Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Semester 2 Review Inverse functions & Restrictions: (check INNER domain on composites) sin sin-1 1,1 2 , 2 cos cos-1 1,1 0, tan , 2 2 , tan-1 Simple Harmonic Motion Equations: At rest: d a sin t At max displacement: d a cos t Max displacement: a Period: 2 frequency: Graphs y = Asin(Bx – C) + D Amplitude = A period = 2 B phase shift = 1 or 2 period y = Acsc(Bx – C) + D first graph the associated sine equation Zeros become asymptotes. C B Key points every period 4 y = Atan(Bx – C) Asymptotes: Bx C 2 2 Zeros: midway between asymptotes points: (midway between asy. and zero, A ) y = Acot(Bx – C) (midway between zero and asy., A ) (midway between zero and asy., - A ) 582752097 Asymptotes: 0 Bx C Zeros: midway between asymptotes points: (midway between asy. and zero, A ) Page 1 of 26 Semester 2______ Review ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ s r s r Revolution x 360 = Degrees opposite hypotenuse adjacent cos hypotenuse opposite tan adjacent sin hypotenuse opposite hypotenuse sec = adjacent adjacent cot = opposite 1 csc 1 cos sec 1 tan cot 1 sin 1 sec = cos 1 cot = tan csc = Quotient Identities (2) sin tan cos ______ ______ ______ ______ ______ Deg 180 cos cot sin r Revolution x 2π = radians csc = Reciprocal Identities (6) sin ______ ______ ______ 0 30 45 sin Θ cos Θ tan Θ Cofunction Identities (6) sin cos 90 cos sin 90 sec csc 90 csc sec 90 tan cot 90 cot tan 90 for radians, replace 90with / 2. Pythagorean Identities (3) sin 2 cos 2 1 1 tan 2 sec 2 1 cot 2 cot 2 582752097 60 Page 2 of 26 90 Semester 2 Review Ch 5 5.1 Find the radian measure of Θ. 1. A central angle, Θ, in a circle of radius 12 in intercepts an arc of length 42 in. 2. A central angle, Θ, in a circle of radius 6 cm intercepts an arc of length 15 cm. 3. A central angle, Θ, in a circle of radius 10 km intercepts an arc of length 45 km. Convert each angle in degrees to radians: 4. 60 5. 270 Convert each angle in radians to degrees: 4 7. 8. 4 3 Draw and label each angle in standard position: 3 10. 11. 12. 4 4 6. -300 9. 6 7 4 Find a positive angle less than 360 that is coterminal with each of the following: 13. 400 14. -135 15. 750 Find a positive angle less than 2π that is coterminal with each of the following: 17 13 17 16. 17. 18. 6 5 3 Find the length of the arc intercepted by the central angle in terms of π. Then round to two decimal places. 19. A circle has a radius of 6 inches. A central angle is 45. 20. A circle has a radius of 10 inches. A central angle is 120. 21. A circle has a radius of 8 feet. A central angle is 270. 22. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center. 23. A wind machine generates electricity using blades that are 10 feet long. These propellers rotate at 4 revolutions per minute. Find the linear speed, in feet per second, of the tips of the blades. 582752097 Page 3 of 26 Semester 2 Review 5.2 Find the value of each of the 6 trigonometric functions of Θ in the figure. 1. 2. c=5 Θ a=1 Θ b=4 a=3 Find each of the following. If a radical appears in the denominator, rationalize. 3. tan 60 4. sin 30 5. cos 45 6. csc 45 7. cot 45 8. sec 60 9. Given sin 5 2 and cos , find the value of the four remaining trig functions. 3 3 10. Given sin 21 2 and cos , find the value of the four remaining functions. 5 5 1 and Θ is an acute angle, find the value of cosΘ using a 2 trigonometric identity. 