1a)

An invertible function f(x) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate (f−1)′(x) at the indicated value.

f(x)=11e^2x, Point=(0,11)

(f^−1)′(11)=

1b)

An invertible function f(x) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate (f−1)′(x) at the indicated value.

f(x)=2x+sin3x.   −π/6 ≤ x ≤ π/6 Point=(π/18, 2π/18+0.5)

(f^−1)′(2π/18+0.5)=

1c)

An invertible function f(x) is given along with a point that lies on its graph. Using Theorem 2.7.1, evaluate (f−1)′(x)at the indicated value.

f(x)=x^2−11x+36, x≥5.5   Point=(8,12)

(f^−1)′(12)=

2- Ordinary Differential Equations

a) y'+y = sen(x)

b)By what technique do you solve an ODE below: (x + yˆ2) dy + (y-xˆ2) dx = 0?

c) Solve the following ODE by Exact Equation: y '= 2x

d) Resolution y '= (2 + exp (xy)) / (2y-xexp (xy))

Consider Matrix A = ([5, 0, 4],[1, -1, 0],[1, 1, 0]). Note that [5, 0, 4] is row 1. [1, -1, 0] is row 2. [1, 1, 0] is row 3.

a) Find all Eigenvalues and Eigenvectors.

Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].

Find the standard normal area for each of the following (Round your answers to 4 decimal places.): Standard normal area a. P( Z ≤ 2.16) b. P(Z ≤ 3.05) c. P(Z ≤ 2.05) d. P(Z > 0.55)

For this Discussion, think of a specific testing scenario. Then consider a reliable test item for that testing scenario and an unreliable item for that same testing scenario. Consider how you might know if these items are reliable or unreliable.

With these thoughts in mind:

Post a brief description of a specific testing scenario. Then describe one reliable test item and one unreliable test item for that testing scenario. Finally, explain what determines whether an item is reliable or unreliable within the scenario you presented.

Identify at least three cash flow model inputs that are significantly stochastic in your opinion. Define appropriate probability distribution to covert these input data to random variables. Estimate the host governments take (HGT) statistic and the earning power of the investment on this project with 50 percent degree of certainty using a stochastic spreadsheet modeling tool of your choice

Problem 3. Let C_ξ and C_ν be two Cantor sets (constructed in previous HW ). Show that there exist a function F : [0, 1] → [0, 1] with the following properties

(a) F is continuous and bijective.

(b)F is monotonically increasing.

(c) F maps C_ξ surjectively onto Cν.

(d) Now give an example of a measurable function f and a continuous function Φ so that f ◦ Φ is non-measurable. One may use function F constructed above (BUT YOU NEED TO PLAY WITH IT). One of the ideas is to take two measurable sets C₁ and C₂ such that m(C₁) > 0 but m(C₂) = 0 and function Φ : C₁ → C₂, continuous. Also take N ⊂ C₁ - non-measurable set and define f = χΦ(N) .

(d) Use the above construction to show that there exists a Lebesque measurable set that is not a Borel set.

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.

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THIS WAS THE PREVIUS HW, WHERE WE DEFINE C_ξ and C_ν

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an interval of length ξ × a), one in each of the remaining intervals after stage 1, and so on. Let C_ξ denote the set which remains after applying the above procedure indefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected and perfect.

(c) Atually, prove that the complement of C_ξ in [0, 1] is the union of open intervals of total length equal to 1.

Problem 2. Let N denote the non-measurable subset of [0, 1], constructed in class and in the book "Real Analysis: Measure Theory, Integration, and Hilbert Spaces" by E. M. Stein, R. Shakarchi.

(a) Prove that if E is a measurable subset of N , then m(E) = 0.

(b) Assume that G is a subset of R with m_∗(G) > 0, prove that there is a subset of G such that it is non-measurable.

(c) Prove that if N^c = [0, 1] \ N , then m_∗(N^c) = 1.

(d) Now, conclude that

m_∗(N ) + m∗(N^c ) ≠ m_∗(N ∪ N^c ).

For each relation below, determine the following.(i) Is it a function? If not, explain why not and stop. Otherwise, answer part (ii).(ii) What are its domain and image?

(a){(x, y) :x, y∈Z, y- 2x}.

(b){(x, y) :x, y∈Z, xy- 0}.

(c){(x, y) :x, y∈Z, y-x2}.

(d){(x, y) :x, y∈Z, x|y}.

(e){(x, y) :x, y∈Z, x+y= 0}.

(f){(x, y) :x, y∈R, x2+y2= 1}.

Juiceco manufactures two products: premium orange juice and regular orange juice. Both
products are made by combining two types of oranges: grade 6 and grade 3. The oranges in
premium juice must have an average grade of at least 5, those in regular juice, at least 4.
During each of the next two months Juiceco can sell up to 1,000 gallons of premium juice and
up to 2,000 gallons of regular juice. Premium juice sells for $1,00 per gallon while regular
juice sells for 80 cents per gallon. At the beginning of month 1 Juiceco has 3,000 gallons of
grade 6 oranges and 2,000 gallons of grade 3 oranges. At the beginning of month 2, Juiceco
may purchase additional grade 3 oranges for 40 cents per gallon and additional grade 6
oranges for 60 cents per gallon. Juice spoils at the end of the month, so it makes no sense to
make extra juice during month 1 in the hopes of using it to meet month 2 demand. Oranges
left at the end of month 1 may be used to produce juice for month 2. At the end of month 1 a
holding cost of 5 cents is assessed against each gallon of leftover grade 3 oranges and 10 cents
against each gallon of leftover grade 6 oranges. In addition to the cost of the oranges it costs
10 cents to produce each gallon of (regular or premium) juice. Formulate an LP that could be
used to maximize the profit (revenues-costs) earned by Juiceco during the next two months.

Please formulate the LP problem and not provide coding.

Using MIL STD 105 E, probability of accepting a lot acceptance number when the lot size is 50,000 units, the inspection is normal general level II. Acceptance quality level is 0.065% and the proportion of defective product in the lots is 0.1%

FIND Pa ( prop of acceptance)?

PLEASE SHOW WORK!

The Patricia Garcia Company Is Currently Producing Seven New Medical Products. Each Of Garcia's ... Question: The Patricia Garcia company is currently producing seven new medical products. Each of Garcia's e... The Patricia Garcia company is currently producing seven new medical products. Each of Garcia's eight plants can add one more product to its current line of medical devices. The unit manufacturing costs for producing the different parts at the eight plants are shown below. How should Garcia assign the new products to the plants in order to minimize manufacturing costs? I need a step by step example*** use linear programming manually not excel

After the Introduction of a new product, the percent of the public that is aware of the product is approximated by

A(t)= 10t²2^-t

where t is time in months.

(a). Calculate A'(t).

(b). Find the rate of change of the percent of the public that is aware of the product after 2 months and after 4 months and include units.

(c). In part (b), one of your answers should have been positive while the other should have been negative. what dose that tell you about how the public awareness of the product has changed?