in the textbook. • Write up a solution including explanations in complete sentences of each step and drawings or computer graphics if helpful. Cite any sources you use and mention how you made any diagrams. • Write at a level that will be comprehensible to someone who is mathematically competent, but may not have taken Calculus 3. Use calculus, but explain your method in simple terms. Your report should consist of 80−90% explanation and 10−20% equations. If you find yourself with more equations than words, then you do not have nearly enough explanation. See the checklist at the end of this document. • One person from each group must present the work orally to Naveed or Ali. Presenters must make an appointment. Visit the Calc 3 tab: http://www.fit.edu/mac/group_projects_presentations.php • Submit written work to the Canvas dropbox for Project 2 by October 7 at 9:55PM. The deadline for the oral presentation is October 7 at 2PM. Problems 1. You probably studied Newton’s method for approximating the roots of a function (i.e. approximating values of x such that f(x) = 0) in Calculus 1: (1) Guess the solution, xj (2) Find the tangent line of f at xj, y = f′(xj)(x−xj) + f(xj) (1) (3) Find the tangent line’s x-intercept, call it xj+1, 0 = f′(xj)(xj+1 −xj) + f(xj) xj+1f ′(xj) = xjf ′(xj) −f(xj) xj+1 = xj − f(xj) f′(xj) (2) (4) If f(xj+1) is sufficiently close to 0, stop, xj+1 is an approximate solution. Otherwise, return to step (2) with xj+1 as the guess. See this animation for a geometric view of the process. It simply follows the tangent line to the curve at a starting point to its x-intercept, and repeats with this new x value until we (hopefully) find a good approximation of the solution. Newton’s method can be generalized to two dimensions to approximate the points (x,y) where the surfaces z = f(x,y) and z = g(x,y) simultaneously touch the xy-plane. (In other words, it can approximate solutions to the system of equations f(x,y) = 0 and g(x,y) = 0.) Here, the method is http://www.fit.edu/mac/group_projects_presentations.php http://upload.wikimedia.org/wikipedia/commons/e/e0/NewtonIteration_Ani.gif (1) Guess the solution (xj,yj) (2) Find the tangent planes to each f and g at this point. z = f(x,y) = z = g(x,y) = (3) Find the line of intersection of the planes. (4) Find the line’s xy-intercept, call this point (xj+1,yj+1), xj+1 = yj+1 = (5) If f(xj+1,yj+1) < ε and g(xj+1,yj+1) < ε for some small number ε (error tolerance), stop, (xj+1,yj+1) is an approximate solution. Otherwise, return to step (2) with (xj+1,yj+1) as the guess. (a) Find equations of the tangent planes for step (2), an equation for their line of intersection for step (3), and find formulas for xj+1 and yj+1 for step (4). (b) What assumptions must we make about f and g in order for the method to work? How might the method fail? Explain in words h ...

1. MTH 2001: Project 2
Instructions
• Each group must choose one problem to do, using material
from chapter 14 in the textbook.
• Write up a solution including explanations in complete
sentences of each step and drawings or computer
graphics if helpful. Cite any sources you use and mention how
you made any diagrams.
• Write at a level that will be comprehensible to someone who is
mathematically competent, but may
not have taken Calculus 3. Use calculus, but explain your
method in simple terms. Your report should
consist of 80−90% explanation and 10−20% equations. If you
find yourself with more equations than
words, then you do not have nearly enough explanation. See the
checklist at the end of this document.
• One person from each group must present the work orally to
Naveed or Ali. Presenters must make an
appointment. Visit the Calc 3 tab:
http://www.fit.edu/mac/group_projects_presentations.php
• Submit written work to the Canvas dropbox for Project 2 by
October 7 at 9:55PM. The
deadline for the oral presentation is October 7 at 2PM.
Problems

