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4 - 26 - 2012. Do Now. 1.) If there are initially 100 fruit flies in a sample, and the number of fruit flies decreases by one-half each hour, How many fruit flies will be present after 5 hours?.

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Do Now

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  1. 4 - 26 - 2012 Do Now 1.) If there are initially 100 fruit flies in a sample, and the number of fruit flies decreases by one-half each hour, How many fruit flies will be present after 5 hours? 2.) Sara bought 4 fish. Every month the number of fish she has doubles. After 6 months she will have how many fish.

  2. 4 - 27 - 2012 Do Now 1.) Simplify. 2.) 3.) Evaluate using this formula when P is 1219, r is 0.12, and t is 5.

  3. 4 - 30 - 2012 Do Now 1.) How many half-lives would it take to have a 700 gram sample of uranium reduce to under 3 grams of uranium ? 2.) If there are initially 10 bacteria in a culture, and the number of bacteria double each hour, find the number of bacteria after 24hours.

  4. 5 - 4 - 2012 Do Now When a person takes a dosage of I milligrams of a medicine, the amount A ( in milligrams) of medication remaining in the person’s bloodstream after t hours can be modeled by the equation . Using the formula, Find the amount of medication remaining in a person’s bloodstream if the dosage was 500 mg and 2.5 hours has lapsed.

  5. 5 - 4 - 2012 Do Now Compound Interest You want to have $ 20,000 in your account after 18 years. Find the amount your initial deposit should be if the account pays 4.5% annual interest compounded monthly. Identify: A = P = r = n = t =

  6. 5 - 4 - 2012 Do Now Compound Interest You want to have $ 20,000 in your account after 18 years. Find the amount your initial deposit should be if the account pays 4.5% annual interest compounded monthly. A = 20,000 P = ? r = .045 n = 12 t = 18

  7. 5 - 8 - 2012 Do Now In the equation , which of the following is true? a) There is a Growth Rate? g) The initial amount is 350? b) There is a Decay Rate? h) The time period is 10 ? I) “y”is the final amount? c) The Decay Rate is 75% ? d) The Decay Rate is 25% ? • This is an Exponential • ….Growth • ….Decay e) The Decay Factor is .25 ? f) The Decay Factor is (1 - .75) ?

  8. 5 - 9 - 2012 Do Now 1.) If you invested $ 2,000 at a rate of 0.6% compounded continuously, find the balance in the account after 5 years, use the formula $ 2,060.91 2. ) Simplify the Expression

  9. 5 - 10 - 2012 Do Now Exponential Growth Exponential Decay Compounded Interest ex) Compounded daily Compounded monthly Compounded quarterly ContinuouslyCompounded Interest

  10. 5 - 11 - 2012 Do Now 1.) RE-Write in Exponential form 2.) RE-Write in Logarithmic form 3.) Evaluate 4.) Graph

  11. Ch 7.1 Exponential Growth Whatyou should learn: 1 Goal Graph and use Exponential Growth functions. 2 Goal Write an Exponential Growth model that describes the situation. p. 478 A2.5.2 7.1 Graph Exponential Growth Functions

  12. Exponential Function • f(x) = bx where the base b is a positive number other than one. • Graph f(x) = 2x Notice the end behavior • As x → ∞ f(x) → ∞ • As x → -∞ f(x) → 0 • y = 0 is an asymptote

  13. What is an Asymptote? • A line that a graph approaches as you move away from the origin The graph gets closer and closer to the line y = 0 ……. But NEVER reaches it 2 raised to any power Will NEVER be zero!! y = 0

  14. Example 1 • Graph • Plot (0, ½) and (1, 3/2) • Then, from left to right, draw a curve that begins just above the x-axis, passes thru the 2 points, and moves up to the right y = 0 What do you think the Asymptote is?

