MAT9004 Assignment 1 Due in week 4 (See Moodle for submission details). 1. A beekeeper estimates that the number of bees in a particular hive at the start of 2019 is 500, and that at the start of any subsequent day the number of bees in the hive has increased by 1% compared to what it was exactly one day earlier. (a) Find a function h : R } (0,00) such that Mt) models the population of the hive in the sense that b(t) agrees with the beekeeper's estimates whenever t E N. Here t E R measures the number of days since the start of the year (e.g. t = 1/2 is noon on New Year's day). Similarly, you should consider the population of bees to be a gradually growing real number (ignore the fact that it is actually restricted to being an integer). [3] (b) Explain why the inverse function 51 exists. [3] (c) Calculate a formula for ()4. [2] (d) The function b can be described as taking the time since the start of this year as input and outputting the population of bees in the hive at that time. Provide a similar description for what the function b'l does. [2] (e) Find the derivative b'(t). Explain what b'(t) is measuring by giving a description of the same type that you gave in part ((1) [2] (f) On which date does the population of bees in the hive reach 10000? Suppose that at the end of that day all but 1000 of the bees leave the hive to start a new colony elsewhere. [2] (g) Write a function 6 : [0, 365] ) (0,00) which gives the population of bees in the hive throughout 2019, given the migration described in (f). Assume that the population continues to increase at the same rate (1% per day) after the migration. [4] (h) What is the population of bees in the hive at the end of the year? [2] 2. READIT is a vendor of ebooks. They provide recommendations to customers based on ratings by other customers. Each customer rates each book that they read, by giving a star rating from 1 star to 5 stars (there are no \"half\" stars). Suppose READIT wants to recommend books to a customer called Wendy. For each customer X other than Wendy, they rst calculate fo ifn