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C-26 Appendix C MINI PROJECT Group Work on Functions of Time In each of the examples and exercises of this section, the goal was to express a certain quantity as a function of a single variable. In the three problems that follow, that single variable will be time t. For each problem, after the group finds a solution, each individual should write up the solution on his or her own. (The problems are in order of increasing difficulty.) 1. (a) This problem is based on Example 1.* So first, have someone in your group present the example. (b) Under the conditions of Example 1, assume the following additional information. The width x is increasing at the rate of 3 cm/min, and at time t 0 min, the width x is 2 cm. Express the area of the rectangle as a function of time t. (c) During the time interval from t 0 min to t 16 min, when is the area of the rectangle increasing? When is it decreasing? Why are we using the value t 16 min? 2. Starting at time t 0 hr, two cars move away from the same intersection. Car A goes north at 40 miles/hr, and car B goes east at 30 miles/hr. Express the distance between the cars as a function of the time t. Is your result a linear function, a quadratic function, or neither? 3. Car C passes through an intersection, heading north at 40 miles/hr. One quarter hour later, car D goes through the intersection, heading east at 30 miles/hr. Express the distance between the two cars as a function of time. Is your result a linear function, a quadratic function, or neither? *Section 4.4, Precalculus: A Problems-Oriented Approach, 7th edition
Appendix C C-27 PROJECT The Least-Squares Line The great advances in mathematical astronomy made during the early years of the nineteenth century were due in no small part to the development of the method of least squares. The same method is the foundation for the calculus of errors of observation now occupying a place of great importance in the scientific study of social, economic, biological, and psychological problems. —A Source Book in Mathematics, by David Eugene Smith (New York: Dover Publications, 1959) Of all the principles which can be proposed [for finding the line or curve that best fits a data set], I think there is none more general, more exact, and more easy of application than . . . rendering the sum of the squares of the errors a minimum. —Adrien Marie Legendre (1752–1833) [Legendre and Carl Friedrich Gauss (1777–1855) developed the leastsquares method independently.] When we worked with the least-squares (or regression) line in Section 4.1,* we said that this is the line that “best fits” the given set of data points. In this project we explain the meaning of “best fits,” and we use quadratic functions to gain some insight into how the least-squares line is calculated. Suggestion: Two or three people could volunteer to study the text here and then present the material to another group or to the class at large. The presentation will be deemed successful if the audience can then work on their own the two exercises at the end of this project. Suppose that we have a data point (x i , y i ) and a line y f(x) mx b (not necessarily the least-squares line). As indicated in Figure A, we define the deviation e i of this line from the data point as e i y i f(x i ) y Data point {x i , y i } e i =y i -f(x i ) y=ƒ Figure A {x i , f(x i )} In Figure A, where the data point is above the line, the deviation e i represents the vertical distance between the data point and the line. If the data point were below the line, then the deviation y i f(x 1 ) would be the negative of that distance. Think of e i as the error made if f(x i ) were used as a prediction for the value of y i . x *Section 4.1, Precalculus: A Problems-Oriented Approach, 7th edition
Page 1 and 2: APPENDIX C Title and Page Number Pr
Page 3 and 4: Appendix C C-3 MINI PROJECT Discuss
Page 5 and 6: Appendix C C-5 MINI PROJECT Thinkin
Page 7 and 8: Appendix C C-7 1. equilateral trian
Page 9 and 10: Appendix C C-9 MINI PROJECT Flying
Page 11 and 12: Appendix C C-11 MINI PROJECT An Ine
Page 13 and 14: Appendix C C-13 (a) The two data po
Page 15 and 16: Appendix C C-15 and http://www.svob
Page 17 and 18: Appendix C C-17 For Figure D we can
Page 19 and 20: Appendix C C-19 MINI PROJECT A Grap
Page 21 and 22: Appendix C C-21 MINI PROJECT Who Ar
Page 23 and 24: Appendix C C-23 MINI PROJECT 1 How
Page 25: Appendix C C-25 either the quadrati
Page 29 and 30: Appendix C C-29 Then, for any line
Page 31 and 32: Appendix C C-31 PROJECT Finding Som
Page 33 and 34: Appendix C C-33 3. When a person co
Page 35 and 36: Appendix C C-35 2. Use part (b) of
Page 37 and 38: Appendix C C-37 PROJECT Coffee Temp
Page 39 and 40: Appendix C C-39 MINI PROJECT More C
Page 41 and 42: Appendix C C-41 The calculations co
Page 43 and 44: Appendix C C-43 PROJECT Transits of
Page 45 and 46: Appendix C C-45 E A ¨ V ¨ Bª d S
Page 47 and 48: Appendix C C-47 PROJECT A Linear Ap
Page 49 and 50: Appendix C C-49 Continuing with the
Page 51 and 52: Appendix C C-51 PROJECT Constructin
Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
Page 59 and 60: Appendix C C-59 PROJECT The Motion
Page 61 and 62: Appendix C C-61 PROJECT The Design
Page 63 and 64: Appendix C C-63 denote the angles o
Page 65 and 66: Appendix C C-65 inverse trigonometr
Page 67 and 68: Appendix C C-67 PROJECT Inverse Sec
Page 69 and 70: Appendix C C-69 Alternatively, usin
Page 71 and 72: Appendix C C-71 PROJECT Snell’s L
Page 73 and 74: Appendix C C-73 vertically downward
Page 75 and 76: 0 0 0 Appendix C C-75 PROJECT Vecto
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Appendix C C-77 Exercise 8 We outli
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Appendix C C-79 y y=2x+5 Îy=2 A (1
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
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Appendix C C-83 PROJECT Parameteriz
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Appendix C C-85 It is worth noting
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Appendix C C-87 6. Show that the li
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Appendix C C-89 2. An altitude of a
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Appendix C C-91 PROJECT The Leontie
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Appendix C C-93 n x units of that
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Appendix C C-95 him/herself through
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Appendix C C-97 PROJECT The Leontie
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Appendix C C-99 As in the solution
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Appendix C C-101 MINI PROJECT 2 Con
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Appendix C C-103 PROJECT Using Hype
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Appendix C C-105 PROJECT Two Method
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Appendix C C-107 (b) Use Method 1 t
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Appendix C C-109 MINI PROJECT An Un
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Appendix C C-111 PROJECT More on Su
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Appendix C C-113 SOLUTION 25 25 a (
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Appendix C C-115 4. Simplify the fo
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