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6.6 The Natural Base, e

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Natural Base, e The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

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**Natural Exponential Function**

The exponential function with base e, f(x)=ex, is called the natural exponential function. And e is called the natural base. 1 You calculator has a FANCY ex button you can use!!! 2

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**Evaluate f(x)=ex to the nearest thousandth for each value of x below:**

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**Exponential Properties**

Are the same for base e!! Simplify the expression. a. e2 e5 = e2 + 5 = e7 b. 12 e4 3 e3 = e4 – 3 4 = 4e (5 ) c. e –3x 2 = 52 ( e –3x ) 2 = 25 e –6x = 25 e6x

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**Try This! (10 ) Simplify the expression. 1. e7 e4 e11 4. e –4x 3 1000**

2. 2 e–3 6e5 12 e2 Use a calculator to evaluate 3. 24 e8 4 e5 6 e3 2.117

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**The Graph of the Exponential Function**

1 3 2

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**Graph natural base functions**

Graph the function. State the domain and range. a. y = 3e 0.25x SOLUTION Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plug in 0 and 1. x y 1 The domain is all real numbers, and the range is y > 0.

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**Graph natural base functions**

Graph the function. State the domain and range. b. y = e –0.75(x – 2) + 1 SOLUTION a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit. e –0.75x The domain is all real numbers, and the range is y > 1.

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**Try This! Graph the function. State the domain and range. y 2e 0.5x =**

domain: all real numbers, range: y>0

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**Try This! Graph the function. State the domain and range. 1 2 f(x) =**

e -x +1 domain: all real numbers, range: y>1

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**Continuous Compounding Formula**

If P dollars are invested at an interest rate, r, that is compounding continuously, then the amount, A, of the investment at time t is given by: A = Pert

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**An investment of $1000 earns an annual interest rate of 7.6%**

Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. Compounded Quarterly Continuously

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Find the value: Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. $716.66

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Homework Green Book!! Page 142 # 1-25odd #27 and 28

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Warm-Up Exercises Evaluate the expression without using a calculator. ANSWER –1 ANSWER 27 1. 1 3 –3 2.– 2 5 0.

Warm-Up Exercises Evaluate the expression without using a calculator. ANSWER –1 ANSWER 27 1. 1 3 –3 2.– 2 5 0.

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