Unit 7 Precalculus review questions
SECTION 7 573 Review Exercises CHAPTER 7 Key Ideas Procedures Properties, Theorems, and Formulas Solving Systems of Equations Algebraically 520, 521, 544, 550 f 1×2 Partial Fraction Decomposition of 557 d1x2 Matrix Operations 530, 532 Theorem Inverses of n * n Matrices 536 Properties of Matrices 537 Theorem Invertible Square Linear Systems 550 CHAPTER 7 Review Exercises Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 1 and 2, find (a) A + B, (b) A – B, (c) – 2A, and (d) 3A – 2B. 1. A = c -1 4 3 d, 0 B = c 2 2. A = C 1 0 3 4 -3 -1 -2 2 2 4 -1 d 3 2 -3 S , 1 -1 B = C 2 -2 2 -1 4 0 3 1 4 3S 3 In Exercises 3–8, find the products AB and BA, or state that a given product is not possible. 3. A = c 4 d, 6 -1 0 B = c -1 -2 5 d 4 -2 B = C 2 -1 3 1 2 3 0 -1 4. A = C 3 4 2 -1 S, 3 5. A = 3 – 1 44, B = c 1 d, 1 3 1 B = D 3 1 -1 6. A = c 0 0 7. A = C 1 0 1 0 0 0 0 S, 1 0 1 8. A = D 0 0 1 0 0 0 0 0 0 1 0 0 T, 1 0 -2 -1 -1 -4 1 0 2 2 -1 10. A = C 2 -1 1 1 0 1 0 S, 2 1 3 T, 2 3 8 2 B = D -1 -2 -0.4 B = C 0.8 – 0.2 1.5 0.5 -0.5 0 0.4 0.2 0.2 0.5 0.5 0.5 0 – 4.5 – 1.5 T 0.5 1 0.2 -0.4 S 0.6 In Exercises 11 and 12, find the inverse of the matrix if it has one. If it does, use multiplication to support your result. 1 2 11. D 2 -1 2 -1 0 1 1 13. C 2 -2 0 1 1 1 -1 2 T 2 4 -1 12. C 2 1 0 -1 1 1 1S 1 -3 4 0 2 -1 S 1 -2 3 14. D 5 1 3 0 2 -1 0 2 -3 2 1 0 T 4 3 In Exercises 15–18, find the reduced row echelon form of the matrix. -4 2 T 1 1 2 B = C 1 -2 1 1 9. A = D 1 2 In Exercises 13 and 14, evaluate the determinant of the matrix. 1 0S -3 -3 d 1 5 2 In Exercises 9 and 10, use multiplication to verify that the matrices are inverses. -3 2 1 -2 3 B = D -1 3 4 -3 S -1 1 0 1 -2 0 2 2 1 1 1 T -1 0 1 15. C 3 1 0 1 -1 1 17. C 2 1 2 3 2 2 5S 3 3 3 4 1 -2 S 6 2 16. C – 3 5 1 -1 2 1 -2 2 1 1S 3 1 18. C -2 2 -2 5 -4 0 3 1 4 -6 S 9 In Exercises 19–22, state whether the system of equations has a solution. If it does, solve the system. 19. 3x – y = 1 x + 2y = 5 20. 21. x + 2y = 1 4y – 4 = – 2x 22. x – 2y = 9 3 3y – x = – 9 2 x – 2y = – 1 -2x + y = 5 574 CHAPTER 7 Systems and Matrices In Exercises 23–28, use Gaussian elimination to solve the system of equations. 23. 25. x + z + w = 2 24. x + w = -2 x + y + z = 3 x + y + z + 2w = – 2 3x + 2y + 3z + w = 8 – x – 2y – 2z – 3w = 2 39. 3x – 2y = 5 2x + y = – 2 40. y = x – 1.5 y = 0.5x 2 – 3 41. y = – 0.5x 2 + 3 42. x 2 + y 2 = 4 x + y – 2z = 2 3x – y + z = 4 – 2x – 2y + 4z = 6 43. y = 2 sin x y = 2x – 3 26. x + y – 2z = 2 3x – y + z = 1 -2x – 2y + 4z = – 4 27. -x – 6y + 4z – 5w = – 13 2x + y + 3z – w = 4 2x + 2y + 2z = 6 -x – 3y + z – 2w = – 7 28. In Exercises 39–44, solve the system of equations graphically. – x + 2y + 2z – w = – 4 y + z = -1 – x + 2y + 2z – 2w = – 6 – x + 3y + 3z – w = – 5 In Exercises 29–32, solve the system of equations by using inverse matrices. 29. x + 2y + z = – 1 x – 3y + 2z = 1 2x – 3y + z = 5 30. x + 2y – z = – 2 2x – y + z = 1 x + y – 2z = 3 31. 2x + y + z – w = 1 2x – y + z – w = – 2 -x + y – z + w = -3 x – 2y + z – w = 1 32. x – 2y + z – w = 2 2x + y – z – w = – 1 x – y + 2z – w = – 1 x + 3y – z + w = 4 2 y = 2x 2 – 3 y = 0.5x – 1 44. y = ln 2x y = 2x 2 – 12x + 15 In Exercises 45 and 46, find the coefficients of the function so that its graph goes through the given points. 45. Curve Fitting ƒ1×2 = ax 3 + bx 2 + cx + d 46. Curve Fitting ƒ1×2 = ax 4 + bx 3 + cx 2 + dx + e 12, 82, 14, 52, 1 -2, -42, 16, 32, 11, 22, 19, 42 13, 62, 14, -22, 17, 82 In Exercises 47–52, find the partial fraction decomposition of the rational function. 