Find the power series solution of the differential equation using the FROBENIUS METHOD xy"+y'+4xy=0 Show all of the steps.

rotate around the z -axis the region :T ={(x,z)'∈ R^2:sin z<x<π -z,0<z<π} to obtain the solid Ω.find its volume and center of mass.

True or False? The data in the table below depict y as an exactly exponential function of x.

Use the graph of the given function f to determine the limit at the indicated value of a, if it exists. (If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. A function labeled y = f(x) with 2 parts is graphed.

• The first part is linear, enters the window in the third quadrant, goes up and right, crosses the x-axis at approximately x = −2, crosses the y-axis at y = 1, and ends at the approximate point (1, 1.5).
• The second part is linear, begins at the open point (1, 3), goes horizontally right, and exits the window in the first quadrant.
• The equation a = 1 is shown.
• A vertical line extends up from the x-axis at x = 1, passes through the approximate point (1, 1.5), and ends at the open circle (1, 3).

Overbooking flights

Eagle Air is a small commuter airline.  Each of their planes holds 15 people. Past records indicate that only 80% of people with reservations (tickets) do show up. Therefore, Eagle Air decides to overbook every flight. Suppose Eagle Air decides that it will accept up to 18 reservations per flight (18 is the maximum number of reservations per flight).

Demand for Eagle Air flights is so strong that 18 reservations are booked for every flight. Everyone knows how popular Eagle Air flights are, and so no one ever shows up without a reservation.

This is the first of 6 questions in this set.

a. Calculate the probability that on any given flight, at least one passenger holding a reservation will not have a seat.  Four decimals

b. What is the probability that there will be one or more empty seats? Four decimals

c. What is the probability that the first person who is bumped from a flight will not get on either of the next two flights? Assume that the number of “no-shows” is independent across flights. Also assume that the first person who is bumped has priority when an empty seat comes up on a subsequent flight.  Four decimals

d. What is the expected number of people who show up for a flight? Reminder: Everyone makes a reservation ahead; no one shows up without a reservation; a maximum of 18 reservations is accepted; and every flight has 18 reservations because Eagle Air is so popular.  One decimal

e. Suppose that each flight costs $1000 to run, considering all costs. If tickets are priced at $75 each, what is Eagle Air’s expected profit per flight? Assume that when sixteen or more people show up for a flight, any overbooked passengers wait until a seat becomes available, so Eagle Air ultimately gets the revenue from everyone who shows up.  One decimal

f. What is the standard deviation of the number of people who show up for a flight? Two decimals

Let R be the real line with the Euclidean topology.

(a) Prove that R has a countable base for its topology.

(b) Prove that every open cover of R has a countable subcover.

1)How many license plates are there if they start with three capital letters followed by 4 digits from 0 to 9. (Note that letters and digits can be repeated?

2)In how many ways can 5 men and 5 women be arranged in a line if the line starts with a man, and the men and women alternate?

Describe inequalities as presented in the assigned reading for this week. Why are some inequalities represented with a solid point and others represented with an open point? Propose another method for representing inequalities.
First ,define an inequality in general terms. Then introduce the various inequality symbols and tell what each means. Give some examples of inequalities and tell which ones might use open or closed points when graphed and explain how you know.
What are some uses of inequalities in real life? One example is a range of target values for cholesterol levels. Another example is a limit on " screen time" for a child each day.

Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern U.S. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow.

An individual retirement account, or IRA, earns tax-deferred interest and allows the owner to invest up to $5000 each year. Joe and Jill both will make IRA deposits for 30 years (from age 35 to 65) into stock mutual funds yielding 9.1%. Joe deposits $5000 once each year, while Jill has $96.15 (which is 5000/52) withheld from her weekly paycheck and deposited automatically. How much will each have at age 65? (Round your answer to the nearest cent.) Joe $ Jill $

In each case, determine a 3 × 3 matrix ? that fits the description. Justify that the conditions are met in each case.

(a) Matrix ? has rank 1 and ? + ? is not invertible.

(b) Matrix A is symmetric and the null space of ? is a point.

What is the remainder of 23451638 divided by 5? What is the remainder of 23451638 divided by 4? [Hint: This is easy, use 4 and 5 divide 100.]

4. Use a proof by contradiction to show that the square root of 3 is irrational. You may use the following fact: For any integer k, if k2 is a multiple of 3, then k is a multiple of 3. Hint: The proof is very similar to the proof that √2 is irrational.

5. Use a direct proof to show that the product of a rational number and an integer must be a rational number.

6. Use a proof by contradiction to show that the sum of an integer and an irrational number must be irrational.

Read Proposition 1.23, the paragraph between the proposition and its proof, and the proof.
a. In your own words, explain the meaning of the two different mathematical ideas of "existence" and "uniqueness".
b. How do the ideas of "existence" and "uniqueness" relate to the phrase "one and only one"?
c. In your own words, explain why the proof of a "one and only one" statement will have two distinct parts.

Prove Proposition 1.26. Proposition 1.26. Let m,n ∈ Z. If mn = 0, then m = 0 or n = 0.

Prove Proposition 2.3. Proposition 2.3. 1 ∈ N.