Chapter 11. Limits and an Introduction to Calculus. Selected Applications


 Malcolm Johnson
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1 Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications Limit concepts ave man reallife applications. Te applications listed below represent a small sample of te applications in tis capter. FreeFalling Object, Eercises 77 and 78, page 799 Communications, Eercise 80, page 800 Market Researc, Eercise 6, page 809 Rate of Cange, Eercise 65, page 809 Average Cost, Eercise 5, page 88 Scool Enrollment, Eercise 55, page 88 Geometr, Eercise 5, page 87 Te it process is a fundamental concept of calculus. In Capter, ou will learn man properties of its and ow te it process can be used to find areas of regions bounded b te graps of functions. You will also learn ow te it process can be used to find slopes of tangent lines to graps. Inga Spence/Inde Stock Americans produce over 00 million pounds of waste eac ear. Man residents and businesses reccle about 8% of te waste produced. Limits can be used to determine te average cost of reccling material as te amount of material increases infinitel. 779
2 780 Capter Limits and an Introduction to Calculus. Introduction to Limits Te Limit Concept Te notion of a it is a fundamental concept of calculus. In tis capter, ou will learn ow to evaluate its and ow te are used in te two basic problems of calculus: te tangent line problem and te area problem. Eample Finding a Rectangle of Maimum Area You are given inces of wire and are asked to form a rectangle wose area is as large as possible. Wat dimensions sould te rectangle ave? Let w represent te widt of te rectangle and let l represent te lengt of te rectangle. Because w l Perimeter is. it follows tat l w, as sown in Figure.. So, te area of te rectangle is A lw ww w w. Formula for area Substitute w for l. Simplif. Wat ou sould learn Use te definition of a it to estimate its. Determine weter its of functions eist. Use properties of its and direct substitution to evaluate its. W ou sould learn it Te concept of a it is useful in applications involving maimization. For instance, in Eercise on page 788, te concept of a it is used to verif te maimum volume of an open bo. w l = w Poto credit Figure. Dick Luria/Gett Images Using tis model for area, ou can eperiment wit different values of w to see ow to obtain te maimum area. After tring several values, it appears tat te maimum area occurs wen w 6, as sown in te table. Widt, w Area, A In it terminolog, ou can sa tat te it of A as w approaces 6 is 6. Tis is written as A w w 6 w 6 w 6. Now tr Eercise.
3 Section. Introduction to Limits 78 Definition of Limit Definition of Limit If f becomes arbitraril close to a unique number L as approaces c from eiter side, te it of f as approaces c is L. Tis is written as f L. c Eample Estimating a Limit Numericall Use a table to estimate numericall te it: Let f. Ten construct a table tat sows values of f for two sets of values one set tat approaces from te left and one tat approaces from te rigt. From te table, it appears tat te closer gets to, te closer f gets to. So, ou can estimate te it to be. Figure. adds furter support to tis conclusion. Now tr Eercise f() ? In Figure., note tat te grap of f is continuous. For graps tat are not continuous, finding a it can be more difficult. 5 5 Figure. (, ) f() = Eample Estimating a Limit Numericall Use a table to estimate numericall te it: Let f. Ten construct a table tat sows values of f for two sets of values one set tat approaces 0 from te left and one tat approaces 0 from te rigt. From te table, it appears tat te it is. Te grap sown in Figure. verifies tat te it is. Now tr Eercise f() ? f ( ) = 0 (0, ) 5 f() = f is undefined at = 0. + Figure.
4 78 Capter Limits and an Introduction to Calculus In Eample, note tat f as a it wen 0 even toug te function is not defined wen 0. Tis often appens, and it is important to realize tat te eistence or noneistence of f at c as no bearing on te eistence of te it of f as approaces c. Eample Estimate te it: Using a Graping Utilit to Estimate a Limit. Numerical Let f. Because ou are finding te it of f as approaces, use te table feature of a graping utilit to create a table tat sows te value of te function for starting at 0.9 and setting te table step to 0.0, as sown in Figure.(a). Ten cange te table so tat starts at 0.99 and set te table step to 0.00, as sown in Figure.(b). From te tables, ou can estimate te it to be. Grapical Use a graping utilit to grap using a decimal setting. Ten use te zoom and trace features to determine tat as gets closer and closer to, gets closer and closer to from te left and from te rigt, as sown in Figure.5. Using te trace feature, notice tat tere is no value given for wen, and tat tere is a ole or break in te grap wen. 5. (a) (b) Figure. Now tr Eercise..7.7 Figure.5. Eample 5 Using a Grap to Find a Limit Find te it of f as approaces, were f is defined as f, 0, Because f for all oter tan and because te value of f is immaterial, it follows tat te it is (see Figure.6). So, ou can write f. Te fact tat f 0 as no bearing on te eistence or value of te it as approaces. For instance, if te function were defined as f,,. te it as approaces would be te same. Now tr Eercise 5. f () =, 0, = Figure.6
