062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated a polnomial function f at 3 and obtained 7, so 7 must be f2. :3 75. I m working with functions f and g for which f2 = 0, g2 = 5, and 3f2 g24 = 5. :4 :4 :4 76. I m working with functions f and g for which f2 f2 = 0, g2 = 5, and :4 g2 = 0. :4 :4 77. I m working with functions f and g for which f2 = 0, :4 :4 g2 = 5, and In Eercises 78 79, find the indicated it. 78. a 79. :0 b fa h2 fa2 In Eercises 80 8, find h:0 h. 80. f2 = 2 2 3, a = 8. f2 =, a = In Eercises 82 83, use properties of its and the following its sin cos =, = 0, :0 :0 sin = 0, cos = :0 :0 to find the indicated it. tan 82. 83. :0 Group Eercises :4 g2 f2 Z 0. a :4 4 ba 4 b 2 sin cos :0 3 84. In the net column is a list of ten common errors involving algebra, trigonometr, and its that students frequentl make in calculus. Group members should eamine each error and describe the mistake. Where possible, correct each error. Finall, group members should offer suggestions for avoiding each error. a. ( h) 3 3 = 3 h 3 3 = h 3 b. a b = a b c. a b = a b d. 2 h 2 = 2 2h 2 = 2h sin 2 e. = sin 2 a b f. = b a 3 g. : = 3 = 0 0 = h. sin( h) sin = sin sin h sin = sin h i. a = b, so a = b 2 9 2 9 j. To find it is necessar to rewrite b :4 3, 3 factoring 2 9. 85. Research and present a group report about the histor of the feud between Newton and Leibniz over who invented calculus. What other interests did these men have in addition to mathematics? What practical problems led them to the invention of calculus? What were their personalities like? Whose version established the notation and rules of calculus that we use toda? Preview Eercises Eercises 86 88 will help ou prepare for the material covered in the net section. In each eercise, use what occurs near 3 and at 3 to graph the function in an open interval about 3. (Graphs will var.) Is it necessar to lift our pencil off the paper to obtain each graph? Eplain our answer. 86. f2 = 5; f32 = 5 87. f2 = 5; f32 = 6 :3 :3 88. :3 f2 = 5; f2 = 6; f32 = 5 :3 Section.3 Objectives Determine whether a function is continuous at a number. Determine for what numbers a function is discontinuous. Limits and Continuit Wh ou should not ski down discontinuous slopes In everda speech, a continuous process is one that goes on without interruption and without abrupt changes. In mathematics, a continuous function has much the same meaning. The graph of a continuous function does not have interrupting breaks, such as holes, gaps, or jumps. Thus, the graph of a continuous function can be drawn without lifting a pencil off the paper. In this section, ou will learn how its can be used to describe continuit.
Section.3 Limits and Continuit 063 Determine whether a function is continuous at a number. Limits and Continuit Figure.2 shows three graphs that cannot be drawn without lifting a pencil from the paper. In each case, there appears to be an interruption of the graph of f at = a. = f() = f() = f() a a (a) (b) (c) Figure.2 Each graph has an interruption at = a. a Eamine Figure.2(a). The interruption occurs because the open dot indicates there is no point on the graph corresponding to = a. This shows that fa2 is not defined. Now, eamine Figure.2(b). The closed blue dot at = a shows that fa2 is defined. However, there is a jump at a. As approaches a from the left, the values of f get closer to the coordinate of the point shown b the open dot. B contrast, as approaches a from the right, the values of f get closer to the coordinate of the point shown b the closed dot. There is no single it as approaches a. The jump in the graph reflects the fact that f2 does not eist. Finall, eamine Figure.2(c). The closed blue dot at = a shows that fa2 is defined. Furthermore, as approaches a from the left or from the right, the values of f get closer to the coordinate of the point shown b the open dot. Thus, f2 eists. However, there is still an interruption at a. Do ou see wh? The it as approaches a, f2, is the coordinate of the open dot. B contrast, the value of the function at a, fa2, is the coordinate of the closed dot. The interruption in the graph reflects the fact that f2 and fa2 are not equal. We now provide a precise definition of what it means for a function to be continuous at a number. Notice how each part of this definition avoids the interruptions that occurred in Figure.2. Definition of a Function Continuous at a Number A function f is continuous at a when three conditions are satisfied.. f is defined at a; that is, a is in the domain of f, so that fa2 is a real number. 