1. d = 1. or Use Only in Pilot Program F Review Exercises 131

Size: px

Start display at page:

Download "1. d = 1. or Use Only in Pilot Program F Review Exercises 131"

Barbara Johnston
6 years ago
Views:

1 or Use Onl in Pilot Program F 0 0 Review Eercises. Limit proof Suppose f is defined for all values of near a, ecept possibl at a. Assume for an integer N 7 0, there is another integer M 7 0 such that f - L 6 >N whenever - a 6 >M. Prove that f = L using the precise definition of a it Proving that f L Use the following definition for the noneistence of a it. Assume f is defined for all values of near a, ecept possibl at a. We sa that f L if for some e 7 0 there is no value of d 7 0 satisfing the condition f - L 6 e whenever a 6 d. 6. For the following function, note that S f. Find a value of e 7 0 for which the preceding condition for noneistence is satisfied Prove that does not eist. 8. Let f = e 0 if is rational if is irrational. Prove that f does not eist for an value of a. (Hint: Assume f = L for some values of a and L and let e =.) 9. A continuit proof Suppose f is continuous at a and assume f a 7 0. Show that there is a positive number d 7 0 for which f 7 0 for all values of in a - d, a + d. (In other words, f is positive for all values of in the domain sufficientl close to a.) QUICK CHECK ANSWERS. d = 0 or smaller. d = 0.6 or smaller. d must decrease b a factor of 00 = 0 (at least). f () 0 CHAPTER REVIEW EXERCISES. Eplain wh or wh not Determine whether the following statements are true and give an eplanation or countereample. a. The rational function - has vertical asmptotes at - = - and =. b. Numerical or graphical methods alwas produce good estimates of f. c. The value of f, if it eists, is found b calculating f a. d. If f = or f = -, then f does not eist. e. If f does not eist, then either f = or f = -. f. If a function is continuous on the intervals a, b and b, c, where a 6 b 6 c, then the function is also continuous on a, c. g. If f can be calculated b direct substitution, then f is continuous at = a.. Estimating its graphicall Use the graph of f in the figure to find the following values, if possible. a. f - b. f c. f d. f S- - S- + S- e. f f. f S i. f j. f S + S 6 g. S f f () h. S - f Copright 0 Pearson Education, Inc.

2 or Use Onl in Pilot Program F 0 0 Chapter Limits. Points of discontinuit Use the graph of f in the figure to determine the values of in the interval -, at which f fails to be continuous. Justif our answers using the continuit checklist. f () T. Computing a it graphicall and analticall sin u a. Graph =. Comment on an inaccuracies in the graph sin u and then sketch an accurate graph of the function. sin u b. Estimate using the graph in part (a). us0 sin u c. Verif our answer to part (b) b finding the value of sin u analticall using the trigonometric identit us0 sin u sin u = sin u cos u. T. Computing a it numericall and analticall cos a. Estimate b making a table of values of Sp> cos - sin cos for values of approaching p>. Round our cos - sin estimate to four digits. cos b. Use analtic methods to find the value of Sp> cos - sin. 6. Long-distance phone calls Suppose a long-distance phone call costs $0.7 for the first minute (or an part of the first minute), plus $0.0 for each additional minute (or an part of a minute). a. Graph the function c = f t that gives the cost for talking on the phone for t minutes, for 0 t. b. Evaluate c. Evaluate ts - ts.9 f t. f t and ts + f t. d. Interpret the meaning of the its in part (c). e. For what values of t is f continuous? Eplain. 7. Sketching a graph Sketch the graph of a function f with all the following properties. f = f = - f = S- - S- + f = f = f = S - S + 8. Evaluating its Evaluate the following its analticall. 8. 8p S 0. hs0 + h -, where is constant h S - -. S S S -. S - a + - t - > b 6. ts> t S S8-8 sin -. Sp> + p>. One-sided its Evaluate S + T. Appling the Squeeze Theorem a. Show that p - 8. ps p - sin u - cos u 0. usp> sin u - cos u - and A - S - - sin - A -. on -,. Confirm this result with a graphing utilit. b. Use part (a) and the Squeeze Theorem to eplain wh sin = 0.. Appling the Squeeze Theorem Assume the function g satisfies the inequalit g sin +, for near 0. Use the Squeeze Theorem to find g. 9. Finding infinite its Evaluate the following its S - S S tan us0 + u - sin u 0. Finding vertical asmptotes Let f = T. - a. Calculate f, f, f, and f. - + S - S + b. Does the graph of f have an vertical asmptotes? Eplain. c. Graph f and then sketch the graph with paper and pencil, correcting an errors obtained with the graphing utilit. 6. Limits at infinit Evaluate the following its or state that the do not eist ae -z + S- zs z b. tan rs ln r + Copright 0 Pearson Education, Inc.

