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integer n. • Recognize patterns and write the th term of a sequence. • Find the sums of powers of n integers. • Find finite differences of sequences. Why you should learn it Finite differences can be used to determine what type of model can be used to represent a sequence. For instance, in Exercise 61 on page 682, you... |
Mathematical induction is one method of proof. n 333202_0904.qxd 12/5/05 11:35 AM Page 674 674 Chapter 9 Sequences, Series, and Probability It is important to recognize that in order to prove a statement by induction, both parts of the Principle of Mathematical Induction are necessary. The Principle of Mathematical In... |
1 P3, P3 P2, P2 P4, P1 FIGURE 9.6 333202_0904.qxd 12/5/05 11:35 AM Page 675 Section 9.4 Mathematical Induction 675 When using mathematical induction to prove a summation formula (such as the one in Example 2), it is helpful to think of Sk1 as Sk1 Sk ak1 where ak1 is the k 1 th term of the original sum. Example 2 Using ... |
/5/05 11:35 AM Page 676 676 Chapter 9 Sequences, Series, and Probability Example 3 Using Mathematical Induction Use mathematical induction to prove the formula 12 22 32 42... n2 Sn nn 12n 1 6 for all integers n ≥ 1. Solution 1. When n 1, 12 123. S1 6 the formula is valid, because 2. Assuming that 12 22 32 42... k2 Sk k... |
n k, you have 2k1 22k > 2k 2k. 2k k k > k 1 Because 2k1 > 2k > k 1 or By assumption for all it follows that k > 1, k 1 < 2k1. Combining the results of parts (1) and (2), you can conclude by mathematical n ≥ 1. induction that for all integers n < 2n Now try Exercise 17. Example 5 Proving Factors by Mathematical Inducti... |
1 1 2 Solution Begin by writing out the first few sums. 3 4 4 5... 1 nn 1 S1 S2 S3 S4 12 1 2 1 3 4 48 60 From this sequence, it appears that the formula for the th sum is k Sk kk 1 k k 1. To prove the validity of this hypothesis, use mathematical induction. Note that so you can begin by assuming you have already verif... |
ences, Series, and Probability For a linear model, the first differences should be the same nonzero number. For a quadratic model, the second differences are the same nonzero number. Finite Differences The first differences of a sequence are found by subtracting consecutive terms. The second differences are found by su... |
an ________ sequence if the first differences are all the same nonzero number. 4. If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Ex... |
0, x2 > 0,..., xn > 0, ln ln x2... xn lnx1x2 x1 then... ln xn. 27. Generalized Distributive Law: xy1... yn a bin and xy2... xyn are complex conjugates for all 28. xy1 y2 a bin n ≥ 1. 29. A factor of 30. A factor of 31. A factor of 32. A factor of 33. A factor of 34. A factor of is 3. is 3. is 2. n3 3n2 2n n3 n 3 n4 n ... |
Use the models found in parts (b) and (c) to estimate the number of individual tax returns filed in 2008. How do these values compare? In Exercises 51–56, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence... |
sketch the graph of the rational function. 71. 72. 73. 74. f x x x 3 gx x2 x2 4 ht t 7 t f x 5 x 1 x 333202_0905.qxd 12/5/05 11:37 AM Page 683 9.5 The Binomial Theorem Section 9.5 The Binomial Theorem 683 What you should learn • Use the Binomial Theorem to calculate binomial coefficients. • Use Pascal’s Triangle to ca... |
r the coefficient of is nCr x n1y r... nxyn1 yn nCr. n! n r!r! n r The symbol is often used in place of nCr to denote binomial coefficients. For a proof of the Binomial Theorem, see Proofs in Mathematics on page 724. 333202_0905.qxd 12/5/05 11:37 AM Page 684 684 Chapter 9 Sequences, Series, and Probability Te c h n o ... |