2. f2 eists. 3. f2 = fa2 If f is not continuous at a, we sa that f is discontinuous at a. Each of the functions whose graph is shown in Figure.2 is discontinuous at a. EXAMPLE Determining Whether a Function Is Continuous at a Number Determine whether the function is continuous: a. at 2; b. at. f2 = 2 2 2
064 Chapter Introduction to Calculus Solution According to the definition, three conditions must be satisfied to have continuit at a. 2 a. To determine whether f() = is continuous at 2, we check the 2 2 conditions for continuit with a = 2. Condition f is defined at a. Is f22 defined? f22 = Because f22 is a real number,, f22 is defined. Condition 2 f2 eists. Does f2 eist? :2 f2 = :2 :2 2 # 2 2 # 2 2 2 = 4 8 2 = 5 5 = = 2 2 2 = 2 2 :2 :2 22 2 2 # 2 2 # 2 2 2 = 4 8 2 = 5 5 = Using properties of its, we see that f2 eists. :2 Condition 3 f2 fa2 Does f2 = f22? We found that :2 f2 = and f22 =. Thus, as gets closer to 2, the corresponding :2 values of f2 get closer to the function value at 2: f2 = f22. :2 Because the three conditions are satisfied, we conclude that f is continuous at 2. 2 b. To determine whether f() = is continuous at, we check the 2 2 conditions for continuit with a =. Condition f is defined at a. Is f2 defined? Factor the denominator of the function s equation: 2 f()= ()(2) 5 4 3 5 4 3 2 3 4 5 2 3 4 5 2 f() = 2 2 Figure.3 f is continuous at 2. It is not continuous at or at 2. Because division b zero is undefined, the domain of f is E ƒ Z, Z 2F. Thus, f is not defined at. Because one of the three conditions is not satisfied, we conclude that f is not continuous at. Equivalentl, we can sa that f is discontinuous at. 2 The graph of f2 = is shown in Figure.3. The graph verifies 2 2 our work in Eample. Can ou see that f is continuous at 2? B contrast, it is not continuous at, where the graph has a vertical asmptote. The graph in Figure.3 also reveals a discontinuit at 2. The open dot indicates that there is no point on the graph corresponding to = Can ou see what is happening as approaches 2? 2. : 2 2 2 2 = : 2 As gets closer to the graph of gets closer to 2 Because is not defined at 2, f the graph has a hole at A 2, 2 3. f 2, 3B. This is shown b the open dot in Figure.3. Check Point Determine whether the function is continuous: a. at ; b. at 2. Denominator is zero at =. 2 2 2 2 f2 = 2 2 4 Denominator is zero at = 2. = : 2 = 2 = 2 3
Section.3 Limits and Continuit 065 Determine for what numbers a function is discontinuous. Determining Where Functions Are Discontinuous We have seen that the it of a polnomial function as approaches a is the polnomial function evaluated at a. Thus, if f is a polnomial function, then f2 = fa2 for an number a. This means that a polnomial function is continuous at ever number. Man of the functions discussed throughout this book are continuous at ever number in their domain. For eample, rational functions are continuous at ever number, ecept an at which the are not defined. At numbers that are not in the domain of a rational function, a hole in the graph or an asmptote appears. Eponential, logarithmic, sine, and cosine functions are continuous at ever number in their domain. Like rational functions, the tangent, cotangent, secant, and cosecant functions are continuous at ever number, ecept an at which the are not defined. At numbers that are not in the domain of these trigonometric functions, an asmptote occurs. Stud Tip Most functions are alwas continuous at ever number in their domain, including polnomial, rational, radical, eponential, logarithmic, and trigonometric functions. Most of the discontinuities ou will encounter in calculus will be due to jumps in piecewise functions. Eample 2 illustrates how to determine where a piecewise function is discontinuous. EXAMPLE 2 Determining Where a Piecewise Function Is Discontinuous Determine for what numbers, if an, the following function is discontinuous: 2 if 0 f2 = c 2 if 0 6. 