Define a function ?∶ ℝ→ℝ by

?(?)={?+1,[?] ?? ??? ?−1,[?]?? ????

where [x] is the integer part function. Is ? injective?

1. (b) Verify if the following function is bijective. If it is bijective, determine its inverse.

?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)

1. Let q and p be natural numbers, and show that the metric on R^q+p is equivalent to the metric it has if we identify R^q+p with R^q × R^p.

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 2 0 e^x/ 1 + x^2 dx, n = 10 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule

Solve the following differential equation: y''+4y'+4y=u(t-1)-u(t-3), y(0)=0, y'(0)=0

Solve the wave equation
∂2u/∂t2 = 4 ∂2u/∂x2 , 0 < x < 2, t > 0

subject to the following boundary and initial conditions.
u(0, t) = 0, u(2, t) = 0,
u(x, 0) = { x, 0 < x ≤ 1
2 − x, 1 < x < 2 , ut(x, 0) = 0

Define our “universe” S as the region in the xy-plane that is described by S = {(x, y) : |x| + |y| ≤ 2 and y > −1} .

(a) Mathematically and graphically describe three sets A1, A2, and A3 taken from S that are mutually exclusive but not collectively exhaustive.

(b) Mathematically and graphically describe three sets B1, B2, and B3 taken from S that are collectively exhaustive but not mutually exclusive.

(c) Mathematically and graphically describe three sets C1, C2, and C3 that form an event space for S. Make the area of C2 equal twice the area of C1, and make the area of C3 equal to three times the area of C1.

Draw a quick but accurate sketch of f(x) = √x²−4 over the interval [−4,0]. This covers the interval of integration.

1. Partition the interval of integration into 10 intervals. Show this on your graph with a right or left Riemann Sum

2. Create a table showing your interval index, i, the value x_i at which you evaluate f(x) in each interval, the values of f(x_i) and ∆x for each interval, and the contribution each rectangle makes toward the Riemann Sum. Evaluate the Riemann Sum for f(x) over the integration interval using your partition.

Using the method of separation of variables and Fourier series, solve the following heat
conduction problem in a rod.
∂u/∂t =∂2u/∂x2
, u(0, t) = 0, u(π, t) = 3π, u(x, 0) = 0

Using variation of parameters, find a particular solution of the given differential equations:

a.) 2y" + 3y' - 2y = 25e^-2t (answer should be: y(t) = 2e^-2t (2e^{5/2 t} - 5t - 2)

b.) y" - 2y' + 2y = 6 (answer should be: y = 3 + (-3cos(t) + 3sin(t))e^t )

Please show work!

Suppose {a1,...,am} is a complete set of representatives for Z/mZ. Show:

(i) If (b,m)=1, then{b*a1,...,b*am}is a complete set of
representatives.

(ii) If (b,m)> 1, then{b*a1,...,b*am}is not a
complete set of representatives.

A plane delivers two types of cargo between two destinations. Each crate of cargo I is 5 cubic feet in volume and 145 pounds in weight, and earns $35 in revenue. Each crate of cargo II is 5 cubic feet in volume and 290 pounds in weight, and earns $45 in revenue. The plane has available at most 425 cubic feet and 13,920 pounds for the crates. Finally, at least twice the number of crates of I as II must be shipped. Find the number of crates of each cargo to ship in order to maximize revenue. Find the maximum revenue.

2. (a) Prove that U45 is generated by the set {14,28}.

(b) Prove that the additive group Z×Z is generated by the set S={(3,1),(−2,−1),(4,3)}.

Please be thorough step by step with details, please.

For this assignment, you will use the provided database in the Unit 5 script file. You will add statements to this file to query the database tables. For your reference, below is a screenshot of the enhanced entity relationship diagram (ERD) as a reference when you write your queries. With the data types listed in the diagram, this will help you identify the types of operators that you can use for particular queries.

Use paiza.io for MySQL to execute and troubleshoot your Structured Query Language (SQL) statements. It is recommended that you use Notepad to write your SQL statements. Then, copy and paste your SQL statements to paiza.io to test your statements to ensure that they execute without errors. Save your work frequently.

The asterisk (*) indicates that a column is not null (i.e., required to have a value).

1. If you would like to see the test SQL statements, click on “Run” to see the statements execute. You should see “Hello” appear in the Output window (see the following screenshot).
2. Because you do not require these test SQL statements for your assignment, delete the sample SQL statements. To do this, select the text, and click on the Delete key on your keyboard. Your MySQL paiza.io screen should now be blank, as you see in the following screenshot. Note that if you ran the test statements, “Hello” will still appear in the Output window.
3. Copy and paste the code from the provided script file into the code editor, and click on “Run” (“Ctrl + Enter”). You should see no error messages displayed in the Output area at the bottom of your paiza.io screen. After you click on “Run,” you should see a green bar with “Success” at the top. You will see this circled in the following screenshot:
4. Next, you will write 1 query to respond to each of the following questions. Note that you can only use the information that is in the question or the enhanced ERD. Write a query to do the following:
   1. Display all of the rows in the publisher table.
   2. Display all of the books in the book table that were written by the author whose author ID is 1504192.
   3. Display all of the purchases in the shopbasket that were ordered on 2020-02-29.
   4. Display the largest quantity ordered in the shopbasket.
   5. Display the authors' first and last names sorted by their last name.
   6. Display a count of the number of orders placed via the shopbasket.
   7. Display the order date and quantity of orders in the shopbasket that have a quantity of more than 25.
   8. Display the first and last names of those authors who have a null Facebook.
   9. Display all of the customers' names, city, and state if they live in GA, FL, or CA.
   10. Display the book titles of those books that were not published by Scholastic Press.
   11. Display the first and last names of the authors and the book titles. Only display those authors who wrote a book with the word database in the title.
   12. Add a multiline comment before the first CREATE TABLE statement that includes your name, the course code (ITCO231), and the date when you are submitting your assignment.
5. Save your script file using your last name, ITCO231, U5, and the file extension .sql, as in “Smith_ITCO231_U5.sql.”
   1. Tip: When saving a Notepad file, you can change the file extension to .sql as required for the assignment by changing the “Save as type” to “All Files” and then typing the .sql extension, as shown in the example.

I didnt understand by what you mean by SQL TEST. If what you are asking me is that an SQL TEST? Yes! It is SQL QUERY TEST!!

Solve the initial value problem: 9y″+18y′+19y=0, y(π/2)=−2, y′(π/2)=−2. Give your answer as y=... . Use x as the independent variable.