(a) Let n = 2k be an even integer. Show that x = r^k is an element of order 2 which commutes with every element of D_n.

(b) Let n = 2k be an even integer. Show that x = r^k is the unique non-identity element which commutes with every element of D_n.

(c) If n is an odd integer, show that the identity is the only element of D_n which commutes with every element of D_n.

The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 700 homeless patients and no more than 620 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 1,800 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of $13,000 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $9,000 per month. How many of each type of patient should it house in order to maximize profit?

Prove that for all real x, we have |x−1|+|x + 2|≥ 3.

The current population of Tanzania is 49.3 million with a population growth rate of 2.53% per year. The average annual agricultural yield in Tanzania is 8950 kg/ha (where "ha" means hectare, which you can think of as a metric acre), a yield that is comprised of both grains (e.g. maize/corn) and tubers (e.g. cassava root) in a 1:1 ratio. The total amount of cropland farmed in Tanzania is 4,500,000 ha. The average agricultural output has increased at a linear rate of about 1.3% per year for the last five years or so. Ideally, one person should have a caloric intake of at least 2000 kcal per day in order to maintain their life. 1 kg grain supplies 3000 kcal; 1 kg tubers supplies 1000 kcal. Use the equations from our mini-lecture and the linear growth equation from the last module's quantitative assignment, to answer the following questions. You will also have to do some conversions for which you won't find specific equations. If Tanzania can currently feed 110 Million people, in what year will they cease to be self-sufficient in food? In other words, when will they need to begin importing food?

Hint: Plot both (i) population growth and (ii) food growth on the same set of axes for a span of 100 years or so (a point every 10 years should be enough for you to make a determination). The point where the two lines cross is the point after which they would need to start importing food. A spreadsheet program such as Excel may help though you could simply plot them on paper. Alternatively, you could use your scientific graphing calculator (if you have one) to determine the answer. The current year is 2,018.

Solve the differential equation

y'''+y''+y'+y=sinx+e^x+e^{-x}

1) find the solution t the non-homogenous DE

y''-16y=3e^5x , y(0)=1 , y'(0)=2

2)find the solution to the DE using cauchy-euler method

x²y''+7xy'+9y=0 , y(1)=2 , y'(1)=3

3)find the solution to the DE using Laplace

y''+8y'+16y=0 , y(0)=-1 , y'(0)=8

1. For each permutationσ of {1,2,··· ,6} write the permutation matrix M(σ) and compute the determinant |m(σ)|, which equals sgn(σ).

(a) The permutation given by 1 → 2, 2 → 4, 3 → 3, 4 → 1, 5 → 6, 6 → 5.

(b)  The permutation given by 1 → 5, 2 → 1, 3 → 2, 4 → 6, 5 → 3, 6 → 4.

 

A.) State Product Rule, Quotient Rule, and Chain Rule.

B.) Prove Power Rule

C.) Prove Product rule:

-By definition of derivative

-By implicit differentiation

D.) Prove the Quotient Rule

-By definition of derivative

-By implicit differentiation

-By product rule and chain rule

Please find f(x') and f(x") for all:

1) f(x) = 3(x² -2x)^3/4(7 - 5x²)⁶

2) f(x) = (y - x)⁴(y² - x)³

3) f(x) = x^7/3 + 16x³ + x

4) = f(x) = (x² - x) / (y² - y)

Newton's Law of Cooling is based on the principle that the rate of change of temperature y'(t) of a body in an environment with ambient temperature A is proportional to the difference b/w the temperature y(t) of the body and the ambient temperature A, so that y'=k(A-y) for some constant k.

(1) Sketch the direction field for the equation y'=K(A-y) for k=1/10 and A=70degrees. Then sketch several solution curves based on starting values of y(0) both greater than and less than A. Discuss the implication of your graph in general, and as it relates to the long-term limiting temperature of a cup of hot coffee initially at 190degreeF and a cup of iced coffee initially at 40degreesF in a room with ambient temperature 70degreesF.

Problem 7. Assume that a subset S of polynomials with real coefficients has a property:
If polynomials a(x), b(x) are from S and n(x), m(x) are any two polynomials with real coefficients, then polynomial a(x)n(x) + m(x)n(x) is again in S. Prove that there is a polynomial d(x) from S, such that any other polynomial from S is a multiple of d(x).

a) Prove by induction that if a product of n polynomials is divisible by an irreducible polynomial p(x) then at least one of them is divisible by p(x). You can assume without a proof that this fact is true for two polynomials.
b) Give an example of three polynomials a(x), b(x) and c(x), such that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x) nor b(x).

Show that, given any 3 integers, at least 2 of them must have the property that their difference is even.

Post your answer to one (1) of the following questions:

• What are the three steps for solving a quadratic equation?
• When do you have to use the quadratic formula?
• When can you use the root method to solve a quadratic equation?

Directly and completely answer the question(s).

Clearly and accurately explain your answer based on factual information.

Include examples, illustrations and/or applications in your answer(s).