CHAPTER 11 Limits and an Introduction to Calculus

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Transcription:

CHAPTER Limits ad a Itroductio to Calculus Sectio. Itroductio to Limits................... 50 Sectio. Teciques for Evaluatig Limits............. 5 Sectio. Te Taget Lie Problem................. 50 Sectio. Limits at Ifiit ad Limits of Sequeces........ 5 Sectio.5 Te Area Problem..................... 50 Review Eercises............................. 55 Practice Test............................... 5

CHAPTER Limits ad a Itroductio to Calculus Sectio. Itroductio to Limits If f becomes arbitraril close to a uique umber L as approaces c from eiter side, te te it of f as approaces c is L: f L. c You sould be able to use a calculator to fid a it. You sould be able to use a grap to fid a it. You sould uderstad ow its ca fail to eist: (a) f approaces a differet umber from te rigt of c ta it approaces from te left of c. f icreases or decreases witout boud as approaces c. (c) f oscillates betwee two fied values as approaces c. You sould kow ad be able to use te elemetar properties of its. Vocabular Ceck. it. oscillates. direct substitutio. (a) V baseeigt (d) 00 ( ) ( ) 0 0 (c) V 0 Maimum at V.5.9..5 5 97.0 0.5 0.5 0.0 0.5 0.5 980.0. 5 f.9.99.999.00.0..5.95.995.005.05.5 Te it is reaced. 50

Sectio. Itroductio to Limits 5 5. 9 f.9.99.999.00.0. 0.95 0.9 0.9 Error 0. 0. 0.9 Te it is ot reaced. 7. si 0 f 0. 0.0 0.00 0 0.00 0.0 0..987.99987.9999987 Error.9999987.99987.987 Te it is ot reaced. 9. e 0 f 0. 0.0 0.00 0 0.00 0.0 0..87.980.9980 Error.000.00.0 Te it is ot reaced.. 0.9 0.99 0.999.0.00.0. 5 f 0.5 0.50 0.50 Error 0.99 0.9 0.9. 5 5 0. 0 Actual it is 5. 0.8 0. 0.0 0.00 0 0.00 0.0 0. f 0.7 0.7 0. Error 0. 0.5 0.5 0.8 5...0.00.0.999.99.9 f 0.7 0.975 0.998 Error 0.500 0.505 0.5

5 Capter Limits ad a Itroductio to Calculus 7. Make sure our calculator is set i radia mode. si 0 f 0. 0.0 0.00 0 0.00 0.0 0. 0.998 0.99998 0.9999998 Error 0.9999998 0.99998 0.998 si 9. 0 0 0. 0.0 0.00 0 0.00 0.0 0. f 0.0997 0.000 0.000 Error 0.000 0.000 0.0997 e..0 0 f 0. 0.0 0.00 0 0.00 0.0 0. 0.90 0.990 0.9990 Error.000.00.070 l. f 0.9 0.99 0.999.00.0...00.000 Error.9980.980.8 5. f,, Te it eists as approaces : f 5 < 8 8 7. f does ot eist. 0 8 8 8 9.. does ot eist. f equals to te left of, ad equals to te rigt of.. Te it does ot eist because f oscillates 5. ta does ot eist. betwee ad.

Sectio. Itroductio to Limits 5 5 7. does ot eist. 9. does ot eist. 0 e cos 0 Te grap oscillates betwee ad.. does ot eist.. 8 8 5. l.9 (Eact it is l 7.) 7. (a) g c f g 9 c f (c) c g (d) f c 9. (a) f 8 g 5 8 (c) f g 8 8 (d) g f 8 8 8 5. 5 0 0 5 5 5. 7 55. 9 9 57. 9 0 5 5 7 59. 9 9 7.. 7 5 57 7 5 5. e e 0.0855 7. si si 09. arcsi arcsi 0.5 7. True 7. Aswers will var. 75. (a) No. Te it ma or ma ot eist, ad if it does eist, it ma ot equal. No. f ma or ma ot eist, ad if f eists, it ma ot equal. 77. 5 5 5 5, 5

