Chapter 1 Solutions Manual - CHAPTER 1 Functions EXERCISE...

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1 CHAPTER 1 Functions EXERCISE SET 1.1 1. (a) 2 . 9 , 2 . 0 , 2 . 35 , 2 . 9 (b) none (c) y = 0 (d) 1 . 75 x 2 . 15 (e) y max = 2 . 8 at x = 2 . 6; y min = 2 . 2 at x = 1 . 2 3. (a) yes (b) yes (c) no (vertical line test fails) (d) no (vertical line test fails) 5. (a) around 1943 (b) 1960; 4200 (c) no; you need the year’s population (d) war; marketing techniques (e) news of health risk; social pressure, antismoking campaigns, increased taxation 7. (a) 1999, $34,400 (b) 1985, $37,000 (c) second year; graph has a larger (negative) slope 9. (a) f (0) = 3(0) 2 2 = 2; f (2) = 3(2) 2 2 = 10; f ( 2) = 3( 2) 2 2 = 10; f (3) = 3(3) 2 2 = 25; f ( 2) = 3( 2) 2 2 = 4; f (3 t ) = 3(3 t ) 2 2 = 27 t 2 2 (b) f (0) = 2(0) = 0; f (2) = 2(2) = 4; f ( 2) = 2( 2) = 4; f (3) = 2(3) = 6; f ( 2) = 2 2; f (3 t ) = 1 / 3 t for t > 1 and f (3 t ) = 6 t for t 1. 11. (a) x = 3 (b) x ≤ − 3 or x 3 (c) x 2 2 x + 5 = 0 has no real solutions so x 2 2 x + 5 is always positive or always negative. If x = 0, then x 2 2 x + 5 = 5 > 0; domain: ( −∞ , + ). (d) x = 0 (e) sin x = 1, so x = (2 n + 1 2 ) π , n = 0 , ± 1 , ± 2 , . . . 13. (a) x 3 (b) 2 x 2 (c) x 0 (d) all x (e) all x 15. (a) Breaks could be caused by war, pestilence, flood, earthquakes, for example. (b) C decreases for eight hours, takes a jump upwards, and then repeats. 17. t h 19. (a) x = 2 , 4 (b) none (c) x 2; 4 x (d) y min = 1; no maximum value 21. The cosine of θ is ( L h ) /L (side adjacent over hypotenuse), so h = L (1 cos θ ). 23. (a) If x < 0, then | x | = x so f ( x ) = x + 3 x + 1 = 2 x + 1. If x 0, then | x | = x so f ( x ) = x + 3 x + 1 = 4 x + 1; f ( x ) = 2 x + 1 , x < 0 4 x + 1 , x 0
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2 Chapter 1 (b) If x < 0, then | x | = x and | x 1 | = 1 x so g ( x ) = x + 1 x = 1 2 x . If 0 x < 1, then | x | = x and | x 1 | = 1 x so g ( x ) = x + 1 x = 1. If x 1, then | x | = x and | x 1 | = x 1 so g ( x ) = x + x 1 = 2 x 1; g ( x ) = 1 2 x, x < 0 1 , 0 x < 1 2 x 1 , x 1 25. (a) V = (8 2 x )(15 2 x ) x (b) 0 x 4 (c) 0 V 91 (d) As x increases, V increases and then decreases; the maximum value could be approximated by zooming in on the graph. 100 0 0 4 27. (a) The side adjacent to the building has length x , so L = x + 2 y . (b) A = xy = 1000, so L = x + 2000 /x . (c) all x = 0 120 80 20 80 (d) L 89 . 44 ft 29. (a) V = 500 = πr 2 h so h = 500 πr 2 . Then C = (0 . 02)(2) πr 2 + (0 . 01)2 πrh = 0 . 04 πr 2 + 0 . 02 πr 500 πr 2 = 0 . 04 πr 2 + 10 r ; C min 4 . 39 cents at r 3 . 4 cm , h 13 . 8 cm 7 4 1.5 6 (b) C = (0 . 02)(2)(2 r ) 2 + (0 . 01)2 πrh = 0 . 16 r 2 + 10 r . Since 0 . 04 π < 0 . 16, the top and bottom now get more weight. Since they cost more, we diminish their sizes in the solution, and the cans become taller. 7 4 1.5 5.5 (c) r 3 . 1 cm, h 16 . 0 cm, C 4 . 76 cents 31. (i) x = 1 , 2 causes division by zero (ii) g ( x ) = x + 1, all x 33. (a) 25 F (b) 13 F (c) 5 F 35. As in the previous exercise, WCT 1 . 4157 T 30 . 6763; thus T 15 F when WCT = 10.
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Exercise Set 1.2 3 EXERCISE SET 1.2 1. (e) seems best, though only (a) is bad. –0.5 0.5 y –1 1 x 3. (b) and (c) are good; (a) is very bad. 12 13 14 y 1 1 x 5. [ 3 , 3] × [0 , 5] 2 y x 7. (a) window too narrow, too short (b) window wide enough, but too short (c) good window, good spacing 400 200 y 5 5 10 20 x (d) window too narrow, too short (e) window too narrow, too short 9. [ 5 , 14] × [ 60 , 40] 60 20 40 y 5 5 10 x 11. [ 0 . 1 , 0 . 1] × [ 3 , 3] 2 2 3 y 0.1 x 0.1 13. [ 250 , 1050] × [ 1500000 , 600000] 500000 y 1000 1000 x 15. [
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