These pictures + text is all the information I have. Its the sixth time I'm posting these questions and they don't get answered. Kindly answer them, at least parts 1a, b, and c. Thank you very much in advance!

Calculus I Write-up Week 7 Derivatives of Exponential Functions Include your name and the names of the people in your group on your lab write-up. Each person will submit their own lab write-up, saved as a PDF to the online system. 1. (80 points) Demonstrate and Explain: Clearly and thoroughly explain your answers to these questions based on the work you did with your group on labs this week. These questions refer to Lab 7: Exploring Derivatives of Exponential Functions. (a) (Part 2) Show the graph of the function m(x) = .5(3") and the derivative function m (r). What is the formula for _ (.5(3*))? How can you confirm this using Desmos? (b) (Part 3) Explain the steps you used to find the function P(x) to model the price of gas in the first 6 months of 2022. What does your function predict for April 2022 (month 3)? How does this compare to the source data? (c) (Part 46) Explain how to find a formula for P'(r), evaluate this derivative for April, and explain in words what this represents. (d) EXTRA CREDIT: (Part 7) What is Euler's constant, e? Explain what is special about the derivative of the function f(x) = er, and how you know this. 2. (10 points) Summarize: What do you think are the big ideas for calculus in the week 7 lab? What do you want to remember from your work on this lab? Be sure to include definitions or examples of important vocabulary related to exponential functions.In the first 6 months of 2022, news outlets reported the skyrocketing price of gas (in dollars per gallon) in California as growing exponentially. You can find these data graphed in Desmos at this link: https://tinyurl. com/CalcLab7Bgas. Month Jan Feb Mar Apr May Jun Price/gallon 4.584 4.660 5.655 5.692 5.871 6.294 Below, we see that it looks like these data follow an exponential curve. January 2022 is month 0 and February 2022 is month 1. 16 14 12 10 Average $/gal of gas in CA 6 2 0 2 4 6 8 10 12 14 16 18 20 22 24 Months since 1/1/2022 Source: https://www.cia. gov/dnav/pet/hist/LeafHandler . ashx?n=PET&s=EMM_EPMO_PTE_SCA_DPG&f=M.2. Now we'll generalize this. (a) Change the starting function to f(x) = 67, and add a slider for b. (b) Discuss with your group what interval and step size make sense for b. (c) Change g(r) so that it will match with the graph of f'(x) for any number b. How is the value of a related to the value of b? (d) Summarize your findings as d ( 6 7 ) = (e) Generalize further to consider any function f(r) = k(67) for some constant number, k. Add a slider for k. What do you need to change so that g(r) matches f'(x)? (f) Use your program to find the derivative of f(x) = 1.5(43). (g) Finally, find an approximate value for b so that the graph of f(x) = ] is the same as the graph of f'(x), that is, do (67 ) = 67. What is this special number? In the first 6 months of 2022, news outlets reported the skyrocketing price of gas (in dollars per gallon) in California as growing exponentially. You can find these data graphed in Desmos at this link: https://tinyurl. com/CalcLab7Bgas.16 14 12 10 Average S/gal of gas in CA 00 6 2 2 6 8 10 12 14 16 18 20 22 24 Months since 1/1/2022 Source: https://www.cia.gov/dnav/pet/hist/LeafHandler . ashx?n=PET&s=EMM_EPMO_PTE_SCA_DPC&f=M. 3. Since the points for January and May are very close to the curve modeling these data, we can use these points to find an equation of the form P(x) = k(67). (a) Explain what the points (0, 4.584) and (4, 5.871) represent in this situation. (b) Use the point (0, 4.584) to find a value for k, and then use the point (4,5.871) to solve for b. (c) Enter your function, P(x), in Desmos to check that it approximates your data. How well does it fit the points? (d) What does your model predict for the price of gas in June? How does that compare to the cost shown in the table for the same month? (e) According to reports, the gas price was growing on average at a rate of about 6.5% per month. How is the 6.5% reflected in the value of b