3 or Use Onl in Pilot Program F End behavior Determine the end behavior of the following functions. 7. f = f + = f = - e - 0. f = ln. Vertical and horizontal asmptotes Find all vertical and horizontal asmptotes of the following functions f = tan -. f = + - Review Eercises. Left- and right-continuit a. Is h = - 9 left-continuous at =? Eplain. b. Is h = - 9 right-continuous at =? Eplain.. Sketching a graph Sketch the graph of a function that is continuous on 0, and continuous on, but is not continuous on 0,. T. Intermediate Value Theorem a. Use the Intermediate Value Theorem to show that the equation = 0 has a solution in the interval -, 0. b. Find a solution to = 0 in -, 0 using a root finder. T 6. Continuit at a point Determine whether the following functions are continuous at a using the continuit checklist to justif our blood t hours after an intravenous line is opened is given b. Antibiotic dosing The amount of an antibiotic (in mg) in the answers. mt = 00e -0.t - e -0.t.. f = - ; a = a. Use the Intermediate Value Theorem to show the amount of drug is 0 mg at some time in the interval 0, and again at some time in the interval,. - 6 if b. Estimate the times at which m = 0 mg.. g = - ; a = c. Is the amount of drug in the blood ever 0 mg? 9 if = 6. Limit proof Give a formal proof that - =.. h = - 9; a = S Limit proof Give a formal proof that if S - = g = - ; a = 8 if = 8. Limit proofs a. Assume f L for all near a and g = 0. Give a 7 0. Continuit on intervals Find the intervals on which the following formal proof that f g = 0. functions are continuous. Specif right- or left-continuit at the endpoints. b. Find a function f for which f - 0. Wh 7. f = - 8. g = e - S doesn t this violate the result stated in (a)? 9. h = 0. g = cos e c. The Heaviside function is defined as - H = e 0 if 6 0. Determining unknown constants Let if Ú 0. - if 6 g = a if = a + b if 6. Determine values of the constants a and b for which g is continuous at =. Eplain wh H = Infinite it proof Give a formal proof that S - =. Copright 0 Pearson Education, Inc.

4 or Use Onl in Pilot Program F 0 0 Chapter Limits AP PRACTICE QUESTIONS The following questions are intended to help ou prepare for the AP eam. The are not questions from actual AP eams. Section Part A, Multiple Choice, no technolog For Questions and, use the graph of the function f shown in the figure below f () 0.. Which of the following statements are true? I. f - does not eist. II. f = S- III. f = f - S- (A) I and II onl (B) II and III onl (C) I onl (D) II onl (E) III onl. Evaluate f. S (A) It does not eist. (B) 0 (C) (D) (E) -. The average rate of change of the function f = - on the interval, is. Find. (A) 6 (B) (C) (D) l (E) 0. Given that f = A, which of the following statements must be true? (A) f = f + - (B) f a = A (C) f is continuous at a. (D) f is not necessaril equal to A. + (E) f = A Evaluate S sin p + cos p. (A) (B) - (C) 8 (D) 6. What are the vertical asmptotes f = - + -? (A) = and = - (B) = - (C) = (D) = - (E) = - and = sin 7. Evaluate -cos. (A) - (B) 0 (C) (D) Evaluate + -. (A) (B) 0 (C) 9. Evaluate S- e - e - e - + e. (D) (A) - (B) (C) 0 (D) (E) 0 0. A graph of the function f = in the standard viewing + window -0, 0 * -0, 0 is shown in the figure. Evaluate f (A) (B) 0 (C) 0 (D). For what constant k is the function if - f = k if = continuous at =? (A) (B) - (C) (D) - (E) No value of k will work.. Evaluate (A) (B) 0 (C) (D). Evaluate a S- + b (A) - (B) 0 (C) (D) Copright 0 Pearson Education, Inc.

5 or Use Onl in Pilot Program F 0 0 Guided Projects. Suppose f and g are continuous functions for all real numbers with values given in the table. Let h = f g +. f g 8 0 Which of the following statements are true? I. h = II. A number c eists with 6 c 6 such that hc =. III. h = 8 (A) I and II onl (B) II and III onl (C) I onl (D) II onl (E) III onl Section Part B, Multiple Choice, Technolog Allowed. Based on a graph, what is the value of + -? (A) 0 (B) (C) - (D) e sin 6. Based on a graph, what is the value of tan? (A) 0 (B) (C) (D) 6 7. Describe the behavior of f = as becomes large and positive. + (A) The function values become large and positive. (B) The function values approach 0. (C) The function values approach /. (D) The function values approach /. (E) The function values approach 9/. 8. Describe the behavior of f = e -> as approaches 0. (A) f = (B) f = and f = (C) f = and f = (D) f = 0 and f = + - (E) f = 0 9. Suppose that f = h = 8 and f g S S h for all values of. Which of the following statements are true if g eists? S I. g = 8 II. g = III. g = 6 S S S (A) I onl (B) II onl (C) III onl (D) I and II onl (E) I, II, and III Section Part A, Free Response, Technolog Allowed e - e. a. Determine the value of. e - e b. Determine the value of. c. Let p and q be real numbers. Based on the results of parts (a) and (b), make a conjecture about the value of e p - e q.. Consider the function f = +, where n is an integer. + n a. Give the equation(s) of the horizontal asmptote(s) when n = and n = -. b. Give equations for the vertical asmptotes of f (if an eist) when n = and n = -. c. Eplain wh there is onl one vertical asmptote when n Ú 0 and wh there are three vertical asmptotes when n -. Section Part B, Free Response, No Technolog. Consider the function f = a. What is the domain of f? b. Find the roots of f. c. Evaluate f, and identif all the horizontal asmptotes of f. d. Where are the vertical asmptotes of f located? e. Eplain wh the vertical asmptotes are not located at all the ecluded points of the domain.. For each step, choose an appropriate scale for the - and -aes. a. Sketch a graph of a function g that has the properties g = and g =. S b. Now sketch a possible graph of a function g that has the properties listed in part (a) and the properties S- +g = - and S- -g =. c. Sketch a possible graph of a function g that has the properties listed in parts (a) and (b) and the properties -g = g0 = - and +g =. d. Finall, sketch a graph with all the properties from parts (a) (c) and the properties g = and g = -. S - Chapter Guided Projects Applications of the material in this chapter and related topics can be found in the following Guided Projects. For additional information, see the Preface. Fied-point iteration Local linearit Copright 0 Pearson Education, Inc.

b. Create a graph that gives a more complete representation of f.

b. Create a graph that gives a more complete representation of f. or Use Onl in Pilot Program F 96 Chapter Limits 6 7. Steep secant lines a. Given the graph of f in the following figures, find the slope of the secant line that passes through, and h, f h in terms of h,