Use stars and bars to solve each counting problem. You may leave your answers as binomial coefficients.

(a) How many collections of 6 (not necessarily distinct) coins can be made from an infinite supply of pennies, nickels, dimes, and quarters?

(b) A social security number is a sequence of 9 digits. How many social security numbers are there

n1n2n3 . . . n9

such that ni ≤ ni+1 for i = 1 to 8?

For example, 024455888 would count but 254180419 would not count

Given the matrix with rows

[1,1,k 1]

[1,k,1 1]

[k,1,1 -2]

Find the reduced row echelon form of M, and explain how it depends on k. (b) Consider the linear system Ax = b for which the augmented matrix is A b = M. i. For what values of k is the system inconsistent? ii. For what values of k does the system have a unique solution?

Homework 1.1. (a) Find the solution of the initial value problem x' = x^(3/8) , x(0)=1 , for all t, where x = x(t). (b) Find the numerical solution on the interval 0 ≤ t ≤ 1 in steps of h = 0.05 and compare its graph with that of the exact solution. You can do this in Excel and turn in a printout of the spreadsheet and graphs.

Write each of the following permutations as a product of disjoint cycles:

(a) (1235)(413)

(b) (13256)(23)(46512)

(c) (12)(13)(23)(142)

1. In this problem, you can use the Matlab program posted on course website and Canvas (also given in the lecture) that computes the interpolation polynomial. We want to see how well a given function can be approximated by the interpolation polynomials. Let f be a function. We divide the the interval [−0.6,0.6] into subintervals of the same length h = 0.02. The gridpoints are −0.6 = x1 < x2 < ... < x61 = 0.6. Take N = 61 points (x1,y1),...(xN,yN) on the graph of f.

   1. (a) For f(x) = sinx, plot the graph of the interpolation P on the interval [−0.6,0.6]. Plot f and all of P on the same graph (for example, by using the command hold on). Does the interpolation polynomial approximate well the function f on the interval [−0.6, 0.6]?

   2. (b) The same questions as in Part (a) but for f (x) = 1+x .

   3. (c) We know that the error between f and P is estimated by

| f (x) − P (x)| ≤ max |f (n) | (∗)

n n−1 [a,b]
Let f (x) = 1 and [a, b] = [−0.6, 0.6]. Use Stirling approximation m√m! ≈ 1 (for large

1+x me m) to show that the right hand side of (∗) goes to infinity as n → ∞.

roblem 3 Find the solutions to the general cubic a x^3 +b x^2+c x +d=0 and the solutions to the general quartic a x^4+b x^3+c x^2+d x+e=0. Remember to put a space between your letters. The solutions to the general quartic goes on for two pages it is a good idea to maximize your page to see it. It is a theorem in modern abstract algebra that there is no solution to the general quintic in terms of radicals.

Please write it clearly! thank you!

A manufacturer is marketing a product made of an alloy material requiring a certain specified composition. The three critical ingredients of the alloy are manganese, silicon and copper. The specifications require 15 pounds of manganese, 22 pounds of silica and 39 pounds of copper for each ton of alloy to be produced. This mix ingredients require the manufacturer to obtain inputs from three different mining suppliers. Ore from the different suppliers has different concentrations of alloy ingredients, as detailed on the Table below:

Given this information, the supplier must determine how much ore to purchase from each supplier so that there is no waste of the alloy ingredients. A solution to the problem can be found by defining the following variables:

X_j = amount of ore purchased from supplier j

C_i = amount of ingredient i required per ton of alloy

A_ij = amount of ingredient i contained in each ton of ore shipped from supplier j

What amount of ore should be purchased from each supplier?

P4.4.2 Give a rigorous proof, using strong induction, that every positive natural has at least one factorization into prime numbers.

1. Are the following statements true or false? For each, explain why.

(a) 15 divides 5

(b) gcd(4, 16) = 2

(c) 46 ≡ 2 (mod 4)

(d) For any positive even integer n, gcd(12, n) = 2.

(e) For all integers x, if 10 does not divide x, then 10 does not divide 22x.

(f) For all integers y, if gcd(9, y) = 3, then gcd(9, y2 ) = 9.

1. Cost Estimation; High-Low and Regression Methods The Mac Davis Company specializes in the purchase, renovation, and resale of older homes. Mac employs several carpenters and painters to do the work for him. It is essential for him to have accurate cost estimates so he can determine total renovation costs before he purchases a piece of property. If estimated renovation costs plus the purchase price of a house are higher than the house’s estimated resale value, it is not a worthwhile investment.
   
   Mac has been using the home’s interior square feet for his exterior paint cost estimations. Recently he decided to include the number of external openings—the total number of doors and windows in a house—as a cost driver. Their cost is significant because they require time-consuming preparatory work and careful brushwork. The rest of the house usually is painted either by rollers or spray guns, which are relatively efficient ways to apply paint to a large area. Mac has kept careful records of these exterior painting costs on his last 12 jobs:

Required

1. Using the high-low cost estimation technique and square feet as the independent variable, determine the cost of painting a 3,300-square-foot house with 14 external openings. Also determine the cost for a 2,400-square-foot house with 8 externals openings.

2. Repeat requirement 1, but use number of external openings as the independent variable.

3. Plot the cost data against square feet and against openings. Which variable is a better cost driver? Why?

4. Create a multiple regression model for predicting cost based on openings and square feet. Comment on the statistical reliability and precision of this model.

5. What are the sustainability issues for this company, and what is the role of cost estimation in this regard?

Abstract Algebra

Let G be a discrete group of isometries of R2.

Prove there is a point p ∈ R2 whose stabilizer is trivial.

Let P₂ be the vector space of all polynomials of degree less than or equal to 2.

(i) Show that {x + 1, x² + x, x − 1} is a basis for P₂.

(ii) Define a transformation L from P₂ into P₂ by: L(f) = (xf)'    . In other words, L acts on the polynomial f(x) by first multiplying the function by x, then differentiating. The result is another polynomial in P₂. Prove that L is a linear transformation.

(iii) Compute the matrix representation of the linear transformation L above with respect to the basis for P₂ from the first part of this problem.

1. Find the Legendre polynomial PL(x) for L = 3,4,5,6 where the polynomian is the series solution for Legendre equation

2. Find the other solution QL(x) for the Legendre equation for L = 0,1,2

Please explain in full.

6) (a) Denote the successive intervals that arise in the bisection method by [a₀,b₀, [a₁,b₁] , [a₂,b₂], and so on

   Show that   b_n – a_n = 2^-n(b₀-a₀)

(b) The bisection method is said to have a linear convergence. Explain as clearly as possible what that means

c) For the bisection method, prove that |c_n – c_n+1| = 2^-n-2 (b₀ – a₀)

   where cn is the midpoint of each interval (ie) c_n = (a_n + b_n) /2