2 2 if 7 Solution First, let s determine whether each of the three pieces of f is continuous. The first piece, f2 = 2, is a linear function; it is continuous at ever number. The second piece, f2 = 2, a constant function, is continuous at ever number. And the third piece, f2 = 2 2, a polnomial function, is also continuous at ever number. Thus, these three functions, a linear function, a constant function, and a polnomial function, can be graphed without lifting a pencil from the paper. However, the pieces change at = 0 and at =. Is it necessar to lift a pencil from the paper when graphing f at these values? It appears that we must investigate continuit at 0 and at. To determine whether the function is continuous at 0, we check the conditions for continuit with a = 0. Condition f is defined at a. Is f02 defined? Because a = 0, we use the first line of the piecewise function, where 0. f2 = 2 f02 = 0 2 = 2 This is the function s equation for 0, which includes = 0. Replace with 0. Because f02 is a real number, 2, f02 is defined. Condition 2 f2 eists. Does f2 eist? To answer this question, we look :0 at the values of f2 when is close to 0. Let us investigate the left and righthand its. If these its are equal, then f2 eists. To find f2, :0 :0
066 Chapter Introduction to Calculus 2 if 0 f2 = c 2 if 0 6 2 2 if 7 The given piecewise function (repeated) the lefthand it, we look at the values of f2 when is close to 0, but less than 0. Because is less than 0, we use the first line of the piecewise function, f2 = 2 if 0. Thus, :0 f2 = 22 = 0 2 = 2. :0 To find f2, the righthand it, we look at the values of f2 when is close :0 to 0, but greater than 0. Because is greater than 0, we use the second line of the piecewise function, f2 = 2 if 0 6. Thus, f2 = 2 = 2. :0 :0 Because the left and righthand its are both equal to 2, f2 = 2. Thus, we see that f2 :0 eists. :0 Condition 3 f () f (a) Does f2 = f02? We found that f2 = 2 :0 :0 and f02 = 2. This means that as gets closer to 0, the corresponding values of f2 get closer to the function value at 0: f2 = f02. :0 Because the three conditions are satisfied, we conclude that f is continuous at 0. Now we must determine whether the function is continuous at, the other value of where the pieces change. We check the conditions for continuit with a =. Condition f is defined at a. Is f2 defined? Because a =, we use the second line of the piecewise function, where 0 6. f2 = 2 f2 = 2 This is the function s equation for 0 6, which includes =. Replace with. f() = 2, 0 7 6 5 4 3 2 5 4 3 3 Because f2 is a real number, 2, f2 is defined. Condition 2 f () eists. Does f2 eist? We investigate left and right : hand its as approaches. To find f2, the lefthand it, we look at : values of f2 when is close to, but less than. Thus, we use the second line of the piecewise function, f2 = 2 if 0 6. The lefthand it is To find f2, the righthand it, we look at values of f2 when is : f() = 2 2, > close to, but greater than. Thus, we use the third line of the piecewise function, f2 = 2 2 if 7. The righthand it is 2 3 4 5 f() = 2, 0 < Figure.4 This piecewise function is continuous at 0, where pieces change, and discontinuous at, where pieces change. The left and righthand its are not equal: f2 = 2 and f2 = 3. This means that f2 : : does not eist. : Because one of the three conditions is not satisfied, we conclude that f is not continuous at. In summar, the given function is discontinuous at onl. The graph of f, shown in Figure.4, illustrates this conclusion. Check Point 2 Determine for what numbers discontinuous: f2 = 2 = 2. : : f2 = : : 2 22 = 2 2 = 3., if an, the following function is 2 if 0 f2 = c 2 if 0 6 2. 7 if 7 2
