# 6.6 The Natural Base, e.

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6.6 The Natural Base, e

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

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

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

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

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

The Graph of the Exponential Function
1 3 2

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.

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.

Try This! Graph the function. State the domain and range. y 2e 0.5x =
domain: all real numbers, range: y>0

Try This! Graph the function. State the domain and range. 1 2 f(x) =
e -x +1 domain: all real numbers, range: y>1

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

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

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

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

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