5 Capter Limits ad a Itroductio to Calculus 79. 5 7 5 5 5 5 5, 8. 7 9 9, Sectio. Teciques for Evaluatig Limits You ca use direct substitutio to fid te it of a polomial fuctio p: p pc. c You ca use direct substitutio to fid te it of a ratioal fuctio r p as log as qc 0: r rc pc q,, qc 0. c qc You sould be able to use cacellatio teciques to fid a it. You sould kow ow to use ratioalizatio teciques to fid a it. You sould kow ow to use tecolog to fid a it. You sould be able to calculate oe-sided its. Vocabular Ceck. dividig out tecique. idetermiate form. oe-sided it. differece quotiet. g, g (a) g 0 g (c) g 5. g g, (a) g g 0 (c) 0 g 0 5. 7. 9. t 8 t t t t t t t t t t. 0 0

Sectio. Teciques for Evaluatig Limits 55. 5. 5 7 7. 9. 5 5 5 5 5 5 0 0 5 5 0 5 5 5 5 0 5 5 5 5 0 7 7 7 7 7 7 7. 0 0 0. sec cos 0 ta 0 cos si, does ot eist 0 si 5. cos cos 0 cot 0 cos si 0 si 0 si 7. 0 9. f. f f 0.887 0 Eact it: f 0

5 Capter Limits ad a Itroductio to Calculus. 5 80 00 5. f f 0, 0.05 0 0 0 8 7. f e f 0 5 9. l 0. 0 si 0. ta 5. 0 0. 7. f f 0.5 0.5.5 9. f (a) Grapicall,. f 0.5 0.9 0.99 0.999 0.7 0.5 0.505 0.500 Error Numericall,. (c) Algebraicall,.

Sectio. Teciques for Evaluatig Limits 57 5. f (a) Grapicall, 0. 0 0. 8. (c) Algebraicall, 8. f.00.0..5 Error 0.50 0.50 0.8 0.0 Numericall, 0.5. 5. f f f Limit does ot eist. 8 0 55. f 57. f 59. f f 5 f,, > f f f does ot eist. 5 5. f() = cos. f() = si 5. f() = si 9 9 9 9 = = = = = = f 0 0 0 f 0 0 f 0 7. (a) Ca be evaluated b direct substitutio: si 0 si 0 0 0 Caot be evaluated b direct substitutio: si 0

58 Capter Limits ad a Itroductio to Calculus 9. f f 7. f f 7. f f 75. f f 77. 8 t 8 t t t t t t t t t t t ft sec

Sectio. Teciques for Evaluatig Limits 59 79. Ct.00 0.5t (a).5.0.5.0 0.5 t C...5..7.50.75.75.75.75.75.75 Ct.75 t.5 0.5 5 (c) t C.5.9..5.5.50.50.50.75.75.75 Ct does ot eist. Te oe-sided its t do ot agree. 8. Aswers will var. As t from te left, f t 9.00. As t from te rigt, f t.80. 8. True 85. Ma aswers possible (a) f() = 5,, > 5 f() = si( ),, = 5 87. Slope of lie troug, ad, : 89. r, e, Parabola cos 9 Slope of perpedicular lie: Equatio: 0 0 9 9. 9 9 r, e 5, 9. r, e, Parabola cos cos si Hperbola 9 8 9 9 9 95. 7,,,, 5 7 8 5 0 ortogoal 97.,,, 9, 8 parallel

50 Capter Limits ad a Itroductio to Calculus Sectio. Te Taget Lie Problem You sould be able to visuall approimate te slope of a grap. Te slope m of te grap of f at te poit, f is give b m f f provided tis it eists. You sould be able to use te it defiitio to fid te slope of a grap. Te derivative of f at is give b f f f provided tis it eists. Notice tat tis is te same it as tat for te taget lie slope. You sould be able to use te it defiitio to fid te derivative of a fuctio. Vocabular Ceck. Calculus. taget lie. secat lie. differece quotiet 5. derivative. Slope is 0 at,.. Slope is at,. 5. m sec g g m 7. m sec g g 5 m 9. m sec g g, 0 m. m sec 9 k 9 k 9 k k 9 k 9 k 9 k 9 k9 k 9 k, k 0 m k 0 9 k

Sectio. Te Taget Lie Problem 5. 5. m sec g g m (a) At 0,, m 0 0. At,, m. m sec, 0 m g g 0,,,,, 0 (a) At m At m 0.. 7. m sec m g g, 0 (a) At m 5,, At 0,, m 0 5.. 9.. (, ). (, ) 9 (, ) 9 Slope at, is. Slope at, is. Slope at, is. 5. f f f 5 5 0 g g 7. g 9 9 9. f f f

5 Capter Limits ad a Itroductio to Calculus. f f f. 5. f f f f f f 7. f f f 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

