6 6 The Natural Base e Natural Base
6. 6 The Natural Base, e
Natural Base, e n The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.
Natural Exponential Function n n The exponential function with base e, f(x)=ex, is called the natural exponential function. And e is called the You calculator has a natural. FANCY base. ex button you can use!!!
Evaluate f(x)=ex to the nearest thousandth for each value of x below: n x=2 n x = -1/3 n x = 0. 25
Exponential Properties Simplify the expression. a. e 2 e 5 = e 2 + 5 = e 7 4 12 e 4– 3 4 b. e 3 e 3 = = 4 e c. ( 2 5 e – 3 x ) = 52 2 – 3 x e ( = 25 e – 6 x = 25 e 6 x )
Try This! Simplify the expression. 1. e 7 e 4 2. 2 e– 3 6 e 5 e 11 3 – 4 x ( 10 4. e ) 1000 e 12 x 12 e 2 5. Use a calculator to evaluate 3. 24 e 8 4 e 5 6 e 3 2. 117
The Graph of the Exponential Function
Graph natural base functions Graph the function. State the domain and range. a. y = 3 e 0. 25 x 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 0 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 = e – 0. 75 x right 2 units and up 1 unit. The domain is all real numbers, and the range is y > 1.
Try This! Graph the function. State the domain and range. y = 2 e 0. 5 x domain: all real numbers, range: y>0
Try This! Graph the function. State the domain and range. f(x) = 1 e -x +1 2 domain: all real numbers, range: y>1
Continuous Compounding Formula n 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% n Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. n Compounded Quarterly Continuously
Find the value: n Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. $716. 66
Homework Green Book!! Page 142 # 1 -25 odd #27 and 28
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