5 Limits Tat Fail to Eist Net, ou will eamine some functions for wic its do not eist. Eample 6 Comparing Left and Rigt Beavior Sow tat te it does not eist. 0 Section. Introduction to Limits 78 Consider te grap of te function given b see tat for positive values, and for negative values, > 0 < 0. In Figure.7, ou can Tis means tat no matter ow close gets to 0, tere will be bot positive and negative values tat ield f and f. Tis implies tat te it does not eist. Now tr Eercise. f. f() = Figure.7 f () = f() = Eample 7 Unbounded Beavior Discuss te eistence of te it. 0 Let f. In Figure.8, note tat as approaces 0 from eiter te rigt or te left, f increases witout bound. Tis means tat b coosing close enoug to 0, ou can force f to be as large as ou want. For instance, f will be larger tan 00 if ou coose tat is witin of 0. Tat is, 0 < < 0 Similarl, ou can force f to be larger tan,000,000, as follows. 0 < < 000 Because f is not approacing a unique real number L as approaces 0, ou can conclude tat te it does not eist. Now tr Eercise. 0 f > 00. f >,000,000 f() = Figure.8
6 78 Capter Limits and an Introduction to Calculus Eample 8 Oscillating Beavior Discuss te eistence of te it. sin 0 Let f sin. In Figure.9, ou can see tat as approaces 0, f oscillates between and. Terefore, te it does not eist because no matter ow close ou are to 0, it is possible to coose values of and suc tat sin and sin, as indicated in te table. f() = sin ( ) sin Limit does not eist. Now tr Eercise. Figure.9 Eamples 6, 7, and 8 sow tree of te most common tpes of beavior associated wit te noneistence of a it. Conditions Under Wic Limits Do Not Eist Te it of f as c does not eist if an of te following conditions is true.. f approaces a different number from te rigt side of c tan it approaces from te Eample 6 left side of c.. f increases or decreases witout bound as Eample 7 approaces c.. f oscillates between two fied values as Eample 8 approaces c. TECHNOLOGY TIP A graping utilit can elp ou discover te beavior of a function near te value at wic ou are tring to evaluate a it. Wen ou do tis, owever, ou sould realize tat ou can t alwas trust te graps tat graping utilities displa. For instance, if ou use a graping utilit to grap te function in Eample 8 over an interval containing 0, ou will most likel obtain an incorrect grap, as sown in Figure.0. Te reason tat a graping utilit can t sow te correct grap is tat te grap as infinitel man oscillations over an interval tat contains Figure.0. f() = sin ( (
7 Section. Introduction to Limits 785 Properties of Limits and Direct Substitution You ave seen tat sometimes te it of f as c is simpl f c. In suc cases, it is said tat te it can be evaluated b direct substitution. Tat is, c f f c. Substitute c for. Tere are man wellbeaved functions, suc as polnomial functions and rational functions wit nonzero denominators, tat ave tis propert. Some of te basic ones are included in te following list. Eploration Use a graping utilit to grap te tangent function. Wat are tan and tan? Wat can ou sa about te eistence of te it tan? 0 Basic Limits Let b and c be real numbers and let n be a positive integer.. b b c. c c. (See te proof on page 85.) c n c n. n c, n for n even and c > 0 c and Trigonometric functions can also be included in tis list. For instance, sin sin 0 cos cos 0 0. B combining te basic its wit te following operations, ou can find its for a wide variet of functions. Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions wit te following its. f L c and g K c. Scalar multiple:. Sum or difference: bf bl c f ± g L ± K c. Product: f g LK c. Quotient: f provided K 0 c g L K, 5. Power: c f n L n
8 786 Capter Limits and an Introduction to Calculus Eample 9 Find eac it. Direct Substitution and Properties of Limits a. b. c. d. e. cos f. 9 You can use te properties of its and direct substitution to evaluate eac it. a. Direct Substitution b. 5 5 Scalar Multiple Propert 0 c. tan tan Quotient Propert d. 9 9 e. cos ) cos Product Propert cos 5 tan Eploration Sketc te grap of eac function. Ten find te its of eac function as approaces and as approaces. Wat conclusions can ou make? a. f b. g c. Use a graping utilit to grap eac function above. Does te graping utilit distinguis among te tree graps? Write a sort eplanation of our findings. f. Sum and Power Properties 7 9 Now tr Eercise 9. TECHNOLOGY TIP Wen evaluating its, remember tat tere are several was to solve most problems. Often, a problem can be solved numericall, grapicall, or algebraicall. Te its in Eample 9 were found algebraicall. You can verif tese solutions numericall and/or grapicall. For instance, to verif te it in Eample 9(a) numericall, use te table feature of a graping utilit to create a table, as sown in Figure.. From te table, ou can see tat te it as approaces is 6. Now, to verif te it grapicall, use a graping utilit to grap. Using te zoom and trace features, ou can determine tat te it as approaces is 6, as sown in Figure.. Figure Figure.
9 Te results of using direct substitution to evaluate its of polnomial and rational functions are summarized as follows. Section. Introduction to Limits 787 Limits of Polnomial and Rational Functions. If p is a polnomial function and c is a real number, ten c p pc. (See te proof on page 85.). If r is a rational function given b r pq, and c is a real number suc tat qc 0, ten c r rc pc qc. Eample 0 Find eac it. a. b. 6 Evaluating Limits b Direct Substitution Te first function is a polnomial function and te second is a rational function wit a nonzero denominator at. So, ou can evaluate te its b direct substitution. a. b Now tr Eercise 5. Eploration Use a graping utilit to grap te function f 0. 5 Use te trace feature to approimate f. Wat do ou tink f 5 equals? Is f defined at 5? Does tis affect te eistence of te it as approaces 5?
