6.6 The Natural Base, e Objectives: Evaluate natural exponential and

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6.6 The Natural Base, e.

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Presentation transcript:

6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

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

The Natural Exponential Function The exponential function with base e, is called the natural exponential function and e is called the natural base. The function is graphed. Notice that the domain is all real numbers The range is all positive numbers.

c. x = -1 e-1 = .368 x= 2 e2 = 7.389 b. x= ½ e1/2 = 1.649 d. x = 6 Ex 1. Evaluate f(x) = ex to the nearest thousandth for each value of x below. c. x = -1 e-1 = .368 x= 2 e2 = 7.389 b. x= ½ e1/2 = 1.649 d. x = 6 e6 = 403.429 e. x = 1/3 e1/3 = 1.396 f. x = -2 e-2 = .135

Continuous Compounding Formula

Continuously Quarterly A = Pert A = P(1+ r/n)nt A = 1000e .076 * 8 Ex 2 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. Quarterly A = P(1+ r/n)nt A = 1000(1+ .076/4)4*8 A = 1826.31 Continuously A = Pert A = 1000e .076 * 8 A = 1836.75 TRY THIS: top of 394

Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. P = 500 t = 4 r = .09 A = 500e .36 = $716.66

Page 396 #7

The Natural Logarithmic Function The natural logarithmic function y = loge x, abbreviated y = In x, is the inverse of the natural exponential function, y = ex. The function y = In x is graphed along with y = ex. y=x y=ex y = In x

What are the domain and range the exponential and natural log functions? All positive real numbers All real numbers All real numbers All positive real numbers

b. x = ½ In ½ = -.693 c. x = -1 In -1 = undefined d. x = 5 Ex 3 Evaluate f(x) = ln x to the nearest thousandth for each value of x below. x = 2 ln 2 = .693 b. x = ½ In ½ = -.693 c. x = -1 In -1 = undefined d. x = 5 In 5 = 1.609 e. x = 0.85 In.85 = -.163 f. x = 1 In 1 = 0

The natural logarithmic function can be used to solve an equation of the form A = Pert for the exponent t in order to find the time it takes for an investment that is compounded continuously to reach a specific amount.

Ex. 4 How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously?

“Try This” – p. 395

“Critical Thinking” – page 395 Think, Pair, Share

p. 403 has a useful summary table:

Try these: p. 397 #13-27 odd

p. 397: #41-47 odd

Homework: Practice 6.6 Chapter 6 Test – Thu 4/29