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(37) and the derivative function m'(x). What is the formula for (.5(37))? 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) = e, and how you know this.In this lab you will use Desmos to explore the relationship between the exponential function, f(:r) 2 H391] where It and b are parameters (constants), and its derivative function, f'[3:}. To do this, you will looir for a function, 9(1), whose graph matches the graph of flat). You will also review limits at a point, to learn more about the special number, 6:, and the exponential function, u?) : e\". Finally, we model the price of gas in 2022 with an exponential function and investigate the rate of change. 1. Graph at) : 21 and its derimtive function y = 1'12"}. (Or you can open this Desmos le: https :Nww . dasmos . com/calculator/jf32ply10h.) (a) Dismiss W113.r it makes settw to model fut) with an exponential function, that is Mm} 2 M21]. (h) Graph y{:::} = (19'), and create a slider for (1. Adjust a to make the graph of at) the same as the graph of f'{;r:]. (c) Sammarimr what you found: it . 733:} = (d) Calculate the numher hi2. Conuiare this with the wine you found for a. (e) Change the function to Hm) r 5'. Adjust hath the function {Am} and the parameter a so that 90:) lines up with f'{:r}. {f} Summarize this as Calculate 1:15 and mmpare this with a. 2. Now mz'il gtllllil'illii'! this. (a) Change the starting function to [(1') = b', and add a slider for b. (h) Dismiss with your group what interval and step size make 511-21150. for b. (c) Change g[:i:) so that it will match with the graph of ts) for any number b. How is the value of a related to the value of b? (d) Smnmarire your ndings as d r a\" l - (e) Generalize further to consider any function r) : Hb'} for some constant number, it. Add a slider for it. What. do you need to change so that 9(3) matches fits]? (f) Use your program to nd the derivative of r) = 1.5M"). (g) Finally, nd an approximate value for b so that the graph of f(::) = b\" is the same as the graph of 1.11\"), that is, d I I Eiblb' 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. 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 S/gal of gas in CA 00 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_DPG&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(r), 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?4. While the percent increase is important, the (instantaneous) rate of change of increase in actual dollars is what really matters to people. (a) Use Desmos to compute P'(4) and explain in words what this quantity represents. Be sure to include the units for P'(4). (b) Now use what you know about the derivatives of exponential functions to find the deriva- tive function, P'(r). Evaluate this function when I is 4. Check your answer with what you found in Part 4a. 5. Let's assume that the same inflation continued after June 2022. Approximate the cost of gasoline in July in two ways: (a) Evaluate the function P(r) for July 2022. (Remember, we begin counting at 0.) (b) Construct a linear approximation line from June. What does this tangent line predict for July 2022? (c) Compare your answers with what actually happened according to the data provided at the source website. 6. It is now March 2023. What does your function, P(x), predict the gas price would be now? Do you think that gas prices continued to grow at 6.5%? Why or why not? 7. A bit more about that special number from Part 2g. (a) Consider the function: k(x) = (1+1)1/1 What is k(0)? Explain why it makes sense to think that the limit lim k(x) = lim (1 + x)1/z I-+0 exists, and find an approximate value for the value. (Use a table of values or a graph to find an approximate value for this limit.) We call this number e, for Euler's number, and it is defined by this limit: e = lim (1 + x)1/I. (b) Write the limit definition of the derivative function for f(x) = e", f'(z) = lim f (I + h) - f(x) h +0 h (c) Use the exponent rule efth = et . ch to simplify the numerator. Factor et out of both terms of the numerator. (d) Now we have a new limit to consider: what is lim -? Use tables or graphs to find h-+0 h this value. (e) Lets put all the pieces together. Add your limit from the previous part to the ( ) below to simplify the derivative. If f(x) = er, then e(Ith) - et f'(x) = lim = lim et (eh - 1) h +0 h e lim h h +0 h ) = 0 * ( ). (f) Summarize what you have learned about the derivative of the function f(x) = ex