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Splash Screen. Five-Minute Check (over Chapter 11) Then/Now New Vocabulary Example 1: Estimate a Limit = f ( c ) Example 2: Estimate a Limit ≠ f ( c ) Key Concept: Independence of Limit from Function Value at a Point Key Concept: One-Sided Limits

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 11) Then/Now New Vocabulary Example 1: Estimate a Limit = f (c) Example 2: Estimate a Limit ≠ f (c) Key Concept: Independence of Limit from Function Value at a Point Key Concept: One-Sided Limits Key Concept: Existence of a Limit at a Point Example 3: Estimate One-Sided and Two-Sided Limits Example 4: Limits and Unbounded Behavior Example 5: Limits and Oscillating Behavior Concept Summary: Why Limits at a Point Do Not Exist Key Concept: Limits at Infinity Example 6: Estimate Limits at Infinity Example 7: Real-World Example: Estimate Limits at Infinity Lesson Menu

  3. Classify the random variable X, where X represents the number of players on a baseball team, as discrete or continuous. Explain your reasoning. A.discrete; the number of baseball players is countable. B.continuous; the number of baseball players is countable. C.discrete; the number of players on a baseball team varies from team to team. D.continuous; the number of players on a baseball team can be anywhere from 9 to 24 players. 5–Minute Check 1

  4. Find X if z = 2.91, μ = 49, and σ = 2.7. A.145.29 B.135.21 C.56.86 D.54.61 5–Minute Check 2

  5. Workers at a candle company test one type of candle and find that 77% burn for more than 14 hours before extinguishing. What is the probability that in a random sample of 35 candles, more than 30 will burn for more than 14 hours? A.7.0% B.15.2% C.84.8% D.93.0% 5–Minute Check 3

  6. A random survey of 40 people in a grocery store showed that the average time spent shopping was 18.7 minutes. Assume the standard deviation from a previous survey was 1.4 minutes. Which of the following represents the maximum error of estimate given a 95% confidence level? A.0.36 minute B.0.43 minute C.0.45 minute D.0.57 minute 5–Minute Check 4

  7. You estimated limits to determine the continuity and end behavior of functions. (Lesson 1-3) • Estimate limits of functions at fixed values. • Estimate limits of functions at infinity. Then/Now

  8. one-sided limit • two-sided limit Vocabulary

  9. Estimate using a graph. Support your conjecture using a table of values. The graph of f(x) = 4x + 1 suggests that as x gets closer to –7, the corresponding function values get closer to –27. Therefore, we can estimate that = –27. Estimate a Limit = f(c) Analyze Graphically Example 1

  10. Estimate a Limit = f(c) Support Numerically Make a table of values for f, choosing x-values that approach –7 by using some values slightly less than –7 and some values slightly greater than –7. Example 1

  11. Estimate a Limit = f(c) The pattern of outputs suggests that as x gets close to–7 from the left or right, f(x) gets closer to –27. This supports our graphical analysis. Answer:–27 Example 1

  12. Estimate using a graph. A. 3, B. 1, C. –1, D. –3, Example 1

  13. Estimate using a graph. Support your conjecture using a table of values. The graph of suggests that as x gets closer to 4, the corresponding function value approaches 8. Therefore, we can estimate that is 8. Estimate a Limit ≠ f(c) Analyze Graphically Example 2

  14. Estimate a Limit ≠ f(c) Support Numerically Make a table of values, choosing x-values that approach 4 from either side. Example 2

  15. Estimate a Limit ≠ f(c) The pattern of outputs suggests that as x gets closer to 4, f(x) gets closer to 8. This supports our graphical analysis. Answer:8 Example 2

  16. Estimate using a graph. A. 0, B. 0, C. 6, D. –6, Example 2

  17. Key Concept 3

  18. Key Concept 3

  19. Key Concept 3

  20. A. Estimate each one-sided or two-sided limit, if it exists. The graph of suggests that f(x) = –2 and f(x) = 3. Because the left- and right-hand limits of f(x) as x approaches 1 are not the same, does not exist. Estimate One-Sided and Two-Sided Limits Example 3

  21. Answer: Estimate One-Sided and Two-Sided Limits Example 3

  22. B. Estimate each one-sided or two-sided limit, if it exists. 0 0 The graph of g(x) suggests that g(x) = –1 and g(x) = –1. Because the left- and right-hand limits of g(x) as x approaches 0 are the same, exists and is –1. Estimate One-Sided and Two-Sided Limits Example 3

