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C H A P T E R Limits and Their Properties Section 1.1 A Preview of Calculus 305 Section 1.2 Finding Limits Graphically and Numerically 305 Section 1.3 Evaluating Limits Analytically 309 Section 1.4 Continuity and One-Sided Limits Section 1.5 Infinite Limits Review Exercises 315 320 324 Problem Solving 327 C H A P T E R Limits and Their Properties Section 1.1 A Preview of Calculus Solutions to Even-Numbered Exercises Precalculus: rate of change ϭ slope ϭ 0.08 Calculus: velocity is not constant Distance Ϸ ͑20 ft͞sec͒͑15 seconds͒ ϭ 300 feet Precalculus: Area ϭ ͑Ί2͒ Precalculus: Volume ϭ ͑3͒26 ϭ 54 ϭ 2 5 ϩ ϩ Ϸ 10.417 5 5 5 ϩ ϩ ϩ ϩ ϩ ϩ Ϸ 9.145 Area Ϸ ϩ 1.5 2.5 3.5 4.5 10 (a) Area Ϸ ϩ (b) You could improve the approximation by using more rectangles Section 1.2 x 1.9 1.99 1.999 2.001 2.01 2.1 f ͑x͒ 0.2564 0.2506 0.2501 0.2499 0.2494 0.2439 lim x→2 xϪ2 Ϸ 0.25 x2 Ϫ Ϫ 3.1 Ϫ 3.01 Ϫ 3.001 Ϫ 2.999 Ϫ 2.99 Ϫ 2.9 f ͑x͒ Ϫ0.2485 Ϫ0.2498 Ϫ0.2500 Ϫ0.2500 Ϫ0.2502 Ϫ0.2516 lim Ί1 Ϫ x Ϫ xϩ3 ͑ Actual limit is Ϫ 14 ͒ Ϸ Ϫ0.25 x 3.9 3.99 3.999 4.001 4.01 4.1 f ͑x͒ 0.0408 0.0401 0.0400 0.0400 0.0399 0.0392 lim x→4 ͑ Actual limit is 14 ͒ x x→Ϫ3 Finding Limits Graphically and Numerically ͓x͑͞x ϩ 1͔͒ Ϫ ͑4͞5͒ Ϸ 0.04 xϪ4 Ϫ 0.1 x f ͑x͒ lim x→0 0.0500 Ϫ 0.01 0.0050 cos x Ϫ Ϸ 0.0000 x ͑ Actual limit is 251 ͒ Ϫ 0.001 0.0005 0.001 0.01 Ϫ 0.0005 Ϫ 0.0050 0.1 Ϫ 0.0500 (Actual limit is 0.) (Make sure you use radian mode.) 305 306 Chapter Limits and Their Properties 10 lim ͑x2 ϩ 2͒ ϭ 12 lim f ͑x͒ ϭ lim ͑x2 ϩ 2͒ ϭ x→1 x→1 does not exist since the xϪ3 function increases and decreases without bound as x approaches x→1 16 lim sec x ϭ 14 lim x→3 18 lim sin͑x͒ ϭ x→0 x→1 20 C͑t͒ ϭ 0.35 Ϫ 0.12͠Ϫ ͑t Ϫ 1͒͡ (a) (b) t C͑t͒ 3.3 3.4 3.5 3.6 3.7 0.59 0.71 0.71 0.71 0.71 0.71 0.71 lim C͑t͒ ϭ 0.71 t→3.5 (c) 2.5 2.9 3.1 3.5 0.47 0.59 0.59 0.59 0.71 0.71 0.71 t C͑t͒ lim C͑t͒ does not exist The values of C jump from 0.59 to 0.71 at t ϭ t→3.5 Խ Խ 22 You need to find ␦ such that < x Ϫ < ␦ implies f ͑x͒ Ϫ ϭ x2 Ϫ Ϫ ϭ x2 Ϫ < 0.2 That is, Խ Խ Խ Ϫ0.2 Ϫ 0.2 3.8 Ί3.8 Ί3.8 Ϫ Խ Խ < x2 Ϫ < x2 < x2 < x < xϪ2 < < < < < Խ 0.2 ϩ 0.2 4.2 Ί4.2 Ί4.2 Ϫ So take ␦ ϭ Ί4.2 Ϫ Ϸ 0.0494 Խ Խ Then < x Ϫ < ␦ implies Ϫ ͑Ί4.2 Ϫ 2͒ < x Ϫ < Ί4.2 Ϫ Ί3.8 Ϫ < x Ϫ < Ί4.2 Ϫ Using the first series of equivalent inequalities, you obtain Խ f ͑x͒ Ϫ 3Խ ϭ Խx2 Ϫ 4Խ < ⑀ ϭ 0.2 24 lim Ϫ x→4 Խ x ϭ2 Խ Խ Խ Խ Խ 4Ϫ x Ϫ < 0.01 2Ϫ x < 0.01 Ϫ ͑x Ϫ 4͒ < 0.01 Խ Խ Hence, if < Խx Ϫ 4Խ < ␦ ϭ 0.02, you have < x Ϫ < 0.02 ϭ ␦ Խ Խ Խ Խ Խ Խ Ϫ ͑x Ϫ 4͒ < 0.01 2Ϫ 4Ϫ x < 0.01 x Ϫ < 0.01 Խ f ͑x͒ Ϫ LԽ < 0.01 Section 1.2 26 lim ͑x2 ϩ 4͒ ϭ 29 28 lim ͑2x ϩ 5͒ ϭ Ϫ1 x→5 x→Ϫ3 Խ͑x ϩ 4͒ Ϫ 29Խ < 0.01 Խ Finding Limits Graphically and Numerically Given ⑀ > 0: Խ x2 Ϫ 25 < 0.01 Խ͑2x ϩ 5͒ Ϫ ͑Ϫ1͒Խ < ⑀ Խ2x ϩ 6Խ < ⑀ 2Խx ϩ 3Խ < ⑀ Խ͑x ϩ 5͒͑x Ϫ 5͒Խ < 0.01 0.01 Խx Ϫ 5Խ < Խx ϩ 5Խ Խx ϩ 3Խ If we assume < x < 6, then ␦ ϭ 0.01͞11 Ϸ 0.0009 Խ 0.01 , you have 11 xϪ5 < 0.01 < ͑0.01͒ 11 xϩ5 Խ Hence, if < x Ϫ < ␦ ϭ Խ Խ Խ Խx Ϫ 5ԽԽx ϩ 5Խ < 0.01 Խx2 Ϫ 25Խ < 0.01 Խ͑x2 ϩ 4͒ Ϫ 29Խ < 0.01 Խ f ͑x͒ Ϫ LԽ < 0.01 ⑀ Hence, if < x ϩ < ␦ ϭ , you have Խ Խ Խ͑ ϩ 9͒ Ϫ Խ Խx Ϫ 1Խ 3x 29 Ϫ ⑀ Խ2x ϩ 6Խ < ⑀ Խ͑2x ϩ 5͒ Ϫ ͑Ϫ1͒Խ < ⑀ Խ f ͑x͒ Ϫ LԽ < ⑀ 32 lim ͑Ϫ1͒ ϭ Ϫ1 x→2 Given ⑀ > 0: Խ 0, you have < ⑀ Խ͑Ϫ1͒ Ϫ ͑Ϫ1͒Խ < ⑀ Խ f ͑x͒ Ϫ LԽ < ⑀ < 32 ⑀ Hence, let ␦ ϭ ͑3͞2͒⑀ Խ Խ Խx ϩ 3Խ < x→1 3x ⑀ ϭ␦ Hence, let ␦ ϭ ⑀͞2 2 29 30 lim ͑ x ϩ 9͒ ϭ ͑1͒ ϩ ϭ Given ⑀ > 0: < Խ Hence, if < x Ϫ < ␦ ϭ 2⑀, you have Խ Խ͑ 3x Խ Խ x Ϫ < 32⑀ 3x Ϫ ϩ 9͒ Ϫ 29 Խ Խ < ⑀ < ⑀ Խ f ͑x͒ Ϫ LԽ < ⑀ 34 lim Ίx ϭ Ί4 ϭ x→4 Given ⑀ > 0: Խ Խ x Ϫ 2Խ Խ Ί Խ x ϩ 2Խ < ⑀Խ Խx Ϫ 4Խ < ⑀Խ x→3 Ίx Ϫ < ⑀ Ί Խ Խ Խ Խ x ϩ 2Խ Խ͑x Ϫ 3͒ Ϫ 0Խ Խx Ϫ 3Խ Ί Խ Խ < x Ϫ < ␦ ϭ 3⑀ ⇒ x Ϫ < ⑀ Ίx ϩ Խ Խ Given ⑀ > 0: Ίx ϩ Assuming < x < 9, you can choose ␦ ϭ 3⑀ Then, Խ Խ 36 lim x Ϫ ϭ Խ ⇒ Ίx Ϫ < ⑀ < ⑀ < ⑀ϭ␦ Hence, let ␦ ϭ ⑀ Խ Խ Hence for < x Ϫ < ␦ ϭ ⑀, you have Խx Ϫ 3Խ < ⑀ ԽԽx Ϫ 3Խ Ϫ 0Խ < ⑀ Խ f ͑x͒ Ϫ LԽ < ⑀ 307 308 Chapter Limits and Their Properties 40 f ͑x͒ ϭ 38 lim ͑x2 ϩ 3x͒ ϭ x→Ϫ3 Given ⑀ > 0: Խ͑x2 ϩ 3x͒ Ϫ 0Խ Խx͑x ϩ 3͒Խ x2 xϪ3 Ϫ 4x ϩ −3 lim f ͑x͒ ϭ < ⑀ x→3 −4 < ⑀ ⑀ Խx ϩ 3Խ < ԽxԽ The domain is all x 1, The graphing utility does not show the hole at ͑ 3, 12 ͒ If we assume Ϫ4 < x < Ϫ2, then ␦ ϭ ⑀͞4 ⑀ Hence for < x Ϫ ͑Ϫ3͒ < ␦ ϭ , you have Խ Խ 1 Խx ϩ 3Խ < 4⑀ < ԽxԽ⑀ Խx͑x ϩ 3͒Խ < ⑀ Խx ϩ 3x Ϫ 0Խ < ⑀ Խ f ͑x͒ Ϫ LԽ < ⑀ 42 f ͑x͒ ϭ xϪ3 x2 Ϫ lim f ͑x͒ ϭ x→3 44 (a) No The fact that f ͑2͒ ϭ has no bearing on the existence of the limit of f ͑x͒ as x approaches −9 (b) No The fact that lim f ͑x͒ ϭ has no bearing on the x→2 value of f at −3 The domain is all x ± The graphing utility does not show the hole at ͑ 3, 16 ͒ 46 Let p͑x͒ be the atmospheric pressure in a plane at altitude x (in feet) 48 0.002 (1.999, 0.001) (2.001, 0.001) limϩ p͑x͒ ϭ 14.7 lb͞in2 x→0 1.998 2.002 Using the zoom and trace feature, ␦ ϭ 0.001 That is, for Խ Խ < x Ϫ < 0.001, 50 True Խ Խ x2 Ϫ Ϫ < 0.001 xϪ2 52 False; let f ͑x͒ ϭ Άx10,Ϫ 4x, x xϭ4 lim f ͑x͒ ϭ lim ͑x2 Ϫ 4x͒ ϭ and f ͑4͒ ϭ 10 x→4 x2 Ϫ x Ϫ 12 ϭ7 x→4 xϪ4 54 lim n ϩ ͓0.1͔n f ͑4 ϩ ͓0.1͔n͒ n Ϫ ͓0.1͔n x→4 f ͑4 Ϫ ͓0.1͔n͒ 4.1 7.1 3.9 6.9 4.01 7.01 3.99 6.99 4.001 7.001 3.999 6.999 4.0001 7.0001 3.9999 6.9999 Section 1.3 56 f ͑x͒ ϭ mx ϩ b, m Խ Let ⑀ > be given Take ␦ ϭ Խ If < x Ϫ c < ␦ ϭ ⑀ m Խ Խ xc Խ Խ Խ ⑀ , then m Խ Խ That is, Ϫ 12L < g͑x͒ Ϫ L < 12L 2L g͑x͒ < < 2L Hence for x in the interval ͑c Ϫ ␦, c ϩ ␦͒, x g͑x͒ > x→c 2L (a) lim g͑x͒ ϭ 2.4 10 x→4 x→0 > (a) lim f ͑t͒ ϭ 10 t→4 (b) lim f ͑t͒ ϭ Ϫ5 −5 t→Ϫ1 10 10 −5 − 10 g͑x͒ ϭ 12͑ Ίx Ϫ 3͒ xϪ9 Խ lim ͑3x ϩ 2͒ ϭ 3͑Ϫ3͒ ϩ ϭ Ϫ7 x→Ϫ3 x→Ϫ2 10 lim ͑Ϫx2 ϩ 1͒ ϭ Ϫ ͑1͒2 ϩ ϭ x→1 x→Ϫ3 18 lim x→3 2 ϭ ϭ Ϫ2 x ϩ Ϫ3 ϩ Ίx ϩ xϪ4 ϭ Ί3 ϩ 3Ϫ4 Խ f ͑t͒ ϭ t t Ϫ lim x3 ϭ ͑Ϫ2͒3 ϭ Ϫ8 14 lim ϭ Ϫ2 22 lim ͑2x Ϫ 1͒3 ϭ ͓2͑0͒ Ϫ 1͔3 ϭ Ϫ1 x→0 12 lim ͑3x3 Ϫ 2x2 ϩ 4͒ ϭ 3͑1͒3 Ϫ 2͑1͒2 ϩ ϭ x→1 16 lim x→3 2x Ϫ 2͑3͒ Ϫ 3 ϭ ϭ xϩ5 3ϩ5 3 x ϩ ϭΊ 4ϩ4ϭ2 20 lim Ί x→4 24 (a) lim f ͑x͒ ϭ ͑Ϫ3͒ ϩ ϭ x→Ϫ3 (b) lim g͑x͒ ϭ 42 ϭ 16 x→4 (c) lim g͑ f ͑x͒͒ ϭ g͑4͒ ϭ 16 x→Ϫ3 26 (a) lim f ͑x͒ ϭ 2͑42͒ Ϫ 3͑4͒ ϩ ϭ 21 x→4 21 ϩ ϭ (b) lim g͑x͒ ϭ Ί 28 lim tan x ϭ tan ϭ x→ x→21 (c) lim g͑ f ͑x͒͒ ϭ g͑21͒ ϭ x→4 30 lim sin x→1 x ϭ sin ϭ 2 lim cos x ϭ cos x→5͞3 c, Evaluating Limits Analytically (b) lim g͑x͒ ϭ 34 Խ such that < x Ϫ < ␦ implies g͑x͒ Ϫ L < ⑀ ϭ 2L which shows that lim ͑mx ϩ b͒ ϭ mc ϩ b 5 ϭ 309 58 lim g͑x͒ ϭ L, L > Let ⑀ ϭ 2L There exists ␦ > ԽmԽԽx Ϫ cԽ < ⑀ Խmx Ϫ mcԽ < ⑀ Խ͑mx ϩ b͒ Ϫ ͑mc ϩ b͒Խ < ⑀ Section 1.3 Evaluating Limits Analytically 32 lim cos 3x ϭ cos 3 ϭ Ϫ1 x→ 36 lim sec x→7 6x ϭ sec 76 ϭ Ϫ23 Ί3 310 Chapter Limits and Their Properties 38 (a) lim ͓4f ͑x͔͒ ϭ lim f ͑x͒ ϭ x→c x→c 32 ϭ 3 lim f ͑x͒ ϭ Ί27 ϭ f ͑x͒ ϭ Ί 40 (a) lim Ί x→c (b) lim ͓ f ͑x͒ ϩ g͑x͔͒ ϭ lim f ͑x͒ ϩ lim g͑x͒ ϭ x→c x→c x→c (c) lim ͓ f ͑x͒g͑x͔͒ ϭ ͓ lim f ͑x͔͓͒ lim g͑x͔͒ ϭ x→c x→c x→c x→c lim f ͑x͒ x→c f ͑x͒ 27 ϭ ϭ ϭ (b) lim x→c 18 lim 18 18 ϩ ϭ2 2 x→c (c) lim ͓ f ͑x͔͒ ϭ ͓lim f ͑x͔͒2 ϭ ͑27͒2 ϭ 729 3212 ϭ 