  15. Example 2 • Graph y = - (3/2)x • Plot (0, -1) and (1, -1.5) • Connect with a curve • Mark asymptote • D = ?? • All reals • R = ??? • All reals < 0 y = 0

  16. Example 3Graph y = 3·2x-1 - 4 • Lightly sketch y = 3·2x • Passes thru (0,3) & (1,6) • h = 1, k = -4 • Move your 2 points to the right 1 and down 4 • AND your asymptote k units (4 units down in this case)

  17. Now…you try one! Example 4 • Graph y = 2·3x-2 +1 • State the Domain and Range! • D = all reals • R = all reals >1 y=1

  18. When a real-life quantity increases by a fixed percent each year, the amount y of the quantity after t years can be modeled by the equation where • a - Initial principal • r – percent increase expressed as a decimal • t – number of years • y – amount in account after t years Notice that the quantity (1 + r) is called the Growth Factor

  19. Example The amount of money, A, accrued at the end of n years when a certain amount, P, is invested at a compound annual rate, r, is given by If a person invests $310 in an account that pays 8% interest compounded annually, find the approximant balance after 5 years. A = $455.49

  20. Compound Interest Consider an initial principal P deposited in an account that pays interest at an annual rate, r, compounded n times per year. • P - Initial principal • r – annual rate expressed as a decimal • n – compounded n times a year • t – number of years • A – amount in account after t years

  21. Compound Interest example • You deposit $1000 in an account that pays 8% annual interest. • Find the balance after 1 year if the interest is compounded with the given frequency. • a) annually b) quarterly c) daily A=1000(1+.08/4)4x1 =1000(1.02)4 ≈ $1082.43 A=1000(1+.08/365)365x1 ≈1000(1.000219)365 ≈ $1083.28 A=1000(1+ .08/1)1x1 = 1000(1.08)1 ≈ $1080

  22. Whatyou should learn: Goal 1 Graph and use Exponential Decay functions. Ch 7.2Exponential Decay Goal 2 Write an Exponential Decay model that describes the situation. p. 486 A2.5.2 7.2 Graph Exponential decay Functions

  23. Discovery Education – Example 3: Exponential Decay-Bloodstream 7.2 Exponential Decay P. 486

  24. Exponential Decay • Has the same form as growth functions f(x) = a(b)x • Where a > 0 • BUT: 0 < b < 1 (a fraction between 0 & 1)

  25. Recognizing growth and decay functions • State whether f(x) is an exponential growth or DECAY function • f(x) = 5(2/3)x • b = 2/3, 0 < b < 1 it is a decay function. • f(x) = 8(3/2)x • b = 3/2, b > 1 it is a growth function. • f(x) = 10(3)-x • rewrite as f(x)= 10(1/3)xso it is decay

  26. Recall from 7.1: • The graph of y= abx • Passes thru the point (0,a) (the y intercept is a) • The x-axis is the asymptote of the graph • a tells you up or down • D is all reals (the Domain) • R is y>0 if a>0 and y<0 if a<0 • (the Range)

  27. Graph: • y = 3(1/4)x • Plot (0,3) and (1,3/4) • Draw & label asymptote • Connect the dots using the asymptote y=0 Domain = all reals Range = reals>0

  28. Graph • y = -5(2/3)x • Plot (0,-5) and (1,-10/3) • Draw & label asymptote • Connect the dots using the asymptote y=0 Domain : all reals Range : y < 0

  29. Now remember: To graph a general Exponential Function: • y = a bx-h + k • Sketch y = a bx • h= ??? k= ??? • Move your 2 points h units left or right …and k units up or down • Then sketch the graph with the 2 new points.

  30. Example graph y=-3(1/2)x+2+1 • Lightly sketch y=-3·(1/2)x • Passes thru (0,-3) & (1,-3/2) • h=-2, k=1 • Move your 2 points to the left 2 and up 1 • AND your asymptote k units (1 unit up in this case)

  31. y=1 Domain : all reals Range : y<1

  32. Using Exponential Decay Models • When a real life quantity decreases by fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by: • y = a(1-r)t • Where a is the initial amount and r is the percent decrease expressed as a decimal. • The quantity 1-r is called the decay factor Discovery Ed - Using functions to Gauge Filter Eff

  33. Ex: Buying a car! • You buy a new car for $24,000. • The value y of this car decreases by 16% each year. • Write an exponential decay model for the value of the car. • Use the model to estimate the value after 2 years. • Graph the model. • Use the graph to estimate when the car will have a value of $12,000.

