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7.3 The Natural Base e

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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. if b>0…growth if b<0…decay Let’s take a look at the graph of a natural base exponential function…

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

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Ex. Graph f(x) = e x x f(x) –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

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Simplify. a. e 3 e 5 b. 8e 6 2e 2 c. (2e –3x ) –2 = e 8 = 4e 4 = 2 –2 e 6x = e 6x 4 To simplify natural base expressions, follow the Properties of Exponents:

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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 = Pe rt 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 = Pe rt = 5000 e.085 5 = $7647.95

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

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Linear vs. exponential growth Linear vs. exponential growth: t = 0 A = 1x(1+1) 0 = 1 A = 1x0 + 1 = 1.

Linear vs. exponential growth Linear vs. exponential growth: t = 0 A = 1x(1+1) 0 = 1 A = 1x0 + 1 = 1.

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