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Extra resources for BS EN 1926:2006 Natural stone test methods. Determination of uniaxial compressive strength

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F ͑Ϫ1͒ ϭ Ϫ1 2. lim f ͑x͒ ϭ Ϫ1 x→Ϫ1 3. f ͑Ϫ1͒ ϭ lim f ͑x͒ x→Ϫ1 ␲x < 1 tan 4 , ϭ ≥ 1 x, Ά Ϫ1 < x < 1 has possible discontinuities at x ϭ Ϫ1, x ϭ 1. x ≤ Ϫ1 or x ≥ 1 f ͑1͒ ϭ 1 lim f ͑x͒ ϭ 1 x→1 f ͑1͒ ϭ lim f ͑x͒ x→1 f is continuous at x ϭ ± 1, therefore, f is continuous for all real x. 4 Continuity and One-Sided Limits 39 51. f ͑x͒ ϭ csc 2x has nonremovable discontinuities at integer multiples of ␲͞2. 53. f ͑x͒ ϭ ͠x Ϫ 1͡ has nonremovable discontinuities at each integer k. 55. limϩ f ͑x͒ ϭ 0 57. f ͑2͒ ϭ 8 50 x→0 lim f ͑x͒ ϭ 0 Find a so that limϩ ax2 ϭ 8 ⇒ a ϭ x→0Ϫ x→2 f is not continuous at x ϭ Ϫ2.

5 Implicit Differentiation . . . . . . . . 6 Related Rates . . . . . . . . . . 85 Review Exercises . . . . . . . . . . . . . 92 Problem Solving . . . . . . . . . . . . . 1 2 The Derivative and the Tangent Line Problem Solutions to Odd-Numbered Exercises 1. (a) m ϭ 0 (c) y ϭ 3. (a), (b) (b) m ϭ Ϫ3 y f )4) 4 f )1) )x 1 f )1) 1) x ϭ 1 y f ͑4͒ Ϫ f ͑1͒ ͑x Ϫ 1͒ ϩ f ͑1) 4Ϫ1 3 ͑x Ϫ 1͒ ϩ 2 3 ϭ 1͑x Ϫ 1͒ ϩ 2 6 f )4) 5 ϭxϩ1 5 )4, 5) 4 f )4) f )1) 3 3 2 f )1) )1, 2) 2 1 x 1 5.

Y ϭ 2 2x 27. y ϭ 29. y ϭ Basic Differentiation Rules and Rates of Change x 31. f ͑x͒ ϭ 3 ϭ 3xϪ2, ͑1, 3͒ x2 fЈ͑x͒ ϭ Ϫ6xϪ3 Ϫ9 Ϫ4 x 8 yЈ ϭ Ϫ 1 7 33. f ͑x͒ ϭ Ϫ ϩ x3, 2 5 Ϫ6 ϭ 3 x 1 2x3͞2 ΂0, Ϫ 21΃ ϭ 4x2 ϩ 4x ϩ 1 21 fЈ͑x͒ ϭ x2 5 yЈ ϭ 8x ϩ 4 yЈ͑0͒ ϭ 4 fЈ͑0͒ ϭ 0 fЈ͑1͒ ϭ Ϫ6 2 Ϫ2 39. f ͑x͒ ϭ x ϩ 5 Ϫ 3x 37. f ͑␪͒ ϭ 4 sin ␪ Ϫ ␪, ͑0, 0͒ 41. g͑t͒ ϭ t2 Ϫ fЈ͑x͒ ϭ 2x ϩ 6xϪ3 ϭ 2x ϩ fЈ͑␪͒ ϭ 4 cos ␪ Ϫ 1 6 x3 4 ϭ t2 Ϫ 4tϪ3 t3 gЈ͑t͒ ϭ 2t ϩ 12tϪ4 ϭ 2t ϩ fЈ͑0͒ ϭ 4͑1͒ Ϫ 1 ϭ 3 43. f ͑x͒ ϭ y ϭ ͑2x ϩ 1͒2, ͑0, 1͒ 35. x3 Ϫ 3x2 ϩ 4 ϭ x Ϫ 3 ϩ 4xϪ2 x2 45.

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