First: read the story. I tell stories too 🙂 Second: read the notes. Third: view the videos. Fourth: solve the questions/solved examples. Fifth: check your solutions with my thoroughly-explained solutions.
You may use the calculators to verify your answers.
You are encouraged to solve the questions first, before you check your answers.
I wrote the codes for the calculators using JavaScript, a client-side scripting language.
I used the AJAX Javascript library for the rest of the codes.
Please use the latest Internet browsers. The calculators should work.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
Thank you for visiting.
Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S
Story
The event will hold in three days.
Well-cooked and delicious: rice and beans
Fresh avocados
Options of fish, chicken, cabeza Son: Dad, what about vegan? Dad: Let's not worry about vegan. Those folks would really be hungry.
They need to eat well. Son: Does it mean that those who are vegan do not eat well? Dad: I did not say that. Do not force words into my mouth. Son: I just asked a question, Dad.
What about al pastor? Dad: No, we do not eat pork.
Why should we give people what we do not eat?
Okay, let's go and order these burritos.
Dad: Buenas tardes, señora
How are you doing today? Son: Good afternoon, Mama Esther. Esther: Good afternoon. I am doing well.
How are you doing too? Dad: We are doing well.
Thank you.
How much are your burritos? Esther: It's $7.00 each.
Guacamole costs extra ¢50 each. Dad: We want to place an order for 1000 burritos. Son: Well-cooked: rice and beans; fresh avocados...everything
No guacamole. Esther: Okay... Dad: 300 pescado; 300 cabeza; 400 chicken
How much is the cost? Esther: How soon do you need them? Dad: In three days. Esther: Would I deliver them, or would you pick them up? Dad: We shall pick them up. Esther: It's going to be $7 each for the first 300 burritos $6 each for the next 300, and $5.00 each for the remaining 400.
Fair enough?
So, the price is ... $2100 for the first 300 + $1800 for the next 300 + $2000 for the remaining 400
Total of $5900
Add 10% tax of $5900
That gives $590
Total cost of $6490.00
Son: Dad, I could calculate the cost another way. Dad: What way?
Is it better than the way Esther did? Son: For many orders, yes!
And for writing a computer program to calculate it. Dad: Interesting. What is it? Son: We learned it in school.
Mama Esther did it as a simple Arithmetic.
Mr. C called it the Manual method.
Then, he taught us to do it algebraically ...
using the Piecewise Function method.
I think it is much better. Dad: Okay, what's the method? Son: We begin by defining the variables involved.
Let the number of burritos = b
and the cost = c
Which variable depends on the other variable? Son: Does the number of burritos depend on the cost, or
does the cost depend on the number of burritos? Dad: The cost depends on the number of burritos. Son: That is correct.
We write it this way: $c = f(b)$
It can also be written as: $c(b)$ Dad: Okay... Son: So,
Dad: How is this better than the way Esther did it? Son: Say you wanted to order $900$ burritos
How much would it cost excluding the taxes? Dad:$7 for the first 300 gives $2100 $6 for the next 300 gives $1800 $5 for the remaining 300 gives $1500
That gives a total of $5400 Son: How many steps did you do before you got the answer? Dad: Three steps... Son: I shall do only two steps before I get that answer.
$c(900) = 5(900) + 900 = 4500 + 900 = 5400$ Dad: You used the 3rd equation because 900 falls in ... Son: that piece...yes...in that domain. Dad: That is interesting! How did you get those equations? Son: Welcome to Piecewise Functions!, Dad.
Overview of Piecewise Functions
Objectives
Students will:
(1.) Define piecewise functions.
(2.) Discuss piecewise functions.
(3.) Discuss real-world applications of piecewise-functions.
(4.) Give examples of mathematical functions that are piecewise functions.
(5.) Graph piecewise functions manually.
(6.) Graph piecewise functions using the Texas Instruments (TI) series graphing calculators.
(7.) Analyze the graphs of piecewise functions.
(8.) Solve problems involving the application of piecewise functions.
Skills Measured/Acquired
(1.) Use of prior knowledge
(2.) Critical Thinking
(3.) Interdisciplinary connections/applications
(4.) Technology
(5.) Active participation through direct questioning
(6.) Project-based learning (PBL) Research.
Homework Assignment
Students should research real-world applications of piecewise functions.
(1.) Write the word problem (real-world application).
(2.) Solve solve questions (numbers) "manually" for each piece/interval.
(3.) Write the piecewise function for the word problem.
(4.) Solve those questions again using the piecewise function.
(5.) Check to ensure that you get the same answers (verify your answers) using both methods (manually and piecewise
function).
(6.) Write a computer program that solves that word problem.
(7.) Verify your answers again with the computer program.
Check for prior knowledge. Ask students about these terms.
Vocabulary Words
Bring it to English: piece, step, hybrid, exception, except, condition
Bring it to Math: piecewise, function, piecewise-defined, hybrid, domain,
range, intercepts, x-intercept, y-intercept, graph, continuity, discontinuity
Bring it to Computer Science: conditional statements, if, if-else, if-elif
Why Study Piecewise Functions?
Ask students to give examples of scenarios where exceptions have been made, or where certain conditions have
been used.
Let us begin with a simple example.
At some Golden Corral restaurants in the U.S, children aged 3 years and under (at most 3 years) eat free.
Let us analyze the Kid's Buffet. The information is found
here.
It is also written here for you:
Ages 3 and under: Free
Ages 4 - 8: $5.99
Ages 9 - 12: $6.99
As you see, there are different prices (costs) for children in a certain age group.
So, you only eat free if you are at most 3 years old. That is a condition that must be satisfied before you eat free.
When writing a computer program for this function, you have to use conditional statements.
Let us write this function as a piecewise function. First: Define the variables.
Two things are involved in this function: age and cost.
Let $a$ be the age of the child.
Let $c$ be the cost of the meal.
As you can see, the cost of the meal depends on the age of the child.
This means that the cost is the dependent variable.
And the age of the child is the independent variable.
So, the cost of the meal is a function of the age of the child.
$c = f(a)$. We can write this as $c(a)$ pronounced as $c \: of \: a$
$$
c(a) =
\begin{cases}
\$0 & \quad 0 < a ≤ 3 \\[3ex]
\$5.99 & \quad 3 < a ≤ 8 \\[3ex]
\$6.99 & \quad 8 < a ≤ 12
\end{cases}
$$
Exercises
Use the piecewise function above and answer these questions.
(1.) Peter is 3 years old.
How much (excluding taxes) is the cost of his meal?
Peter eats free because he is $3$ years old. His age is included in the first "piece".
The cost of his meal is $\$0.00$
(2.) Mary is 7 years old.
How much (including taxes) is the cost of her meal?
Assume Golden Corral charges a tax rate of 7.5%.
Mary is $7$ years old. So, her age is included in the second "piece".
The cost of her meal is $\$5.99$
Students should give more examples as seen in national parks, car rentals, and retail stores among others.
They should cite their sources "properly".
Teachers should emphasize the importance of citing sources accurately, and citing them in the correct format.
What is a Piecewise Function?
A piecewise function is a function consisting of sub-functions (pieces) defined on a sequence of intervals (domains).
It is also known as a piecewise-defined function or hybrid function.
It is function defined by different equations/functions on different parts of its domain.
The domain of each function must not overlap.
Ask students why the domains must not overlap. What would happen if the domains of any two sub-functions overlap?
Some students may ask you that question first. If they do, ask them for their opinions.
Residential Service Rates by Georgia Power; State of Georgia, USA
As of the 30th day of December, 2018; the power rates for the Residential Service during the Summer Period (June - September) by Georgia Power is found here.
We shall focus on the Residential Service Rates. These costs exclude taxes and other costs.
It is also written here for you:
Basic Service: $10.00
First 650KWh: ¢5.7 per KWh
650 - 1000KWh: ¢9.4 per KWh
Over 1000KWh: ¢9.7 per KWh
KWh means kilowatt hour.
$1$ KWh = $3600000$ J
$1$ kilowatt hour is equivalent to $3600000$ Joules.
The S.I (System International) unit of power is the Joule, $J$
The basic service fee is in dollars.
The rate of consumption is in cents.
We have to convert to the same unit.
We shall use the dollars (not the cents).
$
\$1 = ¢100 \\[2ex]
¢100 = \$1 \\[2ex]
¢5.7 = \$0.057 \\[2ex]
¢9.4 = \$0.094 \\[2ex]
¢9.7 = \$0.097
$
But, please wait a minute!
That information is not "really" correct.
The "Interactive Sample Bill" (hmmmm....it is not interactive though ☺☺☺)
found here
is also incorrect.
The correct information is found in the Georgia Power Bill Calculator of the Georgia Public Service Commission.
The correct information is here
It is great to know that the Georgia Public Service Commission developed a Power Bill calculator.
We shall check our work with their calculator.
And of course...we shall also check our work with my calculator.
The correct information is written here for you:
Basic Service: $10.00
Tier Usage Cost per KWh
1st tier up to 650KWh $0.056582
2nd tier next 350KWh $0.093983
3rd tier over 1000KWh $0.097273
Teaching Moments (to promote critical thinking)
Ask students to explain the differences between the information found in the Georgia Power website (the first one)
and that found in the Georgia Public Service Commission (the second one).
Students should also explain the differences based on the domains of each piece in both websites.
What are the errors in the first one?
Please specify the importance of not rounding intermediate calculations.
Please specify the importance of rounding only the final answer to two decimal place (because it is dollars and cents).
Calculate the power costs for the following consumption of power.
This application has three pieces.
The basic service fee is not a piece. It must be paid whether one consumed any power or not.
Let $p$ = power (in KWh)
Let $r$ = cost (in dollars per KWh)
(1.) $0$ KWh falls in the first piece.
Basic service fee = $$10.00$
cost for $0$ KWh @ $$0.056582$ per KWh = $0 * 0.056582 = 0$
$10.00 + 0.00 = 10.00$
cost for $0$ KWh = $$10.00$
(2.) $300$ KWh falls in the first piece.
Basic service fee = $$10.00$
cost for $300$ KWh @ $$0.056582$ per KWh = $300 * 0.056582 = 16.9746$
$10.00 + 16.9746 = 26.9746$
cost for $300$ KWh = $$26.97$
(3.) $700$ KWh falls in the second piece.
Basic service fee = $$10.00$
Before we use the second piece, we have to go through the first piece first. First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$700 - 650 = 50$
We need to find the cost for the remaining $50$ KWh
That takes us to the second piece. Second Piece for $50$ KWh
cost for $50$ KWh @ $$0.093983$ per KWh = $50 * 0.093983 = 4.69915$
$10.00 + 36.7783 + 4.69915 = 51.47745$
cost for $700$ KWh = $$51.48$
(4.) $1000$ KWh falls in the second piece.
