For this Critical Thinking Assignment, you will be creating your own real-world scenario and exploring the meaning of differentiation of exponential functions in that context.

Part I: Complete the following steps:

1. Select a real-world scenario that can be modeled by an exponential function and define the functionF(x)based on your scenario. (Ex.: Suppose thatF(x)computes the rabbit population on a game reserve that doubles every 6 months. Suppose there were 120 rabbits initially.)
2. Write a mathematical expression forF(x).
3. Find the domain and range ofF(x).
4. FindF(x) andF '(x) at any point.
5. Find allxvalues for whichF '(x) = 0.

Part II: Based on your work in Part I, discuss the following:

1. Discuss if your functionF(x) is differentiable and why. If it is not differentiable, select another function that is and discuss the change you made.
2. Discuss the domain and range ofF(x) and why they make sense in the context of your problem.
3. Discuss the physical meaning ofF(x) andF '(x) at a point in the context of your problem.
4. Discuss what it means forF '(x) = 0 in the context of your problem.
5. Reflect on any adjustments you had to make to your original problem context and discuss what characteristics of a real-world context make it a valid choice for applying differentiation concepts to it.
6. Provide at least two other real-world situations where differentiation can be applied and respond to the following:
7. What common characteristics do the real-world scenarios you chose share?
8. What did you look for in the way that the real-world scenario can be modeled?
9. Discuss the criteria for selecting a real-world scenario that would change if you were seeking to model it with a logarithmic function instead. What key similarities and differences would you find?

Requirements:

You must submit two files for this assignment. The first file should contain the computations, graphs, diagrams, etc., associated with the questions in Part I. This file may be formatted as a numbered list of answers. Unless stated in the problem, a narrative discussion is not required, but you must provide enough information to show how you arrived at the answer.

The second file should be a 2- to 3-page narrative paper, written in APA format, associated with the situation described in Part II. Specific requirements for the paper are provided below:

1. Your paper should be 2-3 pages in length (not counting the title page and references page) and should cite and integrate at least two credible outside sources. The CSU Global Library is a great place to find resources. Your textbook is a credible resource.
2. Include a title page, introduction, body, conclusion, and a reference page.
3. The introduction should describe or summarize the topic or problem. It might discuss the general applications of the topic or it might introduce the unique terminology associated with the topic.
4. The body of your paper should address the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.
5. The conclusion should summarize your thoughts about what you have determined from your analysis in completing the assignment. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.
6. Include any tables of data or calculations, calculated values, and/or graphs referenced in the paper. (Note: The minimum required length excludes any tables, graphs, etc.)