4 0 7 nCr nCnr What characteristic of Pascal’s Triangle is illustrated by this table? Section 9.5 The Binomial Theorem 685 Pascal’s Triangle There is a convenient way to remember the pattern for binomial coefficients. By arranging the coefficients in a triangular pattern, you obtain the following array, which is calle... |
8C1 28 8C2 56 8C3 70 8C4 56 8C5 28 8C6 8 8C7 1 8C8 Now try Exercise 11. 333202_0905.qxd 12/5/05 11:37 AM Page 686 686 Chapter 9 Sequences, Series, and Probability Historical Note Precious Mirror “Pascal’s”Triangle and forms of the Binomial Theorem were known in Eastern cultures prior to the Western “discovery” of the ... |
the same viewing window. The graphs should coincide as shown below. 200 Example 6 Expanding a Binomial Write the expansion for x2 43. Solution Use the third row of Pascal’s Triangle, as follows. x2 43 1x23 3x224 3x242 143 x 6 12x 4 48x2 64 Now try Exercise 29. −5 5 Sometimes you will need to find a specific term in a ... |
expansion are called ________ ________. 2. To find binomial coefficients, you can use the ________ ________ or ________ ________. 3. The notation used to denote a binomial coefficient is ________ or ________. 4. When you write out the coefficients for a binomial that is raised to a power, you are ________ a ________. ... |
x y10 3x 2y9 2x 3y8 x 2 y10 z 2 t10 47. 48. 49. 50. 51. 52. 53. 54. Term ax5 ax8 ax8y 2 ax 2y8 ax4y5 ax 6y 2 ax8y 6 az4t8 In Exercises 55–58, use the Binomial Theorem to expand and simplify the expression. 55. 56. 57. 58. x 34 2t 13 x 23 y133 u35 25 In Exercises 59–62, expand the expression in the difference quotient a... |
x f x 3 n In Exercises 75–78, consider Probability independent trials of an experiment in which each trial has two possible outcomes: “success” or “failure.” The probability of a success on each trial is and the probability of a failure is q 1 p. in the successes expansion of in the n In this context, the term trials o... |
:37 AM Page 690 690 Chapter 9 Sequences, Series, and Probability 80. Child Support The amounts (in billions of dollars) of child support collected in the United States from 1990 to 2002 can be approximated by the model f t 0.031t 2 0.82t 6.1, 0 ≤ t ≤ 12 f t represents the year, with corresponding to (Source: U.S. Depar... |
ify your answer. In Exercises 81– 83, determine whether the 81. The Binomial Theorem could be used to produce each row of Pascal’s Triangle. 82. A binomial that represents a difference cannot always be accurately expanded using the Binomial Theorem. 83. The x10 x2 312 -term and the x14 -term of the expansion of have id... |
this list, you can see that a sum of 12 can occur in five different ways. Now try Exercise 5. Example 2 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two pieces of paper are drawn from the box at the same time, and the numbers on the pieces of paper are writte... |
ing Principle Telephone numbers in the United States currently have 10 digits. The first three are the area code and the next seven are the local telephone number. How many different telephone numbers are possible within each area code? (Note that at this time, a local telephone number cannot begin with 0 or 1.) Soluti... |
. Example 6 Counting Horse Race Finishes Eight horses are running in a race. In how many different ways can these horses come in first, second, and third? (Assume that there are no ties.) Solution Here are the different possibilities. Win (first position): Eight choices Place (second position): Seven choices Show (thir... |
3 n n1 of a third kind, and so on, with n n1... nk. n2 n 3 of one kind of object, n2 of a second kind, Then the number of distinguishable permutations of the objects is n n! n1! n2! n3!... nk!. Example 7 Distinguishable Permutations In how many distinguishable ways can the letters in BANANA be written? Solution This wo... |