Basic service fee = $$10.00$
Before we use the second piece, we have to go through the first piece first. First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$1000 - 650 = 350$
We need to find the cost for the remaining $350$ KWh
That takes us to the second piece. Second Piece for $350$ KWh
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$10.00 + 36.7783 + 32.89405 = 79.67235$
cost for $1000$ KWh = $$79.67$
(5.) $1200$ KWh falls in the third piece.
Basic service fee = $$10.00$
Before we use the third piece, we have to go through the first piece and also through the second piece. First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$1000 - 650 = 350$
We need to find the cost for $350$ KWh
That takes us to the second piece. Second Piece for $350$ KWh
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$1200 - 1000 = 200$
We need to find the cost for the remaining $200$ KWh
That takes us to the third piece. Third Piece for $200$ KWh
cost for $200$ KWh @ $$0.097273$ per KWh = $200 * 0.097273 = 19.4546$
$10.00 + 36.7783 + 32.89405 + 19.4546 = 99.12695$
cost for $1200$ KWh = $$99.13$
Some students may ask if it is possible to have just one function that will find the cost for any consumption of power.
Or is it possible to find the cost for the consumption of power that falls in the second piece, without having to go through the first piece?
Those are really interesting questions!
That is one of the reasons for studying piecewise functions ☺☺☺
Please specify the importance of not rounding intermediate calculations.
Please specify the importance of rounding only the final answer to two decimal place (because it is dollars and cents).
What if we have to calculate the power rates for "several" consumption of power?
Do we have to solve this manually all the time? That will be time consuming!
We can write it as a piecewise function and use each function for the consumption of power that correspond to that piece.
Besides, writing it as a piecewise function helps us to write a computer program that will find the rate for any consumption of power.
This application has three pieces.
Let $p$ = power consumed(in KWh)
Let $c$ = cost per KWh or power consumed (in dollars)
$c = f(p)$
This can be written as: $c(p)$
For the first piece;
Basic service fee = $$10.00$
cost for $p$ KWh @ $$0.056582$ per KWh = $p * 0.056582 = 0.056582p$
$10 + 0.056582p = 0.056582p + 10$
$c(p) = 0.056582p + 10$
For the second piece;
Basic service fee = $$10.00$
We have to "finish" with the first piece first
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$10 + 36.7783 = 46.7783$
Then, we can multiply the remaining consumption of power by $0.093983$
$c(p) = 46.7783 + 0.093983(p - 650)$
$c(p) = 46.7783 + 0.093983p - 61.08895$
$c(p) = 0.093983p - 14.31065$
For the third piece;
Basic service fee = $$10.00$
We have to "finish" with the first piece first
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$10 + 36.7783 = 46.7783$
Then we have to finish with the second piece next
$1000 - 650 = 350$
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$46.7783 + 32.89405 = 79.67235$
Then, we can multiply the remaining consumption of power by $0.097273$
$c(p) = 79.67235 + 0.097273(p - 1000)$
$c(p) = 79.67235 + 0.097273p - 97.273$
$c(p) = 0.097273p - 17.60065$
We can now write the piecewise function as:
$$
c(p) =
\begin{cases}
0.056582p + 10; & \quad 0 \leq p \leq 650 \\[3ex]
0.093983p - 14.31065; & \quad 650 \lt p \leq 1000 \\[3ex]
0.097273p - 17.60065; & \quad p \gt 1000
\end{cases}
$$
Let us recalculate all the questions using the Piecewise Function method.
Which of the two methods do you prefer?
What are your reasons?
What are the pros and cons that you see for each method?
Do you have any other method for solving Piecewise Function applications?
Calculator for the Summer Residential Power Rates by Georgia Power, State of Georgia, USA
Power Consumption and Monthly Bill Statement
Please test the calculator and see the output.
Complete all fields. Then, click "Bill Statement".
Power Consumption
Monthly Bill Statement
Calculator for the Winter Residential Power Rates by Georgia Power, State of Georgia, USA
Power Consumption and Monthly Bill Statement
Please test the calculator and see the output.
Complete all fields. Then, click "Bill Statement".
Power Consumption
Monthly Bill Statement
Student Project: Power Bill
Power Usage and Monthly Bill Statement
Please review the example calculations and test the calculator before you begin.
For Mathematics and Information Technology/Computer Science Students: Midterm Project
(1.) You may work as a group (peer tutor to teach/correct one another).
However, this is an individual project.
In other words, you will submit your own project.
(2.) No two projects should be the same.
(a.) You may not do any of my examples even if the rates of the company have changed. Use my examples as guides to completing your own.
(b.) You may not do any of the projects of my previous students. (Student Project samples are provided below. Use them as guides to completing your own.)
(3.) Research any real-world application of piecewise function.
(a.) All information used for this project should be verifiable on the direct website of the company/organization.
(b.) Textbook examples are NOT allowed.
(c.) Find any application that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(4.) Write the complete address of the direct page of the website where you found the application.
(a.) *If the "direct" web address is too long, please shorten it by pasting the "complete web address" into
www.tinyurl.com *
*This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
(b.) For online students, please copy and paste the link as is.
(c.) Please set the link to open in a new window.
(d.) Please submit the name of the company and the direct web page that has the application, in the Projects: Company Names and Websites page in the Canvas course.
I shall review and respond.
(5.) As a BRCC/VCCS student, you have access to Microsoft Office suite of apps.
(a.) You can download and install these apps on your laptop/desktop. Please contact the IT/Tech support in your college for assistance if you do not know.
In that regard, the project is to be typed using Microsoft Word (not Google Docs).
As at the time of this writing (05/05/2023), the reason for using Microsoft Word over Google Docs is because Microsoft Word has the opening brace for piecewise functions
but Google Docs does not.
(b.) For all English terms (entire project): use Times New Roman; font size of 14; line spacing of 1.5
(c.) For all Math terms: symbols, variables, numbers, formulas, expressions, equations and fractions among others, please use the Math Equation Editor.
(i.) Set the font to Cambria Math; font size of 14; and align accordingly
(ii.) Insert a space after each each equation as applicable. Just make a good work that is organized and spacious.
(iii.) Align the functions in each piece of the piecewise function accordingly.
(d.) Include page numbers. You may include at the top of the pages or at the bottom of the pages but not both.
(6.) Mr. C (SamDom For Peace) wants you to do this real-world project very well.
Hence, he highly recommends that you submit a draft so he can give you feedback.
It is important that you submit your draft to me via email or in the Projects: Drafts page in the Canvas course at least one week
prior to the due date.
Please do not submit any draft if the due date is one week or less.
When you submit your draft, I shall review and provide feedback.
When everything is fine (after you make changes as applicable based on my feedback), please submit your work in the appropriate area in the Canvas course.
Only projects submitted in the Canvas course are graded.
Draft projects are not graded. In other words, projects submitted via email and/or in the Projects: Drafts page are not graded because they are drafts.
They are only for feedback and should be submitted at least one week prior to the due date.
If it is past one week before the project is due, please do not submit it as a draft to me. Just do it very well according to the requirements and submit it in the
appropriate area (Assignments page: Piecewise Function Project) in the Canvas course.
Submitting drafts is highly recommended. If your professor gives you an opportunity to submit a draft, please use that opportunity.
Submitting drafts is not required. It is highly recommended because I want to give you the opportunity to do your project very well and make an excellent grade in it.
(7.) Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.
(8.) Writing Skills: Write or type the main application entirely.
(9.) Mathematical Skills: Arithmetic This is the Arithmetic method.
Use random numbers to test the real-world application manually for each piece.
Show all work including intermediate calculations/values.
Write down your results.
(10.) Mathematical Skills: Algebra: This is the Algebraic Method (Piecewise Function Method).
Write a piecewise function for that application.
Test the piecewise function with the same random numbers that you used for the Arithmetic method. Please NOTE:
(a.) The intermediate results should be the same for both methods.
(b.) The final results should be the same for both methods.
If either of the results are not the same, there are issues. Please fix them.
These are the required information.
You may or may not not use a table format.
Please be creative.
Required Information
Name:
Your name
Date:
The date
Instructor:
Samuel Chukwuemeka
Project:
Power Bill: Residential Rates
Company:
Georgia Public Service Commission (for Georgia Power)
(http://www.psc.state.ga.us/calc/electric/GPcalc.asp)
(Name of the company and the direct Website of company that has the piecewise function)
Objectives:
(1.) Calculate the winter/summer power bill of the residents of the State of Georgia within each range of specific power usage manually using Arithmetic method.
(2.) Write a piecewise function of the residential rates.
(3.) Recalculate the same power bill of the residents of the State of Georgia within each range
of specific power usage algebraically using the Piecewise Function method.
Information:
Write the direct information from the direct website
Arithmetic Method:
Test each piece manually
Piecewise Function:
Write the piecewise function of the information
Piecewise Function Method:
Test the same piece algebraically
Citation:
Indicate the type of citation format.
Cite your source(s) accordingly.
Checklist for Piecewise Function Project
(1.) Project is a piecewise function that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(2.) Title of project.
(3.) Name of Student.
(4.) Name of Instructor.
(5.) Objectives of project.
(6.) Name of company.
(7.) Direct website of company.
(8.) Information on website written as is
(9.) Arithmetic Method:
(a.) Numbers in each piece, including 0 are tested.
(b.) All work including conversions are shown.
(c.) Intermediate calculations and values (before rounding) are shown.
(d.) Final results (rounded) are written and unit (currency) is included.
(10.) Piecewise Function:
(a.) Variables are defined.
(b.) All work is shown to determine the algebraic function for each piece.
(c.) Piecewise function is developed and written well using beginning brace.
(d.) The function in each piece and the corresponding domain is written well using any of the approved formats including the use of semicolon or comma.
(e.) The functions in the piecewise function are aligned properly.
(11.) Algebraic Method (Piecewise Function Method):
(a.) The same numbers including 0 tested using the Arithmetic Method, are also tested using the Algebraic Method.
Testing each number implies that each number within the domain of each piece in the piecewise function is tested in the function for that piece.
(b.) Intermediate calculations and values (before rounding) are shown. Intermediate values must be the same as the Arithmetic Method.
(c.) Final results (rounded) are the same as the Arithmetic Method and the unit is included.
(12.) References:
(a.) The reference style is specified.
(b.) Sources are cited according to the specified reference style.
(13.) Page numbers are included in every page.
(14.) Times New Roman font, font size of 14, and line spacing of 1.5 is used for the project.
(15.) Cambria Math font, font size of 14, and the Math Equation Editor is used for all Math terms including numbers, arithmetic operations, symbols, variables,
fractions, formulas, expressions, and equations among others.
Students Projects (Math Part):
First Sample: Karan: Harrisonburg Electric Commission: Power Rates
Second Sample: Sekani: Alabama Power: Summer: Power Rates
Third Sample: Kayla: Shenandoah Valley Electric Cooperative: Power Rates
Fourth Sample: Avry: Mississippi Power: Summer: Single Phase: Power Rates
Fifth Sample: Ryan: Georgia Power: Winter: Power Rates
The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.