11. Given that sin 3 and Θ is an acute angle, find the value of cosΘ using a 5 trigonometric identity. 12. Given that sin Find a cofunction with the same value as the given expression: cot csc 13. sin 46 14. 15. 16. 12 3 Use a calculator to find the value to four decimal places. 17. sin 72.8 18. csc 1.5 19. cos 48.2 20. Find the value of the variable. 21. a 24 a 750 yd 22. c 10 a sin 72 cot 1.2 500 ft 23. A flagpole 14 meters tall has a shadow 10 meters long. Find the angle of elevation of the sun to the nearest degree. 582752097 Page 4 of 26 Semester 2 Review 5.3 1. Let P(1, -3) be a point on the terminal side of Θ. Find each of the six trig functions of Θ. 2. Let P(-3, -5) be a point on the terminal side of Θ. Find each of the six trig functions of Θ. Evaluate, if possible, the cosine and cosecant functions at the four following quadrantal angles: 3 3. 0 0 4. 90 5. 180 6. 270 2 2 7. 8. If sin 0 and cos 0 , name the quadrant in which Θ lies. If tan 0 and cos 0 , name the quadrant in which Θ lies. 9. Given tan 10. 1 and cos 0 , find sinΘ and secΘ. 3 2 Given tan and cos 0 , find cosΘ and cscΘ. 3 Find the reference angle, Θ’, for each of the following angles: 7 11. Θ = 210 12. 13. Θ = -240 14. Θ = 3.6 4 15. Θ = 665 16. 15 4 17. 11 3 18. Θ = 580 Use reference angles to find the exact value of the following trig functions: 5 17 19. sin 300 20. tan 21. sec 22. cos 4 6 6 22 23. sin 3 582752097 24. tan 14 3 17 25. sec 4 26. sin 135 Page 5 of 26 Semester 2 Review 5.4 Find the values of the trig functions at t on the unit circle. 1. 1 3 2. 0,1 , 2 2 • • t t Find the exact value of each trig function: tan 3. 4. cos 60 3 5. tan 6 6. cos 45 7. cos 420 8. sin 9 4 9. cos 405 10. sin 7 3 11. csc 7 6 12. 5 sin 6 13. tan17 14, sin 47 4 5.5 Determine the amplitude, period, and phase shift, then graph. 1 y sin x y 3sin x y 2sin x 1. 2. 3. 2 4. y 3sin 2 x 582752097 5. y 2 sin 1 x 2 6. y 3cos 2 x Page 6 of 26 7. y 4 cos x 10. y 1 cos 4 x 2 Semester 2 Review 2 8. y 4sin 2 x 3 11. y 3 cos 2 x 2 9. y 3sin 2 x 3 12. y 1 cos x 1 2 Write an equation of the form y A sin( Bx C ) . 13. 14. Window Xmin=-π/4 Xmax=π/2 Xscl=π/8 Ymin=-4 Ymax=4 Yscl=1 15. The depth of water at a boat dock varies with the tides. The depth is 5 feet at low tide and 13 feet at high tide. On a certain day, low tide occurs at 4a.m. and high tide at 10 a.m. If y represents the depth of water, in feet, x hours after midnight, use a sine function of the form y A sin( Bx C ) D to model the water’s depth. 16. A region that is 30 north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y A sin( Bx C ) D to model the hours of daylight. 582752097 Page 7 of 26 Semester 2 Review 5.6 Graph. 17. y 2 tan x for - x 3 2 18. y 3 tan 2 x for 4 x 3 4 19. y tan x 4 20. y tan x 2 21. y 3cot 2 x 22. y 1 cot x 2 2 23. y 2 csc 2 x 24. y csc x 4 25. y 3sec 2 x for –π<x<5π 5.7 Find the exact values. 2 sin 1 27. 28. 2 30. 2 sin 1 2 582752097 31. 26. sin 1 3 2 3 cos 1 2 y 2sec 2 x for 3 3 x 4 4 29. 1 sin 1 2 32. 1 cos 1 2 Page 8 of 26 33. tan 36. 3 sin 1 sin 2 39. cos cos 1 1.2 42. 1 Semester 2 Review 34. tan 1 1 3 cos cos 1 0.6 37. cos cos 1 1.5 38. sin 1 sin 40. 5 cos tan 1 12 41. 3 sin tan 1 4 1 cot sin 1 3 43. 1 cos sin 1 2 35. Write as an algebraic expression in x. 44. If 0 x 1 , cos sin 1 x 45. Find the value to four decimal places. 1 sin 1 46. 47. tan 1 9.65 4 If x>0, sec tan 1 x 48. cos 1 1 3 5.8 Solve the triangle. Round to two decimal places. 49. A = 34.5 , and b = 10.5 50. A = 62.7 , and a = 8.4. 