elements are listed is not important. So, you would count only one of the two sets. A common example of how a combination occurs is a card game in which the player is free to reorder the cards after they have been dealt. Example 8 Combinations of n Elements Taken r at a Time In how many different ways can three letter... |
,598,960 Now try Exercise 63. Example 10 Forming a Team You are forming a 12-member swim team from 10 girls and 15 boys. The team must consist of five girls and seven boys. How many different 12-member teams are possible? Solution There are boys. By the Fundamental Counting Principal, there are choosing five girls and ... |
2. An even integer 4. An integer that is greater than 9 5. An integer that is divisible by 4 6. An integer that is divisible by 3 7. Two distinct integers whose sum is 9 8. Two distinct integers whose sum is 8 9. Entertainment Systems A customer can choose one of three amplifiers, one of two compact disc players, and ... |
under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be less than 5000. (d) The leading digit cannot be zero and the number must be even. 19. Combination Lock A combination lock ... |
44, find the number of distinguishable permutations of the group of letters. 41. A, A, G, E, E, E, M 42. B, B, B, T, T, T, T, T 43. A, L, G, E, B, R, A 44. M, I, S, S, I, S, S, I, P, P, I 45. Write all permutations of the letters A, B, C, and D. 46. Write all permutations of the letters A, B, C, and D if the letters B ... |
. Interpersonal Relationships The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3, (b) 8, (c) 12, and (d) 20. 57. Poker Hand You are dealt five cards from an ordinary deck of... |
the alphabet (A–M) is an example of a permutation. 68. The number of permutations of n elements can be deter- mined by using the Fundamental Counting Principle. 69. What is the relationship between nCr and nCnr? 70. Without calculating the numbers, determine which of the following is greater. Explain. (a) The number o... |
5, x2 2, (b) f 1 (a) f 4 (c) f 5 (c) gx 1 (c) x ≤ 4 x > 4 (c) f 11 f 20 In Exercises 81– 84, solve the equation. Round your answer to two decimal places, if necessary. (b) Number of different arrangements of three types of flowers from an array of 20 types (c) Number of three-digit pin numbers for a debit card 81. x 3... |
HH, HT, TH, TT. Note that this list distinguishes between the two cases though these two outcomes appear to be similar. HT and TH, even c. Following the notation of part (b), the sample space is S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Note that this list distinguishes among the cases HHT, HTH, and THH, and among the... |
cards. What is the probability that it is an ace? Solution a. Following the procedure in Example 1(b), let E HH and S HH, HT, TH, TT. The probability of getting two heads is PE nE nS 1 4. b. Because there are 52 cards in a standard deck of playing cards and there are four aces (one in each suit), the probability of dr... |
, Series, and Probability Example 5 The Probability of Winning a Lottery In the Arizona state lottery, a player chooses six different numbers from 1 to 41. If these six numbers match the six numbers drawn (in any order) by the lottery commission, the player wins (or shares) the top prize. What is the probability of win... |
Probability of a Union of Events One card is selected from a standard deck of 52 playing cards. What is the probability that the card is either a heart or a face card? Hearts Solution Because the deck has 13 hearts, the probability of selecting a heart (event A ) is PA 13 52. Similarly, because the deck has 12 face ca... |
of choosing an employee who has 9 or fewer years of service is about 0.465. Now try Exercise 47. 333202_0907.qxd 12/5/05 11:41 AM Page 707 Section 9.7 Probability 707 Independent Events Two events are independent if the occurrence of one has no effect on the occurrence of the other. For instance, rolling a total of 12... |
