Derive the Catmull-Rom Spline blending function in your own words step by step.

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from Dept. II, and 2 hours of work from Dept. III.)

How many units of each product should the company produce in order to maximize its profit?

What is the largest profit the company can realize?
$

Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows.

For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory is not to exceed 212,000 hr; and the amount of labor for on-site work is to be less than or equal to 237,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture.

Undetermined Coefficients:

a) y'' + y' - 2y = x^2

b) y'' + 4y = e^3x

c) y'' + y' - 2y = sin x

d) y" - 4y = xe^x + cos 2x

e) Determine the correct form of a particular solution, do not solve

y" + y = sin x

Use the simplex method to solve the linear programming problem.

The maximum is P =  at

(x, y, z) =

Use the simplex method to solve the linear programming problem.

The maximum is P =  at

(x, y)

Use the simplex method to solve the linear programming problem.

The maximum is P =  at

(x, y)

Find y as a function of x if y′′′−16y′′+63y′=144ex, y(0)=16, y′(0)=11, y′′(0)=15. y(x)=

Consider the IVP: y'=ty-2, y(1)=1.5

Use the following numerical methods to approximate y(2.5), using a stepsize of h=0.5.

a. Euler's method.

b. Euler's improved method.

c. Runge-Kutta method

Supply proofs for the following miscellaneous propositions from the course in a metric space context:

(1) A compact set (you may use either definition) is closed and bounded.

(2) An epsilon-neighborhood is an open set.

(3) A set is open if and only if its complement is closed.

(a) Let n = 2k be an even integer. Show that x = r^k is an element of order 2 which commutes with every element of D_n.

(b) Let n = 2k be an even integer. Show that x = r^k is the unique non-identity element which commutes with every element of D_n.

(c) If n is an odd integer, show that the identity is the only element of D_n which commutes with every element of D_n.