582752097 Page 9 of 26 Semester 2 Review 51. From a point on level ground 125 feet from the base of a tower, the angle of elevation is 57.2 . Approximate the height of the tower to the nearest foot. 52. From a point on level ground 80 feet from the base of the Eiffel Tower, the angle of elevation is 85.4 . Approximate the height of the Eiffel Tower to the nearest foot. 53. A kite flies at a height of 30 feet when 65 feet of string is out. If the string is in a straight line, find the angle it makes with the ground to the nearest tenth of a degree. 54. A guy wire is 13.8 yards long and is attached from the ground to a pole 6.7 yards above the ground. Find the angle, to the nearest tenth of a degree, that the wire makes with the ground. 55. You are taking your first hot-air balloon ride. Your friend is standing on the ground 100 feet away from your point of launch. At one instant, the angle of elevation to you is 31.7. One minute later, the angle of elevation is76.2. How far did you travel, to the nearest tenth of a foot, during that minute? 56. You are standing on level ground 800 feet from Mt. Rushmore, looking at the sculpture of Abraham Lincoln’s face. The angle of elevation to the bottom of the sculpture is 32 and the angle of elevation to the top is 35. Find the height of the sculpture of Lincoln’s face to the nearest tenth of a foot. N 57. a. b. c. d. Find each of the following: The bearing from O to B. The bearing from O to A. The bearing from O to D. The bearing from O to C. 582752097 B A 40 W 75 C 20 O 25 S E D Page 10 of 26 Semester 2 Review 58. A boat leaves the entrance to a harbor and travels 25 miles on a bearing of N 42 E. The captain then turns the boat 90 clockwise and travels 18 miles on a bearing of S 48 E. At this time: a. How far is the boat, to the nearest tenth of a mile, from the harbor? b. What is the bearing, to the nearest tenth of a degree, of the boat from the harbor? 59. You leave the entrance to a system of hiking trails and hike 2.3 miles on a bearing of S31 W. Then the trail turns 90 clockwise and you hike 3.5 miles on a bearing of N 59 W. At that time: a. How far are you, to the nearest tenth of a mile, from the entrance of the trail system? b. What is your bearing, to the nearest tenth of a degree, from the entrance of the trail system? 60. A ball on a spring is pulled 4 inches below its rest position and then released. The period of motion is 6 seconds. Write the equation for the ball’s simple harmonic motion. 61. A ball on a spring is pulled 6 inches below its rest position and then released. The period of motion is 4 seconds. Write the equation for the ball’s simple harmonic motion. 62. If a mass moves in simple harmonic motion described by d 10 cos 6 t , with t measured in seconds and d in centimeters. Find each of the following: a. the maximum displacement b. the frequency c. the time required for one cycle. 63. A mass moves in simple harmonic motion described by d 12 cos 4 measured in seconds and d in centimeters. Find each of the following: a. the maximum displacement b. the frequency c. the time required for one cycle. 582752097 t , with t Page 11 of 26 6.1 Verify the identity. sec x cot x csc x 1. Semester 2 Review 2. csc x tan x sec x 3. sin x tan x cos x sec x 4. cos x cot x sin x csc x 5. cos x cos x sin 2 x cos3 x 6. sin x sin x cos 2 x sin 3 x 7. 1 sin x sec x tan x cos x 8. 