The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number.

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Presentation transcript:

The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately, The number e is called the natural base. The function f (x) = e x is called the natural exponential function. f (x) = e x f (x) = 2 x f (x) = 3 x (0, 1) (1, 2) (1, e) (1, 3)

Exponential Growth or Decay (in terms of e ) N = N 0 e kt, where N is the total amount, N 0 is the beginning amount, k is the constant, and t is the time. y = e x is one of the most famous exponential functions.

Formulas for Compound Interest After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1.For n compoundings per year: 2.For continuous compounding: A = Pe rt.

Example:Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? SolutionThe better investment is the one with the greater balance in the account after 6 years. Let’s begin with the account with monthly compounding. We use the compound interest model with P = 8000, r = 7% = 0.07, n = 12 (monthly compounding, means 12 compoundings per year), and t = 6. The balance in this account after 6 years is $12, more

Example:Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? SolutionFor the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = , and t = 6. The balance in this account after 6 years is $12,066.60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option.

Example Use A= Pe rt to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously Solution: A= Pe rt A = 2000e (.07)(8) A = 2000 e (.56) A = 2000 * 1.75 A = $3500