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C-110 Appendix C MINI PROJECT Perspective and Alternative Scenarios for Population Growth Models As background for this mini project, someone in your group or in the class at large should review and summarize Example 5 for everyone else. 1. Refer to Figure 2 on page 1002,* which shows a scatter plot for the size of the rabbit population in Example 5* at the end of years t 0, 1,..., 5. In the text it was mentioned that the picture suggests exponential growth. Another possibility is quadratic growth. Explore this as follows. Using a graphing utility, enter the six data points specified in Figure 2 and create a scatter plot similar to the one shown in Figure 2. Then use the exponential regression option on the graphing utility to find the exponential function that best fits the data. Add the graph of the exponential function to your scatter plot. Next, add a quadratic regression curve to the picture. Which curve seems to fit the data points better? 2. In Example 5, suppose we make a change in the model so that the given recursive sequence is P 0 16 P t 0.0003P t1 (3000 P t1 ) (t 1) Use a graphing utility to investigate the population trend or trends. On paper, in complete sentences, compare and/or contrast your results with those in Example 5. 3. In Example 5, instead of beginning with 16 rabbits, suppose that we start with the much larger initial population size P 0 3200. (Assume, however, that the recursion equation from Example 5 remains the same.) Use a graphing utility to investigate the population trend or trends. On paper, in complete sentences, compare and/or contrast your results with those in Example 5. 4. (a) Follow Exercise 3, assuming that the size of the initial population is P 0 6330. (Compare and/or contrast your results with those in Exercise 3, rather than Example 5.) (b) Find out what happens if P 0 6334. In terms of the rabbit population, can you suggest a possible interpretation of the results? *Precalculus: A Problems-Oriented Approach, 7th edition
Appendix C C-111 PROJECT More on Sums In this project we develop some algebra for simplifying sums. First, here are four important properties of summation. Properties of Summation Let a 1 , a 2 , . . . a n , b 1 , b 2 , . . . b n , and c be real numbers. Then n n n n n n 1. a (a k b k ) a a k a b k 2. a (a k b k ) a a k a k1 k1 k1 k1 k1 n n n 3. a ca k c a a k 4. a c nc Proofs 1. To prove Property 1 we write out the sum term-by-term then use the associative and commutative properties of addition to remove parentheses then rearrange and regroup terms. n a (a k b k ) (a 1 b 1 ) (a 2 b 2 ) p (a n b n ) k1 k1 Definition of summation notation Remove parentheses Commutative and associative properties of addition Summation notation 2. The proof of Property 2 is similar to that for Property 1. We leave it for you as Exercise 1. 3. Property 3 is a statement of the distributive property or, equivalently, factoring out a common factor from a sum of terms. n a ca k ca 1 ca 2 p ca n k1 k1 a 1 b 1 a 2 b 2 p a n b n (a 1 a 2 p a n ) (b 1 b 2 p b n ) c(a 1 a 2 p a n ) n c a a k k1 n a a k a k1 Summation notation Factor out a common factor Summation notation 4. Property 4 is tricky. The summation notation says that there are n terms and each term is the same number c: n k1 n b k k1 a c c c p c k1 n terms nc ∂ b k k1
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APPENDIX C Title and Page Number Pr
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Appendix C C-3 MINI PROJECT Discuss
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Appendix C C-5 MINI PROJECT Thinkin
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Appendix C C-7 1. equilateral trian
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Appendix C C-9 MINI PROJECT Flying
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Appendix C C-11 MINI PROJECT An Ine
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Appendix C C-13 (a) The two data po
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Appendix C C-15 and http://www.svob
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Appendix C C-17 For Figure D we can
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Appendix C C-19 MINI PROJECT A Grap
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Appendix C C-21 MINI PROJECT Who Ar
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Appendix C C-23 MINI PROJECT 1 How
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Appendix C C-25 either the quadrati
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Appendix C C-27 PROJECT The Least-S
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Appendix C C-29 Then, for any line
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Appendix C C-31 PROJECT Finding Som
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Appendix C C-33 3. When a person co
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Appendix C C-35 2. Use part (b) of
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Appendix C C-37 PROJECT Coffee Temp
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Appendix C C-39 MINI PROJECT More C
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Appendix C C-41 The calculations co
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Appendix C C-43 PROJECT Transits of
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Appendix C C-45 E A ¨ V ¨ Bª d S
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Appendix C C-47 PROJECT A Linear Ap
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Appendix C C-49 Continuing with the
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Appendix C C-51 PROJECT Constructin
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Appendix C C-53 1 y Figure A A grap
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Appendix C C-55 PROJECT Fourier Ser
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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
Page 77 and 78: Appendix C C-77 Exercise 8 We outli
Page 79 and 80: Appendix C C-79 y y=2x+5 Îy=2 A (1
Page 81 and 82: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
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Page 85 and 86: Appendix C C-85 It is worth noting
Page 87 and 88: Appendix C C-87 6. Show that the li
Page 89 and 90: Appendix C C-89 2. An altitude of a
Page 91 and 92: Appendix C C-91 PROJECT The Leontie
Page 93 and 94: Appendix C C-93 n x units of that
Page 95 and 96: Appendix C C-95 him/herself through
Page 97 and 98: Appendix C C-97 PROJECT The Leontie
Page 99 and 100: Appendix C C-99 As in the solution
Page 101 and 102: Appendix C C-101 MINI PROJECT 2 Con
Page 103 and 104: Appendix C C-103 PROJECT Using Hype
Page 105 and 106: Appendix C C-105 PROJECT Two Method
Page 107 and 108: Appendix C C-107 (b) Use Method 1 t
Page 109: Appendix C C-109 MINI PROJECT An Un
Page 113 and 114: Appendix C C-113 SOLUTION 25 25 a (
Page 115: Appendix C C-115 4. Simplify the fo
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