people have different birthdays and then find the complement. Now, determine the proba- bility that in a room with 50 people at least two people have the same birthday. The Complement of an Event The complement of an event space that are not in PA A 1 PA PA 1. The complement of event A A So, the probability of is the ... |
. 5. If two events from the same sample space have no outcomes in common, then the two events are ________ ________. 6. If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________. 7. The ________ of an event A is the collection of all outcomes in the sample space that... |
more than 7. 20. The sum is odd or prime. Tossing a Coin In Exercises 7–10, find the probability for the experiment of tossing a coin three times. Use S {HHH, HHT, HTH, H T T, THH, TH T, the sample space T TH, T T T}. Drawing Marbles In Exercises 21–24, find the probability for the experiment of drawing two marbles (w... |
budget amendment to the Constitution. The results of the study are listed in the table, represents Republican. where represents Democrat and D R Favor Not Favor Unsure Total D R Total 23 32 55 25 9 34 7 4 11 55 45 100 (c) A Democrat who favors the amendment 35. Graphical Reasoning The figure shows the results of a rece... |
b) the card is a heart or a diamond, and (c) the card is a nine or a face card. 46. Poker Hand Five cards are drawn from an ordinary deck of 52 playing cards. What is the probability that the hand drawn is a full house? (A full house is a hand that consists of two of one kind and three of another kind.) 47. Defective U... |
Jenkins are candidates for public office. It is estimated that Moore and Jenkins have about the same probability of winning, and Taylor is believed to be twice as likely to win as either of the others. Find the probability of each candidate winning the election. 40. Winning an Election Three people have been nominated... |
flight (a) both systems function satisfactorily, functions (b) at satisfactorily, and (c) both systems fail? least one system 52. Backup Vehicle A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available whe... |
red pocket on three consecutive spins. (f) European roulette does not contain the 00 pocket. Repeat parts (a)–(e) for European roulette. How do the probabilities for European roulette compare with the probabilities for American roulette? 333202_0907.qxd 12/5/05 11:41 AM Page 713 56. Estimating is dropped onto a paper ... |
. 64. 65. 66. 67. 68. 69. 70. 6x2 8 0 4x2 6x 12 0 x 3 x2 3x 0 x 5 x 3 2x 0 12 x 3 32 x 2x 2 x 5 3 2x 3 3 x 2 2 x 4 1 2x 3 1 4 x x 2 5 x 2 13 x2 2x In Exercises 71–74, sketch the graph of the solution set of the system of inequalities. 71. 72 5x 2y ≥ 10 x2 y ≥ 2 y ≥ x 4 74. x2 y2 ≤ 4 x y ≥ 2 73. 333202_090R.qxd 12/5/05 ... |
permutations to solve counting problems (p. 693). Use combinations to solve counting problems (p. 696). Section 9.7 Find the probabilities of events (p. 701). Find the probabilities of mutually exclusive events (p. 705). Find the probabilities of independent events (p. 707). Find the probability of the complement of a... |
sequence. Use a graphing utility to construct a bar graph that represents the sequence. (Source: U.S. Administration for Children and Families) In Exercises 9–12, simplify the factorial expression. In Exercises 27–30, determine whether the sequence 9.2 is arithmetic. If so, find the common difference. 9. 5! 11. 3! 5! ... |
80 (inclusive). 333202_090R.qxd 12/5/05 11:43 AM Page 716 716 Chapter 9 Sequences, Series, and Probability 47. Job Offer The starting salary for an accountant is $34,000 with a guaranteed salary increase of $2250 per year. Determine (a) the salary during the fifth year and (b) the total compensation through 5 full yea... |