For Information Technology/Computer Science Students: Midterm Project
Minimum Requirements for Beginning C++, Python
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a console application.
(3.) The Input/Output feature should be used.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(5.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(6.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(8.) Use only tax in percent. Do not worry about tax in decimal.
(9.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(10.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the Blackboard course.
Minimum Requirements for Beginning VB.NET, Beginning C#, Beginning Java, Python
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a desktop application.
(3.) Set the MinimumBox property of the form to True
We want to accommodate users with laptop/desktop smaller screen sizes too.
They should have the ability to adjust your application to fit their screen sizes.
(4.) The Input/Output feature should be used.
Error Handling
(5.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(6.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(8.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(9.) Use only tax in percent. Do not worry about tax in decimal.
(10.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder, and
the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project forum of the
Blackboard course.
Minimum Requirements for Advanced C++
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information on the output.
(2.) The application should be a console application.
(3.) Input/Output feature should be used.
(4.) Class should be used.
(5.) Constructor(s) should be used.
(6.) Property/Properties should be used.
The class members: Initial Reading and the Final Reading should be private.
(7.) Method(s) should be used.
Error Handling
(8.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(9.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(10.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(11.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(12.) Use only tax in percent. Do not worry about tax in decimal.
(13.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(14.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder, and
the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for Advanced VB.NET, Advanced C#, Advanced Java
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information on the output.
(2.) The application should be a desktop application.
(3.) Set the MinimumBox property of the form to True
We want to accommodate users with laptop/desktop smaller screen sizes too.
They should have the ability to adjust your application to fit their screen sizes.
(4.) Input/Output feature should be used.
(5.) Class should be used.
(6.) Constructor(s) should be used.
(7.) Property/Properties should be used.
The class members: Initial Reading and the Final Reading should be private.
(8.) Method(s) should be used.
Error Handling
(9.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(10.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(11.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(12.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
For the requirements for Numbers (9.), (10.), and (11.)
Because the power consumption values are only nonnegative integers:
Alternatively; you may use the NumericUpDown Control
(www.docs.microsoft.com/en-us/dotnet/desktop/winforms/controls/numericupdown-control-windows-forms?view=netframeworkdesktop-4.8)
The control will ensure that the user enters only nonnegative values (zero and positive values).
(13.) Use only tax in percent. Do not worry about tax in decimal.
(14.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(15.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and
the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the
Blackboard course.
Minimum Requirements for ASP.NET
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a web application.
You may use Functional Programming or Object-oriented Programming.
You may use ASP.NET Web Forms, ASP.NET Web Pages, ASP.NET MVC, or any other ASP.NET web platform.
You may use C# or VB.NET
(3.) The Input/Output feature should be used.
Use nice HTML elements and CSS.
Functionality is the most important factor.
However, aesthetics and user-friendliness are also important factors in Web Design.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(5.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(6.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(8.) Use only tax in percent. Do not worry about tax in decimal.
(9.) Include the URL of the web application (web address) in the documentation of your Math work.
Please make sure the web application is accessible.
The draft project should contain: documentation of Math work, the web application address, and the Project
folder.
The Project folder is the folder that is created when you create a new project. It contains all folders and
files.
Include all these in OneDrive, create a shareable link, and post the link in the Midterm Project Drafts forum of
the
course on Blackboard or send to me via email.
I shall review and provide feedback.
(10.) When you are done (everything works well), please zip
the entire project: (documentation of the Math that includes the URL, and the Project folder) into one folder;
and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for JavaScript
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a web application.
(3.) The Input/Output feature should be used.
Use nice HTML elements and CSS.
Functionality is the most important factor.
However, aesthetics and user-friendliness are also important factors in Web Design.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(5.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(6.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(8.) Use only tax in percent. Do not worry about tax in decimal.
(9.) Include the URL of the web application (web address) in the documentation of your Math work.
Please make sure the web application is accessible.
Ensure that you follow the best practices: link to the external CSS file in the head element and the link to the
external
JS file right before the closing body element.
The draft project should contain: documentation of Math work that includes the URL.
Include the draft project in OneDrive, create a shareable link, and post the link in the Midterm Project Drafts
forum of the
course on Blackboard or send to me via email.
I shall review and provide feedback.
(10.) When you are done (everything works well), please submit the entire project:
(documentation of the Math that includes the URL) in the actual Midterm Project forum of the
Blackboard course.
Optional
Further Work Writing Skills: Submit a Reflection Journal.
Include your challenges, and how you overcame those challenges. Technology Skills:
Include penalties for late payments in your project.
Write a mobile application for the project.
Develop a mobile application that sends reminders to customers so they do not miss payments.
Develop a rewards program that rewards customers who make payments on time.
Water Rates by Calhoun County Water Authority; State of Alabama, USA
As of the 7th day of July, 2018; the water rates by the Calhoun County Water Authority is found here
We shall focus on the Residential Rates. These rates exclude taxes.
It is also written here for you:
First 3,000 Gallons: $17.35 Minimum Per Month
Next 2,000 Gallons: $5.24 per 1,000 Gallons
Next 5,000 Gallons: $3.96 per 1,000 Gallons
All over 10,000 Gallons: $3.10 per 1,000 Gallons
Calculate the water rates for the following consumption of water.
This application has four pieces.
Let $g$ = gallons of water (in gallons)
Let $r$ = rate (in dollars per gallons)
(1.) $2500$ gallons falls in the first piece.
rate for $2500$ gallons @ $$17.35$ = $$17.35$
(2.) $3700$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$3700 - 3000 = 700$
We need to find the rate for the remaining $700$ gallons
The first piece (for $3000$ gallons) is a constant rate. It is a flat fee.
However, the second piece is $$5.24$ per $1000$ gallons
The second piece is $$5.24$ per $1000$ (not per $700$) gallons
But, we have $700$ (not $1000$ gallons remaining). What do we do?
We have to use Proportional Reasoning Method to calculate the rate per gallons (rather than using the rate per thousand gallons)
Remember: per gallon means for $1$ gallon; and per $1000$ gallons means for a thousand gallons
Let us set up the Proportional Reasoning
Let $x$ be the cost per gallon
dollarsgallons
5.24 1000
x 1
This means that:
$$
\dfrac{5.24}{x} = \dfrac{1000}{1} \\[5ex]
1000 * x = 5.24 * 1 \\[3ex]
1000 * x = 5.24 \\[3ex]
x = \dfrac{5.24}{1000} \\[5ex]
x = 0.00524
$$
Similarly;
$$
\dfrac{3.96}{x} = \dfrac{1000}{1} \\[5ex]
1000 * x = 3.96 * 1 \\[3ex]
1000 * x = 3.96 \\[3ex]
x = \dfrac{3.96}{1000} \\[5ex]
x = 0.00396
$$
And;
$$
\dfrac{3.10}{x} = \dfrac{1000}{1} \\[5ex]
1000 * x = 3.1 * 1 \\[3ex]
1000 * x = 3.1 \\[3ex]
x = \dfrac{3.1}{1000} \\[5ex]
x = 0.0031
$$
Can we re-write the application?
First 3,000 Gallons: $17.35 Minimum Per Month
Next 2,000 Gallons: $0.00524 per gallon
Next 5,000 Gallons: $0.00396 per gallon
All over 10,000 Gallons: $0.0031 per gallon
Ask students if they understood how the rates per gallon were calculated.
So, back to completing the second question:
rate for $700$ gallons @ $0.00524 per gallon = $0.00524 * 700$ = $$3.668$
Please do not approximate intermediate calculations especially if it deals with money!
rate for $3700$ gallons @ = 17.35 + 3.668 = 21.018
Now, you can round your final answer to the nearest cent. NOTE: If your professor does not want you to round, do not round.
rate for $3700$ gallons = $$21.02$
(3.) $5000$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$5000 - 3000 = 2000$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
rate for $5000$ gallons @ = 17.35 + 10.48 = 27.83
rate for $5000$ gallons = $$27.83$
(4.) $7500$ gallons falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$7500 - 3000 = 4500$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$4500 - 2000 = 2500$
rate for $2500$ gallons @ $0.00396 per gallon = $0.00396 * 2500$ = $$9.90$
rate for $7500$ gallons @ = 17.35 + 10.48 + 9.90 = 37.73
rate for $7500$ gallons = $$37.73$
(5.) $12000$ gallons falls in the fourth piece.
Before we use the third piece, we have to go through the first, second, and third pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$12000 - 3000 = 9000$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$9000 - 2000 = 7000$
rate for $5000$ gallons @ $0.00396 per gallon = $0.00396 * 5000$ = $$19.80$
$7000 - 5000 = 2000$
rate for $2000$ gallons @ $ per gallon = $0.0031 * 2000$ = $$6.20$
rate for $12000$ gallons @ = 17.35 + 10.48 + 19.80 + 6.20 = 53.83
rate for $12000$ gallons = $$53.83$
(6.) $4692$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$4692 - 3000 = 1692$
rate for $1692$ gallons @ $0.00524 per gallon = $0.00524 * 1692$ = $$8.86608$
rate for $4692$ gallons @ = 17.35 + 8.86608 = 26.21608
rate for $4692$ gallons = $$26.22$
(7.) $6456$ gallons falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$6456 - 3000 = 3456$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$3456 - 2000 = 1456$
rate for $1456$ gallons @ $0.00396 per gallon = $0.00396 * 1456$ = $$5.76576$
rate for $6456$ gallons @ = 17.35 + 10.48 + 5.76576 = 33.59576
rate for $6456$ gallons = $$33.60$
Some students may ask if it is possible to have just one function that will find the rate for any gallon(s) of water?.
Or is it possible to find the rate for gallons of water that is contained in the second piece, without having to go through the first piece?
What if we have to calculate the water rates for "several" gallons of water?
Do we have to solve this manually all the time? That will be time consuming!
We can write it as a piecewise function and use each function for the number of gallons that correspond to that piece.
Besides, writing it as a piecewise function helps us to write a computer program that will find the rate for any gallon of water.
This application has four pieces.
Let $g$ = gallons of water (in gallons)
Let $r$ = rate (in dollars per gallons)
$r = f(g)$
This can be written as: $r(g)$
For the first piece;
$r(g)$ = $$17.35$
For the second piece;
We have to "finish" with the first piece first
Then, we can multiply the remaining gallons of water by $0.00524$
$r(g) = 17.35 + 0.00524(g - 3000)$
$r(g) = 17.35 + 0.00524g - 15.72$
$r(g) = 0.00524g + 1.63$
For the third piece;
We have to "finish" the second piece first
Then, we can multiply the remaining gallons of water by $0.00396$
What is the complete gallons for the second piece? - It is $5000$
In other words, $5000$ is the end point for the second piece. $5000$ is included.
