AEP 8: Finding total change

--- tags: mth201, templates --- # AEP 8: Finding total change ## What this AEP is about This AEP will get you to apply the Total Change Theorem, using real data or models that you find in the world. ## Prerequisites and tech requirements You will be ready to begin this AEP following **Module 12.** **Technology used in this AEP:** Some combination of the tools you are familiar with --- Desmos, Wolfram|Alpha, and possibly a spreadsheet. ## Special offer! Earning an "E" mark on this AEP will level you up on Learning Target 15 (*I can explain the meaning of each part of the definition of the definite integral in terms of a graph, and interpret the definite integral in terms of areas, net change, and displacement*). ## Background and setup We learned in class that the **Total Change Theorem** says: :::info If $f$ is a continuously differentiable function defined on an interval $[a,b]$ with derivative $f'$, then $$f(b) - f(a) = \int_a^b f'(x) \, dx$$ ::: In English, the Total Change Theorem says that *the total change in a function over an interval is the integral of its rate of change on that interval.* So, just as integrating a moving object's velocity tells us the total change in the position of object, integrating the rate of change in *any* function will give us the total change in that function. In this AEP, you will be **creating your own application problem for the Total Change Theorem** rather than using pre-packaged problems from a textbook. The specific tasks are in the next section, but generally speaking, this AEP will ask you to **find a real life example of a rate of change, and then apply the Total Change Theorem to find a total amount of change on an interval.** To do the tasks, do the following first. **Read carefully** because there are some specific requirements for what you'll need to set up. 1. On the internet, or in a book or article, or in some other source, **find a function in the form of data, a graph, or a formula that describes the rate at which something is changing**. You can often tell that the function describes a rate by looking at the units of the output and looking for the word "per": Miles per hour, people per year, inches per dollar, etc. 2. Make sure that the function is **two-variable** (one input variable, one output variable). 3. VERY importantly: **The units of the input variable must be the same as the *denominator* of the units of the output variable.** For example, you could use a function whose input is in "days" and the output is in "dollars per day". Or a function whose input is "milligrams" and whose output is "people per milligram". But you could not use a function whose input is "people" and whose output is "cases per day". 4. **Do not use "rate" functions that are not rates of change**. Sometimes functions tell you "rates" that are not really rates of change. For example [this function](https://www.researchgate.net/figure/Nominal-interest-rates_fig4_334012195) gives *interest rates*, and [this one](https://www.heritage.org/data-visualizations/public-health/covid-19-death-rates-by-state/) gives death rates. Neither of these works for this AEP; what we are looking for is a function that is **the rate of change in a value**. (Also note that both these functions fail the previous point --- the units of the input aren't the same as the denominator of the units of the output. In fact it's not clear what the units of the output really are.) 5. Finally, **your function must be authentically real and not made up, either by you or by anyone else**. For example you may not pull made-up functions from a textbook. Although, if you find real data, graphs, or models --- something not made up --- in a textbook, you can use those provided there is a source given in the textbook where you found it. **The hard part of this AEP is finding this function.** [Here is an example of a function that fits all these criteria](https://i.imgur.com/cVjwsbt.jpg). (Source: http://freerangestats.info/blog/2018/12/01/number-births) You may not use this one in your work! You can tell this function will work because of the units: The input units are "years" and the output units are "people per year". Again, your function can be a graph, dataset, or formula. If you're having issues finding a function that works, try searching for data, graphs, or models about birth/death rates, velocity, or acceleration. ## AEP Tasks There's really just one big task for this AEP: **Take the function you found in the setup phase, and use the Total Change Theorem to estimate how much change occurred over an interval.** For example, I might take [the function about birth rates](https://i.imgur.com/cVjwsbt.jpg) and ask, *How many births were there from 1900 to 2000?* Then answer that question with calculus. (No spoilers given here about how that would work; figuring out how is one of the main learning objectives for this assignment.) Your work must contain the following: - A link to the source of your data so it can be checked for authenticity. Data that are made up will not be accepted. If your function is authentic but you found it in a textbook, you'll need to provide a link to both the book and the original source. - A narrative explanation for what your function tells you, and a clear statement of what question you are answering or what value you are computing. (Example: *How many births were there from 1900 to 2000?*) - A complete description of the process you use to find the quantity you are finding. **You can use a computer to do all calculations on this AEP**, but **you also must provide access to your computations**. (See below for details on that.) ## Grading criteria and submission instructions Please refer to the [overall quality standards for AEPs](https://hackmd.io/@rtalbert235/HkSbMs2Av) at the link (and posted to the *AEPs* area on Blackboard) for grading standards. For this AEP, the following particular rules are in place: - The work will receive an "X" mark if there is no way for me to check the authenticity of your function. For example, if you pull data from the internet, provide a link. If you get your function from a book, cite the book *and* the original source that the book used. For example if you find a textbook that includes a graph that came from an article, give a link or a citation for both the book and the article. - The work will receive an "X" mark if you do computations with a computer (Desmos, Wolfram|Alpha, spreadsheets, etc.) but do not provide access to the calculations (links to Desmos worksheets ad Wolfram|Alpha computations, links to Google Sheets, etc.) Also please [see the syllabus](https://hackmd.io/@rtalbert235/SJ5fDZIAv#How-are-individual-assignments-graded) for how grades of **P** (Progressing) and **X** (Not Assessable) are assigned. To submit this AEP: - Create a **typed document** using a word processor utilizing an equation editor for any significant mathematical notation. **Submissions that are handwritten or contain any handwritten work without prior permission will be graded "X" and returned without comment.** - Save the document as a PDF. Please **do not submit Word (.doc, .docx) or Open Office (.odt) files** as these do not always render mathematical notation correctly on screen. - Upload the PDF to the appropriate AEP assignment in the *AEPs* area on Blackboard. **Remember to hit the "Submit" button after uploading**, otherwise your work is merely on the Blackboard server and not in the instructor's grading queue. Once graded, the grade (E, M, P, X) will appear in your gradebook. You can then open the submission to view feedback left by the professor. These appear as text comments on the page or as general comments next to the grade. Grades of E or M may not have much feedback. Grades of P or X *always* have feedback, so please look carefully for this. For information on revisions of AEPs, please [consult the syllabus](https://hackmd.io/@rtalbert235/SJ5fDZIAv#How-do-I-revise-and-resubmit-my-work). [![hackmd-github-sync-badge](https://hackmd.io/0hGsJdDIQ32XTMVuy2I1tw/badge)](https://hackmd.io/0hGsJdDIQ32XTMVuy2I1tw)