1. 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? Do Now 4 - 26 - 2012 2.) Sara bought 4 fish. Every month the number of fish she has doubles. After 6 months she will have how many fish.

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

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

4. Do Now 5 - 4 - 2012 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. 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. Do Now 5 - 4 - 2012 Identify: A = P = r = n = t =

6. 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. Do Now 5 - 4 - 2012 A = 20,000 P = ? r =.045 n = 12 t = 18

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

8. Do Now 5 - 9 - 2012 continuously 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. ) Simplify the Expression $ 2,060.91

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

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

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

12. Exponential Function f(x) = b x b where the base b is a positive number other than one. Graph f(x) = 2 x 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 y = 0 2 raised to any power Will NEVER be zero!!

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 What do you think the Asymptote is? y = 0

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 3 Graph y = 3·2 x-1 - 4 Lightly sketch y = 3·2 x 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! Graph y = 2·3 x-2 +1 State the Domain and Range! D = all reals R = all reals >1 y=1 Example 4

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 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+.08/1) 1x1 = 1000(1.08) 1 ≈ $1080 A=1000(1+.08/4) 4x1 =1000(1.02) 4 ≈ $1082.43 A=1000(1+.08/365) 365x1 ≈1000(1.000219) 365 ≈ $1083.28

22. Ch 7.2 Exponential Decay p. 486 What you should learn: Goal1 Goal2 Graph Graph and use Exponential Decay functions. Write an Exponential Decay model that describes the situation. 7.2 Graph Exponential decay Functions A2.5.2

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

24. Exponential Decay Has the same form as growth functions f(x) = a(b) x Where a > 0 BUT: fraction 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) x so it is decay

26. Recall from 7.1: The graph of y= ab x 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 Domain = all reals Range = reals>0 y=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 b x-h + k Sketch y = a b x 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 arWhere 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 EffFilter

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. 7.3 Use Functions Involving e p. 492 What you should learn: Goal1 Goal2 Will study functions involving the Natural base e Simplify and Evaluate expressions involving e 7.3 Use Functions Involving e A3.2.2 Goal3 Graph functions with 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 e 3 · e 4 e7e7 10e 3 5e 2 2e 3-2 2e2e (3e -4x ) 2 9e (-4x)2 9e -8x 9 e 8x 7.3 Use Functions Involving e

41. More Examples! 24e 8 8e 5 3e33e3 (2e -5x ) -2 2 -2 e 10x e 10x 4 7.3 Use Functions Involving e

42. Using a calculator Evaluate e 2 using a graphing calculator Locate the e x button you need to use the second button 7.389 7.3 Use Functions Involving e

43. Evaluate e -.06 with a calculator 7.3 Use Functions Involving e

44. Graphing f(x) = a e rx is 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 = e x Remember the rules for graphing exponential functions! The graph goes thru (0,a) and (1,e) (0,1) (1,2.7) 7.3 Use Functions Involving e

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

47. Graphing Example Graph y = 2 e 0.75 x 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. (0,2) (1,4.23) 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 = Pe rt 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? 7.3 Use Functions Involving e P = 1000, r =.08, and t = 1 A = Pe rt = 1000e.08*1 ≈ $1083.29

50. 7.4 Logarithms p. 499 What you should learn: Goal1 Goal2 Evaluate logarithms Graph logarithmic functions 7.4 Evaluate Logarithms and Graph Logarithmic Functions A3.2.2 mathbook

51. Evaluating Log Expressions We know 2 2 = 4 and 2 3 = 8 But for what value of y does 2 y = 6 ? Because 2 2 < 6 < 2 3 you would expect the answer to be between 2 & 3. To answer this question exactly, mathematicians defined logarithms.

52. Definition of Logarithm to base a Let a & x be positive numbers & a ≠ 1. The logarithm of x with base a is denoted by log a x and is defined: log a x = y iff a y = x This expression is read “log base a of x” The function f(x) = log a x is the logarithmic function with base a.

53. The definition tells you that the equations log a x = y and a y = x are equivilant. Rewriting forms: To evaluate log 3 9 = x ask yourself… “Self… 3 to what power is 9?” 3 2 = 9 so…… log 3 9 = 2

54. Log form Exp. form log 2 16 = 4 log 10 10 = 1 log 3 1 = 0 log 10.1 = -1 log 2 6 ≈ 2.585 2 4 = 16 10 1 = 10 3 0 = 1 10 -1 =.1 2 2.585 = 6

55. Evaluate without a calculator log 3 81 = Log 5 125 = Log 4 256 = Log 2 (1/32) = 3 x = 81 5 x = 125 4 x = 256 2 x = (1/32) 4 3 4 -5

56. Evaluating logarithms now you try some! Log 4 16 = Log 5 1 = Log 4 2 = Log 3 (-1) = (Think of the graph of y=3 x ) 2 0 ½ ( because 4 1/2 = 2) undefined

57. You should learn the following general forms!!! Log a 1 = 0 because a 0 = 1 Log a a = 1 because a 1 = a Log a a x = x because a x = a x

58. Natural logarithms log e x = ln x ln means log base e

59. Common logarithms log 10 x = log x Understood base 10 if nothing is there.

60. Common logs and natural logs with a calculator log 10 button ln button

61. g(x) = log b x is the inverse of f(x) = b x f(g(x)) = x and g(f(x)) = x Exponential and log functions are inverses and “undo” each other

62. So: g(f(x)) = log b b x = x f(g(x)) = b log b x = x 10 log2 = Log 3 9 x = 10 logx = Log 5 125 x = 2 Log 3 (3 2 ) x =Log 3 3 2x =2x x 3x

63. Finding Inverses Find the inverse of: y = log 3 x By definition of logarithm, the inverse is y=3 x OR write it in exponential form and switch the x & y! 3 y = x 3 x = y

64. Finding Inverses cont. Find the inverse of : Y = ln (x +1) X = ln (y + 1) Switch the x & y e x = y + 1 Write in exp form e x – 1 = y solve for y