10 788 Capter Limits and an Introduction to Calculus. Eercises See for workedout solutions to oddnumbered eercises. Vocabular Ceck Fill in te blanks.. If f becomes arbitraril close to a unique number L as approaces c from eiter side, te of f as approaces c is L.. Te it of f as c does not eist if f between two fied values.. To evaluate te it of a polnomial function, use.. Geometr You create an open bo from a square piece of material, centimeters on a side. You cut equal squares from te corners and turn up te sides. (a) Draw and label a diagram tat represents te bo. (b) Verif tat te volume of te bo is given b V. (c) Te bo as a maimum volume wen. Use a graping utilit to complete te table and observe te beavior of te function as approaces. Use te table to find V V (d) Use a graping utilit to grap te volume function. Verif tat te volume is maimum wen.. Geometr You are given wire and are asked to form a rigt triangle wit a potenuse of 8 inces wose area is as large as possible. (a) Draw and label a diagram tat sows te base and eigt of te triangle. (b) Verif tat te area of te triangle is given b A 8. (c) Te triangle as a maimum area wen inces. Use a graping utilit to complete te table and observe te beavior of te function as approaces. Use te table to find A A (d) Use a graping utilit to grap te area function. Verif tat te area is maimum wen inces. In Eercises 0, complete te table and use te result to estimate te it numericall. Determine weter or not te it can be reaced f? f? f? f? f 7. 0 sin f? f
11 Section. Introduction to Limits tan f? f e f? f ln f? In Eercises 5 8, grap te function and find te it (if it eists) as approaces. 5. f, < 6., f,, > f,, In Eercises 9 6, use te grap to find te it (if it eists). If te it does not eist, eplain w > f 8,, > In Eercises, use te table feature of a graping utilit to create a table for te function and use te result to estimate te it numericall. Use te graping utilit to grap te corresponding function to confirm our result grapicall sin cos sin tan e e ln. ln. cos tan 6. π π π π sin sec π π π π
12 790 Capter Limits and an Introduction to Calculus In Eercises 7 6, use a graping utilit to grap te function and use te grap to determine weter or not te it eists. If te it does not eist, eplain w f cos, 0. f sin,. f,. 5 f,.. f 7, f f ln, f ln7, f In Eercises 7 and 8, use te given information to evaluate eac it (a) (c) (a) (c) (b) (d) (b) (d) In Eercises 9 and 50, find (a) f, (b) (c) [ fg]. and (d) [g f]. 9. f, 50. f 5 e, f e, f, f, c c f g c f c g f 5, c c f g 5g f, f 0 f 0 f 0 f f f f g 6 c g c g 5 g sin f g c f c 6f g c c f g, In Eercises 5 70, find te it b direct substitution e 66. ln e 67. sin 68. tan 69. arcsin 70. Sntesis True or False? In Eercises 7 and 7, determine weter te statement is true or false. Justif our answer. 7. Te it of a function as approaces c does not eist if te function approaces from te left of c and from te rigt of c. 7. Te it of te product of two functions is equal to te product of te its of te two functions. 7. Tink About It From Eercises to 0, select a it tat can be reaced and one tat cannot be reaced. (a) Use a graping utilit to grap te corresponding functions using a standard viewing window. Do te graps reveal weter or not te it can be reaced? Eplain. (b) Use a graping utilit to grap te corresponding functions using a decimal setting. Do te graps reveal weter or not te it can be reaced? Eplain. 7. Tink About It Use te results of Eercise 7 to draw a conclusion as to weter or not ou can use te grap generated b a graping utilit to determine reliabl if a it can be reaced. 75. Tink About It (a) If f, can ou conclude anting about f? Eplain our reasoning. (b) If f, can ou conclude anting about f? Eplain our reasoning. 76. Writing Write a brief description of te meaning of te notation f. 5 Skills Review arccos In Eercises 77 8, simplif te rational epression
13 . Tecniques for Evaluating Limits Section. Tecniques for Evaluating Limits 79 Dividing Out Tecnique In Section., ou studied several tpes of functions wose its can be evaluated b direct substitution. In tis section, ou will stud several tecniques for evaluating its of functions for wic direct substitution fails. Suppose ou were asked to find te following it. 6 Direct substitution fails because is a zero of te denominator. B using a table, owever, it appears tat te it of te function as is Anoter wa to find te it of tis function is sown in Eample. Eample Dividing Out Tecnique Find te it: ? Begin b factoring te numerator and dividing out an common factors. 6 5 Now tr Eercise 7. Factor numerator. Divide out common factor. Simplif. Direct substitution Simplif. Wat ou sould learn Use te dividing out tecnique to evaluate its of functions. Use te rationalizing tecnique to evaluate its of functions. Approimate its of functions grapicall and numericall. Evaluate onesided its of functions. Evaluate its of difference quotients from calculus. W ou sould learn it Man definitions in calculus involve te it of a function. For instance, in Eercises 77 and 78 on page 799, te definition of te velocit of a freefalling object at an instant in time involves finding te it of a position function. Peticolas Megna/Fundamental Potograps Prerequisite Skills To review factoring tecniques, see Appendi G, Stud Capsule. Tis procedure for evaluating a it is called te dividing out tecnique. Te validit of tis tecnique stems from te fact tat if two functions agree at all but a single number c, te must ave identical it beavior at c. In Eample, te functions given b f 6 and g agree at all values of oter tan. So, ou can use g to find te it of f.
14 79 Capter Limits and an Introduction to Calculus Te dividing out tecnique sould be applied onl wen direct substitution 0 produces 0 in bot te numerator and te denominator. Te resulting fraction, 0, as no meaning as a real number. It is called an indeterminate form because ou cannot, from te form alone, determine te it. Wen ou tr to evaluate a it of a rational function b direct substitution and encounter tis form, ou can conclude tat te numerator and denominator must ave a common factor. After factoring and dividing out, ou sould tr direct substitution again. Eample Dividing Out Tecnique Find te it. Begin b substituting into te numerator and denominator. 0 Numerator is 0 wen. 0 Denominator is 0 wen. Because bot te numerator and denominator are zero wen, direct substitution will not ield te it. To find te it, ou sould factor te numerator and denominator, divide out an common factors, and ten tr direct substitution again. Factor denominator. Divide out common factor. Simplif. Direct substitution Simplif. f () = (, ) + f is undefined wen =. Tis result is sown grapicall in Figure.. Now tr Eercise 9. Figure. In Eample, te factorization of te denominator can be obtained b dividing b or b grouping as follows.
15 Rationalizing Tecnique Anoter wa to find te its of some functions is first to rationalize te numerator of te function. Tis is called te rationalizing tecnique. Recall tat rationalizing te numerator means multipling te numerator and denominator b te conjugate of te numerator. For instance, te conjugate of is. Eample Find te it: Rationalizing Tecnique. 0 Section. Tecniques for Evaluating Limits 79 B direct substitution, ou obtain te indeterminate form Indeterminate form In tis case, ou can rewrite te fraction b rationalizing te numerator., 0 Now ou can evaluate te it b direct substitution. Multipl. Simplif. Divide out common factor. Simplif You can reinforce our conclusion tat te it is b constructing a table, as sown below, or b sketcing a grap, as sown in Figure Prerequisite Skills To review rationalizing of numerators and denominators, see Appendi G, Stud Capsule. f () = ( ) 0, + f is undefined wen = 0. Figure f() ? Now tr Eercise 7. Te rationalizing tecnique for evaluating its is based on multiplication b a convenient form of. In Eample, te convenient form is.
