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 | 1. If the
distance from point A to point B plus the distance from point B to point C equals the
distance from point A to point C, explain why the points A, B and C must be
colinear. |
| | 2. If the
slope between points A and B equals the slope between points B and C, explain why the points A, B and C are
colinear. |
| 3. If a graph fails the
vertical line test, it is not the graph of a function. Explain this result in terms of the definition of a
function. |
| | 4. You should not automatically write the equation of a line in slope-intercept form. Compare the following forms of the same line: y = 2.4(x -1.8) + 0.4 and y = 2.4x -3.92. Given x = 1.8 , which equation would you rather use to compute y? How about if you are given x = 0? For x = 8 , is there any advantage to one equation over the other? Can you quickly read off the
slope from either equation? Explain why neither form of the equation is “better. ” |
| In exercises 5-12, determine whether or not the points are
colinear.
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| 5. (1,2), (3,6), (0,0) | 6. (1,1), (3,5), (0, -1) |
| 7. (2,1), (0,2), (4,0) | 8. (3,2), (4,0), (1,6) |
| 9. (3,1), (4,4), (5,8) | 10. (1,2), (2,5), (4,8) |
| 11. (4,1), (3,2), (1,3) | 12. (3,2), (0,1), (6,3)
|
| In exercises 13-20, find the
slope of the line through the given points.
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| 13. (1,2), (3,6) | 14. (0,1), (2,7) |
| 15. (1,2), (3,3) | 16. (0,1), (3,8) |
| 17. (3, -6), (1, -1) | 18. (1, -2), ( -1, -3) |
| 19. (0.3, -1.4), ( -1.1, -0.4) | 20. (1.2,2.1), (3.1,2.4)
|
| In exercises 21-30, find a second point on the line with
slope m and point P, graph the line and find an equation of the line.
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| 21. m = 2, P = (1,3) | 22. m = 2, P = (3,1) |
| 23. m = -2, P = (1,4) | 24. m = -1, P = (2,1) |
| 25. m = 0, P = ( -1,1) | 26. m = 0, P = (2, -1) |
| 27. m =  , P = (2, 1) | 28. m = -  , P = ( -2, 1) |
| 29. m = 1.2, P = (2.3,1.1) | 30. m = 0.8, P = (1.2,0.4)
|
| In exercises 31-40, determine if the lines are
parallel, perpendicular, or neither.
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| 31. y = 3(x -1) + 2 and y = 3(x + 4) - 1
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| 32. y = 2(x -3) + 1 and y = 4(x -3) + 1
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| 33. y = - 2(x + 1) - 1 and y = (x -2) + 3
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| 34. y = 3(x -1) + 1 and y = -  (x + 2) - 2
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| 35. y = 2x - 1 and y = - 2x + 2
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| 36. y = 4x + 3 and y = 4x - 5
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| 37. y = 4x - 2 and y = x + 1
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| 38. y = 3x + 1 and y = - x + 2
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| 39. x + 2y = 1 and 2x + 4y = 3
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| 40. x + 2y = 1 and 2x - y = 3
|
| In exercises 41-46, find an equation of a line through the given point and (a)
parallel to and (b) perpendicular to the given line.
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| 41. y = 2(x + 1) - 2 at (2,1) | 42. y = 3(x -2) + 1 at (0,3) |
| 43. y = 1 at (0, -1) | 44. x = 2 at (0,1) |
| 45. y = 2x + 1 at (3,1) | 46. y = - x + 2 at (1,2)
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| In exercises 47-52, find an equation of the line through the given points and compute the y - coordinate of the point on the line corresponding to x=4.
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| 47.
| 48.
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| 49.
| 50.
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| 51.
| 52.
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| In exercises 53-58, use the
vertical line test to determine whether or not the curve is the graph of a function.
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| 53.
| 54.
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55.
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56.
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57.
| 58.
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| In exercises 59-66, identify the given
function as polynomial, rational, both or neither.
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| 59. f (x) = x3 - 4x + 1 | 60. f (x) = - 2x4 - 3x2 + 1 |
| 61.  | 62.  |
| 63.  | 64. f (x) = x3 - 3 + 1 |
| 65. f (x) = 3-2x + x4 | 66. f (x) = 2x - x2/3 - 6
|
| In exercises 67-74, find the
domain of the function.
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| 67.  | 68.  |
| 69.  | 70. 
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| 71.  | 72.  |
| 73.  | 74. 
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| In exercises 75-78, find the indicated
function values.