43 x→c lim f ͑x͒ 3͞2 f ͑x͒ x→c ϭ ϭ ϭ3 x→c g͑x͒ lim g͑x͒ 1͞2 x→c (d) lim ͓ f ͑x͔͒2͞3 ϭ ͓lim f ͑x͔͒2͞3 ϭ ͑27͒2͞3 ϭ (d) lim x→c x→c x→c 42 f ͑x͒ ϭ x Ϫ and h͑x͒ ϭ x2 Ϫ 3x agree except at x ϭ x 44 g͑x͒ ϭ x and f ͑x͒ ϭ agree except at x ϭ xϪ1 x Ϫx (a) lim h͑x͒ ϭ lim f ͑x͒ ϭ Ϫ5 (a) lim f ͑x͒ does not exist (b) lim h͑x͒ ϭ lim f ͑x͒ ϭ Ϫ3 (b) lim f ͑x͒ ϭ Ϫ1 x→Ϫ2 x→1 x→Ϫ2 x→0 x→0 x→0 2x2 Ϫ x Ϫ and g͑x͒ ϭ 2x Ϫ agree except at xϩ1 x ϭ Ϫ1 x3 ϩ and g͑x͒ ϭ x2 Ϫ x ϩ agree except at xϩ1 x ϭ Ϫ1 46 f ͑x͒ ϭ 48 f ͑x͒ ϭ lim f ͑x͒ ϭ lim g͑x͒ ϭ Ϫ5 x→Ϫ1 lim f ͑x͒ ϭ lim g͑x͒ ϭ x→Ϫ1 x→Ϫ1 x→Ϫ1 −8 −4 50 lim x→2 2Ϫx Ϫ ͑x Ϫ 2͒ ϭ lim x2 Ϫ x→2 ͑x Ϫ 2͒͑x ϩ 2͒ ϭ lim x→2 54 lim x ϭ lim x→0 Ίx ϩ Ϫ x→3 xϪ3 x→4 x2 Ϫ 5x ϩ ͑x Ϫ 4͒͑x Ϫ 1͒ ϭ lim x2 Ϫ 2x Ϫ x→4 ͑x Ϫ 4͒͑x ϩ 2͒ ϭ lim x→3 ϭ lim x→4 Ί2 ϩ x Ϫ Ί2 x x→0 ϭ lim 56 lim 52 lim Ϫ1 ϭϪ xϩ2 Ί2 ϩ x Ϫ Ί2 x→0 −1 −8 2ϩxϪ2 и Ί2 ϩ x ϩ Ί2 Ί2 ϩ x ϩ Ί2 ͑ Ί2 ϩ x ϩ Ί2͒ x Ίx ϩ Ϫ xϪ3 и ͑x Ϫ 1͒ ϭ ϭ ͑x ϩ 2͒ ϭ lim x→0 Ί2 ϩ x ϩ Ί2 Ίx ϩ ϩ Ίx ϩ ϩ ϭ lim x→3 ϭ Ί2 ϭ 2Ί2 xϪ3 1 ϭ lim ϭ ͑x Ϫ 3͓͒Ίx ϩ ϩ 2͔ x→3 Ίx ϩ ϩ 1 Ϫ ͑x ϩ 4͒ Ϫ xϩ4 4͑x ϩ 4͒ Ϫ1 ϭ lim ϭ lim ϭϪ 58 lim x→0 x→0 x→0 4͑x ϩ 4͒ x x 16 60 lim ⌬x→0 ͑x ϩ ⌬x͒2 Ϫ x2 x2 ϩ 2x⌬x ϩ ͑⌬x͒2 Ϫ x2 ⌬x͑2x ϩ ⌬x͒ ϭ lim ϭ lim ϭ lim ͑2x ϩ ⌬x͒ ϭ 2x ⌬x→0 ⌬x→0 ⌬x→0 ⌬x ⌬x ⌬x Section 1.3 62 lim ⌬x→0 Evaluating Limits Analytically 311 ͑x ϩ ⌬x͒3 Ϫ x3 x3 ϩ 3x2⌬x ϩ 3x͑⌬x͒2 ϩ ͑⌬x͒3 Ϫ x3 ϭ lim ⌬x→0 ⌬x ⌬x ϭ lim ⌬x→0 64 f ͑x͒ ϭ ⌬x͑3x2 ϩ 3x⌬x ϩ ͑⌬x͒2͒ ϭ lim ͑3x2 ϩ 3x⌬x ϩ ͑⌬x͒2͒ ϭ 3x2 ⌬x→0 ⌬x Ϫ Ίx x Ϫ 16 x 15.9 15.99 15.999 16 16.001 16.01 16.1 f ͑x͒ Ϫ 1252 Ϫ 125 Ϫ 125 ? Ϫ 125 Ϫ 125 Ϫ 1248 20 −1 It appears that the limit is Ϫ0.125 Ϫ Ίx ͑ Ϫ Ίx͒ ϭ lim x→16 x Ϫ 16 x→16 ͑ Ίx ϩ 4͒͑ Ίx Ϫ 4͒ Analytically, lim ϭ lim x→16 Ϫ1 Ίx ϩ ϭϪ x5 Ϫ 32 ϭ 80 x→2 x Ϫ 100 66 lim x f ͑x͒ 1.9 1.99 72.39 79.20 1.999 79.92 1.9999 2.0 79.99 ? 2.0001 80.01 2.001 80.08 2.01 2.1 80.80 88.41 x5 Ϫ 32 ͑x Ϫ 2͒͑x4 ϩ 2x3 ϩ 4x2 ϩ 8x ϩ 16͒ ϭ lim x→2 x Ϫ x→2 xϪ2 Analytically, lim ϭ lim ͑x4 ϩ 2x3 ϩ 4x2 ϩ 8x ϩ 16͒ ϭ 80 x→2 (Hint: Use long division to factor x5 Ϫ 32.) 68 lim x→0 3͑1 Ϫ cos x͒ Ϫ cos x ϭ lim x→0 x x ΄ ΄ ΅ ϭ ͑3͒͑0͒ ϭ sin x tan2 x sin2 x ϭ lim ϭ lim x→0 x→0 x cos x x→0 x x 72 lim и cos2 x΅ sin x ϭ ͑1͒͑0͒ ϭ 76 lim x→ ͞4 Ϫ tan x cos x Ϫ sin x ϭ lim sin x Ϫ cos x x→͞4 sin x cos x Ϫ cos2 x Ϫ ͑sin x Ϫ cos x͒ ϭ lim x→ ͞4 cos x͑sin x Ϫ cos x͒ Ϫ1 ϭ lim x→ ͞4 cos x ϭ lim ͑Ϫsec x͒ x→ ͞4 ϭ Ϫ Ί2 78 lim x→0 ΄ sin 2x sin 2x ϭ lim sin 3x x→0 2x 13sin3x3x ϭ 2͑1͒13͑1͒ ϭ 32 70 lim →0 cos tan sin ϭ lim ϭ1 →0 74 lim sec ϭ ͑Ϫ1͒ ϭ Ϫ → −4 −25 312 Chapter Limits and Their Properties 80 f ͑h͒ ϭ ͑1 ϩ cos 2h͒ Ϫ 0.1 h f ͑h͒ Ϫ 0.01 1.98 Ϫ 0.001 1.9998 0.001 0.01 0.1 ? 1.9998 1.98 −5 −4 Analytically, lim ͑1 ϩ cos 2h͒ ϭ ϩ cos͑0͒ ϭ ϩ ϭ The limit appear to equal h→0 82 f ͑x͒ ϭ sin x x Ί −3 Ϫ 0.1 x f ͑x͒ Ϫ 0.01 0.215 0.0464 Ϫ 0.001 0.001 0.01 0.1 0.01 ? 0.01 0.0464 0.215 −2 The limit appear to equal sin x sin x ϭ lim Ί x ϭ ͑0͒͑1͒ ϭ x→0 Ί x→0 x x Analytically, lim 84 lim h→0 Ίx ϩ h Ϫ Ίx Ίx ϩ h Ϫ Ίx f ͑x ϩ h͒ Ϫ f ͑x͒ ϭ lim ϭ lim h→0 h→0 h h h ϭ lim h→0 и Ίx ϩ h ϩ Ίx Ίx ϩ h ϩ Ίx xϩhϪx 1 ϭ lim ϭ h→0 h͑ Ίx ϩ h ϩ Ίx͒ 2Ίx Ίx ϩ h ϩ Ίx f ͑x ϩ h͒ Ϫ f ͑x͒ x2 ϩ 2xh ϩ h2 Ϫ 4x Ϫ 4h Ϫ x2 ϩ 4x ͑x ϩ h͒2 Ϫ 4͑x ϩ h͒ Ϫ ͑x2 Ϫ 4x͒ ϭ lim ϭ lim h→0 h→0 h→0 h h h 86 lim ϭ lim h→0 Խ Խ h͑2x ϩ h Ϫ 4͒ ϭ lim ͑2x ϩ h Ϫ 4͒ ϭ 2x Ϫ h→0 h Խ Խ 88 lim ͓b Ϫ x Ϫ a ͔ ≤ lim f ͑x͒ ≤ lim ͓b ϩ x Ϫ a ͔ x→a x→a x→a Խ Խ 90 f ͑x͒ ϭ x sin x b ≤ lim f ͑x͒ ≤ b x→a Therefore, lim f ͑x͒ ϭ b x→a − 2 2 −2 Խ Խ lim x sin x ϭ x→0 