16 79 Capter Limits and an Introduction to Calculus Using Tecnolog Te dividing out and rationalizing tecniques ma not work well for finding its of nonalgebraic functions. You often need to use more sopisticated analtic tecniques to find its of tese tpes of functions. Eample Approimating a Limit Approimate te it: 0. Numerical Let f. Because ou are finding te it wen 0, use te table feature of a graping utilit to create a table tat sows te values of f for starting at 0.0 and setting te table step to 0.00, as sown in Figure.5. Because 0 is alfwa between 0.00 and 0.00, use te average of te values of f at tese two coordinates to estimate te it as follows Te actual it can be found algebraicall to be e.788. Grapical To approimate te it grapicall, grap te function, as sown in Figure.6. Using te zoom and trace features of te graping utilit, coose two points on te grap of f, suc as ,.785 and ,.78 as sown in Figure.7. Because te coordinates of tese two points are equidistant from 0, ou can approimate te it to be te average of te coordinates. Tat is, Te actual it can be found algebraicall to be e f() = ( + )/.78 Figure.5 Now tr Eercise 7. 0 Figure.6 Figure Eample 5 Approimating a Limit Grapicall Approimate te it: 0 sin. Direct substitution produces te indeterminate form 0. To approimate te it, begin b using a graping utilit to grap f sin, as sown in Figure.8. Ten use te zoom and trace features of te graping utilit to coose a point on eac side of 0, suc as , and , Finall, approimate te it as te average of te coordinates of tese two points, sin It can be sown 0 algebraicall tat tis it is eactl. Now tr Eercise. 0 Figure.8 f() = sin
17 Section. Tecniques for Evaluating Limits 795 TECHNOLOGY TIP Te graps sown in Figures.6 and.8 appear to be continuous at 0. But wen ou tr to use te trace or te value feature of a graping utilit to determine te value of wen 0, tere is no value given. Some graping utilities can sow breaks or oles in a grap wen an appropriate viewing window is used. Because te oles in te graps in Figures.6 and.8 occur on te ais, te oles are not visible. TECHNOLOGY SUPPORT For instructions on ow to use te zoom and trace features and te value feature, see Appendi A; for specific kestrokes, go to tis tetbook s Online Stud Center. OneSided Limits In Section., ou saw tat one wa in wic a it can fail to eist is wen a function approaces a different value from te left side of c tan it approaces from te rigt side of c. Tis tpe of beavior can be described more concisel wit te concept of a onesided it. c f L c f L Eample 6 or f L as c Limit from te left or f L as c Limit from te rigt Evaluating OneSided Limits Find te it as 0 from te left and te it as 0 from te rigt for f. From te grap of f, sown in Figure.9, ou can see tat f for all < 0. Terefore, te it from te left is. 0 Limit from te left Because f for all > 0, te it from te rigt is. Limit from te rigt 0 Now tr Eercise 5. f() = Figure.9 f() = f () = In Eample 6, note tat te function approaces different its from te left and from te rigt. In suc cases, te it of f as c does not eist. For te it of a function to eist as c, it must be true tat bot onesided its eist and are equal. Eistence of a Limit If f is a function and c and L are real numbers, ten c f L if and onl if bot te left and rigt its eist and are equal to L.
18 796 Capter Limits and an Introduction to Calculus Eample 7 Finding OneSided Limits Find te it of f as approaces. f,, Remember tat ou are concerned about te value of f near rater tan at. So, for <, f is given b, and ou can use direct substitution to obtain For >, f is given b, and ou can use direct substitution to obtain f. f. Because te onesided its bot eist and are equal to, it follows tat f. < > Te grap in Figure.0 confirms tis conclusion. Prerequisite Skills For a review of piecewisedefined functions, see Section f() =, < f() =, > 5 6 Figure.0 Now tr Eercise 57. Eample 8 Comparing Limits from te Left and Rigt To sip a package overnigt, a deliver service carges $7.80 for te first pound and $.0 for eac additional pound or portion of a pound. Let represent te weigt of a package and let f represent te sipping cost. Sow tat te it of f as does not eist. f 7.80, 9.0, 0.60, Te grap of f is sown in Figure.. Te it of f as approaces from te left is f 9.0 wereas te it of f as approaces from te rigt is f < < < Because tese onesided its are not equal, te it of f as does not eist. Sipping cost (in dollars) Overnigt Deliver For <, f() = For <, f() = For 0 <, f() = Weigt (in pounds) Figure. Now tr Eercise 8.
19 A Limit from Calculus In te net section, ou will stud an important tpe of it from calculus te it of a difference quotient. Eample 9 Evaluating a Limit from Calculus For te function given b f, find 0 Direct substitution produces an indeterminate form. 0 f f. f f Section. Tecniques for Evaluating Limits 797 B factoring and dividing out, ou obtain te following. So, te it is f f Now tr Eercise Note tat for an value, te it of a difference quotient is an epression of te form 0 Direct substitution into te difference quotient alwas produces te indeterminate 0 form For instance, 0. f f. f f f 0 f f f Prerequisite Skills For a review of evaluating difference quotients, refer to Section..
