By Stamatis Cambanis, Aleksander Weron

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**Extra resources for Probability Theory on Vector Spaces IV**

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But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

9. 10. 11. 12. from an observer up to the highest point in a rainbow is 42°? ) How can we explain the shapes of cans on supermarket shelves? ) Where is the best place to sit in a movie theater? ) How can we design a roller coaster for a smooth ride? ) How far away from an airport should a pilot start descent? ) How can we fit curves together to design shapes to represent letters on a laser printer? ) How can we estimate the number of workers that were needed to build the Great Pyramid of Khufu in ancient Egypt?

F ͑x͒ 48. f ͑x͒ 49. f ͑x͒ f ͑x͒ Ϫ f ͑a͒ xϪa 30. f ͑x͒ xϩ3 , xϩ1 50. f ͑x͒ Խ Խ Խ 3x ϩ x x ͭ ͭ ͭ xϩ2 1Ϫx 3 Ϫ 12 x 2x Ϫ 5 ͭ 4 Ϫ t2 2Ϫt Խ Խ Խ 46. t͑x͒ x Ϫ x if x Ͻ 0 if x ജ 0 if x ഛ 2 if x Ͼ 2 x ϩ 2 if x ഛ Ϫ1 x2 if x Ͼ Ϫ1 xϩ9 Ϫ2x Ϫ6 if x Ͻ Ϫ3 if x ഛ 3 if x Ͼ 3 Խ Խ 51–56 Find an expression for the function whose graph is the given curve. 51. The line segment joining the points ͑1, Ϫ3͒ and ͑5, 7͒ 52. The line segment joining the points ͑Ϫ5, 10͒ and ͑7, Ϫ10͒ f ͑3 ϩ h͒ Ϫ f ͑3͒ h 1 29. f ͑x͒ , x uϩ1 1 1ϩ uϩ1 37.