ԽԽ 92 f ͑x͒ ϭ x cos x 94 h͑x͒ ϭ x cos − 2 0.5 2 − 0.5 0.5 −6 − 0.5 ԽԽ lim x cos x ϭ x→0 x lim x cos x→0 ϭ0 x Section 1.3 x2 Ϫ and g͑x͒ ϭ x ϩ agree at all points xϪ1 except x ϭ 96 f ͑x͒ ϭ Evaluating Limits Analytically 98 If a function f is squeezed between two functions h and g, h͑x͒ ≤ f ͑x͒ ≤ g͑x͒, and h and g have the same limit L as x → c, then lim f ͑x͒ exists and equals L x→c 100 f ͑x͒ ϭ x, g͑x͒ ϭ sin2 x, h͑x͒ ϭ sin2 x x When you are “close to” the magnitude of g is “smaller” than the magnitude of f and the magnitude of g is approaching zero “faster” than the magnitude of f Thus, g ͞ f Ϸ when x is “close to” g −3 ԽԽԽԽ h f −2 102 s͑t͒ ϭ Ϫ16t2 ϩ 1000 ϭ when t ϭ s lim t→5Ί10͞2 5 210 Ϫ s͑t͒ 10 ϭ seconds Ί1000 16 Ί Ί 5Ί10 Ϫt ϭ ϭ ϭ 104 Ϫ4.9t2 ϩ 150 ϭ when t ϭ lim t→5Ί10͞2 Ϫ ͑Ϫ16t2 ϩ 1000͒ 5Ί10 Ϫt 16 t2 Ϫ lim t→5Ί10͞2 lim t→5Ί10͞2 125 5Ί10 Ϫ t Ϫ16 t ϩ ϭ t Ϫ 210 Ί Ϫ t Ϫ 10 16 t ϩ lim t→5Ί10͞2 5Ί10 Ί 5Ί10 ϭ Ϫ80Ί10 ft͞sec Ϸ Ϫ253 ft͞sec 1500 ϭΊ Ϸ 5.53 seconds Ί150 4.9 49 The velocity at time t ϭ a is s͑a͒ Ϫ s͑t͒ ͑Ϫ4.9a2 ϩ 150͒ Ϫ ͑Ϫ4.9t2 ϩ 150͒ Ϫ4.9͑a Ϫ t͒͑a ϩ t͒ ϭ lim ϭ lim t→a t→a t→a aϪt aϪt aϪt lim ϭ lim Ϫ4.9͑a ϩ t͒ ϭ Ϫ2a͑4.9͒ ϭ Ϫ9.8a m͞sec t→a Hence, if a ϭ Ί1500͞49, the velocity is Ϫ9.8Ί1500͞49 Ϸ Ϫ54.2 m͞sec 106 Suppose, on the contrary, that lim g͑x͒ exists Then, since lim f ͑x͒ exists, so would lim ͓ f ͑x͒ ϩ g͑x͔͒, which is a x→c x→c contradiction Hence, lim g͑x͒ does not exist x→c x→c 108 Given f ͑x͒ ϭ x n, n is a positive integer, then lim x n ϭ lim ͑x x nϪ1͒ ϭ ͓lim x͔͓lim x nϪ1͔ x→c x→c x→c x→c ϭ c͓ lim ͑x x nϪ2͔͒ ϭ c͓lim x͔͓lim x nϪ2͔ x→c x→c ϭ c͑c͒lim ͑ x→c x→c ͒ ϭ ϭ c n x x nϪ3 110 Given lim f ͑x͒ ϭ 0: x→c Խ Խ Խ Խ For every ⑀ > 0, there exists ␦ > such that f ͑x͒ Ϫ < ⑀ whenever < x Ϫ c < ␦ Խ Խ Խ Խ ԽԽ Now f ͑x͒ Ϫ ϭ f ͑x͒ ϭ Խ Խ 313 Խ Խ Խ Խ f ͑x͒ Ϫ < ⑀ for x Ϫ c < ␦ Therefore, lim f ͑x͒ ϭ x→c Section P.3 21 f ͑x͒ ϭ Ϫ x 23 h͑x͒ ϭ Ίx Ϫ y Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Functions and Their Graphs 15 y Domain: ͓1, ϱ͒ Range: ͓0, ϱ͒ x 2 x 2 25 f ͑x͒ ϭ Ί9 Ϫ x2 27 g͑t͒ ϭ sin t y Domain: ͓Ϫ3, 3͔ Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓0, 3͔ 2 Range: ͓Ϫ2, 2͔ x y t −1 29 x Ϫ y ϭ ⇒ y ϭ ± Ίx y is not a function of x Some vertical lines intersect the graph twice 33 x2 ϩ y2 ϭ ⇒ y ϭ ± Ί4 Ϫ x2 31 y is a function of x Vertical lines intersect the graph at most once 35 y2 ϭ x2 Ϫ ⇒ y ϭ ± Ίx2 Ϫ y is not a function of x since there are two values of y for some x ԽԽ Խ y is not a function of x since there are two values of y for some x Խ 37 f ͑x͒ ϭ x ϩ x Ϫ If x < 0, then f ͑x͒ ϭ Ϫx Ϫ ͑x Ϫ 2͒ ϭ Ϫ2x ϩ ϭ 2͑1 Ϫ x͒ If ≤ x < 2, then f ͑x͒ ϭ x Ϫ ͑x Ϫ 2͒ ϭ If x ≥ 2, then f ͑x͒ ϭ x ϩ ͑x Ϫ 2͒ ϭ 2x Ϫ ϭ 2͑x Ϫ 1͒ Thus, Ά 2͑1 Ϫ x͒, f ͑x͒ ϭ 2, 2͑x Ϫ 1͒, x < 0 ≤ x < x ≥ 39 The function is g͑x͒ ϭ cx2 Since ͑1, Ϫ2͒ satisfies the equation, c ϭ Ϫ2 Thus, g͑x͒ ϭ Ϫ2x2 41 The function is r͑x͒ ϭ c͞x, since it must be undefined at x ϭ Since ͑1, 32͒ satisfies the equation, c ϭ 32 Thus, r͑x͒ ϭ 32͞x 43 (a) For each time t, there corresponds a depth d 45 (b) Domain: ≤ t ≤ 27 Range: ≤ d ≤ 30 (c) d 18 d 30 25 t1 20 15 10 t t2 t3 t 16 Chapter P Preparation for Calculus y 47 (a) The graph is shifted units to the left y (b) The graph is shifted unit to the right 4 −6 −4 x −2 2 −2 −4 −4 −6 −6 y (c) The graph is shifted units upward x −2 −2 −4 6 −4 −6 x −2 −2 −8 y −4 −2 (e) The graph is stretched vertically by a factor of x −2 −4 y (d) The