20 798 Capter Limits and an Introduction to Calculus. Eercises See for workedout solutions to oddnumbered eercises. Vocabular Ceck Fill in te blanks.. To evaluate te it of a rational function tat as common factors in its numerator and denominator, use te. 0. Te fraction 0 as no meaning as a real number and terefore is called an.. Te it f L is an eample of a. c f f. Te it of a is an epression of te form. 0 In Eercises, use te grap to determine eac it (if it eists). Ten identif anoter function tat agrees wit te given function at all but one point.. g.. g. (a) g (b) (c) 6 6 (a) 0 (b) (c) g g 0 6 (a) (b) 0 (c) f (a) f (b) f (c) f In Eercises 5 8, find te it (if it eists). Use a graping utilit to verif our result grapicall t 8 t t sec 0 tan. 5. cos 0 cot 6. sin a 6 a a z 7 z 0 z 8 sin cos sin 0 sin cos In Eercises 9 6, use a graping utilit to grap te function and approimate te it
21 Section. Tecniques for Evaluating Limits In Eercises 7 8, use a graping utilit to grap te function and approimate te it. Write an approimation tat is accurate to tree decimal places. 7. e ln 0 0. sin. 0. tan Grapical, Numerical, and Algebraic Analsis In Eercises 9 5, (a) grapicall approimate te it (if it eists) b using a graping utilit to grap te function, (b) numericall approimate te it (if it eists) b using te table feature of a graping utilit to create a table, and (c) algebraicall evaluate te it (if it eists) b te appropriate tecnique(s) In Eercises 5 60, grap te function. Determine te it (if it eists) b evaluating te corresponding onesided its f were 58. f were 59. f were f,, f,, f,, e 0 0 ln sin 0 cos 0 0 > < > 60. f were 0 In Eercises 6 66, use a graping utilit to grap te function and te equations and in te same viewing window. Use te grap to find 0 6. f cos 6. f sin f sin 65. f sin 66. f cos In Eercises 67 and 68, state wic it can be evaluated b using direct substitution. Ten evaluate or approimate eac it. 67. (a) (b) 0 sin 68. (a) (b) 0 cos In Eercises 69 76, find 69. f 70. f f 7. f 7. f 7. f 75. f 76. f FreeFalling Object In Eercises 77 and 78, use te position function st 6t 8, wic gives te eigt (in feet) of a freefalling object. Te velocit at time t a seconds is given b sa st. t a a t f,, 0 0 > 0 sin 0 f f. 77. Find te velocit wen t second. 78. Find te velocit wen t seconds. f cos cos Communications Te cost of a cellular pone call witin our calling area is $.00 for te first minute and $0.5 for eac additional minute or portion of a minute. A model for te cost C is given b Ct t, were t is te time in minutes. (Recall from Section. tat f te greatest integer less tan or equal to.) (a) Sketc te grap of C for 0 < t 5. (b) Complete te table and observe te beavior of C as t approaces.5. Use te grap from part (a) and te table to find Ct. t.5 t C?
22 800 Capter Limits and an Introduction to Calculus (c) Complete te table and observe te beavior of C as t approaces. Does te it of Ct as t approaces eist? Eplain. (c) Complete te table and observe te beavior of C as t approaces. Does te it of Ct as t approaces eist? Eplain. 8. Salar Contract A union contract guarantees a 0% salar increase earl for ears. For a current salar of $,500, te salar ft (in tousands of dollars) for te net ears is given b ft.50, 9.00, 6.80, were t represents te time in ears. Sow tat te it of f as t does not eist. 8. Consumer Awareness Te cost of sending a package overnigt is $.0 for te first pound and $.90 for eac additional pound or portion of a pound. A plastic mailing bag can old up to pounds. Te cost f of sending a package in a plastic mailing bag is given b f.0, 8.0,.0, t C? 80. Communications Te cost of a cellular pone call witin our calling area is $.5 for te first minute and $0.5 for eac additional minute or portion of a minute. A model for te cost C is given b Ct.5 0.5t, were t is te time in minutes. (Recall from Section. tat f te greatest integer less tan or equal to.) (a) Sketc te grap of C for 0 < t 5. (b) Complete te table and observe te beavior of C as t approaces.5. Use te grap from part (a) and te table to find Ct. t.5 t C? t C? 0 < t < t < t 0 < < < were represents te weigt of te package (in pounds). Sow tat te it of f as does not eist. Sntesis True or False? In Eercises 8 and 8, determine weter te statement is true or false. Justif our answer. 8. Wen our attempt to find te it of a rational function 0 ields te indeterminate form 0, te rational function s numerator and denominator ave a common factor. 8. If fc L, ten c f L. 85. Tink About It (a) Sketc te grap of a function for wic f is defined but for wic te it of f as approaces does not eist. (b) Sketc te grap of a function for wic te it of f as approaces is but for wic f. 86. Writing Consider te it of te rational function pq. Wat conclusion can ou make if direct substitution produces eac epression? Write a sort paragrap eplaining our reasoning. p p (a) (b) c q c q 0 p p (c) (d) c q 0 c q 0 0 Skills Review 87. Write an equation of te line tat passes troug 6, 0 and is perpendicular to te line tat passes troug, 6 and,. 88. Write an equation of te line tat passes troug, and is parallel to te line tat passes troug, and 5,. In Eercises 89 9, identif te tpe of conic algebraicall. Ten use a graping utilit to grap te conic. 89. r cos 90. r 9. 9 r cos 9. r 9. 5 r sin 9. r In Eercises 95 98, determine weter te vectors are ortogonal, parallel, or neiter ,,,,, , 5, 0, 0, 5, 97.,, 6,, 9, 8 98.,,,,, sin cos 6 sin
23 Section. Te Tangent Line Problem 80. Te Tangent Line Problem Tangent Line to a Grap Calculus is a branc of matematics tat studies rates of cange of functions. If ou go on to take a course in calculus, ou will learn tat rates of cange ave man applications in real life. Earlier in te tet, ou learned ow te slope of a line indicates te rate at wic a line rises or falls. For a line, tis rate (or slope) is te same at ever point on te line. For graps oter tan lines, te rate at wic te grap rises or falls canges from point to point. For instance, in Figure., te parabola is rising more quickl at te point, tan it is at te point,. At te verte,, te grap levels off, and at te point,, te grap is falling. (, ) Wat ou sould learn Use a tangent line to approimate te slope of a grap at a point. Use te it definition of slope to find eact slopes of graps. Find derivatives of functions and use derivatives to find slopes of graps. W ou sould learn it Te derivative, or te slope of te tangent line to te grap of a function at a point, can be used to analze rates of cange. For instance, in Eercise 65 on page 809, te derivative is used to analze te rate of cange of te volume of a sperical balloon. (, ) (, ) (, ) Figure. To determine te rate at wic a grap rises or falls at a single point, ou can find te slope of te tangent line at tat point. In simple terms, te tangent line to te grap of a function f at a point P, is te line tat best approimates te slope of te grap at te point. Figure. sows oter eamples of tangent lines. P Ricard Hutcings/Corbis Prerequisite Skills For a review of te slopes of lines, see Section.. P P Figure. From geometr, ou know tat a line is tangent to a circle if te line intersects te circle at onl one point. Tangent lines to noncircular graps, owever, can intersect te grap at more tan one point. For instance, in te first grap in Figure., if te tangent line were etended, it would intersect te grap at a point oter tan te point of tangenc.
