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7.3 The Natural Base e

7.3 The Natural Base e. The Natural Base e. A famous constant (similar to π) is the natural base, e (also known as Euler’s number). e ≈ 2.718281828459 e is irrational (non-repeating, not-terminating). A function in the form f(x) = Ce bx is a natural base exponential function.

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7.3 The Natural Base e

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  1. 7.3 The Natural Base e

  2. The Natural Base e A famous constant (similar to π) is the natural base, e (also known as Euler’s number). e ≈ 2.718281828459 e is irrational (non-repeating, not-terminating). A function in the form f(x) = Cebx is a natural base exponential function. if b>0…growth if b<0…decay Let’s take a look at the graph of a natural base exponential function…

  3. –2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Ex. Graph f(x) = ex x f(x)

  4. –2 -1.5 -1 -0.5 0 0.5 1 1.5 2 .14 .22 .37 .61 1 1.65 2.72 4.48 7.39 Ex. Graph f(x) = ex x f(x)

  5. To simplify natural base expressions, follow the Properties of Exponents: Simplify.a. e3 • e5b. 8e6 2e2c. (2e–3x)–2 = e8 = 4e4 =2–2e6x = e6x 4

  6. Recall that we have studied the formula for the value of an investment in which interest is compounded n times per year: A = P(1 + r/n)nt Here is another common investment formula: A = Pert A is the balance P is the initial principal r is the interest rate t is the number of years In this formula, the interest is compounded continuously (about every second) Example: For a deposit of $5000 into an account that pays 8.5% interest, find the balance after 5 years if the interest is compounded continuously. A = Pert = 5000 • e.085 • 5 = $7647.95

  7. Homework: p.495-497 #3-39 multiples of 3, #56 and #57

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