1 cos x csc x cot x sin x 9. cos x 1 sin x 2sec x 1 sin x cos x 10. sin x 1 cos x 2 csc x 1 cos x sin x 11. sin x 1 cos x 1 cos x sin x 12. cos x 1 sin x 1 sin x cos x 13. 15. tan x sin x 1 cos x tan x 1 1 2 2 cot 2 x (Both) 1 cos x 1 cos x 582752097 14. 16. sec x csc x sec x csc x sin x cos x 1 1 2 2 tan 2 x 1 sin x 1 sin x Page 12 of 26 Semester 2 Review 6.2 Find the exact values. 17. cos 15 (Use 60-45=15) 18. cos 30 (Use 90-60=30) 19. cos80 cos20 + sin80 sin20 20. cos70 cos40 +sin70 sin40 21. sin 22. sin 7 , 12 if 7 12 3 4 Verify the identity. cos 23. cot tan sin cos 25. tan x 1 tan x 4 tan x 1 24. 26. 5 , 12 if cos cos cos 5 12 6 4 1 tan tan tan x tan x 12 3 for α in quadrant II and sin for β in quadrant I. 13 5 cosβ 27. cos(α+β) 28. sin(α+β) Find the exact values if sin 25. cosα 26. 4 1 for α in quadrant II and sin for β in quadrant I. 5 2 cosβ 31. cos(α+β) 32. sin(α+β) Find the exact values if sin 29. cosα 582752097 30. Page 13 of 26 Semester 2 Review 5 and x lies in quadrant II, find the exact values: 13 sin 2x 34. cos 2x 35. tan 2x 6.3 If sin x 33. 4 and x lies in quadrant II, find the exact values: 5 sin 2x 37. cos 2x 38. tan 2x If sin x 36. Find the exact values: 2 tan15 39. 1 tan 2 15 40. cos 2 15 sin 2 15 42. tan x sin 2 x 1 cos 2 x 6.5 Solve the equation. 1. 3sin x 2 5sin x 1 2. 5sin x 3sin x 3 3. tan 3x 1, 4. tan 2 x 3, 5. sin 6. sin Verify the identity. 1 cos 2 x tan x 41. sin 2 x 582752097 x 3 , 2 2 0 x 2 0 x 2 x 1 , 3 2 0 x 2 0 x 2 Page 14 of 26 Semester 2 Review 7. 2cos2 x cos x 1 0, 0 x 2 8. 2sin 2 x 3sin x 1 0, 0 x 2 9. 4sin 2 x 1 0, 0 x 2 10. 4cos2 x 3 0, 11. tan x sin 2 x 3tan x, 0 x 2 12. sin x tan x sin x, 13. 2cos2 x 3sin x 0, 0 x 2 14. 2sin 2 x 3cos x 0, 15. cos 2 x 3sin x 2 0, 16. cos 2 x sin x 0, 0 x 2 17. 1 sin x cos x , 2 0 x 2 18. 1 sin x cos x , 2 0 x 2 19. sin x cos x 1, 0 x 2 20. cos x sin x 1, 0 x 2 0 x 2 0 x 2 0 x 2 0 x 2 Solve each equation, correct to four decimal places, for 0 ≤ x < 2π. 21. tan x = 12.8044 22. cos x = -0.4317 23. tan x = 3.1044 24. sin x = -0.2315 25. sin 2 x sin x 1 0 26. cos 2 x 5cos x 3 0 582752097 Page 15 of 26 Semester 2 Review 7.1 Solve triangle ABC. Round to the nearest tenth. 27. A = 46 , C = 63 , and c = 56 in 28. A = 64 , C = 82 , and c = 14 cm 29. A = 50 , C = 33.5 , and b = 76 30. A = 40 , C = 22.5 , and b = 12 31. A = 43 , a = 81, and b = 62 32. A = 57 , a = 33, and b = 26 33. A = 75 , a = 51, and b = 71 34. A = 50 , a = 10, and b = 20 35. A = 40 , a = 54, and b = 62 36. A = 35 , a = 12, and b = 16 Find the area of a triangle rounded to the nearest square meter. 37. side lengths 24m and 10m and an included angle of 62. 38. side lengths 8m and 12m and an included angle of 135. 39. Two fire-lookout stations are 20 miles apart, with station B directly east of station A. Both stations spot a fire on a mountain to the north. The bearing from station A to the fire is N50E. The bearing from station B to the fire is N36W. How far, to the nearest tenth of a mile, is the fire from station A? 40. Two fire-lookout stations are 13 miles apart, with station B directly east of station A. Both stations spot a fire. The bearing from station A is N35E and the bearing of the fire from station B to the fire is N49W. How far, to the nearest tenth of a mile, is the fire from station B? 582752097 Page 16 of 26 Semester 2 Review 7.2 Solve the triangle. Round lengths to the nearest tenth and angles to the nearest degree. 41. A = 60, b = 20, and c = 30. 