prove that if an even integer n is subtracted from an odd integer m. then m - n is odd.

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) .

Problem 1

Larry has a hot pot of water initially at 95 degrees celsius and in just 5 minutes it cools to 80 degrees celsius while he is watching TV in a room of temperature 21 degrees celsius. Use Newton's law of cooling to determine when the temperature of the pot will reach 50 degrees celsius.

Problem 2

After 1 year from 2015, a country's cumulative exports were estimated to be $8 million. Economist in that country are now trying to estimate the country's cumulative exports in year 2026, if the country's cumulative exports grow in proportion to its average size over the last t years, plus a fixed growth rate of 10.

Can you help these economists to figure out the country's cumulative exports in year 2026 by first finding the general formula for the country's cumulative exports y(t) (in millions of dollars)? (Assume that the constant of proportionality is equal to 1)

Suppose you borrowed $400,000 for a home mortgage on January 1, 2010 with an annual interest rate of 3.5% per year compounded monthly.

(a) If you didn't make any payments and were only charged the interest (and no late fees), how much would you owe on the mortgage on January 1, 2030?

(b) Suppose the balance on the mortgage is amortized over 20 years with equal monthly payments at the end of each month. (This means the unpaid balance on January 1, 2030 should be $0). What are the monthly payments?

(c) How much interest was paid during the 20 years of the mortgage?

(d) What is the unpaid balance on the mortgage on January 1, 2015?

Using the carrying capacity model recursively, k= 800 million, the population of the United States for each year between 1950-2017. 1950 population was 158804 thousand. What is the model’s predicted U.S. population for the year 1999 (in thousands)? the growth rate of 0.01078. Give your answer correct to the nearest whole number.

Question: Can you please break this proof down to a level where a high school student can understand it? Thank you.

Example:

Prove that the product of n successive integers is always divisible by n!

If the product of n successive integers is always divisible by n!, then we would only need to prove it is true for positive integers. If one of the integers in the product is zero, it will always be true. If the integers are negative, n! will divide by their absolute value.

Proof by contradiction:

If there is a number of n successive positive integers whose product is not divisible by n!, then we can choose the smallest and call it N. N must be greater than 2 because the product of any two successive integers is always even. Therefore, there must be an integer, m, such that (m+1)(m+2)...(m+N) is not divisible by N! Of these numbers, m, let M be the smallest. M must be positive since N! is divisible by N!. So, we are supposing that (M +1)(M+2) …(M+N) is not divisible by N!

(M+1)(M+2) …(M+N-1)(M+N) = M[(M+1)(M+2)...(M+N-1)]+N[(M+1)(M+2)...(M+N-1)]

Using our choice of M, n! divides into M[(M+1)(M+2)...(M+N-1)]. Using our choice of N, (N-1)! divides (M+1)(M+2)...(M+N-1) and therefore N! divides N[(M+1)(M+2)...(M+N-1)].

In combination, N! Divides the right side of the last equation. This contradiction establishes the result.

Can someone explain it with detail explanation?

Two clothing stores in a shopping center compete for the weekend trade. On a clear day the larger store gets 60% of the business and on a rainy day the larger store gets 80% of the business. Either or both stores may hold a sidewalk sale on a given weekend, but the decision must be made a week in advance and in ignorance of the competitor’s plans. If both have a sidewalk sale, each gets 50% of the business. If, however, one holds the sale and the other doesn’t, the one conducting the sale gets 90% of the business on a clear day and 10% on a rainy day. It rains 40% of the time. How frequently should each retailer conduct sales?

A ball is thrown eastward into the air from the origin (in the direction of the positive x-axis). The initial velocity is

50 i + 80 k,

with speed measured in feet per second. The spin of the ball results in a southward acceleration of 8 ft/s², so the acceleration vector is a = −8 j − 32 k. Where does the ball land? (Round your answers to one decimal place.)

_______________ft from the origin at an angle of ________________ ° from the eastern direction toward the south.

With what speed does the ball hit the ground? (Round your answer to one decimal place.)
___________ ft/s

1.) (15 pts.) Miamiville Interiors is a small shop that produces a variety of items that are sold through home improvement departments at local hardware stores. Among the items MI produces are replacement cabinet facings for kitchen and bathroom cabinets. The production manager must decide how many of each of two types of cabinet should be produced in the next production period. The net revenues and demand for each type of cabinet are listed in the table below. Also, in the table are production cost per unit and the units available for each of three types of wood used in the cabinet construction. Revenue per unit Maple Pine Walnut Demand Cabinet A $8 6 11 400 Cabinet B 13 4 9 600 Cost/unit $3.00 $1.80 $3.75 Units Available 400 600 200 Formulate a linear program to determine how many of each of the types of cabinet should be produced, using each type of wood. You seek to maximize profit (revenue – cost), while satisfying demand for the two cabinet styles.

Solve it step by step and please with clear handwritten. Be ready to follow the comment

Integrability Topic

Question1. Let g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0

Use the sequential criterion for continuity to prove that g is discontinuous at every rational number in[0,1]

Question.2 g is continuous at any irrational point in[0,1]. Explain why g is Riemann integrable on[0,1] based on the following fact that

Suppose h:[a,b]→R is continuous everywhere except at a countable number of points in[a,b]. Then h is Riemann integrable on[a,b]

Question.3

Letf:[0,1]→R be defined by f(x)=0 if x=0; f(x)=1 if 0<x<=1 we know that f is integrable on [0,1] Suppose c is a rational number in [0,1]. Compute(f◦g)(c). Now suppose c is an irrational number in[0,1]. Compute(f◦g)(c). Can you recognize the function f◦g:[0,1]→R?

8. A plane with an airspeed of 550 mph is headed at a bearing of S35^degreeE. The wind is blowing at a bearing of N45^degreeE at 82mph. the wind is moving the plane off of the direction it planned to fly on. Find the actual speed the plane is flying at (ground speed) and the direction is actually traveling in.

answer should be , speed: 541.83 and direction 543.57^degreeE how do I get these two answers?

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ and ?(?) = ? 2 if ? ∈ ℝ ∖ ℚ. Find all points at which ? is continuous

Can someone please explain step by step the following solution. i am going to post the solution i just need someone to please explain why each step is taking and what formula is being used. please explain it to me thank tyou

Suppose A is a set and {Bi | i ∈ I} is an indexed family of sets with I ≠∅.

1. Prove that A \ ( ∪i∈I Bi ) = ∩i∈I (A \ Bi).

• A be a set
• Bi | i ∈ I} is an indexed family of sets with I ≠∅
• Let x∈A - ∪i∈I Bi
• => x∈A & x∉∪i∈I Bi
• => x∈A & x∉ Bi for each i∈I
• => x∈A - Bi for each i∈I
• => x∈ ∩i∈I (A - Bi)
• so A - ( ∪i∈I Bi ) = ∩i∈I (A \ Bi) proven

2. Interpret this theorem when I = {1, 2}.

• To show that A-(B₁∪B₂)=(A-B₁) ∩(A-B₂)
• Let y∈A - (B₁∪B₂)
• => y∈A & y∉ B₁∪B₂
• => y∈A & y∉ B₁ & y∉B₂
• => y∈A & y∉ B₁ & y∈A & y∉B₂
• => y∈A - B₁ & y∈A - B₂
• => y∈ (A - B₁) ∩ y∈ (A - B₂)
• so our claim is justified and hence the proof.