Unformatted text preview: Section 1.1 1 Chapter 1 Prerequisites for Calculus
s Section 1.1 Lines (pp. 19)
Quick Review 1.1
1. y 2. 3 3 2x x 3. m 4. m 2 3 3 2 1
2 5 2 3 3 4 ( 3) ( 1) 1 1 5 4 1 4 Section 1.1 Exercises
1. 2. 3. x y x y x y x y 1 0 2 1 1 1 2 8 1 0 4 1 2 2 3 ( 3) 2 2 4 ( 3) 0 0 6
y 5 5 4(3 2(x 2x 3) 1) 2 2 4(0) 2 0 2 4. 5. (a, c) 1
B 5 A x 5. (a) 3(2) 6 (b) 3(3) 6. (a) 7 7 (b) 1 1 7. d 4 5 5 5 5 Yes (b) m No 6. (a, c)
5 1 4( 1) 13 1 2 ( 2) 1 3 1
y 3 2( 1) 5 2 5 Yes 9 (x2 (0 2 x1)2 2)2 y1)2
2 2( 2) No x1)2 1)2 5
x (y2 (1 y1)2 0)2 5 A B 8. d = (x2 (1 (y2
1 3 4 2 3 (b) m 7. (a, c) 2 ( 1) 1 ( 2)
y 5 B 1 3 1 3 1 ( 1)2 1
25 9 5 3 16 9 A 5 x 9. 4x 3y 3y y 7 4x
4 x 3 7 3 (b) m 7 8. (a, c) 3 3 1 2 0 3
y 5 A 0 10. 2x 5y 5y y 3 2x
2 x 5 3
3 5
B 5 x 2 Section 1.1
26. The line contains (0, 0) and (5, 2).
3 2 1 1 5 (undefined) 0 8. continued (b) m m y 27. 3x This line has no slope. 9. (a) x (b) y 10. (a) x (b) y 11. (a) x (b) y 12. (a) x (b) y 13. y 14. y y 15. y 16. y y 17. m y y 2y 3x 18. m y y 19. m 1(x 1[x 1(x 2(x 2[x 2(x 0) 0 1) 1 ( 1)] 1 1) 1 3 ( 4)] 0 4) 0
4 3 2 3 1 0 2 2 5 2 x 5 0 0 2 5 4y 3x
3 x 4 12 12 3
3 4 4y y (a) Slope: (c) (b) yintercept: 3 [ 10, 10] by [ 10, 10] 28. x y y x 2 2 1 3 0 3 2 0 2 3 (x 0) 0 2 3 x 2 (a) Slope: (c) (b) yintercept: 2 3x 2y
1 2 1 1 0
0 1
[ 10, 10] by [ 10, 10] 0 29. 1 0(x 1
2 2 1) x 3 y 4 y 4 x 3 4 x 3 1 1 4
4 3 0 ( 2) 2 (undefined) 0 y Vertical line: x 20. m y 4y 4y 3x 21. y 22. y 23. y 24. y
2 2 1 ( 2) 2
3 4 3 4 (a) Slope: (b) yintercept: 4 (c) 3 [x 4 ( 2)] 2) 2 2 2 2 or y 3 1 4 1 3(x 3x 4y 3x 1x
1 x 2 1 x 3 [ 10, 10] by [ 10, 10] 30. y 2x 4 (a) Slope: 2 x 2 (b) yintercept: 4 (c) [ 10, 10] by [ 10, 10] 25. The line contains (0, 0) and (10, 25). m y
25 10 5 x 2 0 0 25 10 5 2 Section 1.1
31. (a) The desired line has slope (0, 0): y 1(x 0) 0 or y (b) The desired line has slope (0, 0): y 1(x 0) 0 or y x. 1 and passes through x.
1 1 3 39. (a) y 0.680x 9.013 1 and passes through (b) The slope is 0.68. It represents the approximate average weight gain in pounds per month. (c) 32. (a) The given equation is equivalent to y 2x 4. The desired line has slope 2 and passes through ( 2, 2): y 2(x 2) 2 or y 2x 2. (b) The desired line has slope ( 2, 2): y
1 (x 2 1 2 1 and passes through 2 [15, 45] by [15, 45] (d) When x 30, y 0.680(30) She weighs about 29 pounds. 40. (a) y 1,060.4233x 9.013 29.413. 2,077,548.669 2) 2 or y 1 x 2 3. (b) The slope is 1,060.4233. It represents the approximate rate of increase in earnings in dollars per year. (c) 33. (a) The given line is vertical, so we seek a vertical line through ( 2, 4): x 2. (b) We seek a horizontal line through ( 2, 4): y 4. 34. (a) The given line is horizontal, so we seek a horizontal line through 1,
1 :y 2 1 . 2 1 1, : x 2 [1975, 1995] by [20,000, 35,000] (b) We seek a vertical line through 35. m f(x)
9 3 2 1 7 (x 2 7 2 1. (d) When x 2000, y 1,060.4233(2000) 2,077,548.669 43,298. In 2000, the construction workers' average annual compensation will be about $43,298. 41. y 1 (x 3) 4 y x 3 4 y x 1 This is the same as the equation obtained in Example 5. 1) 2 Check: f(5) Since f(x) 36. m f(x)
4 4 7 (5) 2 7 x 2 3 , 2 7 3 x 2 2 3 16, as expected. 2 7 we have m and b 2 3 2 3 x 2 3 . 2 42. (a) When y When x 0, we have ( 1) 2 3 (x 2 3 2 2) ( 1) 2 2 (b) When y When x 2. Check: f(6) Since f(x) 37.