24 80 Capter Limits and an Introduction to Calculus Slope of a Grap Because a tangent line approimates te slope of a grap at a point, te problem of finding te slope of a grap at a point is te same as finding te slope of te tangent line at te point. Eample Visuall Approimating te Slope of a Grap Use te grap in Figure. to approimate te slope of te grap of f at te point,. From te grap of f, ou can see tat te tangent line at, rises approimatel two units for eac unit cange in. So, ou can estimate te slope of te tangent line at, to be Slope. cange in cange in Because te tangent line at te point, as a slope of about, ou can conclude tat te grap of f as a slope of about at te point,. Now tr Eercise. f() = 5 Figure. Wen ou are visuall approimating te slope of a grap, remember tat te scales on te orizontal and vertical aes ma differ. Wen tis appens (as it frequentl does in applications), te slope of te tangent line is distorted, and ou must be careful to account for te difference in scales. Eample Approimating te Slope of a Grap Figure.5 grapicall depicts te montl normal temperatures (in degrees Fareneit) for Dallas, Teas. Approimate te slope of tis grap at te indicated point and give a psical interpretation of te result. (Source: National Catic Data Center) From te grap, ou can see tat te tangent line at te given point falls approimatel 6 units for eac twounit cange in. So, ou can estimate te slope at te given point to be cange in Slope degrees per mont. cange in 6 8 Tis means tat ou can epect te montl normal temperature in November to be about 8 degrees lower tan te normal temperature in October. Now tr Eercise. Temperature ( F) Figure.5 Montl Normal Temperatures (0, 69) Mont
25 Section. Te Tangent Line Problem 80 Slope and te Limit Process In Eamples and, ou approimated te slope of a grap at a point b creating a grap and ten eeballing te tangent line at te point of tangenc. A more sstematic metod of approimating tangent lines makes use of a secant line troug te point of tangenc and a second point on te grap, as sown in Figure.6. If, f is te point of tangenc and, f is a second point on te grap of f, te slope of te secant line troug te two points is given b m sec f f. Slope of secant line Te rigt side of tis equation is called te difference quotient. Te denominator is te cange in, and te numerator is te cange in. Te beaut of tis procedure is tat ou obtain more and more accurate approimations of te slope of te tangent line b coosing points closer and closer to te point of tangenc, as sown in Figure.7. ( +, f ( + )) (, f()) Figure.6 f ( + ) f ( ) ( +, f( + )) ( +, f( + )) ( +, f( + )) (, f()) f ( + ) f ( ) (, f()) f ( + ) f ( ) (, f()) f ( + ) f ( ) Tangent line (, f()) As approaces 0, te secant line approaces te tangent line. Figure.7 Using te it process, ou can find te eact slope of te tangent line at, f. Definition of te Slope of a Grap Te slope m of te grap of f at te point, f is equal to te slope of its tangent line at, f, and is given b m 0 m sec 0 f f provided tis it eists. From te above definition and from Section., ou can see tat te difference quotient is used frequentl in calculus. Using te difference quotient to find te slope of a tangent line to a grap is a major concept of calculus.
26 80 Capter Limits and an Introduction to Calculus Eample Finding te Slope of a Grap Find te slope of te grap of f at te point,. Find an epression tat represents te slope of a secant line at,. m sec f f, Net, take te it of as approaces 0. m 0 m sec 0 m sec Set up difference quotient. Substitute into f. Epand terms. Simplif. Factor and divide out. Simplif. 0 0 Te grap as a slope of at te point,, as sown in Figure.8. Now tr Eercise 5. Tangent line at (, ) m = Figure.8 5 f() = Eample Finding te Slope of a Grap Find te slope of f. m 0 f f 0 ( 0 0 Set up difference quotient. Substitute into f. Epand terms. Divide out. Simplif. You know from our stud of linear functions tat te line given b f as a slope of, as sown in Figure.9. Tis conclusion is consistent wit tat obtained b te it definition of slope, as sown above. Now tr Eercise 7. f() = + m = Figure.9
27 Section. Te Tangent Line Problem 805 It is important tat ou see te difference between te was te difference quotients were set up in Eamples and. In Eample, ou were finding te slope of a grap at a specific point c, f c. To find te slope in suc a case, ou can use te following form of te difference quotient. m 0 f c f c Slope at specific point In Eample, owever, ou were finding a formula for te slope at an point on te grap. In suc cases, ou sould use, rater tan c, in te difference quotient. m 0 f f Formula for slope Ecept for linear functions, tis form will alwas produce a function of, wic can ten be evaluated to find te slope at an desired point. Eample 5 Finding a Formula for te Slope of a Grap Find a formula for te slope of te grap of f. Wat are te slopes at te points, and, 5? m sec Net, take te it of as approaces 0. Set up difference quotient. Substitute into f. Epand terms. Simplif. Factor and divide out. Simplif. Using te formula m for te slope at, f, ou can find te slope at te specified points. At,, te slope is and at, 5, te slope is m. f f, m 0 m sec m 0 m sec 0 0 Te grap of f is sown in Figure.0. Now tr Eercise. TECHNOLOGY TIP Tr verifing te result in Eample 5 b using a graping utilit to grap te function and te tangent lines at, and, 5 as in te same viewing window. Some graping utilities even ave a tangent feature tat automaticall graps te tangent line to a curve at a given point. If ou ave suc a graping utilit, tr verifing te solution of Eample 5 using tis feature. For instructions on ow to use te tangent feature, see Appendi A; for specific kestrokes, go to tis tetbook s Online Stud Center. Tangent line at (, ) f() = + Tangent line at (, 5) Figure.0
28 806 Capter Limits and an Introduction to Calculus Te Derivative of a Function In Eample 5, ou started wit te function f and used te it process to derive anoter function, m, tat represents te slope of te grap of f at te point, f. Tis derived function is called te derivative of f at. It is denoted b f, wic is read as f prime of. Definition of te Derivative Te derivative of f at is given b f 0 f f provided tis it eists. Remember tat te derivative f is a formula for te slope of te tangent line to te grap of f at te point, f. STUDY TIP In Section., ou studied te slope of a line, wic represents te average rate of cange over an interval. Te derivative of a function is a formula wic represents te instantaneous rate of cange at a point. Eample 6 Finding a Derivative Find te derivative of f. f 0 f f So, te derivative of f is f 6. Now tr Eercise 9. Eploration Use a graping utilit to grap te function f. Use te trace feature to approimate te coordinates of te verte of tis parabola. Ten use te derivative of f to find te slope of te tangent line at te verte. Make a conjecture about te slope of te tangent line at te verte of an arbitrar parabola. Note tat in addition to f, oter notations can be used to denote te derivative of f. Te most common are d d,, d d f, and D.
