You are planning on flying out of an airport on a trip. The airport parking garage charges $6 per day for the first four days, $4 per day for the next three days and $2 per day thereafter. A parking garage just outside the airport charges $5 per day and provides a free shuttle to the airport. When is it more cost-effective to park at the airport parking garage?

Your solution MUST include responses to ALL four parts. a) Understand the problem. Restate the problem in your own words. What do you know from the reading the problem? What are you looking for? What type of problem is this? What is needed in order to solve the problem?

b) Make a plan. State your plan for solving this problem. You may use words or diagrams. You might want to consider making a table, drawing a diagram, looking for a pattern, or building an equation or model.

c) Implement your plan. Once you have articulated your plan, carry out your plan. Using the information given, create mathematical model, an equation that you can use to determine when it will be more cost-effective to park at the airport parking garage. Clearly indicate how you arrived at that answer. Show your work! (*Even if using a calculator, discuss how and why you took the steps you did, not just what buttons you pushed.)

d) Look back. Is your answer reasonable? Can you find a way to check your work? Interpret your results. Remember that you have multiple representations – words, tables, graphs, and equations. Can you find another way to look at this problem that would allow you to check that your solution is correct? Interpret the answers in the context of the original application.

Find a subgroup G in symmetric (permutation) group Sn such that

(1) n = 4 and G is abelian noncyclic group

(2) n = 8 and G is dihedral group.

Scott secured a 5-year car lease at 6.20% compounded annually that required him to make payments of $881.83 at the beginning of each month. Calculate the cost of the car if he made a downpayment of $3,000.

Starting from mynewton write a function program mysymnewton that takes as its input a symbolic function f and the ordinary variables x0 and n. Let the program take the symbolic derivative f ′ , and then use subs to proceed with Newton’s method. Test it on f(x) = x 3 − 4 starting with x0 = 2. Turn in the program and a brief summary of the results

Let f be a function with domain the reals and range the reals. Assume that f has a local minimum at each point x in its domain. (This means that, for each x ∈ R, there is an E = Ex > 0 such that, whenever | x−t |< E then f(x) ≤ f(t).) Do not assume that f is differentiable, or continuous, or anything nice like that. Prove that the image of f is countable. (Hint: When I solved this problem as a student my solution was ten pages long; however, there is a one-line solution due to Michael Spivak.)

1. Must be nicely written up AS A PROOF.

a. Show that gcd(m + n, m) = gcd(m, n).

b. If n | k(n + 1), show that n | k.

c. Show that any two consecutive odd integers are relatively prime.

A body of mass 7 kg is projected vertically upward with an initial velocity 18 meters per second. The gravitational constant is g=98m/s². The air resistance is equal to k|v| where k is a constant.

Find a formula for the velocity at any time ( in terms of k ):

Find the limit of this velocity for a fixed time t₀ as the air resistance coefficient k goes to 0.

Given a scalar fifield φ(x, y, z)=3x²-yz and a vector field F(x, y, z)=3xyz²+2xy³j-x²yzk.

Find:

(i)F.∇φ.

(ii)F×∇φ.

(iii)∇(∇.F).

100 kg of R-134a at 320 kPa are contained in a piston-cylinder device whose volume is 7.530 m³. The piston is now moved until the volume is one-half its original size. This is done such that the pressure of the R-134a does not change. Determine the final temperature and the change in the total internal energy of the R-134a. (Round the final answers to two decimal places.)

Analyzing Transactions Using the Financial Statement Effects Template Following are selected transactions of Mogg Company. Record the effects of each using the financial statement effects template. Shareholders contribute $10,000 cash to the business in exchange for common stock. Employees earn $500 in wages that have not been paid at period-end. Inventory of $3,000 is purchased on credit. The inventory purchased in transaction 3 is sold for $4,500 on credit. The company collected the $4,500 owed to it per transaction 4. Equipment is purchased for $5,000 cash. Depreciation of $1,000 is recorded on the equipment from transaction 6. The Supplies account had a $3,800 balance at the beginning of this period; a physical count at period-end shows that $800 of supplies are still available. No supplies were purchased during this period. The company paid $12,000 cash toward the principal on a note payable; also, $500 cash is paid to cover this note's interest expense for the period. The company receives $8,000 cash in advance for services to be delivered next period. Use negative signs with your answers, when appropriate. Hint: For transaction 4, enter the net effect amount for balance sheet answers.

Question 1. State the prove The Density Theorem for Rational Numbers.

Question 2. Prove that irrational numbers are dense in the set of real numbers.

Question 3. Prove that rational numbers are countable

Question 4. Prove that real numbers are uncountable

Question 5. Prove that square root of 2 is irrational

Question 1. State the prove The Density Theorem for Rational Numbers.

Question 2. Prove that irrational numbers are dense in the set of real numbers.

Question 3. Prove that rational numbers are countable

Question 4. Prove that real numbers are uncountable

Question 5. Prove that square root of 2 is irrational