Date Per r Remember the compound interest formula At () = P 1+ The investment of $1 is going to earn 100% annual interest over a period of 1 year.

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1 Advanced Algebra Name CW 38: The Natural Base Date Per r Remember the compound interest formula At () = P + where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years? Use the compound interest formula to investigate what happens to an investment as it is compounded more frequently. The investment of $ is going to earn 00% annual interest over a period of year. Compounding Schedule n n Amount at the End of the Year Annually Semiannually 2 Quarterly 4 Monthly 2 Daily 365 Hourly Every Minute Every Second Continuously (Choose a really big number for n!) nt What happens to the values of the amount at the end of the year as the value for n increases? This is the same method that the Swiss mathematician Jacob Bernoulli ( ) used to discover this new irrational constant. Leonard Euler popularized using the symbol e. (Find e on your calculator now if you haven t already!) rt Continuously Compounding Formula: A() t = Pe. Find the value of $500 after 4 years invested at an annual rate of 9% compounded continuously. 2. $2000 is invested at an annual interest rate of 9%. Compare the final amounts after 2 years for interest compounded quarterly and continuously. 3. Use a calculator to evaluate the function f ( ) = e to the nearest thousandth for each value of. a. = 3 b. = c. = 3

2 The Natural Logarithm Function Since ( ) f = e is the natural eponential function, its inverse is the natural logarithm function f ( ) = log e. It is typically written as f ( ) = ln. Use your calculator to help you graph the following functions: y = e y = ln y = State the domain and range of y = e and y ln =. Why does the graph prove that are inverses of each other? y = e and y = ln Since they are inverses, what is the value of ln e? Confirm your answer on a calculator. Eamples:. Use a calculator to evaluate the function ( ) ln f = to the nearest thousandth for each value of. a. = 5 b. = 0.85 c. = 2. How long does it take for an investment to triple at an annual interest rate of 7.5% compounded continuously? Radioactive Decay When a plant or animal dies, the amount of carbon-4 it contains decays in such a way that eactly half of its initial amount is present after 5730 years t The function N( t) = Ne 0 models the decay of carbon-4, where N 0 is the initial amount of carbon-4 and N(t) is the amount present t years after the plant or animal dies.. Suppose that a jar containing grain is found at an archaeological dig and that archaeologists claim that it is 6500 years old. Tests indicate that the grain contains 62% of its original carbon-4. Could the grain be 6500 years old? Why or why not? 2. How much of a 20 milligram sample of carbon-4 will remain after 50 years?

3 Advanced Algebra KEY Name The Natural Base Date Per r Remember the compound interest formula At () = P + where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years? Use the compound interest formula to investigate what happens to an investment as it is compounded more frequently. The investment of $ is going to earn 00% annual interest over a period of year. Compounding Schedule n n Amount at the End of the Year Include 9 decimal places! Annually 2.00 Semiannually Quarterly nt Monthly Daily Hourly Every Minute Every Second Continuously (Choose a really big number for n!) What happens to the values of the amount at the end of the year as the value for n increases? The values approach which is a constant called e

4 This is the same method that the Swiss mathematician Jacob Bernoulli ( ) used to discover this new irrational constant. Leonard Euler popularized using the symbol e. Continuously Compounding Formula: A = Pe rt Eamples: 4. Find the value of $500 after 4 years invested at an annual rate of 9% compounded continuously. $ $2000 is invested at an annual interest rate of 9%. Compare the final amounts after 2 years for interest compounded quarterly and continuously. Quarterly: $ Continuously: $ Compounding continuously yields about $70 more than compounding quarterly. 6. Use a calculator to evaluate the function f ( ) = e to the nearest thousandth for each value of. a. = b. = c. = The Natural Logarithm Function Since f ( ) e = is the natural eponential function, its inverse is the natural logarithm function f ( ) = log e. It is typically written as f ( ) = ln. Use your calculator to help you graph the following functions: y = e f ( ) = ln y = State the domain and range of y = e and y = ln. y = e domain: all real numbers range: all positive real numbers y = ln domain: all positive real numbers range: all real numbers Why does the graph prove that y = e and y = ln are inverses of each other? They are reflections of each other over the line y =. Since they are inverses, what is the value of ln e? Confirm your answer on a calculator. Eamples: 3. Use a calculator to evaluate the function f ( ) = ln to the nearest thousandth for each value of. a. = 5 b. = 0.85 c. =

5 4. How long does it take for an investment to triple at an annual interest rate of 7.5% compounded continuously? 4 years and 8 months Radioactive Decay When a plant or animal dies, the amount of carbon-4 it contains decays in such a way that eactly half of its initial amount is present after 5730 years t The function N( t) = Ne 0 models the decay of carbon-4, where N 0 is the initial amount of carbon-4 and N(t) is the amount present t years after the plant or animal dies.. Suppose that a jar containing grain is found at an archaeological dig and that archaeologists claim that it is 6500 years old. Tests indicate that the grain contains 62% of its original carbon-4. Could the grain be 6500 years old? Why or why not? No, if you solve for t, t = , so the grain is about 4000 years old. 2. How much of a 20 milligram sample of carbon-4 will remain after 50 years? 9.88 mg