Creating Your Own Real-World Scenario Involving Derivatives of Exponential Functions

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 function F(x) based on your scenario. (Ex.: Suppose that F(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 for F(x).
3. Find the domain and range of F(x).
4. Find F(x) and F ‘(x) at any point.
5. Find all x values for which F ‘(x) = 0.

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

1. Discuss if your function F(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 of F(x) and why they make sense in the context of your problem.
3. Discuss the physical meaning of F(x) and F ‘(x) at a point in the context of your problem.
4. Discuss what it means for F ‘(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:
   1. What common characteristics do the real-world scenarios you chose share?
   2. What did you look for in the way that the real-world scenario can be modeled?
7. 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?