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7.6 The Natural Base, e

7.6 The Natural Base, e. Compound Interest. The compound interest formula is: Where A is the total amount, P is the principal (original amount), r is the annual interest rate, and n is the number of times the interest is compounded per year, and t is the time in years.

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7.6 The Natural Base, e

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  1. 7.6 The Natural Base, e

  2. Compound Interest • The compound interest formula is: • Where A is the total amount, P is the principal (original amount), r is the annual interest rate, and n is the number of times the interest is compounded per year, and t is the time in years.

  3. $1 invested at 100% interest compounded n times for one year: • As n gets very large, interest is continuously compounded (meaning you’re earning interest on the interest you’ve already earned, etc.) • Play around with this function on your calculator for large values of n. (Graph it, substitute in values, try n = 1000, n = 10000, n = 100000.

  4. As n gets really large, what happens? • It approaches a specific number. • Look at the graph. There is a horizontal asymptote. Where is it located? • This number is called e. • e = 2.718281828459045…… • e is an irrational number, just like pi.

  5. Exponential functions with base e. • Exponential functions with base e have the same properties as all other exponential functions we’ve studied. • The function looks like f(x) = ex. • Graph this on your calculator. You can find e next to the 4 button. Push 2ndln. • The domain of f(x) = ex is all real numbers, but the range is y > 0.

  6. Graphing WITHOUT using the graph key on your calculator. • Graph f(x) = ex – 3 by MAKING A TABLE OF VALUES!!!!

  7. The natural Logarithm • A logarithm whose base is e (so loge ) is called the natural logarithm. • It is “ln”. • Again, natural logarithms have the same properties as all other logs, they just have this special name. • So, that means that the inverse of f(x) = ex is?

  8. Simplify (using the properties of logs you already know) • ln e0.15t • e3 ln (x + 1) • ln e2x + ln ex

  9. Continuous Compounded Interest • The formula for continuously compounded interest is A = Pert • A is the total amount • P is the principal (original amount) • r is annual interest rate (make sure it’s changed to a decimal) • t is time in years

  10. Try this: • What is the total amount for an investment of $500 invested at 5.25% for 40 years, compounded continuously?

  11. Half-life • Scientists are able to determine the age of a really old fossil or other substance by measuring a half-life. • The half-life of a substance is the time it takes for half the substance to breakdown or convert to another substance during the process of decay. • Natural decay equation: N(t) = N0e-kt

  12. Plutonium • Plutonium-239 (Plu-239) has a half-life of 24110 years. How long does it take for a 1 g sample of Plu-239 to decay to 0.1 g? • What do we know? • What do we need to find?

  13. Another one: • Determine how long it will take for 650 mg of a sample of chromium-51, which has a half-life of about 28 days, to decay to 200 mg.

  14. Simplify: • ln e1 • ln ex-y • ln e(-x/3) • eln2x • e3lnx

  15. Emma receives $7750 and invests it in an account that earns 4% interest compounded continuously. What is the total amount of her investment after 5 years?

  16. Graph: • f(x) = -e1-x

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