Let us find the rate for that end point, $5000$
$r(g) = 0.00524(5000) + 1.63$
$r(g) = 26.2 + 1.63 = 27.83$
So, $$27.83$ is the most that can be charged for the second piece.
Any remaining gallons over $5000$ would be multiplied by $0.00396$
$r(g) = 27.83 + 0.00396(g - 5000)$
$r(g) = 27.83 + 0.00396g - 19.8$
$r(g) = 0.00396g + 8.03$
For the fourth piece;
We have to "finish" the third piece first
Then, we can multiply the remaining gallons of water by $0.0031$
What is the complete gallons for the second piece? - It is $10000$
In other words, $10000$ is the end point for the third piece. $10000$ is included.
Let us find the rate for that end point, $10000$
$r(g) = 0.00396(10000) + 8.03$
$r(g) = 39.6 + 8.03 = 47.63$
So, $$47.63$ is the most that can be charged for the third piece.
Any remaining gallons over $10000$ would be multiplied by $0.0031$
$r(g) = 47.63 + 0.0031(g - 10000)$
$r(g) = 47.63 + 0.0031g - 31$
$r(g) = 0.0031g + 16.63$
We can now write the piecewise function as:
$$
r(g) =
\begin{cases}
$17.35; & \quad 0 \leq g \leq 3000 \\[2ex]
0.00524g + 1.63; & \quad 3000 \lt g \leq 5000 \\[2ex]
0.00396g + 8.03; & \quad 5000 \lt g \leq 10000 \\[2ex]
0.0031g + 16.63; & \quad g \gt 10000
\end{cases}
$$
Let us recalculate all the questions using the Piecewise Function method.
(1.) $2500$ gallons falls in the first piece.
The rate for the first piece is $$17.35$
$\therefore$ the rate for $2500$ gallons of water = $$17.35$
(2.) $3700$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 3700$, $r = 0.00524(3700) + 1.63$
$r = 19.388 + 1.63$
$r = 21.018$
$\therefore$ the rate for $3700$ gallons of water = $$21.02$
(3.) $5000$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 5000$, $r = 0.00524(5000) + 1.63$
$r = 26.2 + 1.63$
$r = 27.83$
$\therefore$ the rate for $5000$ gallons of water = $$27.83$
(4.) $7500$ gallons falls in the third piece.
The rate for the third piece is $0.00396g + 8.03$
For $g = 7500$, $r = 0.00396(7500) + 8.03$
$r = 29.7 + 8.03$
$r = 37.73$
$\therefore$ the rate for $7500$ gallons of water = $$37.73$
(5.) $12000$ gallons falls in the fourth piece.
The rate for the fourth piece is $0.0031g + 16.63$
For $g = 12000$, $r = 0.0031(12000) + 16.63$
$r = 37.2 + 16.63$
$r = 53.83$
$\therefore$ the rate for $12000$ gallons of water = $$53.83$
(6.) $4692$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 4692$, $r = 0.00524(4692) + 1.63$
$r = 24.58608 + 1.63$
$r = 26.21608$
$\therefore$ the rate for $4692$ gallons of water = $$26.22$
(7.) $6456$ gallons falls in the third piece.
The rate for the third piece is $0.00396g + 8.03$
For $g = 6456$, $r = 0.00396(6456) + 8.03$
$r = 25.56576 + 8.03$
$r = 33.59576$
$\therefore$ the rate for $6456$ gallons of water = $$33.60$
Which of the two methods do you prefer?
What are your reasons?
What are the pros and cons that you see for each method?
Do you have any other method for solving Piecewise Function applications?
Calculator for the Water Rates by Calhoun County Water Authority, State of Alabama, USA
Water Usage and Monthly Bill Statement
Please test the calculator and see the output.
Complete all fields. Then, click "Bill Statement".
Water Usage
Monthly Bill Statement
Student Project: Water Bill
Water Usage and Monthly Bill Statement
Please review the example calculations and test the calculator before you begin.
For Mathematics and Information Technology/Computer Science Students: Midterm Project
(1.) You may work as a group (peer tutor to teach/correct one another).
However, this is an individual project.
In other words, you will submit your own project.
(2.) No two projects should be the same.
(a.) You may not do any of my examples even if the rates of the company have changed. Use my examples as guides to completing your own.
(b.) You may not do any of the projects of my previous students. (Student Project samples are provided below. Use them as guides to completing your own.)
(3.) Research any real-world application of piecewise function.
(a.) All information used for this project should be verifiable on the direct website of the company/organization.
(b.) Textbook examples are NOT allowed.
(c.) Find any application that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(4.) Write the complete address of the direct page of the website where you found the application.
(a.) *If the "direct" web address is too long, please shorten it by pasting the "complete web address" into
www.tinyurl.com *
*This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
(b.) For online students, please copy and paste the link as is.
(c.) Please set the link to open in a new window.
(d.) Please submit the name of the company and the direct web page that has the application, in the Projects: Company Names and Websites page in the Canvas course.
I shall review and respond.
(5.) As a BRCC/VCCS student, you have access to Microsoft Office suite of apps.
(a.) You can download and install these apps on your laptop/desktop. Please contact the IT/Tech support in your college for assistance if you do not know.
In that regard, the project is to be typed using Microsoft Word (not Google Docs).
As at the time of this writing (05/05/2023), the reason for using Microsoft Word over Google Docs is because Microsoft Word has the opening brace for piecewise functions
but Google Docs does not.
(b.) For all English terms (entire project): use Times New Roman; font size of 14; line spacing of 1.5
(c.) For all Math terms: symbols, variables, numbers, formulas, expressions, equations and fractions among others, please use the Math Equation Editor.
(i.) Set the font to Cambria Math; font size of 14; and align accordingly
(ii.) Insert a space after each each equation as applicable. Just make a good work that is organized and spacious.
(iii.) Align the functions in each piece of the piecewise function accordingly.
(d.) Include page numbers. You may include at the top of the pages or at the bottom of the pages but not both.
(6.) Mr. C (SamDom For Peace) wants you to do this real-world project very well.
Hence, he highly recommends that you submit a draft so he can give you feedback.
It is important that you submit your draft to me via email or in the Projects: Drafts page in the Canvas course at least one week
prior to the due date.
Please do not submit any draft if the due date is one week or less.
When you submit your draft, I shall review and provide feedback.
When everything is fine (after you make changes as applicable based on my feedback), please submit your work in the appropriate area in the Canvas course.
Only projects submitted in the Canvas course are graded.
Draft projects are not graded. In other words, projects submitted via email and/or in the Projects: Drafts page are not graded because they are drafts.
They are only for feedback and should be submitted at least one week prior to the due date.
If it is past one week before the project is due, please do not submit it as a draft to me. Just do it very well according to the requirements and submit it in the
appropriate area (Assignments page: Piecewise Function Project) in the Canvas course.
Submitting drafts is highly recommended. If your professor gives you an opportunity to submit a draft, please use that opportunity.
Submitting drafts is not required. It is highly recommended because I want to give you the opportunity to do your project very well and make an excellent grade in it.
(7.) Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.
(8.) Writing Skills: Write or type the main application entirely.
(9.) Mathematical Skills: Arithmetic This is the Arithmetic method.
Use random numbers to test the real-world application manually for each piece.
Show all work including intermediate calculations/values.
Write down your results.
(10.) Mathematical Skills: Algebra: This is the Algebraic Method (Piecewise Function Method).
Write a piecewise function for that application.
Test the piecewise function with the same random numbers that you used for the Arithmetic method. Please NOTE:
(a.) The intermediate results should be the same for both methods.
(b.) The final results should be the same for both methods.
If either of the results are not the same, there are issues. Please fix them.
These are the required information.
You may or may not not use a table format.
Please be creative.
Required Information
Name:
Your name
Date:
The date
Instructor:
Samuel Chukwuemeka
Project:
Water Bill: Residential Rates
Company:
Calhoun County Water Authority
(https://calhouncwa.com/rates.html)
(Name of the company and the direct Website of company that has the piecewise function)
Objectives:
(1.) Calculate the water bill of the residents of Calhoun County in the State of Alabama within each
range of specific water usage manually using Arithmetic method.
(2.) Write a piecewise function of the residential rates.
(3.) Recalculate the same water bill of the residents of Calhoun County in the State of Alabama
within each range of specific water usage algebraically using the Piecewise Function method.
Information:
Write the direct information from the direct website
Arithmetic Method:
Test each piece manually
Piecewise Function:
Write the piecewise function of the information
Piecewise Function Method:
Test the same piece algebraically
Citation:
Indicate the type of citation format.
Cite your source(s) accordingly.
Checklist for Piecewise Function Project
(1.) Project is a piecewise function that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(2.) Title of project.
(3.) Name of Student.
(4.) Name of Instructor.
(5.) Objectives of project.
(6.) Name of company.
(7.) Direct website of company.
(8.) Information on website written as is
(9.) Arithmetic Method:
(a.) Numbers in each piece, including 0 are tested.
(b.) All work including conversions are shown.
(c.) Intermediate calculations and values (before rounding) are shown.
(d.) Final results (rounded) are written and unit (currency) is included.
(10.) Piecewise Function:
(a.) Variables are defined.
(b.) All work is shown to determine the algebraic function for each piece.
(c.) Piecewise function is developed and written well using beginning brace.
(d.) The function in each piece and the corresponding domain is written well using any of the approved formats including the use of semicolon or comma.
(e.) The functions in the piecewise function are aligned properly.
(11.) Algebraic Method (Piecewise Function Method):
(a.) The same numbers including 0 tested using the Arithmetic Method, are also tested using the Algebraic Method.
Testing each number implies that each number within the domain of each piece in the piecewise function is tested in the function for that piece.
(b.) Intermediate calculations and values (before rounding) are shown. Intermediate values must be the same as the Arithmetic Method.
(c.) Final results (rounded) are the same as the Arithmetic Method and the unit is included.
(12.) References:
(a.) The reference style is specified.
(b.) Sources are cited according to the specified reference style.
(13.) Page numbers are included in every page.
(14.) Times New Roman font, font size of 14, and line spacing of 1.5 is used for the project.
(15.) Cambria Math font, font size of 14, and the Math Equation Editor is used for all Math terms including numbers, arithmetic operations, symbols, variables,
fractions, formulas, expressions, and equations among others.
Students Projects (Math Part):
First Sample: Omar: Blue Ridge Rural Water Company: Water Rates
Second Sample: Michelle: City of Akron, Ohio: Water Rates
Third Sample: John: City of Akron, Ohio: Water Rates
The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.
For Information Technology/Computer Science Students: Midterm Project
Minimum Requirements for Beginning C++, Python
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a console application.
(3.) The Input/Output feature should be used.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(5.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(6.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(8.) Use only tax in percent. Do not worry about tax in decimal.
