Exponential functions are useful in business and economics. PowerPoint Lesson $7$ discusses them. Show how the values are entered into your functions and also calculate the amounts of each of the following:

a$1.$ You learn on the business channel that inflation was about $0\mathrm{.}85\%$ last month. Assume this rate is maintained each month for a year. What will the annualized rate be$?$ EXAMPLE: A rate of $0\mathrm{.}1\%$ per month represents $(1+0\mathrm{.}001)12-1=0\mathrm{.}0121$ or $1\mathrm{.}21\%$ annually.

a$2.$ The price of beef increased by about $8\%$ last year. What average monthly growth rate is implied? EXAMPLE: A $2\%$ growth rate for the year would require $1\mathrm{.}02=(1+$ r$)12.$ Solve this for r: ; r $=\mathrm{.}00165$ or $\mathrm{.}165\%$ per month on average.

b$1.$ F $=$ Pert $,$ which assumes continuous compounding, says that the Future value $($F$)$ of an amount $($P$)$ invested today at an annual rate $($r$,$ expressed as a decimal for the time $($t$,$ expressed in years$)$ is given by the function. e is usually a key on a scientific calculator or it may be considered to be a constant approximately equal to $2\mathrm{.}7183.$ EXAMPLE: invest $$100$ at the annual rate of $51/2\%$ for $6$ years and $3$ months and you should get back $($at the end of the time$),$ F $=$ $$100$e$(0\mathrm{.}055)(6\mathrm{.}25)=$ $$100$e$(0\mathrm{.}3438)=$ $$100(1\mathrm{.}4102)=$ $$141\mathrm{.}02.$ YOUR PROBLEM: What amount would you expect to get back at maturity in $31/2$ years if you invested $$30000$ today in a note paying $3\%$ annually?

b$2.$ Alternatively, if a borrower tells you that he needs a loan for $6$ years and $3$ months and will pay you an annual rate of $51/2\%$ for the loan, but will only give you $$141\mathrm{.}02$ back at the end of the loan term, you should only loan him $$100$ today. YOUR PROBLEM: A borrower agrees to pay you $6\%$ annually for $5$ years and $3$ months. At the end of the term he will make a balloon payment of $$20000$ to repay the loan and interest. What amount $($P$)$ does the formula P $=$ F$/$ert indicate you should loan this prospect?