42. A = 120, b = 7, and c = 8 43. a = 6, b = 9, and c = 4 44. a =8, b = 10, and c = 5 45. Two airplanes leave an airport at the same time on different runways. One flies on a bearing of N66W at 325 miles per hour. The other plane flies on a bearing of S26W at 300 miles per hour. How far apart will the planes be after 2 hours? 46. Two airplanes leave an airport at the same time on different runways. One flies directly north at 400 miles per hour. The other plane flies on a bearing of N75E at 350 miles per hour. How far apart will the planes be after 2 hours? Find the area of the triangle. Round to the nearest whole unit. 47. a= 12 yd, b = 16 yd, c = 24 yd 48. a = 6m, b = 16m, and c = 18m 582752097 Page 17 of 26 Semester 2 Review 4.1 The exponential function f(x) = 13.49(0.967)x - 1 describes the number of O-rings expected to fail, f(x), when the temperature is xF. Find the number of O-rings expected to fail at the given temperature. 1. 31F 2. 60F 3. -10F Graph. 4. f x 2 7. 1 f x 3 x x 5. g x 3 6. 1 h x 2 8. g x 3x1 9. h x 2x 3 x x 10. The function f(x) = 3.6e0.02x describes world population, f(x), in billions, x years after 1969. Find the world population in 2020. 11. The function f(x) = 6.4e0.0123x describes world population, f(x), in billions, x years after 2004. Find the world population in 2050. 12. You decide to invest $8000 for 6 years. How much will have if you invest at 7% per year, compounded monthly? at 6.85% per year, compounded continuously? 13. You decide to invest $10000 for 5 years at an annual rate of 8%. How much will have if it is compounded quarterly? compounded continuously? 582752097 Page 18 of 26 Semester 2 Review 4.2 Write each equation in its equivalent exponential form: 14. 2 = log5x 15. 3 = logb64 16. log37 = y Write each equation in its equivalent logarithmic form: 17. 122 = x 18. b3 = 8 19. ey = 9 Evaluate. 20. log216 21. log39 22. log255 23. log77 24. log51 25. log778 26. 6log69 27. log445 28. log81 33. f(x) = ln (3 - x) Graph. 29. f(x) = log2x Find the domain. 31. f(x) = log4 (x + 3) 34. 30. 32. g(x) = log3x g(x) = log4 (x - 5) g(x) = ln (x - 3)2 35. The percentage of adult height attained by a boy who is x years old can be modeled by f(x) = 29 + 48.8 log (x + 1), where x represents the boy’s age and f(x) represents the percentage of his adult height. An 8 year old boy has attained approximately what percentage of his adult height? 36. The function f(x) = 13.4 ln x - 11.6 models the temperature increase, f(x), in degrees Fahrenheit, after x minutes in an enclosed vehicle. Find the temperature increase after 50 minutes. 582752097 Page 19 of 26 4.3 Expand each expression. 37. log4(7• 5) 38. Semester 2 Review log (10x) 39. log574 42. ln x 45. 3x log 6 4 36 y 40. 19 log 7 x 41. e5 ln 11 43. log (4x)5 44. log b x 2 y Write as a single logarithm. 1 log x 4 log x 1 2 46. log42 + log432 47. log (4x - 3) - logx 48. 49. 3 ln (x + 7) - ln x 50. 1 4 log b x 2 log b 6 log b y 2 Evaluate. 51. log5140 52. log72506 4.4 Solve. 53. 23x-8 = 16 54. 27x+3 = 9x-1 55. 4x = 15 56. 40e0.6x - 3 = 237 57. 5x-2 = 42x+3 58. e2x - 4ex + 3 = 0 59. log4 (x + 3) = 2 60. 3 ln (2x) = 12 61. log2 x + log2 (x - 7) = 3 62. 1 ln x 2 ln 4 x 3 ln x 582752097 Page 20 of 26 Semester 2 Review 63. The risk of having a car accident while under the influence of alcohol can be modeled by R = 6e12.77x, where x is the blood alcohol concentration and R, given as a percent, is the risk of a car accident. What blood alcohol level corresponds to a 20% risk of a car accident? 