geometric sequence. 7 3i1 2i1 5 61. 62. i1 i1 63. 65. 4 i1 5 i1 i 1 2 2i1 64. 66. i1 1 3 6 i1 4 i1 63i In Exercises 67–70, use a graphing utility to find the sum of the finite geometric sequence. 67. 69. 10 i1 25 i1 103 5 i1 1001.06i1 68. 70. 15 i1 200.2i1 i1 86 5 20 i1 In Exercises 71–76, find the sum of the infinite... |
333202_090R.qxd 12/5/05 11:43 AM Page 717 In Exercises 95–98, use the Binomial Theorem to 9.5 calculate the binomial coefficient. 95. 97. 6C4 8C5 96. 98. 10C7 12C3 In Exercises 99–102, use Pascal’s Triangle to calculate the binomial coefficient. 101. 102. 100. 99. 9 4 5 3 7 3 8 6 In Exercises 103–108, use the Binomial... |
history courses. How many schedules are possible? 113. Bike Race There are 10 bicyclists entered in a race. In how many different ways could the top three places be decided? 114. Jury Selection A group of potential jurors has been narrowed down to 32 people. In how many ways can a jury of 12 people be selected? 115. A... |
nPr is the domain of the function? 131. Think About It How do the two sequences differ? (a) an (b) an 1n n 1n1 n 132. Graphical Reasoning The graphs of two sequences are shown below. Identify each sequence as arithmetic or geometric. Explain your reasoning. (a) an 4 −2 −8 −12 −16 −20 2 86 10 n (b) an 100 80 60 40 20 −... |
!, 6 4!, 7 5!,... 3. Find the next three terms of the series. Then find the fifth partial sum of the series. 6 17 28 39... 4. The fifth term of an arithmetic sequence is 5.4, and the 12th term is 11.0. Find the n th term. 5. Write the first five terms of the sequence an 52n1. n (Assume that begins with 1.) i1 41 2 i. I... |
from earlier chapters. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 4, solve the system by the specified method. 1. Substitution y 2 y 2 3 x2 x 1 3. Elimination 2x 4y z x 2y 2z x 3y z 3 6 1 2. Elimination x 3y 2x 4y 1 0 4. Gauss-Jordan Elimination 7 5 3 x 3y... |
.06 0.12 Gym Jogging Walking shoes shoes 0.09 0.10 0.25 shoes 0.03 0.05 0.12 MATRIX FOR 18 y 6 5 2 1 − ( 2, 3) (1, 5) (4, 1) −2 −1 1 2 3 4 x FIGURE FOR 21 Cumulative Test for Chapters 7–9 721 18. The percents (by age group) of the total amounts spent on three types of footwear in a recent year are shown in the matrix. ... |
number of distinguishable permutations of the group of letters. 34. B, A, S, K, E, T, B, A, L, L 35. A, N, T, A, R, C, T, I, C, A 36. A personnel manager at a department store has 10 applicants to fill three different sales positions. In how many ways can this be done, assuming that all the applicants are qualified fo... |
Series The study of infinite series was considered a novelty in the fourteenth century. Logician Richard Suiseth, whose nickname was Calculator, solved this problem. If throughout the first half of a given time interval a variation continues at a certain intensity; throughout the next quarter of the interval at double... |
723 333202_090R.qxd 12/5/05 11:43 AM Page 724 The Binomial Theorem (p. 683) In the expansion of x yn x yn xn nx n1y... x nryr the coefficient of is nCr x nry r... nxy n1 y n nCr n! n r!r!. Proof The Binomial Theorem can be proved quite nicely using mathematical induction. The steps are straightforward but look a littl... |
to graph the first 10 terms of the sequence. (b) Use the graph from part (a) to estimate the value of an as approaches infinity. n (c) Complete the table. 1 10 100 1000 10,000 n an (d) Use the table from part (c) to determine (if possible) the value of an as approaches infinity. n 3. Consider the sequence 3 1n. an (a)... |
be 10 feet ahead. Then, when Achilles runs 10 feet, the tortoise will be 5 feet ahead. Achilles will keep cutting the distance in half but will never catch the tortoise. The table shows Zeno’s reasoning. From the table you can see that both the distances and the times required to achieve them form infinite geometric s... |
