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The Number e

PreCalculus NYOS Charter School Quarter 4 "Nature's great book is written in mathematics." ~Galileo. The Number e. The Number e. Leonhard Euler found an interesting irrational number named e . The number is the sum of + …

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The Number e

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  1. PreCalculusNYOS Charter SchoolQuarter 4"Nature's great book is written in mathematics." ~Galileo The Number e

  2. The Number e • Leonhard Euler found an interesting irrational number named e. • The number is the sum of + … • We can round this off to two decimal places as an estimation of e.

  3. The Number e • We use e in exponential growth and decay problems. N is the final amount N0 is the initial amount k is a constant t is time

  4. The Number e Example: DDT is effective against insects, but was found to be harmful to humans in 1973. More than 1 * 109 kg of DDT had already been used before the risk was identified. How much will remain in the environment in 2020, assuming k = -0.0211, if we stopped using DDT in 1973?

  5. The Number e Example: 1 * 109 kg; k = -0.0211, 1973 - 2020? kg

  6. The Number e • We also use e in problems involving continuously compounding interest. A is the final amount P is the initial amount r is the annual rate t is time in years

  7. The Number e Example: Compare the balance after 25 years of a $10,000 investment earning 6.75% interest compounded continuously to the same investment compounded semiannually.

  8. The Number e Example: t = 25; P = $10,000; r = 6.75%; semiannuallyvs. continuously

  9. The Number e Example: t = 25; P = $10,000; r = 6.75%; semiannuallyvs. continuously A = $52,574.62 A = $54,059.49

  10. The Number e Example: t = 25; P = $10,000; r = 6.75%; semiannuallyvs. continuously A = $52,574.62 A = $54,059.49

  11. The Number e • If we were to invest the same amount each month in an account with continuously compounding interest, our formula would be A is the final amount M is the monthly payment amount r is the monthly rate t is time in months

  12. The Number e Example: If we invest $1000 per month in an account that has 6% continuous compounding, how much will we have at the end of one year?

  13. The Number e Example: M = $1000; t = 12; r = .06/12

  14. The Number e Example: If we invest $500 per month in an account that has 4% continuous compounding, how much will we have at the end of 10 years?

  15. The Number e Example: M = $500; t = 120; r = .04/12

  16. The Number e Example: Make a graph of the account balance over the 10 years. This is part of #5 on the rubric for the project.

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