8. This problem is about the definition of periodic function. We assume you already know intuitively what periodic means, and now we want a formal definition. For simplicity, we will restrict ourselves to functions with domain R. A naive (but incorrect) definition of periodic function with period T is

   f (x + T ) = f (x)

   Without accompanying words, this is not a good definition because it does not introduce the variables x and T and it does not explain their role. For which values of x and T does the above definition have to be valid?

   Here is an attempt at a definition, with various ways to complete it:Definition. Let f be a function with domain (−∞, ∞). We say that

   f is periodic when...
   (a) Foreveryx∈(−∞,∞)andforeveryT >0,f(x+T)=f(x).
   (b) For every x ∈ (−∞,∞) there exists T > 0 such that f(x+T) = f(x). (c) There exists T > 0 such that x ∈ (−∞, ∞) =⇒ f (x + T ) = f (x). (d) There exists T > 0 such that for every x ∈ (−∞,∞), f(x+T) = f(x). (e) For every T > 0 there exists x ∈ (−∞,∞) such that f(x+T) = f(x).

9. One or more of the above are valid ways to complete the definition of periodic function. Identify which ones are correct and which ones are wrong. For any property which is wrong, show it by giving an example of a function which satisfies the property but is not periodic, or an example of a function which is periodic but does not satisfy the property. It is okay to give your examples as equations or graphs.

Prove that a subharmonic function remains subharmonic if the independent variable is subjected to a conformal mapping.

A company needs to purchase several new machines to meet its future production needs. It can purchase three different types of machines A,B,andC. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs$50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production.

Let                       Xi= number of machines to be purchased

MAX:                         200X1+250X2+300X3

Subject to:            2X1+3X2+5X3≤20

80X1+50X2+40X3≤500

X1,X2,X3≥0

How many of each machine should they purchase?

A floor-refinishing company charges $1.83 per square foot to strip and refinish a tile floor for up to 1000 square feet. There is an additional charge of $350 for any job over 150 square feet.

a) Create a piecewise model to express the cost, C, of refinishing a floor as a function of the number of square feet, s, to be refinished.
b) Graph the function. Be sure to label your axis and use an appropriate scale.
c) Give the domain and range.

Will Rogers spun a lasso in a vertical circle. The diameter of the loop was 6 ft, and the loop spun 50 times each minutes. If the lowest point on the rope was 6 inches above the ground, write an equation to describe the height of this point above the ground after t seconds.

Please write nicely.

Make a code in matlab to know the determinant of a matrix n x n, using the sarrus rule.

Maximize or minimize the following functions. Be sure to check your second-order conditions.

(a) maxQ Π = (20 − Q)Q − 4Q

(b) maxQ Π = (12 − Q)Q − 2Q2

(c) maxL Π = AL1/2 − 4L

(d) maxL Π = ALα − wL (where A and α are exogenous parameters and w is the wage rate on labor, L)

For the following functions are they concave or convex and what does that depend on? If the answer is possibly both, over what ranges are the functions convex or concave?

(a) Y = AKαL 1−α with respect to K? with respect to L?

(b) Y = (αKρ + βLρ ) 1/ρ with respect to K? with respect to L?

(c) T C = 3Q3 + 2Q2 − Q + 10

The population of Americans age 55 and older as a percentage of the total population is approximated by the function f(t) = 10.72(0.9t + 10)0.3 (0 ≤ t ≤ 20) where t is measured in years, with t = 0 corresponding to the year 2000.† (Round your answers to one decimal place.) At what rate was the percentage of Americans age 55 and older changing at the beginning of 2003? % per year At what rate will the percentage of Americans age 55 and older be changing in 2018? % per year What will be the percentage of the population of Americans age 55 and older in 2018? %

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is the discrete logarithm of 4 base 11 (with prime modulus 13)?

Question 3

Briefly discuss the implications of the Capital Asset Pricing Model for the relationship between the current spot price of an asset and the discount offered by the seller of a futures contract. (100 words)

On April 11, 2014, Cynthia received a loan of $45,000 at 5.55% compounded monthly. On May 13, 2015, the interest rate on

the loan changed to 5.75% compounded quarterly and remained constant thereafter. What will be the accumulated value

of the loan on December 31, 2017?