1. are all true. (b) (c) (d) P3 is true and P1, P2, P3,..., P50 P1, P2, and P3 Pk1 imply that P2 P2k is true and f1, f2,..., fn,... is true. implies P2k2. 11. Let be the Fibonacci sequence. are all true, but the truth of Pk does not (a) Use mathematical induction to prove that f1 f2... fn fn2 1. (b) Find the sum of the... |
different numbers chosen from the numbers 1 to 47, plus one Mega number chosen from the numbers 1 to 27. You purchase a ticket for $1. If the jackpot for the next drawing is $12,000,000, what is the expected value for the ticket?... pnxn. V p1x1 (b) You are playing a dice game in which you need to score 60 points to w... |
a line can be used to determine the change in elevation from the base to the top of the Johnstown Inclined Plane. Inclination of a Line In Section 1.3, you learned that the graph of the linear equation y mx b 0, b. is a nonvertical line with slope There, the slope of a line x. was described as the rate of change in wi... |
intersecting. If they intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure 10.3, you can use the inclinations of the two lines to find the angl... |
is Remember that the values of to the general equation of a line, between a point and a line, see Proofs in Mathematics on page 806. B,A, C Ax By C 0. in this distance formula correspond For a proof of the distance and Example 3 Finding the Distance Between a Point and a Line Find the distance between the point 4, 1 a... |
the following statement is true or false: “The inclination of a line is the angle between the line and the -axis.” Explain. 2, x 333202_1001.qxd 12/8/05 11:05 AM Page 732 732 Chapter 10 Topics in Analytic Geometry 10.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all... |
radians and 15. 16. 17. 18. 6, 1, 10, 8 4, 3 12, 8, 2, 20, 10, 0 0, 100, 50, 0 y 2 1 −1 −3 −2 −1 25. 3x 2y 0 3x 2y 1 26. 2x 3y 22 4x 3y 24 y 2 1 θ x 1 2 −2 −1 −1 − 27. 29. x 2y 7 6x 2y 5 x 2y 8 x 2y 2 28. 30. 5x 2y 16 3x 5y 1 3x 5y 3 3x 5y 12 333202_1001.qxd 12/8/05 8:54 AM Page 733 31. 32. 0.05x 0.03y 0.21 0.07x 0.02... |
. 43. 44. Line 4x 3y 0 2x y 4 4x 3y 10 x y 2 x 1 0 y 4 0 6x y 0 x y 20 52. Road Grade A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road. 53. Pitch of a Roof A roof has a rise of 3 feet for every horizo... |
problem. 60. Exploration Consider a line with slope m and -intercept y 0, 4. (a) Write the distance d between the point m. line as a function of 3, 1 and the Model It 56. Inclined Plane The Johnstown Inclined Plane in Johnstown, Pennsylvania is an inclined railway that was designed to carry people to the hilltop commu... |
the quadratic function. 73. 75. f x x 42 3 gx 2x2 3x 1 f x 6 x 12 74. 76. gx x2 6x 8 333202_1002.qxd 12/8/05 9:00 AM Page 735 10.2 Introduction to Conics: Parabolas Section 10.2 Introduction to Conics: Parabolas 735 What you should learn • Recognize a conic as the intersection of a plane and a double-napped cone. • Wr... |
equidistant from a fixed point h, k. x h 2 y k 2 r 2. Equation of circle 333202_1002.qxd 12/8/05 9:00 AM Page 736 736 Chapter 10 Topics in Analytic Geometry Parabolas In Section 2.1, you learned that the graph of the quadratic function f x ax2 bx c is a parabola that opens upward or downward. The following definition ... |
) Vertex: (, )h k Directrix: x = h − p p < 0 Focus: (h + p, k) Axis: y = k Vertex: (h, k) (a) x h2 4py k p > 0 Vertical axis: (b) x h2 4py k Vertical axis: p < 0 (c) y k2 4px h Horizontal axis: p > 0 (d) y k2 4px h Horizontal axis: p < 0 FIGURE 10.11 333202_1002.qxd 12/8/05 9:00 AM Page 737 Te c h n o l o g y Use a gr... |
4p y k Add 1 to each side. Complete the square. Combine like terms. Standard form you can conclude that is negative, the parabola opens downward, as shown in Figure 10.13. So, the focus of the parabola is h, k p 1, 1 Because and. h 1, k 1, p 1 2. p 2 Now try Exercise 21. 333202_1002.qxd 12/8/05 9:00 AM Page 738 738 Ch... |
