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6.6 The Natural Base, e

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

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6.6 The Natural Base, e

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  1. 6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

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

  3. 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.

  4. 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

  5. Continuous Compounding Formula

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

  7. 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

  8. Page 396 #7

  9. 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

  10. 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

  11. 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

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

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

  14. “Try This” – p. 395

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

  16. p. 403 has a useful summary table:

  17. Try these: p. 397 #13-27 odd

  18. p. 397: #41-47 odd

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

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