2. 1. You probably studied Newton’s method for approximating
the roots of a function (i.e. approximating
values of x such that f(x) = 0) in Calculus 1:
(1) Guess the solution, xj
(2) Find the tangent line of f at xj,
y = f′(xj)(x−xj) + f(xj) (1)
(3) Find the tangent line’s x-intercept, call it xj+1,
0 = f′(xj)(xj+1 −xj) + f(xj)
xj+1f
′(xj) = xjf
′(xj) −f(xj)
xj+1 = xj −
f(xj)
f′(xj)
(2)
(4) If f(xj+1) is sufficiently close to 0, stop, xj+1 is an
approximate solution. Otherwise, return
to step (2) with xj+1 as the guess.
See this animation for a geometric view of the process. It
simply follows the tangent line to the curve
at a starting point to its x-intercept, and repeats with this new x
value until we (hopefully) find a
good approximation of the solution.
Newton’s method can be generalized to two dimensions to
approximate the points (x,y) where the

3. surfaces z = f(x,y) and z = g(x,y) simultaneously touch the xy-
plane. (In other words, it can
approximate solutions to the system of equations f(x,y) = 0 and
g(x,y) = 0.) Here, the method is
http://www.fit.edu/mac/group_projects_presentations.php
http://upload.wikimedia.org/wikipedia/commons/e/e0/NewtonIte
ration_Ani.gif
(1) Guess the solution (xj,yj)
(2) Find the tangent planes to each f and g at this point.
z = f(x,y) =
z = g(x,y) =
(3) Find the line of intersection of the planes.
(4) Find the line’s xy-intercept, call this point (xj+1,yj+1),
xj+1 =
yj+1 =
(5) If f(xj+1,yj+1) < ε and g(xj+1,yj+1) < ε for some small
number ε (error tolerance), stop,
(xj+1,yj+1) is an approximate solution. Otherwise, return to
step (2) with (xj+1,yj+1) as
the guess.
(a) Find equations of the tangent planes for step (2), an equation
for their line of intersection for step
(3), and find formulas for xj+1 and yj+1 for step (4).

4. (b) What assumptions must we make about f and g in order for
the method to work? How might
the method fail? Explain in words how the method works.
For the following parts of the problem, it is recommended to
use a spreadsheet or write some
code.
(c) Use the method to approximate the solutions to the system
of equations
(x− 1) cos(x− 1) + (y − 1) cos(y − 1) =
3
4
(x−π)2 + (y −π)2 = 8
In other words, find the intersections of the blue and orange
curves.
(d) Use the method to approximate the location of the critical
points of f(x,y) = 10x2y−5x2−4y2−
x4 − 2y4 given its contour map,
For each of the critical points, use enough steps in Newton’s
method to approximate the their
locations such that the partial derivatives at the approximation
have absolute values below 10−4.
2. One method for finding extreme values of a function f of two
(or more) variables is gradient descent
(or ascent). The idea is to guess a point in the domain that could
be a minimum, and follow a certain

5. path in the domain to reach a much more precise approximation
of the location of a minimum. In
order to do this, we find the direction of steepest decline, take a
small step in that direction, and repeat
over and over until we reach what seems to be a minimum.
The method would look like the following on a contour map
(i.e. taking the most direct path between
level curves).
Gradient ascent uses the same idea to find maxima, but follows
the steepest incline rather than decline.
(a) Assuming we are able to find partial derivatives of f, use
ideas from §14.6 to write down the steps
for carrying out a gradient descent strategy. How can we know
when to stop (i.e. how do we
know if we have reached a minimum)? What is different for the
method of gradient ascent?
[Hint. The inputs to our method will be f, its partial derivatives,
and a point in the domain we
guess is a minimum.]
(b) Suppose we have function f with continuous partial
derivatives, but we are not able to calculate
them. How can we modify the process in part (a) so that it can
still work? (How could we
determine in which direction f has a steepest decline without the
partial derivatives?)
(c) Suppose our method converges to a relative minimum that is
not an absolute minimum. How can