(9.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(10.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder, and
the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for Beginning VB.NET, Beginning C#, Beginning Java, Python
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a desktop application.
(3.) Set the MinimumBox property of the form to True
We want to accommodate users with laptop/desktop smaller screen sizes too.
They should have the ability to adjust your application to fit their screen sizes.
(4.) The Input/Output feature should be used.
Error Handling
(5.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(6.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(8.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(9.) Use only tax in percent. Do not worry about tax in decimal.
(10.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder, and
the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for Advanced C++
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information on the output.
(2.) The application should be a console application.
(3.) Input/Output feature should be used.
(4.) Class should be used.
(5.) Constructor(s) should be used.
(6.) Property/Properties should be used.
The class members: Initial Reading and the Final Reading should be private.
(7.) Method(s) should be used.
Error Handling
(8.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(9.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(10.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(11.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
For the requirements for Numbers (9.), (10.), and (11.)
Because the power consumption values are only nonnegative integers:
Alternatively; you may use the NumericUpDown Control
(www.docs.microsoft.com/en-us/dotnet/desktop/winforms/controls/numericupdown-control-windows-forms?view=netframeworkdesktop-4.8)
The control will ensure that the user enters only nonnegative values (zero and positive values).
(12.) Use only tax in percent. Do not worry about tax in decimal.
(13.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(14.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the Blackboard course.
Minimum Requirements for Advanced VB.NET, Advanced C#, Advanced Java
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information on the output.
(2.) The application should be a desktop application.
(3.) Set the MinimumBox property of the form to True
We want to accommodate users with laptop/desktop smaller screen sizes too.
They should have the ability to adjust your application to fit their screen sizes.
(4.) Input/Output feature should be used.
(5.) Class should be used.
(6.) Constructor(s) should be used.
(7.) Property/Properties should be used.
The class members: Initial Reading and the Final Reading should be private.
(8.) Method(s) should be used.
Error Handling
(9.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(10.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(11.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(12.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(13.) Use only tax in percent. Do not worry about tax in decimal.
(14.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(15.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the Blackboard course.
Minimum Requirements for ASP.NET
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a web application.
You may use Functional Programming or Object-oriented Programming.
You may use ASP.NET Web Forms, ASP.NET Web Pages, ASP.NET MVC, or any other ASP.NET web platform.
You may use C# or VB.NET
(3.) The Input/Output feature should be used.
Use nice HTML elements and CSS.
Functionality is the most important factor.
However, aesthetics and user-friendliness are also important factors in Web Design.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(5.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(6.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(8.) Use only tax in percent. Do not worry about tax in decimal.
(9.) Include the URL of the web application (web address) in the documentation of your Math work.
Please make sure the web application is accessible.
The draft project should contain: documentation of Math work, the web application address, and the Project
folder.
The Project folder is the folder that is created when you create a new project. It contains all folders and
files.
Include all these in OneDrive, create a shareable link, and post the link in the Midterm Project Drafts forum of
the
course on Blackboard or send to me via email.
I shall review and provide feedback.
(10.) When you are done (everything works well), please zip
the entire project: (documentation of the Math that includes the URL, and the Project folder) into one folder;
and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for JavaScript
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a web application.
(3.) The Input/Output feature should be used.
Use nice HTML elements and CSS.
Functionality is the most important factor.
However, aesthetics and user-friendliness are also important factors in Web Design.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the Final Reading is less than the
Initial Reading.
(5.) Alert the user/Display an appropriate error message to the user, if the Initial Reading is negative.
(6.) Alert the user/Display appropriate error message to the user, if the Final Reading is negative.
(7.) Alert the user/Display appropriate error message to the user, if the Tax is negative.
(8.) Use only tax in percent. Do not worry about tax in decimal.
(9.) Include the URL of the web application (web address) in the documentation of your Math work.
Please make sure the web application is accessible.
Ensure that you follow the best practices: link to the external CSS file in the head element and the link to the
external
JS file right before the closing body element.
The draft project should contain: documentation of Math work that includes the URL.
Include the draft project in OneDrive, create a shareable link, and post the link in the Midterm Project Drafts
forum of the
course on Blackboard or send to me via email.
I shall review and provide feedback.
(10.) When you are done (everything works well), please submit the entire project:
(documentation of the Math that includes the URL) in the actual Midterm Project forum of the
Blackboard course.
Optional
Further Work Writing Skills: Submit a Reflection Journal.
Include your challenges, and how you overcame those challenges. Technology Skills:
Include penalties for late payments in your project.
Write a mobile application for the project.
Develop a mobile application that sends reminders to customers so they do not miss payments.
Develop a rewards program that rewards customers who make payments on time.
Federal Income Tax Rates: 2018
Another important real-world application of Piecewise Functions is in the filing of federal income taxes.
We shall illustrate this application by filing the "Single" option for the $2018$ federal tax return.
We shall design a calculator for these taxes.
We shall verify our calculations with the calculators of some tax companies.
For the simple calculations (only taxes), we shall verify it using the:
calculator from TaxAct
Let us define some important terms used in Income Taxes.
The gross income is the total yearly income. This is the total income earned in a year.
It is the total pre-tax earnings for the year.
Let the gross income = $GI$
Sometimes, there are certain portions of the gross income that is not taxed.
In other words, there are certain untaxed portions of the gross income.
Those untaxed portions of the gross income are known as deductions or adjustments.
Let the adjustments = $A$
The difference between the gross income and the adjustments is known as the adjusted gross income
Let the adjusted gross income = $AGI$
The adjusted gross income is the income after allowable tax deductions.
The information used for the 2018 tax rates, standard deductions, and exemptions is found in these websites:
(1.) Bankrate
(2.) Forbes
Ask students to compare and contrast the two tables.
The information from the first website (Bankrate) is summarized here.
We shall work with this one first.
Tax Rate
Single
Head of household
Married Filing Jointly
Or Qualifying Widow
Married Filing Separately
$10\%$
Up to $\$9,525$
Up to $\$13,600$
Up to $\$19,050$
Up to $\$9,525$
$12\%$
$\$9,526 \:\:to\:\: \$38,700$
$\$13,601 \:\:to\:\: \$51,800$
$\$19,051 \:\:to\:\: \$77,400$
$\$9,526 \:\:to\:\: \$38,700$
$22\%$
$\$38,701 \:\:to\:\: \$82,500$
$\$51,801 \:\:to\:\: \$82,500$
$\$77,401 \:\:to\:\: \$165,000$
$\$38,701 \:\:to\:\: \$82,500$
$24\%$
$\$82,501 \:\:to\:\: \$157,500$
$\$82,501 \:\:to\:\: \$157,500$
$\$165,001 \:\:to\:\: \$315,000$
$\$82,501 \:\:to\:\: \$157,000$
$32\%$
$\$157,501 \:\:to\:\: \$200,000$
$\$157,501 \:\:to\:\: \$200,000$
$\$315,001 \:\:to\:\: \$400,000$
$\$157,001 \:\:to\:\: \$200,000$
$35\%$
$\$200,001 \:\:to\:\: \$500,000$
$\$200,001 \:\:to\:\: \$500,000$
$\$400,001 \:\:to\:\: \$600,000$
$\$200,001 \:\:to\:\: \$300,000$
$37\%$
$\$500,001 \:\:or\:\: more$
$\$500,001 \:\:or\:\: more$
$\$600,001 \:\:or\:\: more$
$\$300,001 \:\:or\:\: more$
Let us calculate the taxes for the filing "Single" option.
We shall test each piece.
Calculate the taxes for single individuals whose taxable incomes are:
(1.) $\$7,000.00$ falls in the first piece.
tax for $\$7000.00$ income @ $10\%$ per $\$$ earned = $7000 * 0.1 = 700$
tax for taxable income of $\$7,000.00$ = $\$700.00$
(2.) $\$12,575.00$ falls in the second piece.
Before we use the second piece, we have to go through the first piece first. First Piece for $\$9525.00$ $9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We have to move on to the second piece.
$12575.00 - 9525.00 = 3050.00$
The rest of the $\$3050.00$ is taxed at $12\%$ Second Piece for $\$3050.00$
tax for $\$3050.00$ income @ $12\%$ per $\$$ earned = $3050.00 * 0.12 = 366.00$
$952.50 + 366.00 = 1318.50$
tax for taxable income of $\$12,575.00$ = $\$1,318.50$
(3.) $\$43,750.00$ falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces. First Piece for $\$9525.00$ $9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We have to move on to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Student: So, do we need to subtract $9525.00$ from $43750.28$
Teacher: No, we do not.
Here is the thing.
We know already that $43750.28$ falls in the third tax bracket.
Student: That is correct.
Teacher: $9,525$ of that money is taxed at $10\%$
Student: Okay...
Teacher: Any money above $9525$ but not exceeding $38700$ is taxed at $12\%$
Not all the money (43750.00 - 9525 = 34225.00) is taxed at $12\%$
Let us look at it this way
For $43750.28$
$[0, 9525]$ is taxed at $10\%$
$(9525, 38700]$ is taxed at $12\%$
$(38700, 43750.28]$ is taxed at $22\%$
Do you realize the mistake that would have been made if $34225.00$ was taxed at $12\%$
Student: Yes, the U,S government would lose money.
Teacher: That is right...and we do not ...
Student: want them to lose money.
Teacher: Correct. That money could be used to help the poor.
Student: I hope they help poor people. There is a high rate of hunger and poverty in the world.
Teacher: I hope and pray so too.
So, as you can see:
The range of $9525 - 0 = 9525$ is taxed at $10\%$
Student: The range of $38700 - 9525 = 29175$ is taxed at $12\%$
Teacher: Correct! Go ahead...
Student: and the range of $43750.00 - 38700 = 5050.00$ is taxed at $22\%$
Teacher: Perfecto!
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We have to move on to the third piece.
$43750.00 - 38700.00 = 5050.00$
The rest of the $\$5050.00$ is taxed at $22\%$ Third Piece for $\$14575.00$
tax for $\$5050.00$ income @ $22\%$ per $\$$ earned = $5050.00 * 0.22 = 1111.00$
$952.50 + 3501.00 + 1111.00 = 5564.50$
tax for taxable income of $\$43,750.00$ = $\$5,564.50$
(4.) $\$120, 327.00$ falls in the fourth piece.
Before we use the fourth piece, we have to go through the first, second, and third pieces. First Piece for $\$9525.00$ $9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
$120327.00 - 82500.00 = 37827.00$
The rest of the $\$37827.00$ is taxed at $24\%$ Fourth Piece for $\$37827.00$
tax for $\$37827.00$ income @ $24\%$ per $\$$ earned = $37827.00 * 0.24 = 9078.48$
$952.50 + 3501.00 + 9636.00 + 9078.48 = 23167.98$
tax for taxable income of $\$120,327.00$ = $\$23,167.98$
(5.) $\$165, 428.00$ falls in the fifth piece.