Sectio. Te Taget Lie Problem 5 9. (a) m sec f f. (a) m sec f f (c), 0 m 5 5 (, ) (c) m, 0 5 (, ) 5. (a) m sec f f 5. (a) m sec f f m 5 5 5, 0 m (c) 5 8 (c) (, ) 0 8 (, ) 8

5 Capter Limits ad a Itroductio to Calculus 7. f f.5 0.5 0 0.5.5.5 0.5 0.5 0 0.5 0.5.5.5 0.5 0 0.5.5 f f Te appear to be te same. 9. f f.5 0.5 0 0.5.5.5..58.7.87.. 0.5 0.08 0.5 0. 0.89 0.7 0.5 0. 0. f f Te appear to be te same. 5. f f f f 0 f as a orizotal taget at,. 5. f f f 9 9 9 9 9 9 9 9 9 9 9 9 9 f 0 9 9 9 ± Horizotal tagets at, ad,

Sectio. Te Taget Lie Problem 55 55. 59. f 057. 0 0,, 0, 0,,,, f e e 0. e 0 0, 0, 0,, e f si 0 si,, 5 f l 0 l e e, e, 5, 5. (a) Pt 0.t.t 88 (c) Usig te it defiitio, Pt.t. P0.0. 8.. (d) Aswers will var.,000 0 00 0 P0 8 At time 00, te populatio is icreasig at approimatel 8,000 per ear. 5. (a) V r 7. Vr Vr Vr r r r r r r r r r V 0.0 (c) Cubic ices per ic; Aswers will var. st t t 80 (a) Usig te it defiitio, st t. s0 80, s 8 (c) (d) (e) Average rate of cage st t 0 t secods Aswers will var. st t t 80 s5 5 9 ftsec 00 8 0 t 5 secods 8 80 ftsec 0 0 9. True. Te slope is wic is differet for all., 7. Matces. (Derivative is alwas positive, but decreasig.)

5 Capter Limits ad a Itroductio to Calculus 7. Matces (d). (Derivative is for < 0, for > 0.) 75. Aswers will var. 77. f Itercept: 0, Vertical asmptotes:, Horizotal asmptote: 0 79. f Lie wit ole at, Itercepts: 0,,, 0 Slat asmptote:, 8. i j k,,,, i j 8., 0, 0,, 0 0,, 0, 0, 0 k 0 Sectio. Limits at Ifiit ad Limits of Sequeces Te it at ifiit f L meas tat f get arbitraril close to L as icreases witout boud. Similarl, te it at ifiit f L meas tat f get arbitraril close to L as decreases witout boud. You sould be able to calculate its at ifiit, especiall tose arisig from ratioal fuctios. Limits of fuctios ca be used to evaluate its of sequeces. If f is a fuctio suc tat f L if a is a sequece suc tat f a, te a L. ad Vocabular Ceck. it, ifiit. coverge. diverge

Sectio. Limits at Ifiit ad Limits of Sequeces 57. Itercept: 0, 0 Horizotal asmptote: Matces (c).. Horizotal asmptote: Vertical asmptote: 0 Matces (d). 5. Vertical asmptotes: ± Horizotal asmptote: Matces (f). 7. Vertical asmptote: Horizotal asmptote: Matces (). 9. 0. 5. 5 9. t t t t t 5 5. 7. does ot eist. t t 8. t. 5. 0 7. t t 5t 0 5 5 t 9. Horizotal asmptote: 8.. Horizotal asmptote: 0 Horizotal asmptote: 5 5. f (a) 0 0 0 0 0 0 0 5 0 f 0.7 0.0995 0.000 0.000.0 0.0 0 5.0 0 f 0 0 f 0

58 Capter Limits ad a Itroductio to Calculus 7. f (a) 0 0 0 0 0 0 0 5 0 f 0.708 0.75 0.795 0.7995 0.79995 0.799995 0.75 f 0.75 5 f 0.75 9. a, 5, 5, 5 7, a 0. a, 5, 7, 9, 5 a. a 5,, 9, 8 7, 5 7 does ot eist. 5.! a!,,, 5,!! does ot eist. 7. a,,,, 5 0 9. 0 0 0 0 0 0 0 5 0 a.55.505.5005.500.500.500 a.5 a a 5. 0 0 0 0 0 0 0 5 0 a. 5. 5. 5. 5. 5. a 5. a 8 a

