457 4.2 Exponential Functions Exponential Models The number e is important as the base of an exponential function in many practical applications. In situations involving growth or decay of a quantity, the amount or number present at time t often can be closely modeled by a function of the form y =y0e kt, where y0 is the amount or number present at time t = 0 and k is a constant. Exponential functions are used to model the growth of microorganisms in a culture, the growth of certain populations, and the decay of radioactive material. EXAMPLE 10 Comparing Interest Earned as Compounding Is More Frequent In Example 7, we found that $1000 invested at 4% compounded quarterly for 10 yr grew to $1488.86. Compare this same investment compounded annually, semiannually, monthly, daily, and continuously. SOLUTION Substitute 0.04 for r, 10 for t, and the appropriate number of compounding periods for n into the formulas A = Pa1 + r nb tn Compound interest formula and A = Pert. Continuous compounding formula The results for amounts of $1 and $1000 are given in the table. Compounded $1 $1000 Annually 11 + 0.04210 ≈1.48024 $1480.24 Semiannually a1 + 0.04 2 b 10122 ≈1.48595 $1485.95 Quarterly a1 + 0.04 4 b 10142 ≈1.48886 $1488.86 Monthly a1 + 0.04 12 b 101122 ≈1.49083 $1490.83 Daily a1 + 0.04 365 b 1013652 ≈1.49179 $1491.79 Continuously e1010.042 ≈1.49182 $1491.82 Comparing the results for a $1000 investment, we notice the following. • Compounding semiannually rather than annually increases the value of the account after 10 yr by $5.71. • Quarterly compounding grows to $2.91 more than semiannual compounding after 10 yr. • Daily compounding yields only $0.96 more than monthly compounding. • Continuous compounding yields only $0.03 more than daily compounding. Each increase in compounding frequency earns less additional interest. S Now Try Exercise 105. LOOKING AHEAD TO CALCULUS In calculus, the derivative allows us to determine the slope of a tangent line to the graph of a function. For the function ƒ1x2 = ex, the derivative is the function ƒ itself: ƒ′1x2 = ex. Therefore, in calculus the exponential function with base e is much easier to work with than exponential functions having other bases.
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