Before we use the fifth piece, we have to go through the first, second, third, and fourth pieces. First Piece for $\$9525.00$ $9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$ Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
$165428.00 - 157500.00 = 7928.00$
The rest of the $\$7928.00$ is taxed at $32\%$ Fifth Piece for $\$7928.00$
tax for $\$7928.00$ income @ $32\%$ per $\$$ earned = $7928.00 * 0.32 = 2536.96$
$952.50 + 3501.00 + 9636.00 + 18000 + 2536.96 = 34626.46$
tax for taxable income of $\$165,428.00$ = $\$34,626.46$
(6.) $\$234,543.00$ falls in the sixth piece.
Before we use the sixth piece, we have to go through the first, second, third, fourth, and fifth pieces. First Piece for $\$9525.00$ $9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$ Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$ Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
$234543.00 - 200000.00 = 34543.00$
The rest of the $\$34543.00$ is taxed at $35\%$ Sixth Piece for $\$34543.00$
tax for $\$34543.00$ income @ $35\%$ per $\$$ earned = $34543.00 * 0.35 = 12090.05$
$952.50 + 3501.00 + 9636.00 + 18000 + 13600 + 12090.05 = 57779.55$
tax for taxable income of $\$234,543.00$ = $\$57,779.55$
(7.) $\$700,712.00$ falls in the seventh piece.
Before we use the seventh piece (last piece), we have to go through the previous six pieces in order. First Piece for $\$9525.00$ $9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$ Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$ Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
We need to calculate the range of taxable income that should be taxed at the sixth tax rate.
$500000.00 - 200000.00 = 300000.00$
$\$300000.00$ is taxed at $35\%$ Sixth Piece for $\$300000.00$
tax for $\$300000.00$ income @ $35\%$ per $\$$ earned = $300000.00 * 0.35 = 105000.00$
We are done with the maximum taxable income that can be taxed at the sixth piece of $35\%$
We move to the seventh piece.
$700712.00 - 500000.00 = 200712.00$
The rest of the $\$200712.00$ is taxed at $37\%$ Seventh Piece for $\$200712.00$
tax for $\$200712.00$ income @ $37\%$ per $\$$ earned = $200712.00 * 0.37 = 74263.44$
$952.50 + 3501.00 + 9636.00 + 18000 + 13600 + 105000 + 74263.44 = 224952.94$
tax for taxable income of $\$700,712.00$ = $\$224,952.94$
Some students may ask if it is possible to have just one function that will calculate the tax for any taxable income.
Or is it possible to find the tax for a taxable income that falls in the second piece, without having to go through the first piece?
Those are really interesting questions!
That is one of the reasons for studying piecewise functions ☺☺☺
Please specify the importance of not rounding intermediate calculations.
Please specify the importance of rounding only the final answer to two decimal place (because it is dollars and cents).
What if we have to calculate the taxes for "several" taxable incomes?
Do we have to solve this manually all the time? That will be time consuming!
We can write it as a piecewise function and use each function to calculate the taxes that corresponds to each taxable income.
Besides, writing it as a piecewise function helps us to write a computer program that will calculate the taxes for any amount of taxable income.
Define the variables.
Let $p$ = taxable income (in $\$$)
Let $t$ = taxes (in $\$$)
The taxes paid is a function of the income earned.
Taxes is the dependent variable.
Taxable income is the independent variable.
$t = f(p)$
We can also write is as $t(p)$
This application has seven pieces.
For the first piece;
tax for $\$p$ income @ $10\%$ per $\$$ earned = $p * 0.1 = 0.1p$
$t(p) = 0.1p$
For the second piece;
We have to "finish" with the first piece first First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece. Second Piece for $\$p - 9525.00$
The remaining income, $(p - 9525)$ is taxed at $12\%$
So, we have to multiply the remaining income by $0.12$
$t(p) = 952.5 + 0.12(p - 9525)$
$t(p) = 952.5 + 0.12p - 1143$
$t(p) = 0.12p - 190.5$
For the third piece;
We have to "finish" with the first and second pieces First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate. Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We have to move on to the third piece. Third Piece for $\$p - 38700.00$
The remaining income, $(p - 38700)$ is taxed at $22\%$
So, we have to multiply the remaining income by $0.22$
$t(p) = 952.5 + 3501 + 0.22(p - 38700)$
$t(p) = 952.5 + 3501 + 0.22p - 8514$
$t(p) = 0.22p - 4060.5$
For the fourth piece;
We have to "finish" with the first, second, and third pieces First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate. Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the second piece of $22\%$
We move to the fourth piece. Fourth Piece for $\$p - 82500.00$
The remaining income, $(p - 82500)$ is taxed at $24\%$
So, we have to multiply the remaining income by $0.24$
$t(p) = 952.5 + 3501 + + 9636 + 0.24(p - 82500)$
$t(p) = 952.5 + 3501 + 9636 + 0.24p - 19800$
$t(p) = 0.24p - 5710.5$
For the fifth piece;
We have to "finish" with the first, second, third, and fourth pieces First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate. Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$ Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece. Fifth Piece for $\$p - 157500.00$
The remaining income, $(p - 157500)$ is taxed at $32\%$
So, we have to multiply the remaining income by $0.32$
$t(p) = 952.5 + 3501 + + 9636 + 18000 + 0.32(p - 157500)$
$t(p) = 952.5 + 3501 + 9636 + 18000 + 0.32p - 50400$
$t(p) = 0.32p - 18310.5$
For the sixth piece;
We have to "finish" with the first, second, third, fourth, and fifth pieces First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate. Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$ Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$ Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece. Sixth Piece for $\$p - 200000.00$
The remaining income, $(p - 200000)$ is taxed at $35\%$
So, we have to multiply the remaining income by $0.35$
$t(p) = 952.5 + 3501 + + 9636 + 18000 + 13600 + 0.35(p - 200000)$
$t(p) = 952.5 + 3501 + 9636 + 18000 + 13600 + 0.35p - 70000$
$t(p) = 0.35p - 24310.5$
For the seventh piece;
We have to "finish" with the previous six pieces in order First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate. Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$ Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$ Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$ Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$ Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
We need to calculate the range of taxable income that should be taxed at the sixth tax rate.
$500000.00 - 200000.00 = 300000.00$
$\$300000.00$ is taxed at $35\%$ Sixth Piece for $\$300000.00$
tax for $\$300000.00$ income @ $35\%$ per $\$$ earned = $300000.00 * 0.35 = 105000.00$
We are done with the maximum taxable income that can be taxed at the sixth piece of $35\%$
We move to the seventh piece. Seventh Piece for $\$p - 500000.00$
The remaining income, $(p - 500000)$ is taxed at $37\%$
So, we have to multiply the remaining income by $0.37$
$t(p) = 952.5 + 3501 + + 9636 + 18000 + 13600 + 105000 + 0.37(p - 500000)$
$t(p) = 952.5 + 3501 + 9636 + 18000 + 13600 + 105000 + 0.37p - 185000$
$t(p) = 0.37p - 34310.5$
We can now write the piecewise function as:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 9525 \\[3ex]
0.12p - 190.5; & \quad 9525 \lt p \leq 38700 \\[3ex]
0.22p - 4060.5; & \quad 38700 \lt p \leq 82500 \\[3ex]
0.24p - 5710.5; & \quad 82500 \lt p \leq 157500 \\[3ex]
0.32p - 18310.5; & \quad 157500 \lt p \leq 200000 \\[3ex]
0.35p - 24310.5; & \quad 200000 \lt p \leq 500000 \\[3ex]
0.37p - 34310.5; & \quad p \gt 500000
\end{cases}
$$
Let us recalculate all the questions using the Piecewise Function method.
(1.) $\$7,000.00$ falls in the first piece.
$
t(p) = 0.1p \\[3ex]
t(7000) = 0.1(7000) \\[3ex]
= 700 \\[3ex]
$
tax for taxable income of $\$7,000.00$ = $\$700.00$
(2.) $\$12,575.00$ falls in the second piece.
$
t(p) = 0.12p - 190.5 \\[3ex]
t(12575) = 0.12(12575) - 190.5 \\[3ex]
= 1509 - 190.5 \\[3ex]
= 1318.5 \\[3ex]
$
tax for taxable income of $\$12,575.00$ = $\$1,318.50$
(3.) $\$43,750.00$ falls in the third piece.
$
t(p) = 0.22p - 4060.5 \\[3ex]
t(43750) = 0.22(43750) - 4060.5 \\[3ex]
= 9625 - 4060.5 \\[3ex]
= 5564.5 \\[3ex]
$
tax for taxable income of $\$43,750.00$ = $\$5,564.50$
(4.) $\$120,327.00$ falls in the fourth piece.
$
t(p) = 0.24p - 5710.5 \\[3ex]
t(120327) = 0.24(120327) - 5710.5 \\[3ex]
= 28878.48 - 5710.5 \\[3ex]
= 23167.98 \\[3ex]
$
tax for taxable income of $\$120,327.00$ = $\$23,167.98$
(5.) $\$165,428.00$ falls in the fifth piece.
$
t(p) = 0.32p - 18310.5 \\[3ex]
t(165428) = 0.32(165428) - 18310.5 \\[3ex]
= 52936.96 - 18310.5 \\[3ex]
= 34626.46 \\[3ex]
$
tax for taxable income of $\$165,428.00$ = $\$34,626.46$
(6.) $\$234,543.00$ falls in the sixth piece.
$
t(p) = 0.35p - 24310.5 \\[3ex]
t(234543) = 0.35(234543) - 24310.5 \\[3ex]
= 82090.05 - 24310.5 \\[3ex]
= 57779.55 \\[3ex]
$
tax for taxable income of $\$234,543.00$ = $\$57,779.55$
(7.) $\$700,712.00$ falls in the seventh piece.
$
t(p) = 0.37p - 34310.5 \\[3ex]
t(700712) = 0.37(700712) - 34310.5 \\[3ex]
= 259263.44 - 34310.5 \\[3ex]
= 224952.94 \\[3ex]
$
tax for taxable income of $\$700,712.00$ = $\$224,952.94$
Ask students their preferred method - solving it manually or solving it by piecewise function. They should give reasons for their answers. Ask students if any of them can come up with another method besides the two methods already discussed.
Before we develop the calculator for the 2018 Federal Income Tax, we need to write the Piecewise Functions for
the other filing options.
We can ask the user to select any of the filing options to calculate the tax.
If you follow the example for the Single filing option, you should get the same results that I get.
Please verify your results with my own. If you find any discrepancy even by one cent, please let me know.