Sectio. Limits at Ifiit ad Limits of Sequeces 59 5. (a) Average cost C C C00 $7 C000 $59.5.50 5,750 (c) C.50 As more uits are produced, te fied costs (5,750) become less domiat. 55. (a) 7 (c) 0.70 Et 78 t 0.009 Te erollmet approaces 78 millio. (d) Aswers will var. 0 0 For 00, t ad E 7.0 millio. For 008, t 8 ad E8 7.8 millio. 57. False. f does ot ave a orizotal asmptote. 59. True. For eample, let f ad g. Te, 0 icreases witout boud, but 0 f g 0.. Coverges to 0 5. Diverges 5 0 0 0 0 0 7. 9. (a) f f (c) f (d) f (a) (d) (c) ) 8 8 8 7. 5 ) 7 0 9 8 5 0 5 0 7 0 9 8 9 8 9 5

50 Capter Limits ad a Itroductio to Calculus 7. f 0 75. 0 5 Real zeros: 0, 0, 5, 5 f Real zero: 0 0 9 9 50 9 77. i 5 7 9 5 0 79. 0 5 05 50 i k Sectio.5 Te Area Problem You sould kow te followig summatio formulas ad properties. (a) (c) c c i i i i i (d) (e) i a i ± b i (f) ka i k i a i i a i ± i b i i i i You sould be able to evaluate a it of a summatio, S. You sould be able to approimate te area of a regio usig rectagles. B icreasig te umber of rectagles, te approimatio improves. Te area of a plae regio above te -ais bouded b f betwee a ad b is te it of te sum of te approimatig rectagles: A f i a b ai b a You sould be able to use te it defiitio of area to fid te area bouded b simple fuctios i te plae. Vocabular Ceck... area. 0 7 70 0. 0 i 0,00 5. i i 0 k k 0 0,00 0,0 7. 5 j j 55 j 5 5850

Sectio.5 Te Area Problem 5 9. (a) S i i S 0 0 0 0 0 0 0.05 0.5505 0.505005 0.5005 (c) S. (a) S i i 9 7 (c) S 0 0 0 0 0 0 S.85.05.005.0005. (a) S i i S 0 0 0 0 0 0 0.85 0.08 0.00 0.000 (c) S 0 5. (a) S i i S 0 0 0 0 0 0 0 0.5 0.5 0.7 0.7 (c) S 7. f,,,, widt Area.5.5 5 5.5.5 square uits 9. Te widt of eac rectagle is. Te eigt is obtaied b evaluatig f at te rigt-ad edpoit of eac iterval. A 8 i f i 8 i i.55 square uits. Widt of eac rectagle is. Te eigt is f i i. A i i Note: Eact area is. 8 0 50 Approimate area 8.8.5. Te widt of eac rectagle is. Te eigt is 9 i. A i 9 i 8 0 50 Approimate area.5.85.8.

5 Capter Limits ad a Itroductio to Calculus 5. f 5, 0, Te widt of eac rectagle is. Te eigt is f i i 8i 5 5. A A i 8i i 8i 5 5 i 0 i 0 A 0 0 8 0 50 00 Area 0 8.8.. 0 7. f,, 5 Te widt of eac rectagle is. Te eigt is f i i 8i. A i A 5 i 8i i 5 5 A 5 5 0 8 0 50 00 Area 8 9. 9.8 9.8 0 9. f 9, 0, Te widt of eac rectagle is. Te eigt is f i 9 i 9 i. A i 9 i 8 0 50 00 Area.5.85 5. 5.58 5.9 A 8 i 8i 8 8 8 A 8 8 8

Sectio.5 Te Area Problem 5. f,, Te widt of eac rectagle is. Te eigt is f i i 7 i. A 7 i A i i 8i 8 A 8 8 0 50 00 Area 9 8.5 8. 8.08 8.0 8. A f i i 5. i i i i A i f i i i i i A square uits A square uits 7. A i i f i i i i i i 8 8 8 A 8 i 0 i 8 i i square uits

5 Capter Limits ad a Itroductio to Calculus 9. A i 7 g i i 8 i i i 7 i i i i i i i i i 7 A 7 7 7 square uits. i. i i i A g i i i i A square uits A i 5 f i i i i i i 7 5 7 A 5 7 i 9 i 5 7 9 5 i i i 7 i i 7 9 8 square uits 5..0 0 0.00.05 00 500 Note tat 0 we 500. Area 05,08. square feet.5 acres 00 00 00 7. True. See Formula, page 80. 9. Aswers will var. 5. ta ta ta ta ta 0 5. u vu, 5,, 5, 5, 0 55. v 5