2 3 2 (6) 3 y 4 3 (6) 2 3 x 2 3 ( 2) 7, as expected.
3 and b 2 x c y 0, we have d x 0, we have c y 0, we have d 1, so x 1, so y 2, so x 2, so y c. d. 2c. 2d. 2, we have m The xintercept is 2c and the yintercept is 2d. 43. (a) The given equations are equivalent to y and y
2 and k 2 x k 2 k 1 , so 1 3 k x 2. 1, respectively, so the slopes are 1, y y y 2
2 x 3 3 1. The lines are parallel when
2 k 4 1 38. 2(x x so k (b) The lines are perpendicular when k 2.
( 2) ( 8) 44. (a) m (b) m (c) m 68 69.5 0.4 0 10 68 4 0.4 5 10 4.7 4 1.5 0.4 58 3.6 5 0.7 3.75 degrees/inch 16.1 degrees/inch 7.1 degrees/inch 8) 8 x 4 2 6 (d) Best insulator: Fiberglass insulation Poorest insulator: Gypsum wallboard The best insulator will have the largest temperature change per inch, because that will allow larger temperature differences on opposite sides of thinner layers. 4 Section 1.1
p d 10.94 1 100 0 9.94 100
(1, 4) (2, 3) (1, 1) (2, 0) 6 x y 6 45. Slope: k 0.0994 atmospheres per meter At 50 meters, the pressure is p (b) 0.0994(50) 45t 1 5.97 atmospheres. 46. (a) d(t) y [0, 6] by [ 50, 300] 6 (c) The slope is 45, which is the speed in miles per hour. (d) Suppose the car has been traveling 45 mph for several hours when it is first observed at point P at time t 0. (e) The car starts at time t 47. (a) y 5632x 11,080,280
(1, 2) (2, 3) (1, 1) (2, 0) 6 x 0 at a point 30 miles past P. (b) The rate at which the median price is increasing in dollars per year (c) y 2732x 5,362,360 50. (d) The median price is increasing at a rate of about $5632 per year in the Northeast, and about $2732 per year in the Midwest. It is increasing more rapidly in the Northeast. 48. (a) Suppose x F is the same as x C. x 1
9 x 32 5 9 x 32 5 4 x 5 y (c, d) W (a, b) Z (g, h) X x Y (e, f) 32 40 40 F is the same as 40 C. Suppose that the vertices of the given quadrilateral are (a, b), (c, d), (e, f ), and (g, h). Then the midpoints of the consecutive sides are W
c b d c e d f , ,X , , 2 2 2 2 e g f h g a h b Y , , and Z , . When these four 2 2 2 2 a x Yes, (b) points are connected, the slopes of the sides of the resulting figure are:
[ 90, 90] by [ 60, 60]
d f 2 e a 2 d 2 c 2 h 2 g a 2 b d 2 a c 2 b e f b 2 c d It is related because all three lines pass through the point ( 40, 40) where the Fahrenheit and Celsius temperatures are the same. 49. The coordinates of the three missing vertices are (5, 2), ( 1, 4) and ( 1, 2), as shown below.
y 6 (2, 3) (5, 2) (1, 1) (2, 0) 6 x WX: c
f f e h g f e h g b a d c b a d c 2 h 2 XY: e g 2 f h 2 ZY: e g 2 h b 2 WZ: g a 2 Opposite sides have the same slope and are parallel. Section 1.1
4 3 0 0 4 . 3 5 51. The radius through (3, 4) has slope The tangent line is tangent to this radius, so its slope is y y y
3 (x 4 3 x 4 3 x 4 1 4/3 3 . We seek the line of slope 4 3 that passes through (3, 4). 4 3)
9 4 25 4 4 4 52. (a) The equation for line L can be written as y y
A x B B (x A C , so its slope is B A . The perpendicular line has slope B 1 A/B B and passes through (a, b), so its equation is A a)
B A b. a) b ABb B2)x x b for y in the equation for line L gives: C AC B2 a
B2 a (b) Substituting (x Ax Ax
2 B
2 B (x A a) a) B (x (A2 AC ABb AC ABb A2 B 2 Substituting the expression for x in the equation for line L gives: A By By By y
B2 a AC ABb By C A2 B 2 A(B2a AC ABb) C(A2 B2) 2 2 A B A2 B 2 2 2 2 2 AB a A C A Bb A C B2C A2 B2 A2Bb A2 b B2C AB2a A2 B2 BC ABa A2 B2 B2a A AC
2 The coordinates of Q are (c) Distance (x
B2a B2a AC A(C A2 ABb A2b B
2 , BC ABa . A2 B2 a)2 (y b)2
2 AC ABb A2 B 2 AC a A2 b B2) 2 BC ABa A2 B 2 A2b BC 2 b
ABa b(A2 A2 B2 B2 ) 2 ABb a(A2 A2 B2 BC B(C A Bb A 2 a 2 B2 A
2 ABa B2b 2 B2 A
2 Bb B2 Aa) 2 Aa A2 B2 Bb) 2 A2(C Aa Bb)2 (A2 B2)2 (A2 (C A C Aa A B2)(C Aa (A2 B2)2 Aa
2 B2(C Aa Bb)2 (A2 B2)2 Bb)2 Bb)2 B
2 Aa A2 Bb
2 Bb B2 C B
2 ...
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