The Piecewise Function for the Head of Household is:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 13600 \\[3ex]
0.12p - 272; & \quad 13600 \lt p \leq 51800 \\[3ex]
0.22p - 5452; & \quad 51800 \lt p \leq 82500 \\[3ex]
0.24p - 7102; & \quad 82500 \lt p \leq 157500 \\[3ex]
0.32p - 19702; & \quad 157500 \lt p \leq 200000 \\[3ex]
0.35p - 25702; & \quad 200000 \lt p \leq 500000 \\[3ex]
0.37p - 35702; & \quad p \gt 500000
\end{cases}
$$
The Piecewise Function for the Married Filing Jointly or Qualifying Widow is:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 19050 \\[3ex]
0.12p - 381; & \quad 19050 \lt p \leq 77400 \\[3ex]
0.22p - 8121; & \quad 77400 \lt p \leq 165000 \\[3ex]
0.24p - 11421; & \quad 165000 \lt p \leq 315000 \\[3ex]
0.32p - 36621; & \quad 315000 \lt p \leq 400000 \\[3ex]
0.35p - 48621; & \quad 400000 \lt p \leq 600000 \\[3ex]
0.37p - 60621; & \quad p \gt 600000
\end{cases}
$$
The Piecewise Function for the Married Filing Separately is:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 9525 \\[3ex]
0.12p - 190.5; & \quad 9525 \lt p \leq 38700 \\[3ex]
0.22p - 4060.5; & \quad 38700 \lt p \leq 82500 \\[3ex]
0.24p - 5710.5; & \quad 82500 \lt p \leq 157000 \\[3ex]
0.32p - 18270.5; & \quad 157000 \lt p \leq 200000 \\[3ex]
0.35p - 24270.5; & \quad 200000 \lt p \leq 300000 \\[3ex]
0.37p - 30270.5; & \quad p \gt 300000
\end{cases}
$$
Which of the two methods do you prefer?
What are your reasons?
What are the pros and cons that you see for each method?
Do you have any other method for solving Piecewise Function applications?
$2018$ Federal Income Tax Calculator
Yearly Tax Statement for $2018$
Please test the calculator and see the output.
Complete all fields. Then, click "Tax Statement".
Taxable Income
$2018$ Tax Statement
Student Project: Yearly Federal Income Tax
Income Taxes
Please review the example calculations and test the calculator before you begin.
For Mathematics and Information Technology/Computer Science Students: Midterm Project
(1.) You may work as a group (peer tutor to teach/correct one another).
However, this is an individual project.
In other words, you will submit your own project.
(2.) No two projects should be the same.
(a.) You may not do any of my examples even if the rates of the company have changed. Use my examples as guides to completing your own.
(b.) You may not do any of the projects of my previous students. (Student Project samples are provided below. Use them as guides to completing your own.)
(3.) Research any real-world application of piecewise function.
(a.) All information used for this project should be verifiable on the direct website of the company/organization.
(b.) Textbook examples are NOT allowed.
(c.) Find any application that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(d.) You may only do the income taxes of any of the filing options for the current year or future year(s).
If you do the income taxes of the filing options for the current year, it should be before the first IRS deadline for filing taxes.
If the IRS deadline is past, then you have to do for future years.
You may NOT do the income taxes of any of the filing options of any previous year(s).
(e.) Please write the direct website of the company/organization where you got the tax information.
(f.) Please note that it is your responsibity to ensure that the tax information is correct.
In that regard, please compare your information with the information across several reputable websites.
(g.) Please work only on Federal Taxes. Do not work on State, City, or County taxes.
(h.) For any filing option you do, please cover all the pieces.
You may file:
(A.) Single OR
(B.) Head of Household OR
(C.) Married Filing Jointly or Qualifying Widow OR
(D.) Married Filing Separately
(4.) Write the complete address of the direct page of the website where you found the application.
(a.) *If the "direct" web address is too long, please shorten it by pasting the "complete web address" into
www.tinyurl.com *
*This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
(b.) For online students, please copy and paste the link as is.
(c.) Please set the link to open in a new window.
(d.) Please submit the name of the company and the direct web page that has the application, in the Projects: Company Names and Websites page in the Canvas course.
I shall review and respond.
(5.) As a BRCC/VCCS student, you have access to Microsoft Office suite of apps.
(a.) You can download and install these apps on your laptop/desktop. Please contact the IT/Tech support in your college for assistance if you do not know.
In that regard, the project is to be typed using Microsoft Word (not Google Docs).
As at the time of this writing (05/05/2023), the reason for using Microsoft Word over Google Docs is because Microsoft Word has the opening brace for piecewise functions
but Google Docs does not.
(b.) For all English terms (entire project): use Times New Roman; font size of 14; line spacing of 1.5
(c.) For all Math terms: symbols, variables, numbers, formulas, expressions, equations and fractions among others, please use the Math Equation Editor.
(i.) Set the font to Cambria Math; font size of 14; and align accordingly
(ii.) Insert a space after each each equation as applicable. Just make a good work that is organized and spacious.
(iii.) Align the functions in each piece of the piecewise function accordingly.
(d.) Include page numbers. You may include at the top of the pages or at the bottom of the pages but not both.
(6.) Mr. C (SamDom For Peace) wants you to do this real-world project very well.
Hence, he highly recommends that you submit a draft so he can give you feedback.
It is important that you submit your draft to me via email or in the Projects: Drafts page in the Canvas course at least one week
prior to the due date.
Please do not submit any draft if the due date is one week or less.
When you submit your draft, I shall review and provide feedback.
When everything is fine (after you make changes as applicable based on my feedback), please submit your work in the appropriate area in the Canvas course.
Only projects submitted in the Canvas course are graded.
Draft projects are not graded. In other words, projects submitted via email and/or in the Projects: Drafts page are not graded because they are drafts.
They are only for feedback and should be submitted at least one week prior to the due date.
If it is past one week before the project is due, please do not submit it as a draft to me. Just do it very well according to the requirements and submit it in the
appropriate area (Assignments page: Piecewise Function Project) in the Canvas course.
Submitting drafts is highly recommended. If your professor gives you an opportunity to submit a draft, please use that opportunity.
Submitting drafts is not required. It is highly recommended because I want to give you the opportunity to do your project very well and make an excellent grade in it.
(7.) Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.
(8.) Writing Skills: Write or type the main application entirely.
(9.) Mathematical Skills: Arithmetic This is the Arithmetic method.
Use random numbers to test the real-world application manually for each piece.
Show all work including intermediate calculations/values.
Write down your results.
(10.) Mathematical Skills: Algebra: This is the Algebraic Method (Piecewise Function Method).
Write a piecewise function for that application.
Test the piecewise function with the same random numbers that you used for the Arithmetic method. Please NOTE:
(a.) The intermediate results should be the same for both methods.
(b.) The final results should be the same for both methods.
If either of the results are not the same, there are issues. Please fix them.
These are the required information.
You may or may not not use a table format.
Please be creative.
Required Information
Name:
Your name
Date:
The date
Instructor:
Samuel Chukwuemeka
Project:
Federal Tax Rates, Brackets, and Standard Deductions
Company:
Bankrate.com or IRS.com or or any other website you used
(https://www.bankrate.com/taxes/tax-brackets/)
(Name of the company and the direct Website of company that has the piecewise function)
Objectives:
(1.) Calculate the federal income taxes of any United States resident/citizen within each taxable income
bracket manually using Arithmetic method for year 2018.
(2.) Write a piecewise function of the federal income tax rates for year 2018.
(3.) Recalculate the same income tax of the United States resident/citizen for those income
brackets algebraically using the Piecewise Function method for year 2018.
Information:
Write the direct information from the direct website
Arithmetic Method:
Test each piece manually
Piecewise Function:
Write the piecewise function of the information
Piecewise Function Method:
Test the same piece algebraically
Citation:
Indicate the type of citation format.
Cite your source(s) accordingly.
Checklist for Piecewise Function Project
(1.) Project is a piecewise function that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(2.) Title of project.
(3.) Name of Student.
(4.) Name of Instructor.
(5.) Objectives of project.
(6.) Name of company.
(7.) Direct website of company.
(8.) Information on website written as is
(9.) Arithmetic Method:
(a.) Numbers in each piece, including 0 are tested.
(b.) All work including conversions are shown.
(c.) Intermediate calculations and values (before rounding) are shown.
(d.) Final results (rounded) are written and unit (currency) is included.
(10.) Piecewise Function:
(a.) Variables are defined.
(b.) All work is shown to determine the algebraic function for each piece.
(c.) Piecewise function is developed and written well using beginning brace.
(d.) The function in each piece and the corresponding domain is written well using any of the approved formats including the use of semicolon or comma.
(e.) The functions in the piecewise function are aligned properly.
(11.) Algebraic Method (Piecewise Function Method):
(a.) The same numbers including 0 tested using the Arithmetic Method, are also tested using the Algebraic Method.
Testing each number implies that each number within the domain of each piece in the piecewise function is tested in the function for that piece.
(b.) Intermediate calculations and values (before rounding) are shown. Intermediate values must be the same as the Arithmetic Method.
(c.) Final results (rounded) are the same as the Arithmetic Method and the unit is included.
(12.) References:
(a.) The reference style is specified.
(b.) Sources are cited according to the specified reference style.
(13.) Page numbers are included in every page.
(14.) Times New Roman font, font size of 14, and line spacing of 1.5 is used for the project.
(15.) Cambria Math font, font size of 14, and the Math Equation Editor is used for all Math terms including numbers, arithmetic operations, symbols, variables,
fractions, formulas, expressions, and equations among others.
Students Projects (Math Part): Please Note: You should do at least one filing option (one or more).
Angel is a diligent student who decided to do all filing options.
I am impressed with his work: both the Math Part (document) and the Programming Part (the website app: link is in the document)
First Sample: Angel: Federal Income Taxes: 2021: Filing in 2022
The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.
For Information Technology/Computer Science Students: Midterm Project
Minimum Requirements for Beginning C++, Python
These are the minimum requirements.
Please be creative.
I understand I did all the four filing options for the $2018$ Federal Income Tax.
However, you are only required to do a minimum of one option.
You may choose to do all options if you prefer.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a console application.
(3.) The Input/Output feature should be used.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the taxable income is negative.
(5.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(6.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the Blackboard course.
Minimum Requirements for Beginning VB.NET, Beginning C#, Beginning Java, Python
These are the minimum requirements.
Please be creative.
I understand I did all the four filing options for the $2018$ Federal Income Tax.
However, you are only required to do a minimum of one option.
You may choose to do all options if you prefer.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a desktop application.
(3.) Set the MinimumBox property of the form to True
We want to accommodate users with laptop/desktop smaller screen sizes too.
They should have the ability to adjust your application to fit their screen sizes.
(4.) The Input/Output feature should be used.
Error Handling
(5.) Alert the user/Display an appropriate error message to the user, if the taxable income is negative.
(6.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(7.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the Blackboard course.
Minimum Requirements for Advanced C++
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information on the output.
(2.) The application should be a console application.
(3.) Input/Output feature should be used.
(4.) Class should be used.
(5.) Constructor(s) should be used.
(6.) Property/Properties should be used.
The class member: Taxable Income should be private.
