For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.

f(x)=0 if x≤0, f(x)=x^a if x>0

For what a is f continuous at x = 0

For what a is f differentiable at x = 0

For what a is f twice differentiable at x = 0

Twenty calculus students are comparing grades on their first two quizzes of the year. The class discovers that every pair of students received the same grade on at least one of the two quizzes. Prove that the entire class received the same grade on at least one of the two quizzes.

Find a particular solution to the non-homogeneous differential equation:

(a) y'' − 4y = 4t^2

(b) y '' − 4y = sin(2t)

(c) y '' + 4y = sin(2t)

(d) y '' + y = cos(t)

Consider an n×n square board, where n is a fixed even positive integer. The board is divided into n 2 unit squares. We say that two different squares on the board are adjacent if they have a common side. N unit squares on the board are marked in such a way that every unmarked square on the board is adjacent to at least one marked square. Determine the smallest possible value of N.

4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1, 7}. Answer the following questions, giving reasons for your answers.

a) Is ? ⊆ ??

b)Is ? ⊆ ??

c) Is ? ⊂ ??

d) Is ? ⊆ ??

e) Is ? ⊆ ??

5) Let ? = {1, 3, 4} and ? = {2, 3, 6}. Use set-roster notation to write each of the following sets, and indicate the number of elements in each set.

a) ? × ?

b) ? × ?

c) ? × ?

d) ? × ?

4-Consider the following problem:

max − 3x1 + 2x2 − x3 + x4

s.t.

2x1 − 3x2 − x3 + x4 ≤ 0

− x1 + 2x2 + 2x3 − 3x4 ≤ 1

− x1 + x2 − 4x3 + x4 ≤ 8

x1, x2, x3, x4 ≥ 0

Use the Simplex method to verify that the optimal objective value is unbounded. Make use of the final tableau to construct an unbounded direction..

y′′+12y′+35y=8cos(8t+0) y(0) = 14; y'(0) = 3

find the particular, complimentary, and total solution.

I have found the complimentary to be. C1e^-5t + C2e^-7t

I'm having trouble finding the particular, so I can't get to the total solution

Give the Laplace transform of the solution to y"+2y'+3y=0 y(0)=-5 y'(0)=4

USING BISECTION METHOD, FIND THE ROOT OF 0.5e^x - 5x + 2 = 0 ON THE INTERVAL [ 0 , 1 ] UP TO 3 DECIMAL PLACES.

USE NEWTON'S METHOD TO APPROXIMATE THE ROOT OF f(x)=x^2-5    IN THE INTERVAL  [ 2 , 3 ] UP TO 4 DECIMAL PLACES.

short paragraphs

1) When factoring a trinomial, is the factored form unique or can there be more than one factored form?  

2) When factoring, what does it mean that a polynomial is prime?

USE NEWTON'S METHOD TO APPROXIMATE THE SOLUTION TO   2COSX = 3X .  LET X₀ =pi/6 .  ANSWER UP TO 3 DECIMAL PLACES.

USING REGULA FALSI METHOD, SOLVE THE EQUATION x^3 - 4x + 1 = 0 UP TO 3 DECIMAL PLACES.

Let A = {a, b, c, d} and B = {b, d, e}. Write out all of the elements of the following sets.

(a) B ∩ ∅

(b) A ∪ B

(c) (A ∩ B) × B

(d) P(A\B)

(e) {X ∈ P(A) | |X| ≤ 3}

2. Firm I has variable cost VCi = yi^2/10 and fixed cost FCi = 2000.

(1) Find total cost Ci(yi), average cost ACi, marginal cost Mci and the firm supply function Si(p)

(2) There are n=50 firms identical to firm I, facing a market demand of D(p) = 1000-250p. Find the market supply function S(p), the market equilibrium price p*, the market equilibrium quantity Y*.

(3) Given price p* you found in part b, what is the profit maximising yi* that firm i produces? How much profit does firm i make?

(4) The government introduces a tax on demand so that D'(p ) = 1000-250(p+t), where t=8. What is the new equilibrium price p? What is the new market equilibrium quantity Y'?

(5) At the new market price p', and assuming that in the short run the number of firms remains n=50, how much will firm I produce and how much will profit be?

(6) Given what you found in part e, will firms enter or exit? What is the long-run equilibrium number of firms n? What is the long run equilibrium price?

Looking at the first 2z integers 1, 2, . . . , 2z. Choose z + 1 of them. Then demonstrate that there is at least one pair of integers from the selection that are relatively prime. (A way to approach this problem is to show that at least two integers from the selection are consecutive.)