parallel to one another, as shown in Figure 10.15. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces. Reflective Propert... |
TV 333202_1002.qxd 12/8/05 9:00 AM Page 740 740 Chapter 10 Topics in Analytic Geometry 10.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A ________ is the intersection of a plane and a double-napped cone. 2. A collection of points satisfying a geometric property can also be referred to as a ________ of points. x... |
4 In Exercises 25–28, find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. 26. 25. x2 4x 6y 2 0 x 2 2x 8y 9 0 y 2 x y 0 27. 28. y 2 4x 4 0 1. Circle 3. Parabola 2. Ellipse 4. Hyperbola In Exercises 5–10, match the equation with its graph. [The graphs are labeled (a), (b)... |
2, 1; 2, 2; 0, 0; focus: focus: directrix: 1, 0 y 2 directrix: directrix: directrix: x 1 x 2 y 8 Section 10.2 Introduction to Conics: Parabolas 741 In Exercises 51 and 52, change the equation of the parabola so that its graph matches the description. 51. 52. y 3 2 6x 1; y 1 2 2x 4; upper half of parabola lower half of... |
of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complet... |
at a height of Projectile Motion In Exercises 67 and 68, consider the path of a projectile projected horizontally with a velocity of feet, where the model for v the path is x2 v2 16 y s. s In this model (in which air resistance is disregarded), the height (in feet) of the projectile and distance (in feet) the projecti... |
3 12x 22 76. 77. 78. f (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the chord passing through the focus and parallel to the directrix (see figure). How can the length of this chord be determined directly from the standard form of the equation of the parabola?... |
of Halley’s comet. Harvard College Observatory/ SPL/Photo Researchers, Inch k (, c a 2 b2 + c 2 = 2a b2 + c 2 = a 2 FIGURE 10.21 Introduction The second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points distances from two distinct fixed points (foc... |
and minor axes are of equal length, and so the graph is a circle. a b or simply the length of the major axis. Now, if you let ellipse, the sum of the distances between That is, x, y x, y and the two foci must also be be any point on the 2a 2a. b2 a2 c 2, which implies that the Finally, in Figure 10.21, you can see tha... |
, 1 −3 + y 4 3 2 1 −5 − 4 −3 (−3, 0) −1 x −1 Example 1 Finding the Standard Equation of an Ellipse Find the standard form of the equation of the ellipse having foci at 4, 1 and a major axis of length 6, as shown in Figure 10.23. 0, 1 and Solution Because the foci occur at the distance from the center to one of the foci... |
Page 747 Section 10.3 Ellipses 747 Example 3 Analyzing an Ellipse Find the center, vertices, and foci of the ellipse 4x 2 y 2 8x 4y 8 0. Solution By completing the square, you can write the original equation in standard form. 4x 2 y 2 8x 4y 8 0 4x 2 8x y 2 4y 8 4x 2 2x y 2 4y 8 Write original equation. Group terms. Fa... |
iptical Orbit The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure 10.26. The major and minor axes of the orbit have lengths of 768,800 kilometers and 767,640 kilometers, respectively. Find the greatest and smallest distances (the apogee and perigee), respectively from Earth’s... |
orbit of the moon has an eccentricity of e 0.0549, and the eccentricities A S A N The time it takes Saturn to orbit the sun is equal to 29.4 Earth years. Venus: Mercury: of the nine planetary orbits are as follows. e 0.2056 e 0.0068 e 0.0167 e 0.0934 e 0.0484 Mars: Jupiter: Earth: Saturn: Uranus: Neptune: Pluto: e 0.0... |
), (e), and (f).] (a) y (b) In Exercises 7–30, identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. x 2 4 −4 x 4 y 4 2 −4 y 6 2 −6 y 4 −4 x 4 6 x 4 2 −4 (c) (e) −4 −6 1. 3. 5. 6 25 4 x 2 2 16 x 2 2 9 (d) x 2 4 −4... |
