Use the Gram-Schmidt process to construct an orthogonal basis of the subspace of V = C ∞[0, 1] spanned by f(x) = 1, g(x) = x, and h(x) = e x where V has the inner product defined by < f, g >= R 1 0 f(x)g(x)dx.

Prove or disprove: two consecutive rotations about two different axis are commutative. That is, is RuRv = RvRu? (Hint: For simplicity, you can assume that the axis u is the x-axis and v is the y-axis without loss of gnerality).

Show your steps! The previous answer posted is wrong...could you do it again)

4. Three Plinko chips fall in such a way that each chip is equally likely to fall into any one of five slots. All chips may fall into the same slot or two may fall into the same slot or all three may fall into different slots. Let ? denote the number of unoccupied slots at the end of the game. What is the moment generating function of ??

Use the Lagrange multiplier method to identify the stationary point(s) of the function ?(?, ?, ?)=10? + 5? + ? + 2?^2 + 4?? + ?^2 + 2?? - ?^2 - ??, subject to the constraint 5? + 3? + ? = 27. Subsequently determine the nature of the stationary point(s) using the bordered Hessian matrix.

Company A and Company B are both wholly owned subsidiaries of Parent, Inc. Parent has no other operations, balance sheet items or income statement items other than its ownership of Company 1 (located in China) and Company 2 (located in US). Company 2 periodically sells goods to Company 1 for resale to end customers. Such goods are sold at the same pricing terms that Company 2 sells to all other customers. Prior to January 1, 2018, there had never been any inventory sales from Company 1 to Company 2 or from Company 2 to Company 1. The following is data for each company for 2018 and 2019:

Company 1 Company 2

     Year ended 12/31/18

Sales to all customers $300 million $150 million

Costs of sales $150 million $100 million

All other non production expenses $ 60 million $ 40 million

Pre tax income $ 90 million $ 10 million

Inventory purchased from Company 1 held

By Company 2 at end of year NONE

Inventory purchased from Company 2 held

By Company 1 at end of year $15 million

    Year ended 12/31/19

Sales to all customers $280 million $160 million

Costs of sales $140 million $120 million

All other non production expenses $ 60 million $ 30 million

Pretax income $ 80 million $ 10 million

Inventory purchased from Company 1 held

By Company 2 at end of year NONE

Inventory purchased from Company 2 held

By Company 1 at end of year $16 million

What would consolidated pretax income be for Parent for 2018 and 2019

Let S={1,2,3,6} and define the relation ~ on S² by (m,n) ~ (k,l) for m+l=n+k

1. Show that it is an equivalent relation
2. Find the number of different equivalent classes and write all of them

Let S = {1,2,3,4} and let A = SxS
Define a relation R on A by (a,b)R(c,d) iff ad = bc

Write out each equivalence class (by "write out" I mean tell me explicitly which elements of A are in each equivalence class)

Hint: |A| = 16 and there are 11 equivalence classes, so there are several equivalence classes that consist of a single element of A.

Prove a connected simple graph G with 16 vertices and 117 edges is not Eulerian.

3. Given the following maximum problem, set up the initial simplex tableau and circle the first pivot element.

Do not solve the maximum problems

MAXIMIZE : P = 3x1+2x2+3x3 subject to constraints : -x1+2x2+2x3<=8 4x1-x2+6x3<=10 x1+2x2+4x3<=12, x1>=0, x2>=0, x3>=0

The forcing function is a linearly combination of ?(?)=3? and ?(?)=25sin(3?).

Solve the Ordinary Differential Equation, ?’’−2?’+?=3?+10sin(3?).

Task: Network planning

A university has a main campus and a remote campus at another location. The default router to the Internet on the main campus has an IP address of 10.0.0.1/24. The main campus network is connected to the Internet through a router called RouterA. The main campus has no more than 15000 staff and students. The remote campus network is connected to the main campus network through a router called RouterB. The remote campus has no more than 200 staff and students. All staff and students should be able to access to the Internet and to the resources on both campuses. Router A should have two interfaces, one goes to the Internet (thus having an address in 10.0.0.0/24 block) and the other goes to the main campus network. Router B should have two interfaces, one goes to the main campus and the other goes to the remote campus.

Your task is to plan the campus networks.

1. a) Propose appropriate subnet ranges for both campuses from the 16‐bit prefix block of IPv4 private addresses. Answer the question with your calculation.

2. b) Draw a diagram to depict the networks with IP addresses notated with CIDR notation assigned to the all interfaces of bother routers for two campuses. Label the interfaces of routers on the diagram using eth0 or eth1.

Show the routing table of the RouterA that meets the requirements of both campuses. Show the columns of Destination, Gateway, Genmask, Flags and Iface in the routing table as shown by the Linux command route.

Angela Fox and Zooey Caulfield were food and nutrition majors at State University, as well as close friends and roommates. Upon graduation Angela and Zooey decided to open a French restaurant in Draperton, the small town where the university was located. There were no other French restaurants in Draperton, and the possibility of doing something new and somewhat risky intrigued the two friends. They purchased an old Victorian home just off Main Street for their new restaurant, which they named “The Possibility.” Angela and Zooey knew in advance that at least initially they could not offer a full, varied menu of dishes. They had no idea what their local customers’ tastes in French cuisine would be, so they decided to serve only two full-course meals each night, one with beef and the other with fish. Their chef, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient, at least until they could assess which menu items were most popular. Pierre indicated that with each meal he could experiment with different appetizers, soups, salads, vegetable dishes, and desserts until they were able to identify a full selection of menu items. The next problem for Angela and Zooey was to determine how many meals to prepare for each night so they could shop for ingredients and set up the work schedule. They could not afford too much waste. They estimated that they would sell a maximum of 60 meals each night. Each fish dinner, including all accompaniments, requires 15 minutes to prepare, and each beef dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. Angela and Zooey believe that because of the health consciousness of their potential clientele, they will sell at least three fish dinners for every two beef dinners. The profit from each fish dinner will be approximately $12, and the profit from a beef dinner will be about $16. - Formulate a linear programming model for Angela and Zooey that will help them estimate the number of meals they should prepare each night and solve this model graphically. * Please show graph and how to optimize step by step*