47. 49. 51. 3x – 2 x 2 – 3x – 4 3x + 5 x 3 + 4x 2 + 5x + 2 5x 2 – x – 2 x3 + x2 + x + 1 48. 50. 52. x – 16 x2 + x – 2 313 + 2x + x 22 x 3 + 3x 2 – 4 -x 2 – 5x + 2 x 3 + 2x 2 + 4x + 8 In Exercises 53–56, match the function with its graph. Do this without using your grapher. In Exercises 33–36, solve the system of equations by finding the reduced row echelon form of the augmented matrix. 33. x + 2y – 2z + w = 8 2x + 3y – 3z + 2w = 13 34. x + 2y – 2z + w = 8 2x + 7y – 7z + 2w = 25 x + 3y – 3z + w = 11 35. x + 2y + 4z + 6w = 6 3x + 4y + 8z + 11w = 11 2x + 4y + 7z + 11w = 10 3x + 5y + 10z + 14w = 15 36. x + 2z – 2w = 5 2x + y + 4z – 3w = 7 4x + y + 7z – 6w = 15 2x + y + 5z – 4w = 9 In Exercises 37 and 38, find the equilibrium point for the demand and supply curves. 37. p = 100 – x 2 p = 20 + 3x 1 2 x 10 p = 5 + 4x 38. p = 80 – (a) (b) (c) (d) 1 2 + x + 3 x – 1 1 2 54. y = – x + 2 + x + 3 x – 1 1 2 + 55. y = – x + 2 + x + 3 x – 1 2 1 56. y = – x + 2 x + 3 x – 1 53. y = – x + 2 – In Exercises 57 and 58, graph the inequality. 57. 2x – y … 1 58. x + 3y 6 2 SECTION 7 In Exercises 59–64, solve the system of inequalities. Give the coordinates of any corner points. 59. 4x + 9y Ú 360 9x + 4y Ú 360 x + y … 90 60. 7x + 10y … 70 2x + y … 10 x + y Ú 3 x Ú 0 y Ú 0 61. x – 3y + 6 6 0 y 7 x 2 – 6x + 7 62. x + 2y Ú 4 y … 9 – x2 63. x 2 + y 2 … 4 y Ú x2 64. y … x2 + 4 x 2 + y2 Ú 4 In Exercises 65–68, find the minimum and maximum, if they exist, of the objective function ƒ, subject to the constraints. 65. Objective function: ƒ = 7x + 6y Constraints: 7x + 5y Ú 100 2x + 5y Ú 50 x Ú 0, y Ú 0 66. Objective function: ƒ = 11x + 5y Constraints: 5x + 2y Ú 60 5x + 8y Ú 120 x Ú 0, y Ú 0 67. Objective function: ƒ = 3x + 7y Constraints: 5x + 2y Ú 100 x + 4y Ú 110 5x + 11y … 460 x Ú 0, y Ú 0 68. Objective function: ƒ = 9x + 14y Constraints: x + y … 120 9x + 2y Ú 240 3x + 10y Ú 360 69. Rotating Coordinate Systems The xy-coordinate system is rotated through the angle 45° to obtain the x¿y¿ -coordinate system. (a) If the coordinates of a point in the xy-coordinate system are 11, 22, what are the coordinates of the rotated point in the x¿y¿ -coordinate system? (b) If the coordinates of a point in the x¿y¿ -coordinate system are 11, 22, what are the coordinates of the point in the xy-coordinate system that was rotated to it? 70. Medicare Disbursements Table 7.7 shows the total Medicare Disbursements in billions of dollars for several years. Let x = 0 stand for 1990, x = 1 for 1991, and so forth. Review Exercises 575 Table 7.7 Total Medicare Disbursements Year Disbursements (billions) 2000 2001 2002 2003 2004 2005 2006 224.3 247.7 265.1 281.5 309.3 338.0 401.3 Source: U.S. Census Bureau, Statistical Abstract of the United States: 2009, Table 125. National Health Expenditures by Type: 1990 to 2006; http://www.census.gov/prod/2008pubs/09statab/health.pdf (a) Find a linear regression model and superimpose its graph on a scatter plot of the data. (b) Find a power regression model and superimpose its graph on a scatter plot of the data. (c) Find when the models in parts (a) and (b) predict the same disbursement amounts. (d) Writing to Learn Which model appears to be a better fit for the data? Explain. Which model would you choose to make predictions beyond 2010? 