29 Section. Te Tangent Line Problem 807 Eample 7 Using te Derivative Find f for f. Ten find te slopes of te grap of f at te points, and, and equations of te tangent lines to te grap at te points. f f f 0 0 Because direct substitution ields te indeterminate form ou sould use te rationalizing tecnique discussed in Section. to find te it. f At te point,, te slope is f. An equation of te tangent line at te point, is m Pointslope form Substitute for m, for, and for.. Tangent line At te point,, te slope is f. An equation of te tangent line at te point, is m Pointslope form. Substitute for m, for, and for Tangent line. Te graps of f and te tangent lines at te points, and, are sown in Figure.. Now tr Eercise , STUDY TIP Remember tat in order to rationalize te numerator of an epression, ou must multipl te numerator and denominator b te conjugate of te numerator. (, ) 5 f() = Figure. = + = + m = (, ) m =
30 808 Capter Limits and an Introduction to Calculus. Eercises See for workedout solutions to oddnumbered eercises. Vocabular Ceck Fill in te blanks.. is te stud of te rates of cange of functions.. Te to te grap of a function at a point is te line tat best approimates te slope of te grap at te point.. A is a line troug te point of tangenc and a second point on te grap.. Te slope of te secant line is represented b te m sec f f. 5. Te of a function f at represents te slope of te grap of f at te point, f. In Eercises, use te figure to approimate te slope of te curve at te point,... (, ) (, ) 5. g 6. (a) 0, (b), 7. g 8. (a) 5, (b) 0, f (a) 0, (b), f (a) 5, (b) 8,.. (, ) In Eercises 5, use te it process to find te slope of te grap of te function at te specified point. Use a graping utilit to confirm our result. 5. g,, 6. f 0,, 7. g 5,, 8. 5,, 9. g, 0., g, (, ),., 9,. 0,, In Eercises 8, find a formula for te slope of te grap of f at te point, f. Ten use it to find te slopes at te two specified points.. g. g (a) 0, (a), (b), (b), 8 In Eercises 9, use a graping utilit to grap te function and te tangent line at te point, f. Use te grap to approimate te slope of te tangent line. 9. f 0. f. f. f. f. f In Eercises 5 8, find te derivative of te function. 5. f 5 6. f 7. f 9 8. f 5 9. f 0. f. f. f. f. f 8 5. f 6. f 5 7. f 8. f 9 In Eercises 9 6, (a) find te slope of te grap of f at te given point, (b) ten find an equation of te tangent line to te grap at te point, and (c) grap te function and te tangent line. 9. f,, 0. f,,
31 Section. Te Tangent Line Problem 809. f,,. f,, 6. f,,. f,, 5. f, 6. f, 5,, In Eercises 7 50, use a graping utilit to grap f over te interval [, ] and complete te table. Compare te value of te first derivative wit a visual approimation of te slope of te grap f f 7. f 8. f 9. f 50. In Eercises 5 5, find te derivative of f. Use te derivative to determine an points on te grap of f at wic te tangent line is orizontal. Use a graping utilit to verif our results. 5. f 5. f 6 5. f 9 5. f f In Eercises 55 6, use te function and its derivative to determine an points on te grap of f at wic te tangent line is orizontal. Use a graping utilit to verif our results. 55. f, f 56. f, f 57. f cos, f sin, over te interval 0, 58. f sin, f cos, over te interval 0, 59. f e, f e e 60. f e, f e e 6. f ln, f ln 6. f ln f ln, 6. Population Te projected populations (in tousands) of New Jerse for selected ears from 00 to 05 are sown in te table. (Source: U.S. Census Bureau) Year Table for 6 Population (in tousands) (a) Use te regression feature of a graping utilit to find a quadratic model for te data. Let t represent te ear, wit t 0 corresponding to 00. (b) Use a graping utilit to grap te model found in part (a). Estimate te slope of te grap wen t 0, and interpret te result. (c) Find te derivative of te model in part (a). Ten evaluate te derivative for t 0. (d) Write a brief statement regarding our results for parts (a) troug (c). 6. Market Researc Te data in te table sows te number N (in tousands) of books sold wen te price per book is p (in dollars). Price, p Number of books, N $0 900 $5 60 $0 96 $5 7 $0 0 $5 6 (a) Use te regression feature of a graping utilit to find a quadratic model for te data. (b) Use a graping utilit to grap te model found in part (a). Estimate te slopes of te grap wen p $5 and p $0. (c) Use a graping utilit to grap te tangent lines to te model wen p $5 and p $0. Compare te slopes given b te graping utilit wit our estimates in part (b). (d) Te slopes of te tangent lines at p $5 and p $0 are not te same. Eplain wat tis means to te compan selling te books. 65. Rate of Cange A sperical balloon is inflated. Te volume V is approimated b te formula Vr r, were r is te radius. (a) Find te derivative of V wit respect to r. (b) Evaluate te derivative wen te radius is inces.
