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5.2 exponential functions

5.2 exponential functions. Quiz. Fill in the blanks below: 2 x+y = __ * __. Exponential function. Standard form: f(x) = a x , where a>0, a ≠ 1. Example: f(x) = 2 x , f(x) = (1/3) x Compare g(x) = x 2 and f(x) = 2 x. properties. Graph f(x) = 2 x and g(x) = (1/2) x.

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5.2 exponential functions

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  1. 5.2 exponential functions

  2. Quiz • Fill in the blanks below: 2x+y = __ * __

  3. Exponential function • Standard form: f(x) = ax , where a>0, a ≠ 1. • Example: f(x) = 2x, f(x) = (1/3)x • Compare g(x) = x2 and f(x) = 2x

  4. properties • Graph f(x) = 2x and g(x) = (1/2)x

  5. Properties • Characteristics of f(x) • Continuous • One to one • Domain: (- ∞, ∞) • Range: (0, ∞) • Increasing if a>1(growth) • Decreasing if 0<a<1(Decay) • Horizontal asymptote at y = 0. • Key points on the graph: (1,a), (0,1)

  6. Graphs by transformations • Describe how each of the following can be obtained from the graph of f(x) = 2x. a. f(x) = 2x+3 b. f(x) = 2x – 1 c. f(x) = 3 + 2-x Example: exercise #37, #69

  7. Exponential Equations ab = ac b = c • Solve for x: 1. 2x-3 = 8 2. (1/4)3 = 8x 3. 274x = 9x+1 Use a graphing calculator to solve 2x-3 > 8 or 2x-3 ≤ 8

  8. Natural Base -- e • e = (1 + 1/k)k as k approaches positive infinite. • e ≈ 2.71828 • Natural exponential function: f(x) = ex

  9. Compound interest • Suppose that a principal of P dollars is invested at an annual interest rate r, compounded n times per year. Then the amount A accumulated after t years is given by the formula A = P(1 + r/n)nt A = Amount accumulated after t years P = principal r = annual interest rate n = compounded times of a year

  10. Typical Compounding periods • Compound annually: n = 1 • Compound semi- annually: n = 2 • Compound quarterly: n = 4 • Compound monthly: n = 12 • Compound weekly: n = 52 • Compound daily: n = 365

  11. example • Suppose that $100,000 is invested at 6.5% interest, compounded semi-annually. 1. Find a function for the amount of money after t years 2. Find the amount of money in the account at t = 1,4,10 years.

  12. Continuous Compounding • As the number of compounding periods increases without bound, the model becomes A = Pert

  13. example • If you put $7000 in an money market account that pays 4.3% a year compounded continuously, how much will be in the account in 15 years? • You have $1500 to invest. Which is better – 2.25% compounded quarterly for 3 years? Or 1.75% compounded continuously for r years?

  14. Homework • PG. 339: 3-18(M3), 38, 39-75(M3), 80 • KEY: 38, 54, 60, 75 • Reading: 5.3 Logarithms and their properties

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