71. Population Table 7.8 gives the population (in thousands) of the states of Hawaii and Idaho for several years. Let x = 0 stand for 1980, x = 1 for 1981, and so forth. Table 7.8 Population Year Hawaii (thousands) Idaho (thousands) 1980 1990 2000 2002 2003 2004 2005 2006 2007 965 1108 1212 1229 1240 1254 1268 1279 1283 944 1007 1294 1342 1364 1392 1426 1464 1499 Source: U.S. Census Bureau, Current Population Reports, P25-1106. (a) Find a linear regression model for Hawaii’s data and superimpose its graph on a scatter plot of Hawaii’s data. (b) Find a linear regression model for Idaho’s data and superimpose its graph on a scatter plot of Idaho’s data. (c) Using the models in parts (a) and (b), when was the population of the two states the same? (d) Writing to Learn Why might you want to use a different model than a linear model for the population growth? What model do you think would be best for Hawaii and then for Idaho—linear, logistic, exponential, power, quadratic, or cubic? Explain your thinking. 576 CHAPTER 7 Systems and Matrices 72. (a) The 2003 population data for three states are listed below. Use the data in the first table of the Chapter 7 Project to create a 3 * 2 matrix that estimates the number of males and females in each state. State Population (millions) California Florida Rhode Island 35.5 17.0 1.1 (b) Write the data from the 2003 Census table below in the form of a 3 * 2 matrix. State California Florida Rhode Island % Pop. Under 18 Years % Pop. 65 Years or Older 26.5 23.1 22.8 10.6 17.0 14.0 (c) Multiply your 3 * 2 matrix in part (b) by the scalar 0.01 to change the values from percentages to decimals. (d) Use matrix multiplication to multiply the transpose of the matrix from part (c) by the matrix from part (a). What information does the resulting matrix provide? (e) How many males under age 18 lived in these three states in 2003? How many females age 65 or older lived in these three states? 73. Using Matrices A stockbroker sold a customer 200 shares of stock A, 400 shares of stock B, 600 shares of stock C, and 250 shares of stock D. The price per share of A, B, C, and D is $80, $120, $200, and $300, respectively. (a) Write a 1 * 4 matrix N representing the number of shares of each stock the customer bought. (b) Write a 1 * 4 matrix P representing the price per share of each stock. (c) Write a matrix product that gives the total cost of the stocks that the customer bought. 74. Basketball Attendance At Whetstone High School 452 tickets were sold for the first basketball game. There were two ticket prices: $0.75 for students and $2.00 for nonstudents. How many tickets of each type were sold if the total revenue from the sale of tickets was $429? 75. Truck Deliveries Brock’s Discount TV has three types of television sets on sale: a 13-in. portable, a 27-in. remote, and a 50-in. console. They have three types of vehicles to use for delivery: vans, small trucks, and large trucks. The vans can carry 8 portable, 3 remote, and 2 console TVs; the small trucks, 15 portable, 10 remote, and 6 console TVs; and the large trucks, 22 portable, 20 remote, and 5 console TVs. On a given day of the sale they have 115 portable, 85 remote, and 35 console TVs to deliver. How many vehicles of each type are needed to deliver the TVs? 76. Investments Jessica invests $38,000; part at 7.5% simple interest and the remainder at 6% simple interest. If her annual interest income is $2600, how much does she have invested at each rate? 77. Business Loans Thompson’s Furniture Store borrowed $650,000 to expand its facilities and extend its product line. Some of the money was borrowed at 4%, some at 6.5%, and the rest at 9%. How much was borrowed at each rate if the annual interest was $46,250 and the amount borrowed at 9% was twice the amount borrowed at 4%? 