64. How long will it take $25,000 to grow to $500,000 at 9% annual interest compounded monthly? r A P 1 n nt 65. The function f(x) = 34.1 ln x + 117.7 models the number of U.S. Internet users, f(x), in millions, x years after 1999. By what year will there be 200 million Internet users in the U.S.? 4.5 66. In 1970, the U.S. population was 203.3 million. By 2003, it had grown to 294 million. Find the exponential growth function that models this data (A = A0ekt). Then find the year when the population will reach 315 million. 67. Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14. Then estimate the age of scrolls found in 1947 that contained 76% of their original carbon-14. 30000 describes the number of people, f(t), who have 1 20e 1.5t become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants. How many people became ill when the epidemic began? How many people were ill by the end of the fourth week? What is the limiting size of f(t), the population that becomes ill? 68. The function f (t ) 69. Rewrite y = 2.557 (1.017)x in terms of base e. 582752097 Page 21 of 26 Semester 2 Review Ch 8 1. Bottled water and medical supplies are to be shipped to victims of an earthquake by plane. Each container of bottled water will serve 10 people and each medical kit will aid 6 people. Let x represent the number of bottles of water to be shipped and y represent the number of medical kits. a. Write the objective function that describes the number of people who can be helped. b. Each plane can carry no more than 80,000 pounds. Bottled water weighs 20 pounds per container and each medical kit weighs 10 pounds. Write an inequality that describes the constraint. c. Each plane can carry a total volume of supplies that does not exceed 6000 cubic feet. Each water bottle is 1 cubic foot and each medical kit also has a volume of 1 cubic foot. Write an inequality describing this second constraint. d. Determine how many bottles of water and how many medical kits should be sent on each plane to maximize the number of earthquake victims who can be helped. 2. A company manufactures bookshelves and desks for computers. Let x represent the number of bookshelves manufactured daily and y the number of desks manufactured daily. The company’s profits are $25 per bookshelf and $55 per desk. a. Write the objective function that describes the company’s total profit, z, from x bookshelves and y desks. b. To maintain high quality, the company should not manufacture more that a total of 80 bookshelves and desks per day. Write an inequality that describes this constraint. c. To meet customer demand, the company must manufacture between 30 and 80 bookshelves per day, inclusive. Furthermore, the company must manufacture at least 10 and no more than 30 desks per day. Write an inequality that describes each of these sentences. d. How many bookshelves and how many desks should be manufactured per day to obtain maximum profit? What is the maximum daily profit? 3. Find the maximum value of the objective function z = 3x + 5y subject to the constraints x ≥ 0, y ≥ 0, x + y ≥ 1, x + y ≤ 6. 582752097 Page 22 of 26 Semester 2 Review Ch 9 Write the solution set for a system of equations represented by the matrix. 8 1 2 5 19 1 1 1 4. 5. 9 0 1 3 0 1 12 15 0 0 1 0 0 4 1 1 Use matrices to solve the system. Gauss (REF) 3 x y 2 z 31 x y 2 z 19 6. 