(2) It's fun to do math finance with the benefit of hindsight - to look back on volatile stocks and imagine what might have been. In the last 15 years, some particularly volatile investments have emerged: cryptocurrencies. If one made frequent trades, the order and timing of these trades would mean the difference between massive profits or losses. (a) (b) Suppose you invested $2000 on January 3, 2009 (Bitcoin's inception date), and have invested an additional $400 on the last day of each month since then. Give a function modelling how much you have invested (not how much the investment has earned) as a function of the number of years that have passed since your initial investment. What are the slope and intercepts of this function? The following table gives the factor by which an investment in either Bitcoin or Ethereum either grew or shrank between certain dates: 01/02/21 05/08/21 06/26/21 11/13/21 01/22/22 04/02/22 06/18/22 Bitcoin 1.742 Ethereum 4.804 0.554 0.474 2.041 2.594 0.546 0.520 1.300 1.423 0.430 0.300 (c) (d) We define functions fi and gi representing each of the above increases/decreases in an investment's value, for Bitcoin and Ethereum respectively. That is, f1(x) = 1.742x, f6(x) = 0.430x and 91(x) = 4.804x, ..., 96(x) = 0.300x Find a composition of 6 functions (that is, one from each time interval) which maximizes the growth of the initial investment. Find another such composition which minimizes its growth. Is there one of the cryptocurrencies that you would characterize as "higher risk, higher reward" over this time period? Suppose that in retirement, you've saved $15000, and earn royalties from your acting career continuously, at a constant rate of $24000/year. However, you also withdraw $3000 at the start of the first day of each month. = Lett represent the start of the tth month (so for example, t 6.5 represents the middle of June). Sketch roughly what the graph of your savings s(t) as a function of time t would look like for the first year of your retirement (that is, let the y-axis be January 1). Do the following limits exist (note t = 3 represents March 1), and if so, what are their values? lim c(t), lim c(t), lim c(t) t→3- t→3+ t→3 If the contents of your savings account in dollars is given by the function s(t) = 5000+6t2, use the limit definition of a derivative to give the rate of growth of your savings as a function of time t.