78. Home Remodeling Sanchez Remodeling has three painters: Sue, Esther, and Murphy. Working together they can paint a large room in 4 hours. Sue and Murphy can paint the same size room in 6 hours. Esther and Murphy can paint the same size room in 7 hours. How long would it take each of them to paint the room alone? 79. Swimming Pool Three pipes, A, B, and C, are connected to a swimming pool. When all three pipes are running, the pool can be filled in 3 hr. When only A and B are running, the pool can be filled in 4 hr. When only B and C are running, the pool can be filled in 3.75 hr. How long would it take each pipe running alone to fill the pool? 80. Writing to Learn If the products AB and BA are defined for the n * n matrix A, what can you conclude about the order of matrix B? Explain. 81. Writing to Learn If A is an m * n matrix and B is a p * q matrix, and if AB is defined, what can you conclude about their orders? Explain. SECTION 7 Project 577 CHAPTER 7 Project Examine the male and female population data from 1990 to 2006 given in Table 7.9. Table 7.9 U.S. Male and Female Population Data from 1990 to 2006 Population (millions) Male Female 1990 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 121.3 130.8 132.0 133.3 138.1 140.1 141.5 143.0 144.5 146.0 147.5 127.5 137.0 138.3 139.4 143.4 145.1 146.4 147.8 149.2 150.5 151.9 female population? Was the male population ever greater than the female population? Are there enough data to create a model for a hundred years or more? Explain your answers. 4. Notice that the data in Table 7.9 give information for a span of only 15 years. This is often not enough information to accurately answer the questions asked above. Data over a small period often appear to be linear and can be modeled with a linear equation that works well over that limited domain. Table 7.10 gives more data. Now use these data to plot the number of males versus time and females versus time, using 1890 as year zero. 5. Notice that these data in Table 7.10 do not seem to be linear. Often, you may remember from Chapter 3, a logistic model is used to model population growth. Find the logistic regression model for each data plot. What is the intersection of the curves and what does it represent? Would any of your responses in question 3 above change? If so, how? Why? Source: http://www.census.gov/popest/national/asrh/NC-EST2006/ NC-EST2006-02.xls 1. Plot the data using 1990 as the year zero. Find a linear re- gression model for each. 6. Go to the U.S. Census Bureau Web site (www.census.gov). How does your model predict the populations for the current year? 7. Use the census data for 2000. What percentage of the population is male and what percentage is female? 2. What do the slope and y-intercept mean in each equation? 3. What conclusions can you draw? According to these models, will the male population ever become greater than the 8. Go to the U.S. Census Bureau Web site (www.census.gov) and use the most recent data along with the concepts from this chapter to collect and analyze other data. Table 7.10 U.S. Male and Female Population Data from 1890 to 2000 Population (millions) Male Female Population (millions) Male Female 1890 1900 1910 1920 1930 1940 32.2 38.8 47.3 53.9 62.1 66.0 30.7 37.2 44.6 51.8 60.6 65.6 1950 1960 1970 1980 1990 2000 75.2 88.3 98.9 110.1 121.3 138.1 76.1 91.0 104.3 116.5 127.5 143.4
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