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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Sullivan 156 Chapter 1 • Limits and Continuity Answers to Chapter 1 AP® Review Problems 1. B 2. B 3. C 4. C 5. C 6. B 7. C 8. C 9. D 10. B 11. D TRM Full Solutions for the Chapter 1 Project CHAPTER 1 AP® REVIEW PROBLEMS u 1. Which line is an asymptote to the graph of f (x) = e 2x ? u 2. (A) x = 0 (B) y = 0 (C) y = 2 (D) y = x 3x 3 + 4x lim x→∞ 5 − 2x 4 = (A) − 3 2 (B) 0 (C) 3 5 (D) 3 2 u 3. If f (x) = 5x 3 − 1, then lim x→0 f (x) − f (0) x 3 = (A) 0 (B) 1 (C) 5 (D) The limit does not exist. θ 2 u 4. lim θ→0 1 − cos θ = (A) 0 (B) 1 (C) 2 (D) 4 u 5. The table gives values of three functions: x –0.15 –0.1 –0.05 0 0.05 0.1 0.15 f(x) 0.075 0.05 0.025 –4 0.025 0.05 0.075 g(x) –8.3 –8.2 –8.1 undefined –7.9 –7.8 –7.7 h(x) 1.997 1.99 1.9975 1 1.005 1.02 1.045 For which of these functions does the table suggest that the limit as x approaches 0 exists? (A) f only (B) h only (C) f and g only (D) f and h only u 6. If a function f is continuous on the closed interval [1, 4] and if f (1) = 6 and f (4) = –1, then which of the following must be true? (A) f (c) = 0 for some number c in the open interval (–1, 6). (B) f (c) = 1 for some number c in the open interval (1, 4). (C) f (c) = 1 for some number c in the open interval (–1, 6). (D) f (c) = –2 for any number c in the open interval (1, 4). u 7. Which are the equations of the asymptotes of the graph of the x function f (x) = x(x 2 − 9) ? (A) x = –3, x = 0, x = 3, y = 0 (B) x = –3, x = 0, x = 3, y = 1 (C) x = –3, x = 3, y = 0 (D) x = –3, x = 3, y = 1 u 8. Find the value of k that makes the function { x f (x) = 2 + 2 if x ≤−1 kx + 4 if x > −1 continuous for all real numbers. (A) –3 (B) –1 (C) 1 (D) 3 u 9. An odd function f is continuous for all real numbers. If lim f (x) =−2, then which of the statements x→∞ must be true? I. f has no vertical asymptotes. II. lim f (x) = 0 x→0 III. The horizontal asymptotes of the graph of f are y = –2 and y = 2. (A) I only (B) III only (C) I and III only (D) I, II, and III u 10. lim x→0 sin(2x) tan(3x) = (A) 0 (B) 2 3 (C) 1 (D) u 11. If a function f is continuous for all real numbers, and if f (x) = x3 + 8 when x = −2, then f (–2) = x + 2 (A) 0 (B) 4 (C) 8 (D) 12 3 2 CHAPTER 1 PROJECT Pollution in Clear Lake AP Photo The Toxic Waste Disposal Company (TWDC) specializes in the disposal of a particularly dangerous pollutant, Agent Yellow (AY). Unfortunately, instead of safely disposing of this pollutant, the company simply dumped AY in (formerly) Clear Lake. Fortunately, they have been caught and are now defending themselves in court. The facts below are not in dispute. As a result of TWDC’s activity, the current concentration of AY in Clear Lake is now 10 ppm (parts per million). Clear Lake is part of a chain of rivers and lakes. Fresh water flows into Clear Lake and the contaminated water flows downstream from it. The Department of Environmental Protection estimates that the level of contamination in Clear Lake will fall by 20% each year. These facts can be modeled as p(0) = 10 p(t + 1) = 0.80p(t) where p = p(t), measured in ppm, is the concentration of pollutants in the lake at time t, in years. 1. Explain how the above equations model the facts. 2. Create a table showing the values of t for t = 0, 1, 2,...,20. 3. Show that p(t) = 10(0.8) t . 4. Use technology to graph p = p(t). 5. What is lim p(t)? t→∞ Lawyers for TWDC looked at the results in 1–5 above and argued that their client has not done any real damage. They concluded that Clear Lake would eventually return to its former clear and unpolluted state. They even called in a mathematician, who wrote the following on a blackboard: lim p(t) = 0 t→∞ and explained that this bit of mathematics means, descriptively, that after many years the concentration of AY will, indeed, be close to zero. 156 Chapter 1 • Limits and Continuity TE_Sullivan_Chapter01_PART II.indd 39 11/01/17 9:57 am
Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Chapter 1 Project • Pollution in Clear Lake 157 Concerned citizens booed the mathematician’s testimony. Fortunately, one of them has taken calculus and knows a little bit about limits. She noted that, although “after many years the concentration of AY will approach zero,” the townspeople like to swim in Clear Lake and state regulations prohibit swimming unless the concentration of AY is below 2 ppm. She proposed a fine of $100,000 per year for each full year that the lake is unsafe for swimming. She also questioned the mathematician, saying, “Your testimony was correct as far as it went, but I remember from studying calculus that talking about the eventual concentration of AY after many, many years is only a small part of the story. The more precise meaning of your statement lim p(t) = 0 is t→∞ that given some tolerance T for the concentration of AY, there is some time N (which may be very far in the future) so that for all t > N, p(t) N, then p(t) N, then p(t)
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