32 80 Capter Limits and an Introduction to Calculus (c) Wat tpe of unit would be applied to our answer in part (b)? Eplain. 66. Rate of Cange An approimatel sperical benign tumor is reducing in size. Te surface area S is given b te formula Sr r, were r is te radius. (a) Find te derivative of S wit respect to r. (b) Evaluate te derivative wen te radius is mileters. (c) Wat tpe of unit would be applied to our answer in part (b)? Eplain. 67. Vertical Motion A water balloon is trown upward from te top of an 80foot building wit a velocit of 6 feet per second. Te eigt or displacement s (in feet) of te balloon can be modeled b te position function st 6t 6t 80, were t is te time in seconds from wen it was trown. (a) Find a formula for te instantaneous rate of cange of te balloon. (b) Find te average rate of cange of te balloon after te first tree seconds of fligt. Eplain our results. (c) Find te time at wic te balloon reaces its maimum eigt. Eplain our metod. (d) Velocit is given b te derivative of te position function. Find te velocit of te balloon as it impacts te ground. (e) Use a graping utilit to grap te model and verif our results for parts (a) (d). 68. Vertical Motion A Sacajawea dollar is dropped from te top of a 0foot building. Te eigt or displacement s (in feet) of te coin can be modeled b te position function st 6t 0, were t is te time in seconds from wen it was dropped. (a) Find a formula for te instantaneous rate of cange of te coin. (b) Find te average rate of cange of te coin after te first two seconds of free fall. Eplain our results. (c) Velocit is given b te derivative of te position function. Find te velocit of te coin as it impacts te ground. (d) Find te time wen te coin s velocit is 60 feet per second. (e) Use a graping utilit to grap te model and verif our results for parts (a) (d). Sntesis True or False? In Eercises 69 and 70, determine weter te statement is true or false. Justif our answer. 69. Te slope of te grap of is different at ever point on te grap of f. 70. A tangent line to a grap can intersect te grap onl at te point of tangenc. Librar of Parent Functions In Eercises 7 7, matc te function wit te grap of its derivative. It is not necessar to find te derivative of te function. [Te graps are labeled (a), (b), (c), and (d).] (a) (c) 5 f 75. Tink About It Sketc te grap of a function wose derivative is alwas positive. 76. Tink About It Sketc te grap of a function wose derivative is alwas negative. Skills Review In Eercises 77 80, sketc te grap of te rational function. As sketcing aids, ceck for intercepts, vertical asmptotes, orizontal asmptotes, and slant asmptotes. Use a graping utilit to verif our grap. (b) (d) 7. f 7. f f 77. f 78. f 79. f 80. In Eercises 8 8, find te cross product of te vectors. 8.,,,,, 8. 0, 0, 6, 7, 0, 0 8., 0, 0,,, , 7,,, 8, 5 5 f 6
33 . Limits at Infinit and Limits of Sequences Section. Limits at Infinit and Limits of Sequences 8 Limits at Infinit and Horizontal Asmptotes As pointed out at te beginning of tis capter, tere are two basic problems in calculus: finding tangent lines and finding te area of a region. In Section., ou saw ow its can be used to solve te tangent line problem. In tis section and te net, ou will see ow a different tpe of it, a it at infinit, can be used to solve te area problem. To get an idea of wat is meant b a it at infinit, consider te function f. Te grap of f is sown in Figure.. From earlier work, ou know tat is a orizontal asmptote of te grap of tis function. Using it notation, tis can be written as follows. Wat ou sould learn Evaluate its of functions at infinit. Find its of sequences. W ou sould learn it Finding its at infinit is useful in igwa safet applications. For instance, in Eercise 56 on page 89, ou are asked to find a it at infinit to predict te number of injuries due to motor veicle accidents in te United States. f Horizontal asmptote to te left f Horizontal asmptote to te rigt AP Potos Tese its mean tat te value of or increases witout bound. f gets arbitraril close to as decreases f() = + Figure. Definition of Limits at Infinit = If f is a function and and are real numbers, te statements and f L f L L L Limit as approaces Limit as approaces denote te its at infinit. Te first statement is read te it of f as approaces is L, and te second is read te it of f as approaces is L. TECHNOLOGY TIP Recall from Section.7 tat some graping utilities ave difficult graping rational functions. In tis tet, rational functions are graped using te dot mode of a graping utilit, and a blue curve is placed beind te graping utilit s displa to indicate were te grap sould appear.
34 8 Capter Limits and an Introduction to Calculus To elp evaluate its at infinit, ou can use te following definition. Limits at Infinit If r is a positive real number, ten 0. r Eample Find te it. Limit toward te rigt Furtermore, if r is defined wen < 0, ten 0. r Limit toward te left Limits at infinit sare man of te properties of its listed in Section.. Some of tese properties are demonstrated in te net eample. Evaluating a Limit at Infinit Eploration Use a graping utilit to grap te two functions given b and in te same viewing window. W doesn t appear to te left of te ais? How does tis relate to te statement at te left about te infinite it? r Algebraic Use te properties of its listed in Section.. 0 So, te it of f as approaces is. Now tr Eercise 9. Grapical Use a graping utilit to grap. Ten use te trace feature to determine tat as gets larger and larger, gets closer and closer to, as sown in Figure.. Note tat te line is a orizontal asmptote to te rigt. 5 = = 0 0 Figure. In Figure., it appears tat te line is also a orizontal asmptote to te left. You can verif tis b sowing tat. Te grap of a rational function need not ave a orizontal asmptote. If it does, owever, its left and rigt asmptotes must be te same. Wen evaluating its at infinit for more complicated rational functions, divide te numerator and denominator b te igestpowered term in te denominator. Tis enables ou to evaluate eac it using te its at infinit at te top of tis page.
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