Let C₀=[0,1]. C_n is obtained by removing the middle open interval of length α/3ⁿ from each interval of C_n-1 where α∈(0,1).

Let C=⋂C_n. Prove that C contains only boundary points, i.e., x∈C is a boundary point of C if every neighborhood of x contains at least one point in C and at least one point not in C.

1) In the month of August, John earned $2750 in commission. His company has provided him with a drawing account of up to $800 per month. If Paul drew $500 on August 15^th what is the amount of Paul’s commission check on August 31?

2) Fred earns an annual salary of $46,000 and she is paid semi-monthly. What is her pay per period?

3) Ben purchases fresh blueberries from local pickers to sell at the market over a 12 week period. For the first 4 weeks, he pays the pickers $8.00 basket and buys 62 baskets; during the next 4 week period he pays the pickers $6.00 basket and buys 95 baskets; during the last 4 weeks he pays the pickers $9.50 per basket and buys 56 baskets. What is the average cost per basket to Seth over the whole period?

Part I

A ball is thrown straight up from the top of a building that is 185ft high with an initial velocity of 64ft/s. The height of the object can be modeled by the equation s(t) = -16t ² + 64t + 185.
In two or more complete sentences explain how to determine the time(s) the ball is higher than the building in interval notation.

Part II

In two or more complete sentences, describe the transformation(s) that take place on the parent function, f(x) = log(x), to achieve the graph of g(x) = log(-3x-6) - 2.

Part III

The parent function f(x) = log₄x has been transformed by reflecting it over the x-axis, stretching it vertically by a factor of two and shifting it down five units. Which function is representative of this transformation?

A. g(x ) = log₄(2x) - 5

B. g(x) = log₄(-2x) + 5

C. g(x) = 2log₄(x) + 5

D. g(x) = -2log₄(x) - 5

Which of the following statements are true? There may be more than one true statement.

A.

B.

C.

D.

E.

F.

Write down all proofs in acceptable mathematical language: make sure that you mark the beginning and end of the proof, state every assumption, define every variable, give a justification for every assertion (e.g., by definition of…), and use complete, grammatically correct sentences.

Definitions:

• An integer ? is even if and only if there exists an integer ? such that ? = 2?. An integer ? is odd if and only if there exists an integer ? such that ? = 2? +1.

• Two integers have the same parity when they are both even or when they are both odd. Two integers have opposite parity when one is even and the other one is odd.

• An integer ? is divisible by an integer ? with ? ≠ 0, denoted ? | ?, if and only if there exists an integer ? such that ? = ??.

• A real number ? is rational if and only if there exist integers ? and ? with ? ≠ 0 such that ? = ?/?.

• For any real number ?, the absolute value of ?, denoted |?|, is defined as follows: |?| = { ? if ? ≥ 0 −? if ? < 0 2.

Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof.

a. For all integers ?, ?, and ?, if ?? is not divisible by ?, then ? is not divisible by ?.

b. For all positive integers ?, ?, and ?, if ? is divisible by ? and ? is divisible by ?, then ? + ? is divisible by ?.

c. The difference of any rational number and any irrational number is irrational

d. There are no integers ? and ? such that ???+ ?? = ?.

e. Any two consecutive integers have opposite parity.

f. For all real numbers ? and ?, ???(?, ?) = ?+?−|?−?| ? and ???(?, ?) = ?+?+|?−?| ? , where ???(?, ?) and ???(?, ?) denote the minimum and the maximum of ? and ?, respectively.

Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a set with more than one
element.

Show the following:

(1) every open interval is open, (2) every closed interval is closed, (3) every open
ray is open, and (4) every closed ray is closed.

Please note: Its a topology question.

Solve the initial value problem: y'' + 4y' + 4y = 0; y(0) = 1, y'(0) = 0.

Solve without the Laplace Transform, first, and then with the Laplace Transform.

1. Prove that if a set A is bounded, then A-bar is also bounded.

2. Prove that if A is a bounded set, then A-bar is compact.

The Clarkstown Central School District covers 4 towns. There are 22 members of the school board, and the 4 towns have the populations shown in the following:

Population

Town A 9,000

Town B 9,100

Town C 25,475

Town D 56,425

The school district uses the Hamilton method to apportion its 22 board members to the 4 towns.

How many board members are assigned to each town, using this method?

The following year, 1,000 people move out of town A and into town D. Now, how many board members does each town have?

Problem #1: On a certain island, there is a population of snakes, foxes, hawks and mice. Their populations at time t are given by s(t),  f (t), h(t), and m(t) respectively. The populations grow at rates given by the differential equations

s′  =    21/8  s −  f −  7/8  h  −  1/8  m

f ′  =    5/8  s +  f −  7/8  h  −  1/8  m

h′  =    5/8  s + 0 f +  1/8  h  −  1/8  m

m′  =    65/4  s −  5  f −  31/4  h  −  5/4  m

Putting the four populations into a vector y(t)  =  [s(t)  f (t) h(t) m(t)]T, this system can be written as y′  =  Ay. Find the eigenvectors and eigenvalues of A. Label the eigenvectors x1 through x4 in order from largest eigenvalue to smallest (the smallest being negative). Scale each eigenvector so that its first component is 1. When you have done so, identify the eigenvector whose fourth component is the largest. What is that largest fourth component? Problem #1: 14 Problem #1 Attempt #1 Attempt #2 Attempt #3 Your Answer: 14 Your Mark: 0/3✘ 3/3 ✔

Problem #2: Continuing from the system of differential equations from Problem 1, each eigenvector represents a grouping of animals that changes with simple exponential growth or decay. The exponential rate of growth or decay is given by the corresponding eigenvalue. Because the matrix A is invertible and diagonalizable, any initial values for the animal population can be written as a combination of these four special groupings that each grow exponentially by their eigenvalue. Consider the initial population y(0)  =  [18 11 4 102]T. Solve for constants c1 through c4 in order to write y(0)  =  c1 x1  +  c2 x2  +  c3 x3  +  c4 x4 where x1 through x4 are the eigenvectors as detailed in Problem 1 (i.e., the eigenvectors in order, and scaled so that the first component is 1). Enter the values of c1, c2, c3, and c4, separated with commas.

Problem #2: Problem #2 Attempt #1 Attempt #2 Attempt #3 Your Answer:

Problem #3: Based on the system of differential equations from Problem 1, with the initial population from problem 2, find the function for the population of hawks, h(t).

1. State the dimension theorem. Explain how it proved.

Find vectors of the Frenet frame of the curve at any point of the curve:

x = a(t-sin(t)), y = a(1-cos(t)), z = 4acos(t), where a is a positive constant.

Please show all steps and relevant formulae!

Jennifer took clothes to the cleaners three times last month.​ First, she brought 3 shirts and 1 pair of slacks and paid ​$9.96. Then she brought 6 ​shirts, 4 pairs of​ slacks, and 2 sports coats and paid ​$40.88. ​Finally, she brought 4 shirts and 2 sports coats and paid ​$20.94. How much was she charged for each​ shirt, each pair of​ slacks, and each sports​ coat? Jennifer was charged ​$ nothing for each​ shirt, ​$ nothing for each pair of​ slacks, and ​$ nothing for each sports coat.

A mass that weight 6lb stretches a spring 4in. The system is acted on by an external force 2sin(4sqrt(6)t)lb.If the mass is pulled down 3in and then released, determine the position of the mass at any time t. Use 32ft/s^2 as the acceleration due to gravity. Pay close attention to the units. Answer must be in inches