graph is shifted units downward −2 y (f) The graph is stretched vertically by a factor of 14 x −2 −4 −4 x −2 −6 −8 −6 − 10 49 (a) y ϭ Ίx ϩ (b) y ϭ Ϫ Ίx y y (c) y ϭ Ίx Ϫ y 4 x 3 1 x 2 3 −2 x −1 Reflection about the x-axis Vertical shift units upward Horizontal shift units to the right 51 (a) T͑4͒ ϭ 16Њ, T͑15͒ Ϸ 23Њ (b) If H͑t͒ ϭ T͑t Ϫ 1͒, then the program would turn on (and off) one hour later (c) If H͑t͒ ϭ T͑t͒ Ϫ 1, then the overall temperature would be reduced degree 53 f ͑x͒ ϭ x2, g͑x͒ ϭ Ίx ͑ f Њ g͒͑x͒ ϭ f ͑g͑x͒͒ ϭ f ͑ Ίx ͒ ϭ ͑ Ίx ͒ ϭ x, x ≥ Domain: ͓0, ϱ͒ ͑g Њ f ͒͑x͒ ϭ g͑ f ͑x͒͒ ϭ g͑x2͒ ϭ Ίx2 ϭ ԽxԽ Domain: ͑Ϫ ϱ, ϱ͒ No Their domains are different ͑ f Њ g͒ ϭ ͑g Њ f ͒ for x ≥ 55 f ͑x͒ ϭ , g͑x͒ ϭ x2 Ϫ x ͑ f Њ g͒͑x͒ ϭ f ͑g͑x͒͒ ϭ f ͑x2 Ϫ 1͒ ϭ Domain: all x ±1 ͑g Њ f ͒͑x͒ ϭ g͑ f ͑x͒͒ ϭ g Domain: all x No, f Њ g g Њ f x2 Ϫ 3x ϭ 3x Ϫ1ϭ 9 Ϫ x2 Ϫ1ϭ x x2 Section P.3 57 ͑A Њ r͒͑t͒ ϭ A͑r͑t͒͒ ϭ A͑0.6t͒ ϭ ͑0.6t͒2 ϭ 0.36t Functions and Their Graphs 59 f ͑Ϫx͒ ϭ ͑Ϫx͒2͑4 Ϫ ͑Ϫx͒2͒ ϭ x2͑4 Ϫ x2͒ ϭ f ͑x͒ ͑A Њ r͒͑t͒ represents the area of the circle at time t Even 61 f ͑Ϫx͒ ϭ ͑Ϫx͒ cos͑Ϫx͒ ϭ Ϫx cos x ϭ Ϫf ͑x͒ Odd 63 (a) If f is even, then ͑ , 4͒ is on the graph (b) If f is odd, then ͑ , Ϫ4͒ is on the graph 3 65 f ͑Ϫx͒ ϭ a2nϩ1͑Ϫx͒2nϩ1 ϩ ϩ a3͑Ϫx͒3 ϩ a1͑Ϫx͒ ϭ Ϫ ͓a2nϩ1x2nϩ1 ϩ ϩ a3x3 ϩ a1x͔ ϭ Ϫf ͑x͒ Odd 67 Let F ͑x͒ ϭ f ͑x͒g͑x͒ where f and g are even Then F ͑Ϫx͒ ϭ f ͑Ϫx͒g͑Ϫx͒ ϭ f ͑x͒g͑x͒ ϭ F ͑x͒ Thus, F ͑x͒ is even Let F ͑x͒ ϭ f ͑x͒g͑x͒ where f and g are odd Then F ͑Ϫx͒ ϭ f ͑Ϫx͒g͑Ϫx͒ ϭ ͓Ϫf ͑x͔͓͒Ϫg͑x͔͒ ϭ f ͑x͒g͑x͒ ϭ F ͑x͒ Thus, F ͑x͒ is even 69 f ͑x͒ ϭ x2 ϩ and g͑x͒ ϭ x4 are even f ͑x͒g͑x͒ ϭ ͑x2 ϩ 1͒͑x4͒ ϭ x6 ϩ x4 is even f ͑x͒ ϭ x3 Ϫ x is odd and g͑x͒ ϭ x2 is even f ͑x͒g͑x͒ ϭ ͑x3 Ϫ x͒͑x2͒ ϭ x5 Ϫ x3 is odd −6 −4 −4 −1 71 (a) x length and width volume V 24 Ϫ 2͑1͒ 484 24 Ϫ 2͑2͒ 800 24 Ϫ 2͑3͒ 972 24 Ϫ 2͑4͒ 1024 24 Ϫ 2͑5͒ 980 24 Ϫ 2͑6͒ 864 1200 The maximum volume appears to be 1024 cm3 (c) V ϭ x͑24 Ϫ 2x͒2 ϭ 4x͑12 Ϫ x͒2 Yes, V is a function of x (d) 1100 −1 12 − 100 Maximum volume is V ϭ 1024 cm3 for box having dimensions ϫ 16 ϫ 16 cm Domain: < x < 12 73 False; let f ͑x͒ ϭ x2 Then f ͑Ϫ3͒ ϭ f ͑3͒ ϭ 9, but Ϫ3 (b) 75 True, the function is even 17 18 Chapter P Preparation for Calculus Section P.4 Fitting Models to Data Quadratic function Linear function (a), (b) (a) d ϭ 0.066F or F ϭ 15.1d ϩ 0.1 y 250 (b) 125 200 150 100 F = 15.13 d + 0.10 50 x 12 10 15 The model fits well Yes The cancer mortality increases linearly with increased exposure to the carcinogenic substance (c) If F ϭ 55, then d Ϸ 0.066͑55͒ ϭ 3.63 cm (c) If x ϭ 3, then y Ϸ 136 (a) Let x ϭ per capita energy usage (in millions of Btu) 11 (a) y1 ϭ 0.0343t3 Ϫ 0.3451t2 ϩ 0.8837t ϩ 5.6061 y ϭ per capita gross national product (in thousands) y2 ϭ 0.1095t ϩ 2.0667 y ϭ 0.0764x ϩ 4.9985 Ϸ 0.08x ϩ 5.0 r ϭ 0.7052 (b) y3 ϭ 0.0917t ϩ 0.7917 (b) 15 y1 + y2 + y3 40 y1 y2 0 420 y = 0.08x + 5.0 y3 For 2002, t ϭ 12 and y1 ϩ y2 ϩ y3 Ϸ 31.06 cents͞mile (c) Denmark, Japan, and Canada (d) Deleting the data for the three countries above, y ϭ 0.0959x ϩ 1.0539 (r ϭ 0.9202 is much closer to 1.) 