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Understanding Compound Interest: e and Continuous Compounding in Math

Discover the concept of the irrational number 'e' and how it relates to continuous compound interest in mathematics. Learn about different compounding periods and how to calculate future values using formulas and practical examples.

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Understanding Compound Interest: e and Continuous Compounding in Math

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  1. 11.1: The Constant e and Continuous Compound Interest

  2. Review (Mat 115) • Just like π, e is an irrational number which can not be represented exactly by any finite decimal fraction. • However, it can be approximated by for a sufficiently large x e e e

  3. The Constant e Reminder: Use your calculator, e = 2.718 281 828 459 … DEFINITION OF THE NUMBER e

  4. Check the limit using table and graph

  5. Review • Simple interest: A = P + Prt • Compound Interest: A = P(1 + r)t or with n = 1 (interest is compounded annually – once per year) • Other compounding periods: semiannually(2), quarterly(4), monthly(12), weekly(52), daily(365), hourly(8760)… • Continuous Compounding:(see page 589 for the proof) A = Pert A: future value P: principal r: interest rate t: number of years

  6. Example 1: Generous Grandma Your Grandma puts $1,000 in a bank for you, at 5% interest. Calculate the amount after 20 years. Simple interest: A = 1000 (1 + 0.0520) = $2,000.00 Compounded annually: A = 1000 (1 + .05)20 =$2,653.30 Compounded daily: Compounded continuously: A = 1000 e(.05)(20) = $2,718.28

  7. Example 2: IRA • After graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year, compounded continuously. He plans to retire in 35 years. • What will be its value at the end of the time period? • A = Pert = 3000 e(.12)(35) =$200,058.99 • The second year he repeated the purchase of an identical Roth IRA. What will be its value in 34 years? • A = Pert = 3000 e(.12)(34) =$177,436.41

  8. Example 3 What amount (to the nearest cent) will an account have after 5 years if $100 is invested at an annual nominal rate of 8% compounded annually? Semiannually? continuously? • Compounded annually • Compounded semiannually • Compounded continuously A = Pert = 100e(.08*5) = 149.18

  9. If $5000 is invested in a Union Savings Bank 4-year CD that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years. Example 4 • Use your graphing calculator: • Press y= • Type in 5000e^(x*0.0561) • Press ZOOM, scroll down, then press ZoomFit • You will see the graph • To find out the amount after 4 years • Press 2ND, TRACE, 1:VALUE • Then type in 4, ENTER

  10. How long will it take an investment of $10000 to grow to $15000 if it is invested at 9% compounded continuously? Formula: A =P ert 15000 = 10000 e .09t 1.5 = e .09t Ln (1.5) = ln (e .09t) Ln (1.5) = .09 t So t = ln(1.5) / .09 t = 4.51 It will take about 4.51 years Example 5

  11. How long will it take money to triple if it is invested at 5.5% compounded continuously? Formula: A =P ert 3P = P e .055t 3 = e .055t Ln 3 = ln (e .055t) Ln 3 = .055t So t = ln3 / .055 t = 19.97 It will take about 19.97 years Example 6

  12. Review on how to solve exponential equations that involves e if needed (materials in MAT 115)

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