Section.3 Limits and Continuit 067 Eercise Set.3 Practice Eercises In Eercises 8, use the definition of continuit to determine whether f is continuous at a.. f2 = 2 5 2. f2 = 3 4 a = a = 3. f2 = 2 3 7 4. f2 = 2 5 6 a = 4 a = 4 5. 7. 9. f2 = 2 4 2 a = 3 f2 = 5 5 a = 5 f2 = 5 5 a = 5 6. 8. 0. f2 = 2 6 5 a = 6 f2 = 7 7 a = 7 f2 = 7 7 a = 7. f2 = 2 5 2 2. f2 = 2 8 5 2 8 a = 0 a = 0 2 4 3. f2 = c 2 if Z 2 5 if = 2 a = 2 2 36 4. f2 = c 6 if Z 6 3 if = 6 a = 6 5. f2 = b 5 if 0 2 5 if 7 0 a = 0 6. f2 = b 4 if 0 2 4 if 7 0 a = 0 if 6 7. f2 = c 0 2 if = if 7 a = 2 if 6 8. f2 = c if = 2 if 7 a = In Eercises 9 34, determine for what numbers, if an, the given function is discontinuous. 9. f2 = 2 4 6 20. f2 = 2 8 0 2. 2 f2 = 22. f2 = 2 42 22 52 23. f2 = sin 24. f2 = cos 25. f2 = p 26. f2 = c 27. f2 = b if 28. 2 if 7 29. 30. 3. 32. 33. 34. Practice Plus In Eercises 35 38, graph each function. Then determine for what numbers, if an, the function is discontinuous. sin if p 6 0 35. f2 = c sin cos if 0 6 p if p 2p cos if p 6 0 36. f2 = c sin if 0 6 p sin if p 2p 37. 38. In Eercises 39 42, determine for what numbers, if an, the function is discontinuous. Construct a table to find an required its. 39. 40. 4. 2 if Z f2 = c 2 if = 2 9 f2 = c 3 if Z 3 6 if = 3 6 if 0 f2 = c 6 if 0 6 2 2 if 7 2 7 if 0 f2 = c 7 if 0 6 3 2 if 7 3 5 if 6 4 f2 = c 2 if = 4 2 4 if 7 4 7 if 6 6 f2 = c 4 if = 6 2 6 if 7 6 f2 = b if is an integer. if is not an integer. 2 if is an odd integer. f2 = b 2 if is not an odd integer. sin 2 if Z 0 f2 = c 2 if = 0 sin 3 if Z 0 f2 = c 3 if = 0 cos p f2 = e 2 if Z p 2 if = p 2 sin 42. f2 = c p if Z p if = p f2 = b 2 if 2 2 if 7 2
068 Chapter Introduction to Calculus Application Eercises 43. The graph represents the percentage of required topics in a precalculus course, pt2, that a student learned at various times, t, throughout the course. At time t 2, the student panicked for an instant during an eam. At time t 3, after working on a set of cumulative review eercises, a big jump in understanding suddenl took place. Percentage of Material Learned p(t) 00 90 80 70 60 50 40 30 20 0 0 Precalculus course begins. = p(t) t t 2 t 3 t 4 Time Precalculus course ends. a. Find pt2 and pt 2. t:t b. Is p continuous at t? Use the definition of continuit to eplain our answer. c. Find pt2 and pt 2 2. t:t 2 d. Is p continuous at t 2? Use the definition of continuit to eplain our answer. e. Find pt2 and pt 3 2. t:t 3 f. Is p continuous at t 3? Use the definition of continuit to eplain our answer. g. Find pt2 and pt 4 2. t:t 4 h. Eplain the meaning of both the it and the function value in part (g) in terms of the time in the course and the percentage of topics learned. 44. The figure shows the cost of mailing a firstclass letter, f2, as a function of its weight,, in ounces, for weights not eceeding 3.5 ounces. Cost (dollars) 0.93 0.76 0.59 0.42 2 Weight (ounces) Source: Lnn E. Baring, Postmaster, Inverness, CA a. Find f2. :3 b. Find f2. :3 c. What can ou conclude about f2? How is this :3 shown b the graph? d. What aspect of costs for mailing a letter causes the graph to jump verticall b the same amount at its discontinuities? 3 t 45. The following piecewise function gives the ta owed, T2, b a single tapaer in 2007 on a taable income of dollars. T2 = f 0.0 if 0 6 7825 782.50 0.5 78252 if 7825 6 3,850 4386.25 0.25 3,8502 if 3,850 6 77,00 5,698.75 0.28 77,002 if 77,00 6 60,850 39,48.75 0.33 60,8502 if 60,850 6 349,700 0,469.25 0.35 349,7002 if 7 349,700. a. Determine whether T is continuous at 7825. b. Determine whether T is continuous at 3,850. c. If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less mone in taable income. Writing in Mathematics 46. Eplain how to determine whether a function is continuous at a number. 47. If a function is not defined at a, how is this shown on the function s graph? 48. If a function is defined at a, but f2 does not eist, how is this shown on the function s graph? 