Use the Cauchy Criterion to prove the Bolzano–Weierstrass Theorem, and find the point in the argument where the Archimedean Property is implictly required. This establishes the final link in the equivalence of the five characterizations of completeness discussed at the end of Section 2.6.

Consider the following subsets of the set of all students:

A = set of all science majors
B = set of all art majors
C = set of all math majors

D = set of all female students

Using set operations, describe each of the following sets in terms of A, B, C, and D:

a) set of all female physics majors

b) set of all students majoring in both science and art

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