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| 75. f (x) = x2 - x - 1; f (0), f (2), f ( -3), f (1/2)
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| 76.  ; f (0), f (2), f ( -2), f (1/2)
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| 77.  ; f (0), f (3), f ( -1), f (1/2)
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| 78.  f (1), f (10), f (100), f (1/3)
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| In exercises 79-82, a brief description is given of a physical situation. For the indicated variable, state a reasonable
domain.
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| 79. A parking deck is to be built; x = width of deck (in feet). |
| 80. A parking deck is to be built on a 200 ' - by - 200 ' lot; x = width of deck (in feet). |
| 81. A new candy bar is to be sold; x = number of candy bars sold in first month. |
| 82. A new candy bar is to be sold; x = cost of candy bar (in cents). |
| In exercises 83-86, discuss whether you think
y would be a function of x.
|
| 83. y = grade you get on an exam, x = number of hours you study |
| 84. y = probability of getting lung cancer, x = number of cigarettes smoked per day |
| 85. y = a person's weight, x = number of minutes exercising per day |
| 86. y = speed at which an object falls, x = weight of object |
| 87. Figure A shows more data from the ski resort of
Example 2.13. Determine the days on which the snow machine was used.
|
Figure A
Depth of snow.
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| 88. Figure B shows the daily profit of a business as a
function of time. In the time period shown, the company received two large orders and paid one large bill to a parts provider. Identify when these events occurred. If another bill is due to the parts provider, estimate the height of the graph after the bill is paid.
| 
Figure B
Business income.
|
| 89. Figure C shows the speed of a bicyclist as a
function of time. For the portions of this graph that are flat, what is happening to the bicyclist's speed? What is happening to the bicyclist's speed when the graph goes up? down? Identify the portions of the graph that correspond to the bicyclist going uphill; downhill.
| 
Figure C
Bicycle speed.
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| 90. Figure D shows the population of a small country as a
function of time. During the time period shown, the country experienced two influxes of immigrants, a war and a plague. Identify these important events.
| 
Figure D
Population.
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| In exercises 91-94, briefly describe a realistic scenario for how each quantity changes as a
function of time and sketch a graph corresponding to your description (label all interesting points).
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| 91. The electrical usage of a city (include day and night). |
| 92. The temperature of a baked potato just placed into an oven. |
| 93. Your level of alertness listening to the radio (include commercials, good songs, and bad songs). |
| 94. Your speed while driving on the interstate (include speed traps and pit stops). |
| 95. The boiling point of water (in degrees Fahrenheit) at elevation h (in thousands of feet above sea level) is given by B(h) = . Find h such that water boils at 98.6°. Why would this height be dangerous to humans? |
| 96. A linear transformation is defined as any
function f satisfying f (a + b) = f (a) + f (b) and f (cx) = cf(x). Show that, according to this definition, f (x) = 3x + 2 is not a
linear transformation but f (x) = 3x is. (The
function 3x + 2 is called an affine transformation.) |
| 97. The spin rate of a golf ball hit with a 9 iron has been measured at 9100 rpm for a 120 - compression ball and at 10,000 rpm for a 60 - compression ball. Most golfers use 90 - compression balls. If the spin rate is a linear
function of compression, find the spin rate for a 90 - compression ball. |
| 98. Professional golfers often use 100 - compression balls. For the data in exercise 97, estimate the spin rate of a 100 - compression ball. |
 | 99. Suppose you have a machine that will proportionally enlarge a photograph. For example, it could enlarge a 4 6 photograph to 8 12 by doubling the width and height. You could make an 8 10 picture by cropping 1 inch off each side. Explain how you would enlarge a 3 5 picture to an 8 10. A friend returns from Scotland with a 3 5 picture showing the Loch Ness monster in the outer '' on the right. If you use your procedure to make an 8 10 enlargement, does Nessie make the cut? |
| | 100. The human body's capacity to perform a variety of tasks decreases with age. William Clark's book Sex and the Origins of Death presents several statistics on this phenomenon. For example, average female fertility drops from 100% at age 30 to 0% at age 50. With age as x and percentage of fertility as y, show that y = - 5(x -50) , for 30 x 50 , assuming y is a
linear function of x. Show that 50% fertility is reached at age 40. In each of the following situations, determine which age corresponds to 50% capacity.
 | (a) Maximum breathing capacity drops from 100% at age 20 to 40% at age 80.
| (b) Maximum heart rate drops from 100% at age 20 to 68% at age 80.
| (c) Maximum oxygen uptake drops from 100% at age 20 to 30% at age 80.
| (d) Nerve conduction velocity drops from 100% at age 20 to 80% at age 80.
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