7. x 3 y 2 z 25 2w x 3 y z 6 w x 2 y 2 z 1 w x y z 4 8. 9. w 2 x 2 y z 7 10. x 3 y 2 z 25 582752097 x 2y z 6 w 3 x 2 y z 3 2w 7 x y 2 z 1 3w 7 x 3 y 3z 5 5w x 4 y 2 z 18 Gauss-Jordan (RREF) 3 x y 2 z 31 x y 2 z 19 2 x y 2 z 18 x y 2z 9 w x y z 4 11. 2w x 2 y z 0 w 2 x y 2 z 2 3w 2 x y 3z 4 Page 23 of 26 Semester 2 Review 11.1 Write the first four terms of the sequence whose general term is given. 1. 3. an = 3n + 4 an = 3an-1 + 2, for a1 = 5 and n≥ 2 Evaluate. 10! 5. 2!8! 8. 6. 7 2 k 5 k 4 n 1! n! 1 n 2. an 4. 2n an n 1! 7. i 6 i 1 9. 3n 1 2 1 5 3 i1 Express in summation notation. 10. 13 + 23 + 33 + … +73 1 1 1 11. 1 3 9 27 1 3n1 11.2 12. Find the first five terms of the arithmetic sequence in which a1 = 77.4 and an = an-1 - 0.67. 13. Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is -7. 582752097 Page 24 of 26 Semester 2 Review 14. According to the U.S. Census Bureau, new one-family houses sold for an average of $159,000 in 1995. This average sales price has increased by approximately $9700 per year. Write a formula for the nth term of the arithmetic sequence that describes the average cost for new one-family houses n years after 1994. How much will new onefamily houses cost, on average, by the year 2010? 15. Find the sum of the first 100 terms of the arithmetic sequence: 1, 3, 5, 7, … 16. Find the following sum: 25 5i 9 i 1 17. Your grandmother has assets of $400,000. One option that she is considering involves an adult residential community for a six-year period beginning in 2006. The model an = 1800n + 58,730 describes yearly adult residential community costs n years after 2005. Does your grandmother have enough to pay for the facility? 11.3 18. Write the first six terms of the geometric sequence with first term 6 and common ratio 1/3. 19. Find the eighth term of the geometric sequence whose first term is -4 and whose common ratio is -2. 20. The population in Florida from 1990 to 1997 is shown in the following table: Year 1990 1991 1992 1993 1994 1995 1996 1997 Population in millions 12.94 13.20 13.46 13.73 14.00 14.28 14.57 14.86 a. b. c. 582752097 Show that the population is increasing geometrically. Write the general term for the sequence describing population growth for Florida n years after 1989. Estimate Florida’s population, in millions, for the year 2000. Page 25 of 26 21. Semester 2 Review Find the sum of the first 18 terms of the geometric sequence: 2, -8, 32, -128, … 22. Find the following sum: 10 6 2 i i1 23. A union contract specifies that each worker will receive a 5% pay increase each year for the next 30 years. One worker is paid $20,000 the first year. What is this person’s total lifetime salary over a 30-year period? 24. To save for retirement, you decide to deposit $1000 into an IRA at the end of each year for the next 30 years. If the interest rate is 10% per year compounded annually, find the value of the IRA after 30 years. 25. Find the sum of the infinite geometric series: 26. Express 0.7 as a fraction in lowest terms. 3 3 3 3 8 16 32 64 . 27. Suppose the government reduces taxes so that each consumer has $2000 more income. The government assumes that each person will spend 70% of this ($1400). The individuals and businesses receiving this $1400 in turn spend 70% of it (980), creating extra income for other people to spend, and so on. Determine the total amount spent on consumer goods from the initial $2000 tax rebate. 582752097 Page 26 of 26