  34. Let t be the number of years since you bought the car. • The model is: y = a(1-r)t • = 24,000(1-.16)t • = 24,000(.84)t • Note: .84 is the decay factor • When t = 2 the value is y=24,000(.84)2 ≈ $16,934

  35. Now Graph The car will have a value of $12,000 in 4 years!!!

  36. Whatyou should learn: 7.3Use Functions Involvinge Goal 1 Will study functions involving the Natural base e Goal 2 Simplify and Evaluate expressions involving e p. 492 Goal 3 Graph functions with e A3.2.2 7.3 Use Functions Involving e

  37. The Natural base e • Much of the history of mathematics is marked by the discovery of special types of numbers like counting numbers, zero, negative numbers, Л, and imaginary numbers. 7.3 Use Functions Involving e

  38. Natural Base e • Like Л and ‘i’, ‘e’ denotes a number. • Called The Euler Number after Leonhard Euler (1707-1783) • It can be defined by: e= 1 + 1 + 1 + 1 + 1 + 1 +… 0! 1! 2! 3! 4! 5! = 1 + 1 + ½ + 1/6 + 1/24 + 1/120+... ≈ 2.718281828459…. 7.3 Use Functions Involving e

  39. The number e is irrational – its’ decimal representation does not terminate or follow a repeating pattern. • The previous sequence of e can also be represented: • As n gets larger (n→∞), (1+1/n)n gets closer and closer to 2.71828….. • Which is the value of e. 7.3 Use Functions Involving e

  40. Examples (3e-4x)2 9e(-4x)2 9e-8x 9 e8x 10e3 5e2 2e3-2 2e e3 · e4 e7 7.3 Use Functions Involving e

  41. More Examples! (2e-5x)-2 2-2e10x e10x 4 24e8 8e5 3e3 7.3 Use Functions Involving e

  42. Using a calculator 7.389 • Evaluate e2 using a graphing calculator • Locate the exbutton • you need to use the second button 7.3 Use Functions Involving e

  43. Evaluate e-.06witha calculator 7.3 Use Functions Involving e

  44. Graphing • f(x) = aerxis a natural base exponential function • If a > 0 & r > 0 it is a growth function • If a > 0 & r < 0 it is a decay function 7.3 Use Functions Involving e

  45. Graphing examples • Graph y = ex • Remember the rules for graphing exponential functions! • The graph goes thru (0,a) and (1,e) (1,2.7) (0,1) 7.3 Use Functions Involving e

  46. Graphing cont. • Graph y = e-x (1,.368) (0,1) 7.3 Use Functions Involving e

  47. Graphing Example • Graph y = 2e0.75x • State the Domain & Range • Because a=2 is positive and r=0.75, the function is exponential growth. • Plot (0,2)&(1,4.23) and draw the curve. (1,4.23) (0,2) 7.3 Use Functions Involving e

  48. Using e in real life. • In 8.1 we learned the formula for compounding interest n times a year. • In that equation, as n approaches infinity, the compound interest formula approaches the formula for continuously compounded interest: A = Pert 7.3 Use Functions Involving e

  49. Example of Continuously compounded interest You deposit $1000.00 into an account that pays 8% annual interest compounded continuously. What is the balance after 1 year? P = 1000, r = .08, and t = 1 A = Pert = 1000e.08*1 ≈ $1083.29 7.3 Use Functions Involving e

  50. mathbook Whatyou should learn: 7.4 Logarithms Goal 1 Evaluate logarithms Graph logarithmic functions Goal 2 p. 499 A3.2.2 7.4 Evaluate Logarithms and Graph Logarithmic Functions

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