  23. Answer: Estimate One-Sided and Two-Sided Limits Example 3

  24. Estimate each one-sided or two-sided limit, if it exists. A. B. C. D. Example 3

  25. A. Estimate , if it exists. Limits and Unbounded Behavior Analyze Graphically The graph of suggests that and because as x gets closer to 2, the function values of the graph increase. Example 4

  26. Neither one-sided limit at x = 2 exists; therefore, we can conclude that does not exist. However, because both sides approach ∞, we describe the behavior of f(x) at 2 by writing . Limits and Unbounded Behavior Support Numerically Example 4

  27. Limits and Unbounded Behavior The pattern of outputs suggests that as x gets closer to 2 from the left and the right, f(x) grows without bound. This supports our graphical analysis. Answer:∞ Example 4

  28. B. Estimate , if it exists. The graph of suggests thatand because as x gets closer to 0, the function values from the left decrease and the function values from the right increase. Limits and Unbounded Behavior Analyze Graphically Example 4

  29. Neither one-sided limit at x = 0 exists; therefore, does not exist. In this case, we cannot describe the behavior of f(x) at 0 using a single expression because the unbounded behaviors from the left and right differ. Limits and Unbounded Behavior Support Numerically Example 4

  30. Limits and Unbounded Behavior The pattern of outputs suggests that as x gets closer to 0 from the left and the right, f(x) decreases and increases without bound, respectively. This supports our graphical analysis. Answer:does not exist Example 4

  31. Use a graph to estimate , if it exists. A. ∞, B. –∞, C. ∞, D. –∞, Example 4

  32. Estimate , if it exists. Limits and Oscillating Behavior The graph of f(x) = xsin x suggests that as x gets closer to 0, the corresponding function values get closer and closer to 0. Example 5

  33. Therefore, . Limits and Oscillating Behavior Answer:0 Example 5

  34. Estimate , if it exists. A. does not exist B. 1 C. 0 D. –1 Example 5

  35. Concept Summary 6

  36. Key Concept 6

  37. A. Estimate , if it exists. The graph of suggests that . As x increases, f(x) gets closer to 1. Estimate Limits at Infinity Analyze Graphically Example 6

  38. Estimate Limits at Infinity Support Numerically The pattern of outputs suggests that as x increases, f(x) approaches 1. Answer:1 Example 6

  39. B. Estimate , if it exists. The graph of suggests that = –1. As x increases, f(x) gets closer to –1. Estimate Limits at Infinity Analyze Graphically Example 6

  40. Estimate Limits at Infinity Support Numerically The pattern of outputs suggests that as x increases, f(x) approaches –1. Answer:–1 Example 6

  41. C. Estimate , if it exists. Analyze Graphically The graph of f(x) = cos x suggests that does not exist. As x increases, f(x) oscillates between 1 and –1. Estimate Limits at Infinity Example 6

  42. Estimate Limits at Infinity Support Numerically The pattern of outputs suggests that as x increases, f(x) oscillates between 1 and –1. Answer:does not exist Example 6

  43. Estimate , if it exists. A. –2 B. 2 C. –∞ D. ∞ Example 6

  44. A. BACTERIA The growth of a certain bacteria can be modeled by the logistic growth function , where t represents time in hours. Estimate , if it exists, and interpret your result. Estimate Limits at Infinity Example 7

  45. Graph using a graphing calculator. The graph shows that when t = 20, B(t) ≈ 674.44. Notice that as t increases, the function values of the graph get closer and closer to 675. So we can estimate that . Estimate Limits at Infinity Example 7

  46. Answer:; Over time, the population of the bacteria will approach a maximum of 675. Estimate Limits at Infinity Interpret the Result Over time, the population of the bacteria will approach a maximum of 675. Example 7

  47. B. POPULATION The population growth of a certain city is given by the function P(t) = 0.7(1.1)t, where t is time in years. Estimate , if it exists, and interpret your result. Graph the function P(t ) = 0.7(1.1)t using a graphing calculator. The graph shows that as t increases the function values increase. So, we can estimate that . Estimate Limits at Infinity Example 7

  48. Answer: ; If the pattern continues, the population will grow without bound over time. Estimate Limits at Infinity Interpret the Result If the pattern continues, the population will grow without bound over time. Example 7

  49. POPULATION The population growth of deer on Fawn Island is given by P(t) = 200(0.81)t, where t is time given in years. Estimate , if it exists, and interpret your results. A. ; Over time, the deer population will grow without bound. B. ; Over time, the deer population will reach 0. C. ; Over time, the deer population will reach 200. D. ; Over time, the deer population will reach 162. Example 7

  50. End of the Lesson

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