MTH 124 Focus Session 2 Exponential models for interest bearing investments Goals cid 136 Study interest bearing accounts using exponential functions and logarithms cid 136 Understand compounded interest cid 136 Practice on Desmos Pre activity Do before class De nitions Let A t denote the number of dollars in an interest bearing account t years after the initial investment cid 136 The interest rate is usually described by the annual percentage rate or APR cid 136 Interest may be calculated and paid annually monthly daily or even continuously in which case interest is said to be compounded annually monthly daily or continuously cid 136 If the interest is compounded annually then where A0 is the amount of the initial investment and r is the decimal representation of the APR example if the APR is 2 then r 02 cid 136 If the interest is compounded monthly so that every year the interest is computed and paid 12 times then A t A0 1 r t cid 16 cid 17 12t r 12 A t A0 1 A t A0ert where r is the decimal representation of the APR cid 136 If the interest is compounding continously then where r is the decimal representation of the APR cid 136 Graph f t 7 2 t and g t 7 3 t in https www desmos com calculator Describe these two exponential functions Are they increasing or decreasing Which one is changing faster Could you get these answers from just the function s equation How cid 136 What is the domain and the range of these two functions cid 136 If you want to compute log3 12 but your calculator only has ln base e and log base 10 what should you do 1 MTH 124 Focus Session 2 Exponential models for interest bearing investments In Class Activity Example 1 An investment of 10000 is made in an account with a 3 6 APR that is compounded annually Let A t denote the account balance t years after the initial investment in dollars 1 What will the account balance be after 1 year 2 years and 5 years 2 Express A t as a function of t Graph this function in Desmos 3 Find to the nearest whole year when the balance of the account will be 20285 94 Solve this problem in two di erent ways 1 Graphically using Desmos and 2 Algebraically using logarithms Example 2 Consider the balance A t of an account with an initial investment of 10000 and with a 3 6 APR Now assume that interest is compounded monthly This means that the 3 6 annual percentage rate rather than being paid in total once a year is paid out in 12 monthly interest payments each of which pays 3 6 12 or 0 3 interest on the current balance of the account 4 Find the balance of the account described in Example 2 after 1 year What percentage interest did the account actually earn in the rst year round the interest percentage to four places 2 MTH 124 Focus Session 2 Exponential models for interest bearing investments 5 If you have a choice of two investment accounts that have the same APR but one is compounded annually and the other is compounded monthly which should you choose Why Notice that although the APR on the account was 3 6 the account actually earned more than 3 6 interest in one year The percentage interest actually earned in one year is referred to as the annual percentage yield or APY In comparing accounts it is more useful to compare the APY rather than the APR because accounts with the same APR but di erent compounding schedules can result in di erent amounts of interest 6 What is the APY of the account in Example 2 round the interest percentage to four places 7 Find a model for A t the balance after t years of an account with initial balance 100 000 and an APR of 3 6 compounded continuously 8 What is the APY of the account in question 7 round the interest percentage to four places 9 Given the situation in question 7 how long to the nearest year does it take for the account balance to double Solve this problem algebraically using logarithms Check your answer in Desmos 10 Use Desmos to plot the A t for interest compounded annually monthly and continuously Compare their graphs 3