1. Welcome to Week 3 College Trigonometry

2. Exponentials Functions that contain exponents: f (x) = x c x is the base c is the exponent

3. Exponentials IN-CLASS PROBLEMS

4. Exponentials Exponential functions: f (x) = c x c is the base x is the exponent

5. Exponentials Limitations: the base must be a constant the base must be >0 the exponent must be a variable

6. Functions containing an exponent or exponential functions? f (w) = w 9 f (y) = 72 y f (x) = -83.2 x f (x) = x 50 f (z) = (8z+1) z f (x) = (x 2 +4)6 x Exponentials IN-CLASS PROBLEMS

7. Exponentials Graphs of Exponential Functions

8. Exponentials If the base is > 1, then the graph goes up to the right If 0<base<1, then the graph goes up to the left

9. Exponentials All pass through the point (0,1) because a 0 =1

10. Exponentials The domain (x s ) is (-∞,∞) The range (y s ) is (0,∞)

11. Questions?

12. Exponentials

14. Iteration Means repeating a process until you reach a goal

15. Iteration In math class, we are trying to solve an equation by closer and closer approximations to the “true” answer

16. Iteration So, we can manually tweak values of the coefficients until we get the best fit to our data This is actually how the computer calculates trig functions!

17. Iteration When we did the West Africa rainfall model, we tried different values for each of the constants in the equation until we got a “fit” we liked

18. Iteration That was iteration!

19. Iteration Iteration is used by your calculator and in computer software to do calculations

20. Iteration Some people think their calculators have a huge look-up table for all the answers

21. Iteration They don’t – it would be impossible for a calculator to have a look-up table for every possible value a user might enter for every possible function the user might want to use!

22. Iteration The calculator has a formula for every function that will zero in on the right answer after a short number of calculations

23. Iteration The calculator company wants a formula that zeros in on a stable number after the fewest number of calculations

24. Which is the better formula? Iteration IN-CLASS PROBLEMS

25. Iteration The one that gets there quickest!

26. Iteration IN-CLASS PROBLEMS

27. Iteration All the functions in your calculator are based on a long series that iterates back and forth and zeros in on the correct answer

28. Iteration

29. The formula selected by your calculator manufacturer determines how fast and accurate your calculator will be

30. Iteration It’s why different people get slightly different answers to calculator problems

31. Exponentials Irrational number e ≈ 2.718281827

32. Exponentials The natural number e was first calculated by the Swiss Mathematician Jacob Bernoulli He called it “b” https://en.wikipedia.org/wiki/Jacob_Bernoulli#/media/File:Jakob_Bernoulli.jpg

33. Exponentials But it’s also called “Euler’s Number” after the Swiss Mathematician Leonhard Euler – he first called it “e” https://en.wikipedia.org/wiki/Leonhard_Euler#/media/File:Leonhard_Euler.jpg

34. Use a calculator to find: e 2.3 e 3.4 e -0.95 e -0.75 e 7 Exponentials IN-CLASS PROBLEMS

35. Exponentials e is used a lot in population models, compound interest, things that are growing "exponentially"

36. Grey Wolf Population f(x) = 1.85x 2 + 9.4x + 783 vs f(x) = 1.26e 0.264x Which is the better model? Exponentials IN-CLASS PROBLEMS

37. Questions?

38. Logarithms Logarithms were invented by Scottish Baron John Napier in 1614

39. Logarithms AND by Swiss craftsman Joost Bürgi in 1620

40. Logarithms Napier defined logarithms as the algebraic ratio of two distances in a geometric form while Bürgi’s definition was purely geometric

41. Logarithms Now we use them totally differently: as an exponent

42. Logarithms Logarithmic Functions y = log b x is the same as b y = x if x>0, b>0 and b≠1

43. logarithmic function with base b ~or~ exponential function with base b Logarithms

44. log 2 16 2 to what power gives 16? Logarithms

45. log 2 16 = 4 because 2 4 = 16 Logarithms

46. Exponentials IN-CLASS PROBLEMS

47. Logarithms The default logarithm has base 10: log 10 x = log x

48. Logarithms Some definitions: log b 1 = 0log 1 = 0 log b b = 1log 10 = 1 log b b x = xlog 10 x = x b logb x = x10 logx = x

49. Logarithms Logarithms are the inverse of exponential functions

50. Write in exponential form: 4 = log 2 166 = log 2 64 2 = log 9 x3 = log x 27 log 5 125 = y Exponentials IN-CLASS PROBLEMS

51. Write in logarithmic form: 2 3 = 8 5 4 = 625 15 2 = x8 y = 300 5 -3 = 1/125 Exponentials IN-CLASS PROBLEMS

52. Evaluate (w/o calc): log 4 16log 7 49 log 6 1/6log 11 11 Exponentials IN-CLASS PROBLEMS

53. Logarithms Natural logarithms: logs with base e written "ln"

54. Logarithms log b 1 = 0ln 1 = 0 log b b = 1ln e = 1 log b b x = xln e x = x b logb x = xe lnx = x

55. Use a calculator to find: ln (7)ln(2.72) ln(-3)ln(896.5) ln e 7 e ln 125 Exponentials IN-CLASS PROBLEMS

56. Logarithms vs e Exponents Graph demo

57. Percentage of college graduates Exponentials IN-CLASS PROBLEMS

58. Change of temperature in an enclosed vehicle (∆ means “change”) Exponentials IN-CLASS PROBLEMS

59. Liberation! Be sure to turn in your assignments from last week to me before you leave Don’t forget your homework due next week! Have a great rest of the week!