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Understanding Compound Interest and Exponential Functions

Learn about compound interest formulas and exponential functions with detailed explanations and examples. Explore continuously compounded interest, effective annual yield, and continuous growth rates.

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Understanding Compound Interest and Exponential Functions

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  1. Section 4.1 Compound Interest and Exponential Functions

  2. If A0 is invested at an annual interest rate r and compounded semiannually, the amount At after t years is given by the formula INTEREST COMPOUNDED SEMIANNUALLY

  3. If A0 is invested at an annual interest rate r and compounded quarterly, the amount At after t years is given by the formula INTEREST COMPOUNDED QUARTERLY

  4. If A0 is invested at an annual interest rate r and compounded n times annually, the amount At after t years is given by the formula INTEREST COMPOUNEDn TIMES A YEAR

  5. THE NUMBER e The number e is an irrational number, one whose decimal expansion never terminates nor repeats. e≈ 2.718281828459045 e ≈ 2.7 1828 1828 45 90 45

  6. THE NATURAL EXPONENTIAL FUNCTION The exponential function with base e f (x) = ex is called the natural exponential function.

  7. Continuously Compounded Interest: If the initial amount A0 is invested at an annual interest rate r compounded continuously, then the resulting amount after t years is given by CONTINUOUSLY COMPOUNDED INTEREST

  8. EFFECTIVE ANNUAL YIELD Effective Annual Yield: The effective annual yield for an investment is the percentage rate that would yield the same amount of interest if the interest were compounded annually.

  9. CONTINUOUS GROWTH WITH RATE r Continuous Growth with Rate r: If the growth of a quantity is described by the function P(t) =P0ert, then we say that it grows continuously and has continuous growth rate r. The function P(t) = P0ert represents continuous grow if r is positive and continuous decay if r is negative.

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