3x 2 4y 2 12 333202_1003.qxd 12/8/05 9:01 AM Page 751 In Exercises 35–42, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 35. y 8 36. (0, 4) (2, 0) 8 4 (0, 4)− x − ( 2, 0) −8 −4 −8 y 4 −4 ( ) 0, 3 2 (2, 0) 4 ( 0, − ) 3 2 x (−2, 0) −4 37. Vertices: 38. Vert... |
, 1 2 vertices: minor axis of length 2 4, 4, 4, 4 foci: 1, 2, 5, 2 endpoints of the minor axis: Section 10.3 Ellipses 751 54. Vertices: 5, 0, 5, 12; 1, 6, 9, 6 endpoints of the minor axis: 55. Find an equation of the ellipse with vertices eccentricity e 3 5. 56. Find an equation of the ellipse with vertices eccentricit... |
tic Geometry 60. Satellite Orbit The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in 1957). Its highest point above Earth’s surface was 947 kilometers, and its lowest point was 228 kilometers (see figure). The center of Earth was the focus of the elliptical orbit, and the... |
and make a conjecture about the shape of the ellipse with maximum area. 8 9 10 11 12 13 a A (d) Use a graphing utility to graph the area function and use the graph to support your conjecture in part (c). 70. Think About It At the beginning of this section it was noted that an ellipse can be drawn using two thumbtacks,... |
ola. The definition of a hyperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is fixed, whereas for a hyperbola the difference of the distances between the foci and a point on the hyperbola is fixed. Definition of Hyperbol... |
b = 1 ( y k− ) Transverse axis is vertical. Example 1 Finding the Standard Equation of a Hyperbola Find the standard form of the equation of the hyperbola with foci 5, 2 and vertices 4, 2. 0, 2 and 1, 2 and When finding the standard form of the equation of any conic, it is helpful to sketch a graph of the conic with t... |
form. y 2 42 Write in standard form. x 2 22 1 2, 0, 0, 4 a 2, 2, 0 0, 4. b 4, and the transverse axis is horiFrom this, you can conclude that and and the endpoints of the conzontal. So, the vertices occur at Using these four points, you are able to and jugate axis occur at sketch the rectangle shown in Figure 10.34. N... |
can conclude that the equations of the asymptotes are 1 3, 0 1, 2, b 3, a 2 and and y 2 3 x 1 and y 2 3 x 1. Finally, you can determine the foci by using the equation have 1, 2 7. c 22 32 7, The hyperbola is shown in Figure 10.36. and the foci are c2 a2 b2. 1, 2 7 So, you and Now try Exercise 13. Te 7 −1, 2 + 4 3 (−1,... |
2 3 2 Now try Exercise 35. As with ellipses, the eccentricity of a hyperbola is e c a Eccentricity c > a, it follows that and because If the eccentricity is large, the branches of the hyperbola are nearly flat, as shown in Figure 10.38. If the eccentricity is close to 1, the branches of the hyperbola are more narrow, ... |
Figure 10.41. Undoubtedly, there have been many comets with parabolic or hyperbolic orbits that were not identified. We only get to see such comets once. Comets with elliptical orbits, such as Halley’s comet, are the only ones that remain in our solar system. is the distance between the vertex and the focus (in meters... |
the graph is an ellipse. d. For the equation A C 2. 2x2 2y2 8x 12y 2 0, you have Circle So, the graph is a circle. Now try Exercise 49. W RITING ABOUT MATHEMATICS Sketching Conics Sketch each of the conics described in Example 6. Write a paragraph describing the procedures that allow you to sketch the conics efficient... |
as an aid. 5. x 2 y 2 1 7. 9. x 2 y 2 81 25 x 12 4 1 y 22 1 1 6. 8. x 2 9 x 2 36 y 2 25 y 2 4 1 1 1 1 x 32 144 y 62 19 y 12 14 y 22 25 x 22 14 x 32 116 9x 2 y 2 36x 6y 18 0 x 2 9y 2 36y 72 0 x 2 9y 2 2x 54y 80 0 16y 2 x 2 2x 64y 63 0 1 10. 11. 12. 13. 14. 15. 16. In Exercises 17–20, find the center, vertices, foci, an... |
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