1. Rates of Change & Limits

2. Limits in a nutshell – The limit (L) of a function is the value the function (y) approaches as the value of (x) approaches a given value Got it? No? Really??

3. Example 1. f(x) = 3x + 1 x11.51.91.99 f(x)45.56.76.97

4. Example 1. f(x) = 3x + 1 x11.51.91.99 f(x)45.56.76.97 x2.0012.12.53 f(x)7.003

5. Example 1. f(x) = 3x + 1 As x approaches 2 from the left and from the right, the function is CONVERGING to 7.

6. So, f(x)

7. Easy, right? Well, we could have simply just plugged the number into the x-value in f(x) and found our answer. So what is all the fuss about? No fuss, if the graph is continuous. f(2) = 3(2) + 1 = 7 There’s lot’s of fuss if there are gaps.

8. xf(x) 0.52.5 1.04 1.55.5 2.0Err. 2.58.5

9. xf(x) 0.52.5 1.04 1.55.5 2.0Err. 2.58.5 What’s the “intent” of the graph as x  2?

10. I guess we could have done the following: But, if we simply substituted 2 into the equation, we get something very bad

11. What went wrong?

12. We have a point of discontinuity at x = 2

13. Many times you are asked, “What if at x = 2, the function does whatever?” … Who cares? As x gets closer and closer to the special x-number, but it never gets there, what happens? … Who cares? These questions have no effect on the limit problem (except for continuous functions) The limit stays the same.

14. WAKE UP!!! If you’ve been sleeping up to now… WAKE UP!!

15. Examples from Page 346

17. OR

18. Examples from Page 346

20. Homework Page 346 Numbers 1 - 4

21. 14B Limits Involving Infinity

22. We need to think about what happens to a function not at a certain value, but at extremes like infinity.

23. Complete the following table for the function: x f(x) 2½ 10 100 200 500 700 1000 10,000 100,000

24. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

25. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

26. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

27. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

28. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

29. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

30. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

31. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

32. What about if it goes towards negative infinity?

33. Let’s look at it graphically

34. If the degree of the denominator is greater than the numerator, then. If the degree is Bigger On Bottom its 0. BOB0. (y = 0)

35. We can ignore the (-9) and the (+ 12). They really do not add anything to the graph when you go to ±∞

36. So, let’s look at this graphically

37. As the graph approaches ±∞, what is the “height” of the graph?

39. If the degree of the denominator is greater than the numerator, then. If the degree is Bigger On Bottom its 0. BOB0. (y = 0)

40. Bigger On Top, there’s No horizontal asymptote. BOTN

44. Degrees of both the numerator and denominator are equal Then divide the leading coefficients. That’s your horizontal asymptote. EATS-D/C.

47. Page 349 1a) i. There are going to be some new symbols. As x  0 - f(x)  -∞ Vertical Asymptote x = 0 As x  0 + f(x)  ∞

48. Page 349 1a) i. As x  ∞ x  -∞ Horizontal Asymptote x = 0 f(x)  0 + f(x)  0 -

49. Page 349 1a) ii.

50. Homework Page 349 Numbers 1 - 3

51. Problem You are hanging out at your girlfriend’s place and she goes to get you a couple of slices of pizza. Her annoying cat is looking at you and you get this great idea for a calculus experiment. Because cats have nine lives and always land on their feet, you figure no harm can come from this. So, you drop her cat out of her second-story room window. Here’s the formula that tells you how far the cat “jumped” after a given number of seconds (ignoring air- resistance): h(t) = 16t 2. (in fps)

52. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5

53. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2

54. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32

55. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32 48

56. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32 48 64

57. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32 48 64 80

58. What if you wanted to determine the cat’s speed exactly 2 seconds after it “jumped”?

59. The cat is traveling about 16 fps. This is nice, but what if I wanted to know the EXACT speed 1 second after the jumped. The table gives an average. I want to know EXACT speed. See, the cat speeds up between 1 and 2 seconds and so on. Let’s look at the speed between 1.5 and 1 second

60. The cat is traveling about 40 fps.

61. The cat is traveling about 38.4 fps.

62. The cat is traveling about 27 fps.

63. The cat is traveling about 35 fps.

64. The cat is traveling about 32.0 fps.

65. As t gets closer and closer to 1 second, the average speed appears to get closer and closer to 32 fps.

66. Making things difficult Of course, mathematicians have to make things fancy, so here’s the formula we used for average speed between 1 second and t seconds: Average Speed =

67. Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

68. A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous speed at 2 seconds?

69. for some very small change in t where h = some very small change in t Evaluate this expression for smaller and smaller values of h using your calculator.

70. 1 80 0.165.6.0164.16.00164.016.0001 64.0016.0000164.0002 We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.)

71. The limit as h approaches zero: 0

72. Consider: What happens as x approaches zero? Graphically: WINDOW Y= GRAPH

73. Looks like y=1

74. Numerically: TblSet You can scroll down to see more values. TABLE Tbl Start-0.3 ΔTbl0.1 XY1Y1 -0.30.98507 -0.20.99335 -0.10.99833 0ERR:.10.99833.20.99335.30.98507

75. It appears that the limit of as x approaches zero is 1

76. Limit notation: “The limit of f of x as x approaches c is L.” So:

77. Find the solutions a.) b.)

78. Having fun with SUBS Find the solutions a.) b.)

79. THE PRODUCT RULE Solve Graphically a.)

80. Product Rule Cont’d Confirm Algebraically

81. Exists or Not?

82. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

83. Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See page 58 for details.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

84. 1234 1 2 At x=1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match!

85. At x=2:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

86. At x=3:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

87. The Sandwich Theorem: Show that: The maximum value of sine is 1, soThe minimum value of sine is -1, soSo:

88. By the sandwich theorem: Y= WINDOW Y1 = X2X2 Y2 = -x 2 Y3X 2 Sin(1/x)

89. 

90. Thought for the Day I hope that someday we will be able to put away our fears and prejudices and just laugh at people.

91. Homework 2.1a 1-9, 15, 16