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Section 6.3 Compound Interest and Continuous Growth

Section 6.3 Compound Interest and Continuous Growth. Let’s say you just won $1000 that you would like to invest. You have the choice of three different accounts: Account 1 pays 12% interest each year Account 2 pays 6% interest every 6 months (this is called 12% compounded semi-annually )

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Section 6.3 Compound Interest and Continuous Growth

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  1. Section 6.3Compound Interest and Continuous Growth

  2. Let’s say you just won $1000 that you would like to invest. You have the choice of three different accounts: • Account 1 pays 12% interest each year • Account 2 pays 6% interest every 6 months (this is called 12% compounded semi-annually) • Account 3 pays out 1% interest every month (this is called 12% compounded monthly) • Do all the accounts give you the same return after one year? What about after t years? • If not, which one should you choose? • NOTE: In each case 1% is called the periodic rate

  3. If an annual interest r is compounded n times per year, then the balance, B, on an initial deposit P after t years is • For the last problem, figure out the growth factors for 12% compounded annually, semi-annually, monthly, daily, and hourly • We’ll put them up on the board • Also note the nominal rate versus the effective rate or annual percentage yield (APY) • The nominal rate for each is 12%

  4. n is the compounding frequency • is called the periodic rate • The growth factor is given by • So to calculate the Annual Percentage Yield we have • Now back to our table This is the base, b, from our exponential function y = abx

  5. This is the base, b, from our exponential function y = abx • Now let’s look at continuously compounded • We get • r is called the continuous rate • The growth factor in this form is er • So to calculate the Annual Percentage Yield we have • Find the APY for 12% • How does it compare to our previous growth rates?

  6. Now 2 < e < 3 so what do you think we can say about the graph of Q(t) = et? • What about the graph of f(t) = e-t • It turns out that the number e is called the natural base • It is an irrational number introduced by Lheonard Euler in 1727 • It makes many formulas in calculus simpler which is why it is so often used

  7. Consider the exponential function Q(t) = aekt • Then the growth factor (or decay factor) is ek • So from y = abt, b = ek • If k is positive then Q(t) is increasing and k is called the continuous growth rate • If k is negative then Q(t) is decreasing and k is called the continuous decay rate • Note: for the above cases we are assuming a > 0

  8. Example • Suppose a lake is evaporating at a continuous rate of 3.5% per month. • Find a formula that gives the amount of water remaining after t months if it begins with 100,000 gallons of water • What is the decay factor? • By what percentage does the amount of water decrease each month?

  9. Example • Suppose that $500 is invested in an account that pays 8%, find the amount after t years if it is compounded • Annually • Semi-annually • Monthly • Continuously • Find the APY for a nominal rate of 8% in each case • From the chapter 6.3 – 11, 23, 25, 31, 43

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