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Unformatted text preview: Math 006 (Lecture 18) Continuous Compound Interest and the Constant e Definition 1. The number e is defined by e = lim n →∞ 1 + 1 n n = lim s → (1 + s ) 1 /s ≈ 2 . 7182818284590 ..... If a principal P is borrowed at an annual rate r , then after t years, the borrower will owe the lender an amount A given by A = P 1 + r n nt . Now we let n go to infinity, we find that lim n →∞ P 1 + r n nt = P lim n →∞ 1 + r n ( n/r ) rt = P [lim s → (1 + s ) 1 /s ] rt = Pe rt . The above formula is called the continuous compound interest formula. Example 1. If $100 is invested at 6%, what amount will be in the account after 2 years, if 1. compounded monthly? 2. compounded continuously? Example 2. How long will it take to double an investment if it is invested at 5% com- pounded continuously? 1 Derivative of e x Definition 2. f ( x ) = e x is called the exponential function with base e . More generally, f ( x ) = a x , where a > 0, is called the exponential function with base a . Exponential....
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- Fall '09