6. we check to see if there are other minima?
(d) Use a gradient ascent method to find the 3 maxima for part
(d) of problem 2 above.
(e) Use a gradient descent and ascent method to approximate the
absolute minimum and maximum
of f(x,y) = sin x sin y
(xy)2+1
where −2π ≤ x ≤ 2π, −2π ≤ y ≤ 2π. Below is f on the domain,
with red
colors corresponding to high points and blue colors low points.
3. You are the Duke of Wellington and are charged, by his high
and most exalted holiness the emperor
of Egghead, to help settle an age old border dispute with the
neighboring state, Bufoonia. Between
your two states is a no mans land that you need to determine
how to most equitably share between
Egghead and Bufoonia. Representing Bufoonia is your arch
rival and nemesis the Arch Duke of Strong
Island. The existing borders of Egghead are determined by the
following set of inequalities:
x + 2y + z ≥ 3
0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2
The Arch Duke of Strong Island has furnished you with his own
border specifications:
4x + 3y + 6z ≥−12
−3 ≤ x ≤ 0,−4 ≤ y ≤ 0,−2 ≤ z ≤ 0

7. Note: I suggest reading sections 5.1.1, 5.1.2, 5.2.1 (about 3
pages with pretty large font) of Convex
Optimization by Boyd and Vandenberghe (available for free
download from authors here: https:
//web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf) before you
start to get a different perspective
on the problem.
(a) Plot, in 3D, these borders and give at least 2 views showing
the separation of the states. Re-
member, the states are formed by the regions enclosed by the
given planes. You may use the
MATLAB code provided to do this.
(b) Find a plane, passing through (0, 1
2
, 2) that separates Egghead and Bufoonia. Is there more than
one such plane? Why? This is the border that the Arch Duke of
Strong Island (your nemesis)
wants.
(c) Find a plane, passing through (0, 0, 0) that separates
Egghead and Bufoonia. Is there more than
one such plane? Why? This is the border that your rival and foe,
the Arch Duke of Strong Island,
thinks you want (he is, after all, a Bufoon).
https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
(d) Find a plane that strictly separates the two states. That is,
find a vector a =< a1,a2,a3 > and a
scalar γ such that a1x + a2y + a3z > γ for all (x,y,z) ∈ Egghead

8. and a1x + a2y + a3z < γ for all
(x,y,z) ∈ Bufoonia. Make a valid mathematical argument for
why the plane you find is the best
such separating plane. Hint: the vector a and scalar γ must
satisfy
max
(x,y,z)∈ Bufoonia
a1x + a2y + a3z < γ < min
(x,y,z)∈ Egghead
a1x + a2y + a3z
(e) Offer an interesting back story that outlines how you, the
Duke of Wellington, and your nemesis,
the Arch Duke of Strong Island, came to be rivals.
(f) Note: There are two additional MATLAB files provided with
the project document to serve as
a guide to plotting the required regions. Feel free to port these
to other languages / tools if you
wish. If you do, please submit the code in your written project
document.
References
[1] Wolfram Alpha http://www.wolframalpha.com
[2] Plotting 3D in MATLAB
http://www.mathworks.com/help/matlab/2-and-3d-plots.html [3]
Google
http://www.google.com
Project Document Checklist
Checklist for writing projects. Based on checklists by Annalisa

9. Crannell and Franklin & Marshall and Tommy
Ratliff at Wheaton College,
Does this paper:
1. clearly (re)state the problem to be solved?
2. provide an explanation as to how the problem will be
approached?
3. state the answer in a few complete sentences which stand on
their own?
4. Give a precise and well-organized explanation of how the
answer was found?
5. clearly label diagrams, tables, graphs, or any other visual
representations of the math?
6. define all variables, terminology, and notation used?
7. clearly state the assumptions which underlie the formulas and
theorems, and explain how each formula
or theorem is derived or where it can be found?
8. give acknowledgement where it is due?
9. use correct spelling, grammar, and punctuation?
10. contain correct mathematics?
11. solve the questions that were originally asked?
http://www.wolframalpha.com
http://www.mathworks.com/help/matlab/2-and-3d-plots.html
http://www.google.com