13 (a) y1 ϭ 4.0367t ϩ 28.9644 (d) y3 ϭ 0.4297t2 ϩ 0.5994t ϩ 32.9745 y2 ϭ Ϫ0.0099t3 ϩ 0.5488t2 ϩ 0.2399t ϩ 33.1414 (b) 70 70 y1 = 4.04t + 28.96 25 25 y2 = −0.01t + 0.55t + 0.24t + 33.14 (c) The cubic model is better (e) The slope represents the average increase per year in the number of people (in millions) in HMOs (f) For 2000, t ϭ 10, and y1 Ϸ 69.3 million (linear) y2 Ϸ 80.5 million (cubic) Review Exercises for Chapter P 15 (a) y ϭ Ϫ1.81x3 ϩ 14.58x2 ϩ 16.39x ϩ 10 (b) 17 (a) Yes, y is a function of t At each time t, there is one and only one displacement y 300 (b) The amplitude is approximately ͑2.35 Ϫ 1.65͒͞2 ϭ 0.35 The period is approximately 2͑0.375 Ϫ 0.125͒ ϭ 0.5 (c) If x ϭ 4.5, y Ϸ 214 horsepower (c) One model is y ϭ 0.35 sin͑4 t͒ ϩ (d) 0.9 0 19 Answers will vary Review Exercises for Chapter P y ϭ 2x Ϫ x ϭ ⇒ y ϭ 2͑0͒ Ϫ ϭ Ϫ3 ⇒ ͑0, Ϫ3͒ 3 y ϭ ⇒ ϭ 2x Ϫ ⇒ x ϭ ⇒ ͑ , 0͒ y ϭ y-intercept x-intercept xϪ1 xϪ2 Symmetric with respect to y-axis since 0Ϫ1 1 ϭ ⇒ 0, xϭ0⇒yϭ 0Ϫ2 2 yϭ0⇒0ϭ ͑Ϫx͒2y Ϫ ͑Ϫx͒2 ϩ 4y ϭ y-intercept xϪ1 ⇒ x ϭ ⇒ ͑1, 0͒ xϪ2 y ϭ Ϫ 12 x ϩ 32 x2y Ϫ x2 ϩ 4y ϭ x-intercept 11 y ϭ Ϫ 6x Ϫ x2 Ϫ x ϩ y ϭ Ϫ 25 x ϩ y ϭ 65 y y y ϭ 25 x ϩ 65 2 Slope: y-intercept: 5 x y x 10 2 x 1 5 19 20 Chapter P Preparation for Calculus 17 3x Ϫ 4y ϭ 15 y ϭ 4x2 Ϫ 25 13 y ϭ Ί5 Ϫ x 4x ϩ 4y ϭ 20 Domain: ͑Ϫ ϱ, 5͔ Xmin = -5 Xmax = Xscl = Ymin = -30 Ymax = 10 Yscl = y ϭ 28 7x xϭ yϭ Point: ͑4, 1͒ x 19 You need factors ͑x ϩ 2͒ and ͑x Ϫ 2͒ Multiply by x to obtain origin symmetry y ϭ x͑x ϩ 2͒͑x Ϫ 2͒ ϭ x3 Ϫ 4x 21 23 y 1Ϫ5 1Ϫt ϭ Ϫ Ϫ ͑Ϫ2͒ 1ϪtϭϪ ( 5, ) tϭ ( 32 , 1) x Slope ϭ ͑5͞2͒ Ϫ 3͞2 ϭ ϭ Ϫ ͑3͞2͒ 7͞2 y Ϫ ͑Ϫ5͒ ϭ 32͑x Ϫ 0͒ 25 y Ϫ ϭ Ϫ 23͑x Ϫ ͑Ϫ3͒͒ 27 y ϭ 32x Ϫ 2y Ϫ 3x ϩ 10 ϭ y ϭ Ϫ 23x Ϫ 3y ϩ 2x ϩ ϭ y y 4 2 −4 −2 (−3, 0) x −2 −4 −8 (0, −5) −8 −6 −4 x −2 −4 −6 −8 Review Exercises for Chapter P yϪ4ϭ 29 (a) (b) Slope of line is ͑x ϩ 2͒ 16 16y Ϫ 64 ϭ 7x ϩ 14 y Ϫ ϭ ͑x ϩ 2͒ ϭ 7x Ϫ 16y ϩ 78 3y Ϫ 12 ϭ 5x ϩ 10 4Ϫ0 ϭ Ϫ2 Ϫ2 Ϫ mϭ (c) 21 ϭ 5x Ϫ 3y ϩ 22 y ϭ Ϫ2x x ϭ Ϫ2 (d) 2x ϩ y ϭ xϩ2ϭ0 31 The slope is Ϫ850 V ϭ Ϫ850t ϩ 12,500 V͑3͒ ϭ Ϫ850͑3͒ ϩ 12,500 ϭ $9950 33 x Ϫ y2 ϭ 35 y ϭ x2 Ϫ 2x y ϭ ± Ίx Function of x since there is one value of y for each x Not a function of x since there are two values of y for some x y y x −1 x −2 −1 1 −2 −2 −3 37 f ͑x͒ ϭ x 39 (a) Domain: 36 Ϫ x2 ≥ ⇒ Ϫ6 ≤ x ≤ Range: ͓0, 6͔ (a) f ͑0͒ does not exist 1 Ϫ ϩ ⌬x 1 Ϫ Ϫ ⌬x f ͑1 ϩ ⌬x͒ Ϫ f ͑1͒ ϭ ϭ (b) ⌬x ⌬x ͑1 ϩ ⌬x͒⌬x Ϫ1 , ⌬x ϭ ϩ ⌬x 41 (a) f ͑x͒ ϭ x3 ϩ c, c ϭ Ϫ2, 0, y c Ϫ1, (b) Domain: all x Range: all y Range: all y or or ͑Ϫ ϱ, 0͒, ͑0, ϱ͒ ͑Ϫ ϱ, ϱ͒ ͑Ϫ ϱ, ϱ͒ (b) f ͑x͒ ϭ ͑x Ϫ c͒3, c ϭ Ϫ2, 0, y c c or ͑Ϫ ϱ, 5͒, ͑5, ϱ͒ (c) Domain: all x or c 2 x 2 x 2 c —CONTINUED— or ͓Ϫ6, 6͔ c 22 Chapter P Preparation for Calculus 41 —CONTINUED— (c) f ͑x͒ ϭ ͑x Ϫ 2͒3 ϩ c, c ϭ Ϫ2, 0, (d) f ͑x͒ ϭ cx3, c ϭ Ϫ2, 0, y y c c 2 c 1 c x 1 x c c 43 (a) Odd powers: f ͑x͒ ϭ x, g͑x͒ ϭ x3, h͑x͒ ϭ x5 Even powers: f ͑x͒ ϭ x2, g͑x͒ ϭ x4, h͑x͒ ϭ x6 g 2 g h h −3 f f −3 −2 The graphs of f, g, and h all rise to the left and to the right As the degree increases, the graph rises more steeply All three graphs pass through the points ͑0, 0͒, ͑1, 1͒, and ͑Ϫ1, 1͒ The graphs of f, g, and h all rise to the right and fall to the left As the degree increases, the graph rises and falls more steeply All three graphs pass through the points ͑0, 0͒, ͑1, 1͒, and ͑Ϫ1, Ϫ1͒ (b) y ϭ x7 will look like h͑x͒ ϭ x5, but rise and fall even more steeply y ϭ x8 will look like h͑x͒ ϭ x6, but rise even more steeply 45 (a) (b) Domain: < x < 12 y x 40 x y 2x ϩ 2y ϭ 24 12 y ϭ 12 Ϫ x A ϭ xy ϭ x͑12 Ϫ x͒ ϭ 12x Ϫ x2 47 (a) (cubic), negative leading coefficient (b) (quartic), positive leading coefficient (c) (quadratic), negative leading coefficient (c) Maximum area is A ϭ 36 In general, the maximum area