456 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions n a1 + 1 nb n (rounded) 1 2 2 2.25 5 2.48832 10 2.59374 100 2.70481 1000 2.71692 10,000 2.71815 1,000,000 2.71828 The Number e and Continuous Compounding The more often interest is compounded within a given time period, the more interest will be earned. Surprisingly, however, there is a limit on the amount of interest, no matter how often it is compounded. Suppose that $1 is invested at 100% interest per year, compounded n times per year. Then the interest rate (in decimal form) is 1.00, and the interest rate per period is 1 n . According to the formula (with P = 1), the compound amount at the end of 1 yr will be A = a1 + 1 nb n . Let P = 1, r = 1, and t = 1 in A = PA1 + r nB t n . A calculator gives the results in the margin for various values of n. The table suggests that as n increases, the value of A1 + 1 nB n gets closer and closer to some fixed number. This is indeed the case. This fixed number is called e. (In mathematics, e is a real number and not a variable.) Value of e e ?2.718281828459045 Figure 21 shows graphs of the functions y = 2x, y = 3x, and y = ex. Because 2 6e 63, the graph of y = ex lies “between” the other two graphs. As mentioned above, the amount of interest earned increases with the frequency of compounding, but the value of the expression A1 + 1 nB n approaches e as n gets larger. Consequently, the formula for compound interest approaches a limit as well, called the compound amount from continuous compounding. x y 0 –1 1 2 3 4 5 6 7 8 1 2 y = 2x y = ex y = 3x Figure 21 Continuous Compounding If P dollars are deposited at a rate of interest r compounded continuously for t years, then the compound amount A in dollars on deposit is given by the following formula. A =Pert EXAMPLE 9 Solving a Continuous Compounding Problem Suppose $5000 is deposited in an account paying 3% interest compounded continuously for 5 yr. Find the total amount on deposit at the end of 5 yr. SOLUTION A = Pert Continuous compounding formula A = 5000e0.03152 Let P = 5000, r = 0.03, and t = 5. A = 5000e0.15 Multiply exponents. A ≈5809.17 Use a calculator. The total amount on deposit is $5809.17. S Now Try Exercise 97(b).
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