State the dual of the Theorem below.

Let a non-degenerate plane conic touch the sides BC, CA, and AB of a triangle ABC in R² at the points P, Q, and R respectively. Then AP, BQ, and CR are concurrent.

Using Runge-Kutta method of order 4 to approximate y(1) with step size h = 0.1 and h = 0.2 respectively (keep 8 decimals):

dy/dx = x + arctan y, y(0) = 0.

Solutions: when h = 0.1, y(1) = 0.70398191. when h = 0.2, y(1) = 0.70394257.

7. OR. If you prefer, calculate Proj_CS(A)b using the approach of Example 8 on page 266 instead. You should get an infinite number of solutions for x to the normal equation. Any solution for x will work, and then calculate Ax to get Proj_CS(A)
   • For either approach, briefly explain why Proj_CS(A)b is the closest vector in your plane to the vector (See Theorem 5.15 if needed.)
8. Choose 5 arbitrary unique vectors in R² that are not on the same line or scalar multiples of each other. What are your 5 vectors in R²?
   • Use normal equations (see page 265, if needed) to determine the equation of the least square regression line for this set of 5 points.
9. Choose any 2 of the 5 vectors in the above problem that will form a nonstandard basis for R², which we will call B’. (12 points for entire problem)
   • What are your two vectors in B’?
   • If B is the standard basis for R² (I.e. B ={(1.0). (0.1)}, determine the transition matrix (called P^-1 in our text) from B to B’.
   • Use P^-1 to calculate the coordinates for (1,5) with respect to the basis B’.
10. Investigate the following linear differential equation: y’’ + 4y = 0; Solutions {sin(2x), cos(2x)}

• Verify that each solution satisfies the differential equation.
• Use the Wronskian to verify that the solution set is linearly independent.
• Write the general solution of the differential equation.

True or False? Why?

Σ n = 1, ∞ fn(x) converges uniformly on A <=> for all n in N (natural numbers), there exists Mn > 0 such that |fn(x)| <= Mn for all x in A and Σ n = 1, ∞ Mn converges.

For p a given prime number, define the p-adic norm | * |p as follows on Q: Given q in Q, we can write it as a product q = (p^m)(a/b) with a,b integers which are not divisible by p, and m an integer which is uniquely determined by q (check that m is indeed uniquely determined by q). Then define |q|p = p^(-m).

Check that Q with distance dp(q1,q2) = |q1 - q2|p is a metric space (here q1-q2 just means the usual operation of subtraction in Q). Show moreover
that |q1 + q2|p <= max(|q1|p, |q2|p).

Finally, consider the following fixed point iteration xk+1 = g(xk) = arccos −1 1 + e 2x and show that finding a fixed point of g(x) is equivalent to finding a root of f(x) = 0. Use the code fixedpt.m to try to approximate the root using an initial guess of x0 = −3. Can you explain why your iteration behaves as it does? Hint: Plot the fixed-point function and think convergence!

Code in fixedpt.m:-

function [xfinal, niter, xlist] = fixedpt( gfunc, xguess, tol )
% FIXEDPT: Fixed point iteration for x=gfunc(x).
%
%  Sample usage:
%     [xfinal, niter, xlist] = fixedpt( gfunc, xguess, tol )
%
%  Input:
%     gfunc   - fixed point function 
%     xguess  - initial guess at the fixed point
%     tol     - convergence tolerance (OPTIONAL, defaults to 1e-6)
%
%  Output:
%     xfinal  - final estimate of the fixed point
%     niter   - number of iterations to convergence
%     xlist   - list of interates, an array of length 'niter'

% First, do some error checking on parameters.
if nargin < 2
  fprintf( 1, 'FIXEDPT: must be called with at least two arguments' );
  error( 'Usage:  [xfinal, niter, xlist] = fixedpt( gfunc, xguess, [tol] )' );
end
if nargin < 3, tol = 1e-6; end

% fcnchk(...) allows a string function to be sent as a parameter, and
% coverts it to the correct type to allow evaluation by feval().
gfunc = fcnchk(gfunc);
x = xguess;
xlist = [ x ];

niter = 0;
done  = 0;
while ~done,
  xnew  = feval(gfunc,  x);
  xlist = [ xlist; xnew ];  % create a list of x-values 
  niter = niter + 1;
  if abs(x-xnew) < tol,     % stopping tolerance for x only
    done = 1;
  end
  x = xnew;
end
xfinal = xnew;

show that if I is uncountable,then 2^I is not metrizable.

hint: Suppose I as index set.

3-How can digital signal processing affect the quality of sound?

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5

As a mathematical function, does U have a maximum or minimum value? What values of Ci correspond to the minimum value of U? What values of Ci correspond to the maximum value of U? Do these values of Ci make sense from an economic standpoint?

Now let us connect the idea of economic utility to actual dollar values. To keep the values more manageable, we will use household income rather than the entire state budget, and retail costs and measures rather than industrial ones. Find the Median Household Income for Mesa, AZ for the most recent year possible. Then find the dollar cost in Mesa, AZ for a Penny, a pound of Ground Beef, a pair of Jeans, fresh Orange Juice, and a Movie Ticket. (Entertainment is often used as a stand-in for Climate.) A Cost-of- Living Index is a good place to find much of this data. Record these prices as P1, P2, P3, P4, and P5 respectively.

Construct an equation using Median Income, the Ci and Pi values that illustrates how much of each resource the Median Household can afford to purchase. Given this restriction, do the maximum or minimum values of U change? Do the values of Ci that give the maximum or minimum values change? What are these new values? How should the Median Household budget its Income so as to maximize its Economic Utility?

Write up your findings in a paper that you could turn in to an employer. Be sure to show all your work. Include any appropriate references as well as any computational devices used.

* Solve the questions, make the table, and the graph.

Let K_n denote the simple graph on n vertices.
(a) Let v be some vertex of K_n and consider K _{n − v}, the graph obtained by deleting
v. Prove that K _{n − v} is isomorphic to K n−1 .
(b) Use mathematical induction on n to prove the following statement:
K _n , the complete graph on n vertices, has n(n-1)/2
edges

The town hall of a city wants to open some recreational centers. It has been analyzed 3 options. The opening cost and the capacity of each center are listed below.

The selected recreational centers must be hosting the students from 5 schools. In the table below is summarized the number of students at each school.

Each school must be assigned to only one recreational center. And the capacity of each center must be respected. What are the recreational centers that must be open in order to minimize the opening cost?

Illustrate the greedy procedure with the following data:

(Modern Algebra) Show that if G is a finite group that has at most one subgroup for each
divisor of its order then G is cyclical.