(7.) Method(s) should be used.
Error Handling
(8.) Alert the user/Display an appropriate error message to the user, if the taxable income is negative.
(9.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can
help you.
(10.) When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project
forum of the Blackboard course.
Minimum Requirements for Advanced VB.NET, Advanced C#, Advanced Java
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information on the output.
(2.) The application should be a desktop application.
(3.) Set the MinimumBox property of the form to True
We want to accommodate users with laptop/desktop smaller screen sizes too.
They should have the ability to adjust your application to fit their screen sizes.
(4.) Input/Output feature should be used.
(5.) Class should be used.
(6.) Constructor(s) should be used.
(7.) Property/Properties should be used.
The class member: Taxable Income should be private.
(8.) Method(s) should be used.
Error Handling
(9.) Alert the user/Display an appropriate error message to the user, if the taxable income is negative.
(10.) Make sure your executable file runs by itself outside the project folder
If it does run, include all these: (documentation of all your Math work and the .exe file)
in OneDrive, create a
shareable link, and post the link in the Midterm Project Drafts forum of the course on Blackboard or send to me
via email.
You may also upload them directly if you prefer.
I shall review and provide feedback.
If your submission works well, I shall ask you to submit the Project folder.
The Project folder is the folder that is created when you create a
new project.
When you submit it, I shall review and provide feedback.
If it does not run, then you need to fix it.
If you cannot fix it and you have reviewed the resources, please attend the Office Hours/Live Sessions so I can help you.
When you are done (everything works well), please zip
the entire project: (documentation of the Math, executable file that runs by itself outside the Project folder,
and the Project folder) into one folder; and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for ASP.NET
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a web application.
You may use Functional Programming or Object-oriented Programming.
You may use ASP.NET Web Forms, ASP.NET Web Pages, ASP.NET MVC, or any other ASP.NET web platform.
You may use C# or VB.NET
(3.) The Input/Output feature should be used.
Use nice HTML elements and CSS.
Functionality is the most important factor.
However, aesthetics and user-friendliness are also important factors in Web Design.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the taxable income is negative.
(5.) Include the URL of the web application (web address) in the documentation of your Math work.
Please make sure the web application is accessible.
The draft project should contain: documentation of Math work, the web application address, and the Project
folder.
The Project folder is the folder that is created when you create a new project. It contains all folders and
files.
Include all these in OneDrive, create a shareable link, and post the link in the Midterm Project Drafts forum of
the
course on Blackboard or send to me via email.
I shall review and provide feedback.
(6.) When you are done (everything works well), please zip
the entire project: (documentation of the Math that includes the URL, and the Project folder) into one folder;
and submit the zipped folder (.zip only) in the actual Midterm Project forum of the Blackboard course.
Minimum Requirements for JavaScript
These are the minimum requirements. Please be creative.
(1.) Please use appropriate naming for your project application.
The application should display at least the same output of my calculator.
That implies that you should test my calculator to see the output.
Your application may include more information.
(2.) The application should be a web application.
(3.) The Input/Output feature should be used.
Use nice HTML elements and CSS.
Functionality is the most important factor.
However, aesthetics and user-friendliness are also important factors in Web Design.
Error Handling
(4.) Alert the user/Display an appropriate error message to the user, if the taxable income is negative.
(5.) Include the URL of the web application (web address) in the documentation of your Math work.
Please make sure the web application is accessible.
Ensure that you follow the best practices: link to the external CSS file in the head element and the link to the
external
JS file right before the closing body element.
The draft project should contain: documentation of Math work that includes the URL.
Include the draft project in OneDrive, create a shareable link, and post the link in the Midterm Project Drafts
forum of the
course on Blackboard or send to me via email.
I shall review and provide feedback.
(6.) When you are done (everything works well), please submit the entire project:
(documentation of the Math that includes the URL) in the actual Midterm Project forum of the
Blackboard course.
Optional
Further Work Writing Skills: Submit a Reflection Journal.
Include your challenges, and how you overcame those challenges. Technology Skills:
Include penalties for late payments in your project.
Write a mobile application for the project.
Develop a mobile application that sends reminders to residents/citizens to file their taxes on time.
Graphing Piecewise Functions
To graph a piecewise function:
(1.) Write the Table of Values for each piece
(2.) Make sure the end points for the domain of each piece are indicated in each Table of Value
(3.) Specify whether each end point is defined or undefined.
(4.) Plot each Table of Values on a graph.
Any end point that is defined should be labeled as a closed circle.
This means that the end point is included in the domain for that piece.
Any end point that is not defined should be labeled as an open circle.
This means that the end point is not included in the domain for that piece.
Let us do an example.
Example 1: ACT Which of the following is the graph of the function f(x) defined below?
$
f(x) =
\begin{cases}
x^2 - 2 \enspace \text{for} \enspace x \le 1 \\[3ex]
x - 7 \enspace \text{for} \enspace 1 \lt x \lt 5 \\[3ex]
4 - x \enspace \text{for} \enspace x \ge 5
\end{cases} \\[7ex]
$
Let us draw the Table of Values for each function/piece
Then, we shall answer the question by going through each option and eliminating the incorrect options until we find
the correct one
First Piece: $x^2 - 2; \enspace x \le 1$
$x$
$-2$
$-1$
$0$
$1$
$y$
$2$
$-1$
$-2$
$-1$
defined
Second Piece: $x - 7; \enspace 1 \lt x \lt 5$
$x$
$1$
$2$
$3$
$4$
$5$
$y$
$-6$
$-5$
$-4$
$-3$
$-2$
undefined
undefined
Third Piece: $4 - x; \enspace x \ge 5$
$x$
$5$
$6$
$7$
$8$
$y$
$-1$
$1$
$2$
$3$
defined
Option F.
First Piece: when $x = 1$, $y = -3$
This is incorrect...as seen from the Table.
Option J.
First Piece: when $x = 1$, $y = 0$
This is incorrect...as seen from the Table.
Option G.
First Piece: when $x = -2$, $y = 0$
This is incorrect...as seen from the Table.
Option K.
This is the correct option because all the Tables of Values corresponds to the graph
This includes where the values are defined (closed circles) and undefined (open circles)
You may STOP here (or if you have time, check the last option)
Option H.
First Piece: when $x = 0$, $y = 0$
This is incorrect...as seen from the Table.
We can sketch the graph:
(1.) Manually on a graph paper
(2.) Using a graphing calculator/technology.
Let us sketch the graph using the Texas Instrument (TI) Graphing Calculator
You may use any of these TI calculators:
TI-83 Plus
TI-84 Plus series
TI-Nspire CX series
TI-89 Titanium
TI-73 Explorer
The first thing we need to do is to reset the Random Access Memory (RAM).
This will clear everything that was initially stored by a previous user.
Also, after each problem, it is recommended that you reset the calculator. NOTE: Please go over each step. Do not skip.
Reset the Calculator
(1.)
(2.)
(3.)
(4.)
Graph Example 1
Step 1:
Step 2:
Enter the function and the domain of each piece
There are at least two approaches that we can use here:
(A.) First Approach: As a quotient of the function and the domain for each piece
This means $\dfrac{function}{quotient}$ for each piece
This is the $\dfrac{n}{d}$ = $\dfrac{numerator}{denominator}$
Let us begin with the first piece
(2a.)
(2b.)
NOTE:
(1.) If you have a TI-84 Plus calculator, the $\dfrac{n}{d}$ is found by: MATH ⟩⟩ NUM ⟩⟩ D:n/d
(2.) If you have a TI-83 calculator, you can use the division button: ÷
(2c.)
(2d.)
(2e.)
(2f.)
Step 3:
Enter the function and the domain of the second piece
(3a.)
The domain of the second piece is a compound inequality
For compound ineuqlities, we use the logical connectives: and and or depending on the compound inequality
For the domain of this piecewise function, we shall use the logical connective: and to join the two simple inequalities
(3b.)
(3c.)
Step 4:
Enter the function and the domain of the third piece
Step 5:
Graph the piecewise function
We can do this another way.
(B.) Second Approach: As a sum of the products of the function and the domain of all the pieces
We write this sum as a single function (in $Y_1$) rather than in three different functions as we did in the first approach. So, let us reset the calculator.
The graph of this approach is the same as the graph of the first approach.
Step 2:
Enter the function and the domain of each piece
Enclose the function in parenthesis
Enclose the domain in parenthesis
Write as a sum of products
This is the equation that was written: $Y_1 = (X^2 - 2) * (X \le 1) + (X - 7) * (1 \lt X \;\;and\;\; X \lt 5) + (4 - X) * (X \ge 5)$ NOTE: The mutiplication, * and the addition, + symbols are required.
Step 3:
Greatest Integer Function
The greatest integer function is also called the floor function.
The greatest integer of a number is defined as the greatest integer less than or equal to that number.
The greatest integer of a number, say x is denoted by ${\lfloor x \rfloor}$ or $[\![ x ]\!]$ or int(x)
$\mathbb{Z}$ is the set of all integers.
Say we have a decimal: $\large x.y$;
$x$ is the integer part.
Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry (7th ed.).
Boston: Pearson.
Sullivan, M. (2020). Precalculus. (11th ed.). Pearson.
How to Graph Piecewise-Defined Functions on the TI-84 Plus. (n.d.). Dummies. Retrieved March 18, 2023, from
https://www.dummies.com/article/technology/electronics/graphing-calculators/how-to-graph-piecewise-defined-functions-on-the-ti-84-plus-161100/
2020 Federal Tax Rates, Brackets, & Standard Deductions. (2020, January 25). IRS.Com. https://www.irs.com/articles/2020-federal-tax-rates-brackets-standard-deductions/
Bankrate.com. (2018, November 28). 2018-2019 Tax Brackets | Bankrate.com. Retrieved from https://www.bankrate.com/finance/taxes/tax-brackets.aspx
Calhoun County Water Association. (n.d.). Calhoun County Water Association. Retrieved from http://www.calhouncwa.com/rates.htm
Georgia Public Service Commission. (n.d.). Georgia Public Service Commission. Retrieved from http://www.psc.state.ga.us/calc/electric/GPcalc.asp
Kelly Phillips Erb. (2019, January 30). New: IRS Announces 2018 Tax Rates, Standard Deductions, Exemption Amounts And More. Retrieved from https://www.forbes.com/sites/kellyphillipserb/2018/03/07/new-irs-announces-2018-tax-rates-standard-deductions-exemption-amounts-and-more/#3c6ec333133d
TaxAct. (n.d.). TaxAct. Retrieved from https://www.taxact.com/tools/tax-bracket-calculator
CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/
TI Products | Calculators and Technology | Texas Instruments. (n.d.). Education.ti.com. Retrieved March 18, 2023, from https://education.ti.com/en/products
51 Real SAT PDFs and List of 89 Real ACTs (Free) : McElroy Tutoring. (n.d.).
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