is attained when the rectangle is a square In this case, x ϭ 49 (a) Yes, y is a function of t At each time t, there is one and only one displacement y (b) The amplitude is approximately ͑0.25 Ϫ ͑Ϫ0.25͒͒͞2 ϭ 0.25 (d) 5, positive leading coefficient The period is approximately 1.1 (c) One model is y ϭ (d) 2 cos t Ϸ cos͑5.7t͒ 1.1 0.5 −0.5 2.2 Review Exercises for Chapter P 15 (a) y ϭ Ϫ1.81x3 ϩ 14.58x2 ϩ 16.39x ϩ 10 (b) 17 (a) Yes, y is a function of t At each time t, there is one and only one displacement y 300 (b) The amplitude is approximately ͑2.35 Ϫ 1.65͒͞2 ϭ 0.35 The period is approximately 2͑0.375 Ϫ 0.125͒ ϭ 0.5 (c) If x ϭ 4.5, y Ϸ 214 horsepower (c) One model is y ϭ 0.35 sin͑4 t͒ ϩ (d) 0.9 0 19 Answers will vary Review Exercises for Chapter P y ϭ 2x Ϫ x ϭ ⇒ y ϭ 2͑0͒ Ϫ ϭ Ϫ3 ⇒ ͑0, Ϫ3͒ 3 y ϭ ⇒ ϭ 2x Ϫ ⇒ x ϭ ⇒ ͑ , 0͒ y ϭ y-intercept x-intercept xϪ1 xϪ2 Symmetric with respect to y-axis since 0Ϫ1 1 ϭ ⇒ 0, xϭ0⇒yϭ 0Ϫ2 2 yϭ0⇒0ϭ ͑Ϫx͒2y Ϫ ͑Ϫx͒2 ϩ 4y ϭ y-intercept xϪ1 ⇒ x ϭ ⇒ ͑1, 0͒ xϪ2 y ϭ Ϫ 12 x ϩ 32 x2y Ϫ x2 ϩ 4y ϭ x-intercept 11 y ϭ Ϫ 6x Ϫ x2 Ϫ x ϩ y ϭ Ϫ 25 x ϩ y ϭ 65 y y y ϭ 25 x ϩ 65 2 Slope: y-intercept: 5 x y x 10 2 x 1 5 19 20 Chapter P Preparation for Calculus 17 3x Ϫ 4y ϭ 15 y ϭ 4x2 Ϫ 25 13 y ϭ Ί5 Ϫ x 4x ϩ 4y ϭ 20 Domain: ͑Ϫ ϱ, 5͔ Xmin = -5 Xmax = Xscl = Ymin = -30 Ymax = 10 Yscl = y ϭ 28 7x xϭ yϭ Point: ͑4, 1͒ x 19 You need factors ͑x ϩ 2͒ and ͑x Ϫ 2͒ Multiply by x to obtain origin symmetry y ϭ x͑x ϩ 2͒͑x Ϫ 2͒ ϭ x3 Ϫ 4x 21 23 y 1Ϫ5 1Ϫt ϭ Ϫ Ϫ ͑Ϫ2͒ 1ϪtϭϪ ( 5, ) tϭ ( 32 , 1) x Slope ϭ ͑5͞2͒ Ϫ 3͞2 ϭ ϭ Ϫ ͑3͞2͒ 7͞2 y Ϫ ͑Ϫ5͒ ϭ 32͑x Ϫ 0͒ 25 y Ϫ ϭ Ϫ 23͑x Ϫ ͑Ϫ3͒͒ 27 y ϭ 32x Ϫ 2y Ϫ 3x ϩ 10 ϭ y ϭ Ϫ 23x Ϫ 3y ϩ 2x ϩ ϭ y y 4 2 −4 −2 (−3, 0) x −2 −4 −8 (0, −5) −8 −6 −4 x −2 −4 −6 −8 Review Exercises for Chapter P yϪ4ϭ 29 (a) (b) Slope of line is ͑x ϩ 2͒ 16 16y Ϫ 64 ϭ 7x ϩ 14 y Ϫ ϭ ͑x ϩ 2͒ ϭ 7x Ϫ 16y ϩ 78 3y Ϫ 12 ϭ 5x ϩ 10 4Ϫ0 ϭ Ϫ2 Ϫ2 Ϫ mϭ (c) 21 ϭ 5x Ϫ 3y ϩ 22 y ϭ Ϫ2x x ϭ Ϫ2 (d) 2x ϩ y ϭ xϩ2ϭ0 31 The slope is Ϫ850 V ϭ Ϫ850t ϩ 12,500 V͑3͒ ϭ Ϫ850͑3͒ ϩ 12,500 ϭ $9950 33 x Ϫ y2 ϭ 35 y ϭ x2 Ϫ 2x y ϭ ± Ίx Function of x since there is one value of y for each x Not a function of x since there are two values of y for some x y y x −1 x −2 −1 1 −2 −2 −3 37 f ͑x͒ ϭ x 39 (a) Domain: 36 Ϫ x2 ≥ ⇒ Ϫ6 ≤ x ≤ Range: ͓0, 6͔ (a) f ͑0͒ does not exist 1 Ϫ ϩ ⌬x 1 Ϫ Ϫ ⌬x f ͑1 ϩ ⌬x͒ Ϫ f ͑1͒ ϭ ϭ (b) ⌬x ⌬x ͑1 ϩ ⌬x͒⌬x Ϫ1 , ⌬x ϭ ϩ ⌬x 41 (a) f ͑x͒ ϭ x3 ϩ c, c ϭ Ϫ2, 0, y c Ϫ1, (b) Domain: all x Range: all y Range: all y or or ͑Ϫ ϱ, 0͒, ͑0, ϱ͒ ͑Ϫ ϱ, ϱ͒ ͑Ϫ ϱ, ϱ͒ (b) f ͑x͒ ϭ ͑x Ϫ c͒3, c ϭ Ϫ2, 0, y c c or ͑Ϫ ϱ, 5͒, ͑5, ϱ͒ (c) Domain: all x or c 2 x 2 x 2 c —CONTINUED— or ͓Ϫ6, 6͔ c 22 Chapter P Preparation for Calculus 41 —CONTINUED— (c) f ͑x͒ ϭ ͑x Ϫ 2͒3 ϩ c, c ϭ Ϫ2, 0, (d) f ͑x͒ ϭ cx3, c ϭ Ϫ2, 0, y y c c 2 c 1 c x 1 x c c 43 (a) Odd powers: f ͑x͒ ϭ x, g͑x͒ ϭ x3, h͑x͒ ϭ x5 Even powers: f ͑x͒ ϭ x2, g͑x͒ ϭ x4, h͑x͒ ϭ x6 g 2 g h h −3 f f −3 −2 The graphs of f, g, and h all rise to the left and to the right As the degree increases, the graph rises more steeply All three graphs pass through the points ͑0, 0͒, ͑1, 1͒, and ͑Ϫ1, 1͒ The graphs of f, g, and h all rise to the right and fall to the left As the degree increases, the graph rises and falls more steeply All three graphs pass through the points ͑0, 0͒, ͑1, 1͒, and ͑Ϫ1, Ϫ1͒ (b) y ϭ x7 will look like h͑x͒ ϭ x5, but rise and fall even more steeply y ϭ x8 will look like h͑x͒ ϭ x6, but rise even more steeply 45 (a) (b) Domain: < x < 12 y x 40 x y 2x ϩ 2y ϭ 24 12 y ϭ 12 Ϫ x A ϭ xy ϭ x͑12 Ϫ x͒ ϭ 12x Ϫ x2 47 (a) (cubic), negative leading coefficient (b) (quartic), positive leading coefficient (c) (quadratic), negative leading coefficient (c) Maximum area is A ϭ 36 In general, the maximum area is attained when the rectangle is a square In this case, x ϭ 49 (a) Yes, y is a function of t At each time t, there is one and only one displacement y (b) The amplitude is approximately ͑0.25 Ϫ ͑Ϫ0.25͒͒͞2 ϭ 0.25 (d) 5, positive leading coefficient The period is approximately 1.1 (c) One model is y ϭ (d) 2 cos t Ϸ cos͑5.7t͒ 1.1 0.5 −0.5 2.2 Problem Solving for Chapter P 23 Problem Solving for Chapter P x2 Ϫ 6x ϩ y2 Ϫ 8y ϭ (a) (b) Slope of line from ͑0, 0͒ to ͑3, 4͒ is Slope of tangent line 3 is Ϫ Hence, ͑x2 Ϫ 6x ϩ 9͒ ϩ ͑y2 Ϫ 8y ϩ 16͒ ϭ ϩ 16 ͑x Ϫ 3͒2 ϩ ͑y Ϫ 4͒2 ϭ 25 Center: ͑3, 4͒ 3 y Ϫ ϭ Ϫ ͑x Ϫ 0͒ ⇒ y ϭ Ϫ x 4 Radius: (c) Slope of line from ͑6, 0͒ to ͑3, 4͒ is 4Ϫ0 ϭϪ 3Ϫ6 3 Slope of tangent line is Hence, 3 (d) Ϫ x ϭ x Ϫ 4 xϭ 2 y Ϫ ϭ ͑x Ϫ 6͒ ⇒ y ϭ x Ϫ 4 Tangent line xϭ3 Intersection: H͑x͒ ϭ Ά10 x ≥ x < Tangent line 3, Ϫ 49 y x −4 −3 −2 −1 −1 −2 −3 −4 (a) H͑x͒ Ϫ (b) H͑x Ϫ 2͒ y y 4 3 2 1 x −4 −3 −2 −1 −1 x −4 −3 −2 −1 −1 4 4 −2 −3 −3 −4 −4 (c) ϪH͑x͒ (d) H͑Ϫx͒ y y 4 3 2 x −4 −3 −2 −1 −1 −2 −2 −3 −3 −4 −4 (e) 2H͑x͒ (f ) ϪH͑x Ϫ 2͒ ϩ y x −4 −3 −2 −1 −1 y 4 3 −4 −3 −2 −1 −1 x −4 −3 −2 −1 −1 −2 −2 −3 −3 −4 −4 x 24 Chapter P Preparation for Calculus (a) x ϩ 2y ϭ 100 ⇒ y ϭ A͑x͒ ϭ xy ϭ x 100 Ϫ x The length of the trip in the water is Ί22 ϩ x2, and the length of the trip over land is Ί1 ϩ ͑3 Ϫ x͒2 Hence, the total time is 1002Ϫ x ϭ Ϫ x2 ϩ 50x Tϭ Domain: < x < 100 (b) Ί4 ϩ x2 ϩ Ί1 ϩ ͑3 Ϫ x͒2 1600 110 Maximum of 1250 m at x ϭ 50 m, y ϭ 25 m (c) A͑x͒ ϭ Ϫ ͑x2 Ϫ 100x͒ ϭ Ϫ ͑x2 Ϫ 100x ϩ 2500͒ ϩ 1250 ϭ Ϫ ͑x Ϫ 50͒2 ϩ 1250 A͑50͒ ϭ 1250 m is the maximum x ϭ 50 m, y ϭ 25 m (a) Slope ϭ 9Ϫ4 ϭ Slope of tangent line is less than 3Ϫ2 (b) Slope ϭ 4Ϫ1 ϭ Slope of tangent line is greater than 2Ϫ1 (c) Slope ϭ 4.41 Ϫ ϭ 4.1 Slope of tangent line is less than 4.1 2.1 Ϫ (d) Slope ϭ f ͑2 ϩ h͒ Ϫ f ͑2͒ ͑2 ϩ h͒ Ϫ ϭ ͑2 ϩ h͒2 Ϫ h ϭ 4h ϩ h2 h ϭ ϩ h, h (e) Letting h get closer and closer to 0, the slope approaches Hence, the slope at ͑2, 4͒ is 11 (a) At x ϭ and x ϭ Ϫ3 the sounds are equal (b) I Ίx2 ϩ y2 ϭ 2I Ί͑x Ϫ 3͒2 ϩ y2 −6 ͑x Ϫ 3͒2 ϩ y2 ϭ 4͑x2 ϩ y2͒ 3x2 ϩ 3y2 ϩ 6x ϭ x2 ϩ 2x ϩ y2 ϭ ͑x ϩ 1͒2 ϩ y2 ϭ Circle of radius centered at ͑Ϫ1, 0͒ −3 hours Problem Solving for Chapter P d1d2 ϭ 13 y ͓͑x ϩ 1͒2 ϩ y2͔͓͑x Ϫ 1͒2 ϩ y2͔ ϭ ͑x ϩ 1͒2͑x Ϫ 1͒2 ϩ y2͓͑x ϩ 1͒2 ϩ ͑x Ϫ 1͒2͔ ϩ y4 ϭ ͑x2 Ϫ 1͒2 ϩ y2͓2x2 ϩ 2͔ ϩ y4 ϭ x4 Ϫ 2x2 ϩ ϩ 2x2y2 ϩ 2y2 ϩ y4 ϭ ͑x4 ϩ 2x2y2 ϩ y4͒ Ϫ 2x2 ϩ 2y2 ϭ ͑x2 ϩ y2͒2 ϭ 2͑x2 Ϫ y2͒ Let y ϭ Then x4 ϭ 2x2 ⇒ x ϭ or x2 ϭ Thus, ͑0, 0͒, ͑Ί2, 0͒ and ͑Ϫ Ί2, 0͒ are on the curve (− , 0) ( , 0) x −2 −1 −2 (0, 0) 25 ... Preview of Calculus Solutions to Even-Numbered Exercises Precalculus: rate of change ϭ slope ϭ 0.08 Calculus: velocity is not constant Distance Ϸ ͑20 ft͞sec͒͑15 seconds͒ ϭ 300 feet Precalculus:... False; f ͑x͒ ϭ ͑sin x͒͞x is undefined when x ϭ From Exercise 7, we have lim x→0 51 False; let f ͑x͒ ϭ sin x ϭ x Evaluating Limits Analytically 31 53 Answers will vary Άx10,Ϫ 4x, x xϭ4 f ͑4͒ ϭ... Precalculus: ͑20 ft͞sec͒͑15 seconds͒ ϭ 300 feet Calculus required: slope of tangent line at x ϭ is rate of change, and equals about 0.16 Precalculus: Area ϭ 12 bh ϭ 12 ͑5͒͑3͒ ϭ 15 sq units Precalculus: