1901_sol1old.pdf - MAT 9004 January 2019 Exercise...

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MAT 9004 January 2019 Exercise sheet (Week 1 and 2) Functions, Basic Calculus We only give partial solutions to this sheet. Your answers are expected to be contain intermediate steps and justifications. 1. You are looking to open a theme park. Initially, the number of visitors on the n -th day can be approximated as f ( n ) = 100 · 1 . 05 n , starting from day zero. You have decided to charge the standard entrance fee of $20 if the number of visitors on a given day exceeds 1000, and a premium fee of $25 if the number of visitors is at most 1000 (under the unrealistic assumption that you know which fee to charge in advance). Formally, if x is the number of visitors on a given day, then the income on that day, viewed as a function g : R R , is g ( x ) = ( 25 x, x 1000 20 x, x > 1000 . (a) Compute the number of visitors on each of the first four days (rounded to the nearest integer). [1] (b) Given the exponential increase of the number of visitors, do you think it is more likely that the main means of advertising for the park were billboards or ‘word of mouth’? Justify your answer in one or two sentences. [1] (c) Show how you can use logarithms to compute on which day the number of visitors will first exceed 1000. [2] (d) Using sigma notation, write an expression for T , the total income from the entrance fees in the first 20 days, starting from day 0. [1] (e) Look up a formula for the sum of a geometric series. Using the formula, write a simpler expression for the total income T . Compute T . [1] (f) Plot the function g . Is the function continuous? Does it have an inverse? Why? [2] (g) Notice that there is a range 1000 < x b for the number of visitors x on which the daily income g ( x ) drops to less than $ 25000 , so that it financially would not pay off to have the additional visitors. Compute the maximal number of visitors b for which the daily income is less than $ 25000 . [1] (h) For how many days is the number of visitors in the ‘unwanted’ interval (1000 , b ] ?
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  • Fall '19
  • Derivative, Continuous function, Wolfram Alpha

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