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# Class Note for MATH 115A with Professor Dawson at UA 2

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Date Created: 02/06/15

Exponential Math 1 1 5A Sprir a3 2 Exponential Properties bu bv buv b Z My bv bur w Logarithm Properties 10g aax x logax a 1nex x lnx 2x e 2x 10gauv10gau10gav Ll 10ga j10gau logav v 10gau vlogau Discrete Compounding Example 1 going to deposit into a savings account You have 10000 that you are You have the following account options 0 o o o 5000 4800 4600 4400 annual annual annual annual interest compounded annually interest compounded quarterly interest compounded monthly interest compounded daily Compounding refers to how often the interest is paid out Which option should you choose Discrete Compounding Each option earns an annual interest rate r This is usually given as a percent 60 o or as a decimal 060 Interest paid for an interval other than a year has the annual rate adjusted 0 Example 2 If the annual interest rate of 4800 is compounded quarterly then the annual interest is paid 4 times a year at 0484 012 or 1200 Simple Interest The simple interest on P dollars after a time of 139 years at an annual rate r is Prt Since the original money is still in the account the total value of the P dollars after 139 years will be P1rt Example 3 If 1000 is deposited in an account which has a 24 o annual interest rate how much money is the account after 3 years assuming interest is only paid out at the end of the 3 years Assume that the interest is not compounded and that no other deposits or withdrawals are made 0 F 100010243 1072 Compound Interst P dollars invested at an annual rate r compounded n times per year has a value of F dollars after 139 years is 7139 FP1Lj n 0 Think of P as the present value of the deposit and F as the future value of the deposit Compound Interest Example 1 contAfter 2 years if you choose the account which has 50 o annual interest compounded annually how much money is in your account 12 F 10000 NUTS 10000 1052 1 1025 Compound Interest 0 Example 1 contHow much is in your account after 2 years if you choose the account which has a 4800 annual interest compounded quarterly 048 4392 8 F10000 1T 100001012 1100130 o 4600 annual interest compounded monthly 122 F 100001 10000 1038324 1096172 0 4400 annual interest compounded daily 3652 F 10000 I 10000 100012730 1091982 Yield 0 You can compare investments with different interest rates and different frequencies of compounding by looking at the values of P dollars at the end of one year and then computing the annual39rates that would produce these amounts without compounding Yield Such a rate is called an effective annual yield annual percentacle yield orjust the yield Letting t1 the yield y can be found by solving P1Ljn P1y Yield Therefore an annual interest rate of r compounded n times per year has a yield y1L 1 n 0 Example 4 What is the yield for the account with 4600 annual interest compounded monthly 12 y 2 1 1 04698 or 4698 Yield 0 There may be times when you need to find the annual rate that would produce a given yield at a specified frequency of compounding In other words you want to solve for r y rquot1 1 11 y11 n y 1 quot 11 1 r ny 1 quot 1 Yield Example 5 What rate r compounded quarterly has a yield of 5200 r 40521 4 1 051 or 51 Continuous Compouding c As n increases y approaches a constant value r m 1 mn 1 m n P 1 j P 1 j P1 j we let n mr n m m 0 But 6 1im1ij z 2718281828 mam m Continuous Compounding The value of P dollars after 139 years when the annual interest rate r is compounded continuously is FPequot Example 6 If you initially deposit 10000 in an account with 42 o interest compounded continuously how much will you have after 2 years F 10000 60422 2 1087629 Yield As before let t1 and solve for y Per P1 y er 1y y er 1 0 Example 7 What is the yield of an investment at 4200 compounded continuously y 042 1 04289 or 4289 Logarithms To solve equations for a variable in the exponent you need to use logarithms u bv ltgtvlogbu or u bv ltgt vlogb 10gu 10gu SO V 10gb Natural Logarithms o The most commonly used base for logarithms is the constant e 10ge x 1nx ueblt2gtb1nu Natural Logarithms Example 8 Find the annual rate r that produces an effective annual yield of 5400 when compounded continuously y 2 er 1 054 2 er 1 er 2 1054 r 1n1054 r z 0526 or 526 Logarithms Example 9 If you deposit 10000 in an account that has an annual interest rate of 45 o compounded monthly how long will it take to grow to 15000 12 15000 10000 1 15000 10000 100375 12 log15 12 t 10g100375 3 903 years 12 log100375 Logarithms Example 10 If you deposit 10000 in an account that has an annual interest rate of 45 o compounded continuously how long will it take to grow to 15000 15000 10000 e045 15000 2 6045 10000 1n15 045z I 1n15 045 z 901 years G rowth If a sum of money invested at an annual rate r compounded continuously has a future value F in 139 years then its present value P is iven b g y P Fe If the interest is compounded n times per year the present value P is given by PF1L n G rowth Example 11 What is the present value of a 30000 payment 3 years from now if the money earns interest at an annual rate of 525 o compounded continuously P 30000 05253 z 2562830 G rowth Example 12 What is the present value of a 30000 payment 3 years from now if the money earns interest at an annual rate of 525 o compounded daily 0525 365 73653 P 30000 1 j 2562859 Objectives Use the properties of exponents and logarithms to simplify exponential and logarithmic expressions and to solve exponential and logarithmic equations Compute the value after 1 years of P dollars invested at an annual rate r compounded n times per year or compounded continuously Compute the present value of an investment that has a value of F dollars after 1 years if interest is earned at an annual rate r compounded n times per year or compounded continuously Compute the time that would be needed for an investment of P dollars to grow to Fdollars if interest is earned at an annual rate r compounded n times per year or compounded continuously Compute the annual rate r that would be needed for an investment of P dollars to grow to F dollars in 1 years if interest is compounded n times per year or compounded continuously Find the effective annual yield y for a given annual rate r compounded n times per year or compounded continuously Find the annual rate r that produces a given effective annual yield y when compounded n times per year or compounded continuously

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