Find the best weights (w0...w4) of the highest possible order finite difference formula of the form f'(x) ~ w0*f(x) + w1*f(x+h) + w2*f(x+2h) + w3*f(x+3h) + w4*f(x+4h) and use Taylor series to predict the convergence order as h is decreased.

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

7. Let m be a fixed positive integer.

(a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m.

(b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.

The following is a payoff table giving costs for various situations. What decision would an optimist make?

Let Σ ⊆ P rop(A). Show that Σ|− p iff Σ ∪ {¬p} is unsatisfifiable.

Using concepts from funding models, how can they refocus its activities to generate funds to continue the work of the organization?

I'm having trouble figuring out the constraints to this problem. I know that I am maximizing 55x + 45y, however the variables are throwing me off. I only need help on question #1 as it would be a great help to understanding the rest of what the questions are asking. The problem is as follows:

NorCal Outfitters manufactures a variety of specialty gear for outdoor enthusiasts. NorCal has decided

to begin production on two new models of crampons: the Denali and Cascade. The company produces

crampons by first stamping steel sheets into the rough design, then assembling the base crampon with

toe and ankle straps. NorCal’s manufacturing plant has 120 hours of stamping time and 80 hours of

assembly time assigned for producing these crampons.

Each set of Denali crampons requires 30 minutes of stamping time and 25 minutes of assembly time,

and each set of Cascade crampons requires 25 minutes of stamping time and 15 minutes of assembly

time. The labor and material cost is $15 and $10 for each set of Denali and Cascade crampons,

respectively. NorCal sells crampons through wholesale distributors for $55 for the Denali model and $45

for the Cascade model. The V.P of Production at NorCal believes that the Denali model, recently

featured in Outside Magazine, could become a bestseller and has determined that at least 60% of the

crampons produced by NorCal should be the Denali model.

1. Solve this problem using the graphic solution technique.

There are eight different kittens at a store. I have three different children - how many ways are there to give each child a different kitten?

At time t = 0 a tank contains 25lb of salt dissolved in 100 gallons of water. assume that water containing 2lb salt/gallon enters the tank at a rate of 5 gal/min and the well-stirred solution is leaving the tank at the same rate.

solve for Q(t) [Amount of salt in tank at time t ]

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true?

4. Prove that the following is not true by finding a counterexample.

“The sum of any 3 consecutive integers is even"

5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd

6.  Show an informal Direct Proof for “The sum of 2 even integers is even.”

Recursive Definitions

7.  The Fibonacci Sequence is defined as follows:                             

            F(1) = 1

            F(2) = 1

            F(n) = F(n-1) + F(n-2) for n>2.

     The first 10 numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

      Find F(13) = ______________ and F(14) =______________

8.  Prove the following property of the Fibonacci numbers directly from the definition.                                                                                                          

            F(n+3) = 2F(n+1) + F(n-3)+ 2F(n-2)

9.  Find the 3^rd, 4^th and 5^th values in the sequence.

            D(1) = 0

            D(n) = 2D(n-1) + 2 for n>1.

            D(2) = _______________________________________________

            D(3) = _______________________________________________

            D(4) = _______________________________________________

10.  Find the 3^rd, 4^th and 5^th values in the sequence.

            D(1) = 1

            D(2) = 4

            D(n) = 2D(n-1) + D(n-2) for n>2.

            D(3) = _______________________________________________

D(4) = _______________________________________________

            D(5) = _______________________________________________

11.  Using the SelectionSort algorithm on p 169 in your textbook, simulate the execution of the algorithm on the following list L; write the list after every exchange.  

L:    9, 2, 4, 8, 6

_______________

_______________

_______________

_______________

Harvesting Fish. A fish farmer has 5000 catfish in a pond. The number of catfish increases by 8% per month and the farmer harvests 420 catfish per month.

a) (4 points) Find a recursive equation for the catfish population Pn for each month.

b) (4 points) Solve the recursive equation to find an explicit equation for the catfish population.

c) (3 points) How many catfish are in the pond after six months?

d) (3 points) Is harvesting 420 catfish per month a sustainable strategy for the fish farmer?

e) (4 points) What is the maximum number of catfish the fish farmer can harvest to sustain his business?

A mass of 8 kg stretches a spring 16 cm. The mass is acted on by an external force of 7sin⁡(t/4)N and moves in a medium that imparts a viscous force of 3 N when the speed of the mass is 6 cm/s.If the mass is set in motion from its equilibrium position with an initial velocity of 4 cm/s, determine the position u of the mass at any time t. Use 9.8 m/s^2 as the acceleration due to gravity. Pay close attention to the units.

Show that |N| = |Z|, where N is the set of natural numbers and Z is the set of all integers.

A house can be purchased for

​$155 comma 000155,000​,

and you have

​$25 comma 00025,000

cash for a down payment. You are considering the following two financing​ options:

bullet•

Option​ 1. Getting a new standard mortgage with a

7.57.5​%

​(APR) interest and a

3030​-year

term.

bullet•

Option​ 2. Assuming the​ seller's old​ mortgage, which has an interest rate of

5.55.5​%

​(APR), a remaining term of

2525

years​ (the original term was

3030

​years), a remaining balance of

​$97 comma 21797,217​,

and payments of

​$597597

per month. You can obtain a second mortgage for the remaining balance

​($32 comma 78332,783​)

from your credit union at

99​%

​(APR) with a

1010​-year

repayment period.

​(a) What is the effective interest rate of the combined​ mortgage?

The effective interest rate of the combined mortgage is

nothing​%.

1.Prove that{2k+1:k∈N}∩{2k2 :k∈N}=∅.

2.Give two examples of ordered sets where the meaning of ” ≤ ” is not the same as the one used with the set of real numbers R.