49. If a function is defined at a, eists, but f2 f2 Z fa2, how is this shown on the function s graph? 50. In Eercises 43 44, functions that modeled learning in a precalculus course and the cost of mailing a letter had jumps in their graphs. Describe another situation that can be modeled b a function with discontinuities. What aspect of this situation causes the discontinuities? 5. Give two eamples of the use of the word continuous in everda English. Compare its use in our eamples to its meaning in mathematics. Technolog Eercises 52. Use our graphing utilit to graph an five of the functions in Eercises 8 and verif whether f is continuous at a. 53. Estimate 2/ b using the TABLE feature of :0 our graphing utilit to create a table of values. Then use the ZOOM IN feature to zoom in on the graph of f near and to the right of = 0 to justif or improve our estimate. Critical Thinking Eercises Make Sense? In Eercises 54 57, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 54. If f2 Z fa2 and eists, I can redefine fa2 to make f continuous at a. 55. If f2 Z fa2 and f2 Z I can redefine fa2 to make f continuous at a. 56. f and g are both continuous at a, although f g is not. 57. f f and g are both continuous at a, although is not. g 58. Define f2 = 2 8 9 at = 9 so that the function becomes continuous at 9.
MidChapter Check Point 069 59. Is it possible to define f2 = at = 9 so that the 9 function becomes continuous at 9? How does this discontinuit differ from the discontinuit in Eercise 58? 60. For the function find A so that the function is continuous at. Group Eercise f2 = b 2 if 6 A 3 if Ú 6. In this eercise, the group will define three piecewise functions. Each function should have three pieces and two values of at which the pieces change. a. Define and graph a piecewise function that is continuous at both values of where the pieces change. b. Define and graph a piecewise function that is continuous at one value of where the pieces change and discontinuous at the other value of where the pieces change. c. Define and graph a piecewise function that is discontinuous at both values of where the pieces change. At the end of the activit, group members should turn in the functions and their graphs. Do not use an of the piecewise functions or graphs that appear anwhere in this book. Preview Eercises Eercises 62 64 will help ou prepare for the material covered in the net section. In each eercise, find the indicated difference quotient and simplif. f2 h2 f22 62. If f2 = 2, find h. 63. If f2 = 3, find f h2 f2 h. sa h2 sa2 64. If st2 = 6t 2 48t 60, find. h Chapter MidChapter Check Point What ou know: We learned that f2 = L means f2, ( approaches a from the left) is not equal to that as gets closer to a, but remains unequal to a, the the righthand it, f2, ( approaches a from the corresponding values of f2 get closer to L. We found right), then its using tables, graphs, and properties of its. The f2 does not eist. Finall, we defined continuit in terms of its. A function f is continuous quotient propert for its did not appl to fractional at a when f is defined at a, f2 eists, and epressions in which the it of the denominator is zero. In these cases, rewriting the epression using factoring or f2 = fa2. If f is not continuous at a, we sa that rationalizing the numerator or denominator was helpful f is discontinuous at a. before finding the it. We saw that if the lefthand it, In Eercises 7, use the graphs of f and g to find the indicated it or function value, or state that the it or function value does not eist. 2 = f() = g() 2 2 2. f2 2. f2 : : 7. 40 f2 3. f2 : 4. 3f2 g24 8. Use the graph of f shown above to determine for what : : numbers the function is discontinuous. Then use the 5. 3f2 g24 6. f g202 :0 definition of continuit to verif each discontinuit. In Eercises 9, use the table to find the indicated it. 0.03 0.02 0.0 0.007 0.007 0.0 0.02 0.03 f2 sin 2 2 0.9433 0.965 0.9804 0.9862.04.02.042.064 g2 e tan cos 2.004.0006.0002.000.000.000.0006.003