Math 115], Autumn 2023 Written Homework 4 Page 2 of 5 Context: You are a CEO of a tech startup. You just gave a very confidential business presentation to a few potential investors on the new secret technology recently developed by your team. One investor proposed to deposit 30 million coins on each day of the month of November, resulting in the total of 30 x 30 = 900 million coins at the end of the month. The other investor proposed that they would deposit 2 coins on the first day, then deposit the previous day's balance everyday for the rest of the month. So the balance will increase exponentially, i.e., 2 coins on Day 1, 2 l 2 = 4 = 22 coins on Day 2, 4+4 : 8 : 23 coins on Day 3, and so on. This results in the total of 230 coins on Day 30. You need to determine whether it is better than the first proposal of worth 900 million coins. Unfortunately, no electronic device was allowed in this secret meeting, so you need to estimate the value of 230 without using a calculator. In the first two problems, we estimate the value of 230 using the linear approximation of two different functions. Problem 1: First Approximation Attempt (17 points) Let f(x) : X30. 3) (2 points) Identify x such that f(x) : 230. b) (2 points) Find the nearby point a at which the function can be easily calculated. c) (4 points) Find the linear approximation function for f(x) at the basepoint you found in the previous part. Denote this linear approximation function by Lf(X). 230 d) (6 points) Using the results of the previous parts, estimate the value of without using a calculator. e) (3 points) Will this approximation be an over-estimate or an underestimate? EXPLAIN. Math 11517 Autumn 2023 Written Homework 4 Page 3 of 5 Problem 2: Second Approximation Attempt (17 points) Now let g(x) : x3. In this part, it is useful to know that 210 : 1024. a) (2 points) Identify x such that g(x) : 230. b) (2 points) Find the nearby point a at which the function can be easily calculated c) (4 points) Find the linear approximation function for g(x) at the basepoint you found in the previous part. Denote this linear approximation function by Lg(x). d) (6 points) Using the results of the previous parts, estimate the value of 230 without using a calculator. e) (3 points) Will this approximation be an overestimate or an under-estimate.7 EXPLAIN. Math 1151, Autumn 2023 Written Homework 4 Page 4 of 5 Problem 3: Analysis and Discussion (6 points) a) (3 points) In Problems 1 and 2. you found two different estimates for 230. Now using your calculator, determine which of the two estimates is better b) (3 points) Discuss why one linear approximation results in a good estimate of 230 while the other does not