CAL 11 Q3 0101 PF FINAL.pptx212233445666

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1. Basic Calculus Science, Technology, Engineering, and Mathematics Lesson 1.1 Evaluating Limits through Table of Values

2. When you ride a car, do you notice its speed? 2

3. 3 Usually, we get the average speed of the car for a certain duration, like one hour or one minute. But is it possible to estimate the speed of the car at a particular instant?

4. 4 To answer this question, we need the concept of limits . In particular, we will talk about limits of functions.

5. 5 Illustrate the limit of a function using a table of values ( STEM_BC11LC-IIIa-1).

6. 6 Define the limit of a function. Define one-sided limits. Define infinite limits. Estimate the limit of a function using tables of values.

7. 7 Is it correct to say that we can only get an estimate of a limit? Why do you say so?

8. 8 Investigate what happens to the values of the linear function as approaches 2.   Limit of a Function

9. 9 Limit of a Function from the left of 2: from the right of 2:    

10. 10 Limit of a Function from the left of 2:  

11. 11 Limit of a Function from the right of 2:  

12. 12 Limit of a Function from the right of 2:   from the left of 2:  

13. 13 Limit of a Function We say that “the limit of as approaches 2 is 6.” In symbols,  

14. 14 Intuitive Definition of a Limit Suppose the function is defined when is near . If gets closer to from both sides as gets closer to , then we say that “ the limit of as approaches is equal to .”  

15. 15 Estimate the limit of the quadratic function as approaches using tables of values.  

16. 16   Estimate the limit of the quadratic function as approaches using tables of values.  

17. 17 17 Estimate the limit of the function as approaches using tables of values.  

18. 18 Estimate using table of values.  

19. 19   Estimate using table of values.  

20. 20 20 Estimate using tables of values.  

21. 21 Estimate using table of values.  

22. 22   Estimate using table of values.  

23. 23 23 Estimate using tables of values.  

24. 24 In finding the limit, we are only concerned about the value being approached by the function as approaches a number . The function need not be defined at .  

25. 25 Does the limit of a function always exist?

26. 26 One-Sided Limits Left-hand Limit: Suppose the function is defined when is near from the left. Then, the limit of as approaches from the left is equal to a number . This can be written as  

27. 27 One-Sided Limits Right-hand Limit: Suppose the function is defined when is near from the right. Then, the limit of as approaches from the right is equal to a number . This can be written as  

28. 28 One-Sided Limits If the left- and right-hand limits of a function as approaches are both equal to a certain real number, then the limit of the function as approaches exists . Otherwise, it does not exist .  

29. 29 Estimate the limit of the signum function as approaches zero from the left.    

30. 30   Estimate the limit of the signum function as approaches zero from the left.    

31. 31 31 Given the piecewise function below, estimate using a table of values.    

32. 32 Given the function below, estimate .    

33. 33   Given the function below, estimate .    

34. 34 34 Estimate the limit of the signum function as approaches zero from the right.    

35. 35 Given the function below, estimate , , and .    

36. 36 does not exist.   Given the function below, estimate , , and .    

37. 37 37 Given the function below, estimate , , and .    

38. 38 Infinite Limits A function may not have a limit as approaches a certain value because it increases or decreases indefinitely. In this case, we will use the concept of infinity .  

39. 39 Infinite Limits Suppose the function is defined when is as near as possible to on both sides.  

40. 40 Infinite Limits Suppose the function is defined when is as near as possible to on both sides. If increases without bound as approaches , then we write    

41. 41 Infinite Limits If decreases without bound as approaches , then we write     In both cases, the limit does not exist .  

42. 42 Estimate using tables of values.  

43. 43 Estimate using tables of values.    

44. 44 44 Estimate using tables of values.  

45. 45 Estimate using tables of values.  

46. 46 d oes not exist Estimate using tables of values.  

47. 47 47 Estimate using tables of values.  

48. 48 Estimate , and using tables of values.  

49. 49 does not exist.   Estimate , and using tables of values.  

50. 50 50 Estimate , , and using tables of values.  

51. 51 Estimate , , and given the table of values below.  

52. 52 For each item, use tables of values to estimate , , and . Then, determine . 1. ; 2. ; 3. ;  

53. 53 Suppose that the function is defined when is near . If gets closer to from both sides as gets closer to , then we say that “the limit of as approaches is equal to .” This is written as .  

54. 54 Suppose that the function is defined when is near from the left. Then, the left-hand limit of as approaches from the left is equal to a number . This can be written as  

55. 55 Suppose that the function is defined when is near from the right. Then, the right-hand limit of as approaches from the right is equal to a number . This can be written as  

56. 56 If the left- and right-hand limits of as approaches a number are equal, then we say that the limit of as approaches exists . If this is NOT the case, then the limit does not exist .  

57. 57 Suppose the function is defined when is as near as possible to on both sides. If increases without bound as approaches , then we say that “the limit of as approaches is infinity.” This is written as  

58. 58 If decreases without bound as approaches , then we say that “the limit of as approaches is negative infinity.” This is written as  

59. 59 To estimate the limit of a function as approaches , follow the steps below:   Step 1: Construct tables with arbitrary values that are very close to from the left and right sides.  

60. 60 To estimate the limit of a function as approaches , follow the steps below:   Step 2: Complete the table by solving the value of for each value.  

61. 61 To estimate the limit of a function as approaches , follow the steps below:   Step 3: Estimate the values that are being approached by from the left and right sides of .  

62. 62 62 Explain what the equation means. Is it possible for the equation to be true and yet ?  

63. 63 Slides 2 and 3 : Eriks airconditioned road trip car by Stig Nygaard is licensed under CC BY 2.0 via Flickr . Edwards, C.H., and David E. Penney.  Calculus: Early Transcendentals . 7th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2008.   Larson, Ron H., and Bruce H. Edwards.  Essential Calculus: Early Transcendental Functions . Boston: Houghton Mifflin, 2008.   Leithold , Louis.  The Calculus 7 . New York: HarperCollins College Publ., 1997.   Smith, Robert T., and Roland B. Milton.  Calculus . New York: McGraw Hill, 2012.   Tan, Soo T.  Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach . Australia: Brooks/Cole Cengage Learning, 2012.