Review Eercises for Capter 55 Review Eercises for Capter..9.99.999.00.0. Te it (7) ca be reaced. f..9.99 7 7.00 7.0 7.. f e e 0 Te it caot be reaced. f 0. 0.0 0.00 0 0.00 0.0 0..057.0050.0005 Error 0.9995 0.9950 0.95 5. 7. 9. (a) f c f g 5 7 c (c) f g 5 0 c f (d) c g 5. 5. 5 5 5 5 7 77 t 5. 9 0 t t 7. 8 9. si si 0. e e e. arcsi arcsi 5. 9. t t t t t t t t t 8 7.. 5 5 5 50 5 5 5 0 5 0 5

5 Capter Limits ad a Itroductio to Calculus. u u u u 0 u u 0 u u u 0 u u u u 0 u 5. 5 5 5 5 5 5 5 7. (a) 9 f.9.99.0. 0.95 0.9 Error 0. 0.9 9 9. (a) Aswers will var. 9 9 0. 0.0 0.00 0 0.00 0.0 0..85 E 8 7. E 8 Error Error 0 E 87. E 9 0 e does ot eist.. (a) 0. 0.0 0.00 0 0.00 0.0 0..97.9995.999995 error.999995.995.97 si 0. (a) 5 f..0.00.000 0.580 0.57 0.577 0.577 0.577 Eact value: 5. f Limit does ot eist because f ad f. 5 7. f Limit does ot eist.

Review Eercises for Capter 57 5 9. does ot eist. 5 5 8 0 5. f does ot eist. 7 5 5 5. f f 55. Slope Aswers will var. 57. 59. 5 (, 0) 7 (, ) Slope at, f is approimatel. Slope is at,.. 0. m f f 0 (, ) 0 0 At, f,, te slope is approimatel.5. (a) At 0, 0, m 0. At 5, 5, m 5. 5. f f m (a) At 7,, m 7. At 8,, m 8.

58 Capter Limits ad a Itroductio to Calculus f f 5 5 7. f 0 9. k 0 k k k 0 5 k 5 k k k 0 k 7. 7. g g g ft t 5 t 5 t 5 t 5 t 5 ft ft t 5 t 5 t 5 t 5 t 5 t 5 t 5 t 5 75. gs s 5 s 5 s 0 s 0 s 5s 5 s 5s 5 s 5s 5 gs gs s 5

Review Eercises for Capter 59 77. g g g 79. 8. 5 0 8. does ot eist. 85. 0 87. a 5 89. a a 9 a 5 a a a a 5 7 9 a 8 a 5 5 a 9 a 7 a 5 a 0 9. a 0.5,.5,.,.55,. a 0

50 Capter Limits ad a Itroductio to Calculus i i 9. (a) i i i (c) S 5 i i 8 5 S 0 0 0 0 0 0 0.99 0.88 0.88 0.85 95. Area 7 5 7 7.75 97. f, b a 0 A i i i i f i i 8 8 0 50 Approimate area 7.5.75 5.7 5.9 Eact area is 5.. 99. A 0i 0 i 0 00 00 i i i 00 00 00 50 0. A i i i i 9i 5 5 8 i i 7 i i i 5 8 7 00 50 50, eact area 5 9 9 5, eact area

Review Eercises for Capter 5 0. f, 0, Te widt of eac rectagle is. Te eigt is f i i. A i i i i A 05. f,, Te widt of eac rectagle is. Te eigt is f i i i A i i i i i A 8 i i. 8 8 07. (a).7 0 7.75 0 0.8.8 50 (c) Area 88,88 square feet; aswers will var. 0 0 000 09. False. Te it does ot eist.

5 Capter Limits ad a Itroductio to Calculus Capter Practice Test. Use a grapig utilit to complete te table ad use te result to estimate te it f 9..9.99.0.?. Grap te fuctio f ad estimate te it 0.. Fid te it e b direct substitutio.. Fid te it aalticall. 5. Use a grapig utilit to estimate te it si 5 0. 7. Use te it process to fid te slope of te grap of f at te poit,.. Fid te it. 8. Fid te derivative of te fuctio f. 9. Fid te its. (a) (c) 0. Write te first four terms of te sequece a ad fid te it of te sequece.. Fid te sum 5 i i i. i i. Write te sum as a ratioal fuctio S, ad fid S.. Fid te area of te regio bouded b f over te iterval 0.

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