Jim Lambers Math 1B Fall Quarter Final Exam Practice Problems

Transcription

1 Jim Lambers Math 1B Fall Quarter Final Exam Practice Problems The following problems are indicative of the types of problems that will appear on the Final Exam, which will be given on Monday, December 6. The exercises are from the textbook, Precalculus, 5th Edition, by Barnett, Ziegler, and Byleen. As they are odd-numbered exercises, you can find the answers at the end of this document. In doing these problems, you should try to simulate exam conditions as closely as possible. It is recommended that you observe the following guidelines: Select a quiet working environment. Try to complete the problems as rapidly as possible. For a useful practice exam, pre-select problems that you have not previously attempted and see how many of them you can complete accurately in 45 minutes. Scientific calculators will be allowed, so be sure to have one while attempting these problems. Do not use your books or notes while attempting these problems. Section Population Growth. Because of its short life span and frequent breeding, the fruit fly Drosophilia is used in some genetic studies. Raymond Pearl of Johns Hopkins University, for example, studied 300 successive generations of descendants of a single pair of Drosophilia flies. In a laboratory situation with ample food supply and space, the doubling time for a particular population is 2.4 days. If we start with 5 male and 5 female flies, how many flies should we expect to have in: (A) 1 week? (B) 2 weeks? 65. Finance. Suppose $4,000 is invested at 11% compounded weekly. How much money will be in the account in: (A) 1 2 year? (B) 10 years? Compute answers to the nearest cent. 1

2 67. A couple just had a new child. How much should they invest now at 8.25% compounded daily in order to have $40,000 for the child s education 17 years from now? Compute the answer to the nearest dollar. Section Sound. What is the decibel level of: (A) The threshold of hearing, watt per square meter? (B) The threshold of pain, 1.0 watt per square meter? Compute answers to 2 significant digits. 55. Sound. If the intensity of a sound from one source is 1,000 times that of another, how much more is the decibel level of the louder sound than the quiet one? 57. Earthquakes. The largest recorded earthquake to date was in Colombia in 1906, with an energy release of joules. What was its magnitude on the Richter scale? Compute the answer to one decimal place. Section Solve to 3 significant digits x = e x = x+5 = e 4 2x = x = x = Solve exactly. 13. log x + log 4 = ln 8 ln x = log(x + 10) + log(x 5) = Solve to 3 significant digits = x 2

3 21. e 0.005x = , 000 = 75e 0.5x 25. e 0.1x2 = Solve exactly. 27. log x log 5 = log 2 log(x 3) 29. ln x = ln(2x 1) ln(x 2) 31. log(2x + 1) = 1 log(x 1) 33. (ln x) 3 = ln x ln(ln x) = x log x = 100x 77. How many years, to the nearest year, will it take a sum of money to double if it is invested at 15% compounded annually? Use the formula A = P [1 + (r/n)] nt. Section Write the value of each circular function in terms of the coordinates (a, b) of the circular point W (x). (A) cos x (B) csc x (C) cot x (D) sec x (E) tan x (F) sin x 3-19 Find the exact value of each expression (if it exists) without the use of a calculator. 3. cos 0 5. sin(π/6) 7. sin(π/2) 9. tan(π/3) 11. tan(π/2) 13. sec sec(π/4) 3

4 17. tan(π/4) 19. csc In which quadrants must W (x) lie so that: 21. cos x < sin x > cot x < Use the basic identities to find the values of the other five circular functions given the indicated information. 75. cos x = sin x = 3 2 and tan x < 0 and cot x < tan x = 3 and sin x < 0 Section Find the degree measure of each angle, keeping in mind that an angle of one complete rotation corresponds to rotation rotation 9-11 Find the radian measure of each angle, keeping in mind that an angle of one complete rotation corresponds to 2π radians rotation rotation Find the exact radian measure, in terms of π, of each angle , 60, 90, 120, 150, , 90, 135, Find the exact degree measure of each angle. 17. π/3, 2π/3, π, 4π/3, 5π/3, 2π 19. π/2, π, 3π/2, 2π 4

5 Section Find the reference angle α for each angle θ. 27. θ = θ = 7π/6 31. θ = 5π/ Evaluate exactly, using reference angles where appropriate, without using a calculator. 33. tan(3π/4) 35. sin( 30 ) 37. sec(5π/6) 39. cot csc( 150 ) 43. cos(13π/6) 45. tan( 7π/3) 47. sec(23π/4) Section Suppose that one of the acute angles of a right triangle is θ. Let a be the length of the side adjacent to θ, let b be the length of the side opposite θ, and let c be the length of the hypotenuse. Write the ratios of sides corresponding to each trigonometric function given. Do not look back at the definitions. 1. sin θ 3. csc θ 5. tan θ 7-11 Consider the right triangle in Problems 1-5. Each ratio defines a trigonometric function of the complement of θ (that is, the other acute angle in the triangle). Indicate which function without looking back at the definitions. 7. b/a 9. a/c 11. b/c 5

6 19-25 Suppose that one of the acute angles of a right triangle is α, and the other acute angle is β. Let a be the length of the side adjacent to β, let b be the length of the side opposite β, and let c be the length of the hypotenuse. For the given angle and given side length, find the lengths of the other two sides. 19. β = 17.8, c = α = 23, a = α = 53.21, b = Section Find exact values without using a calculator. 1. cos sin 1 ( 3/2) 7. sin 1 ( 2/2) 9. cos sin 1 (1/2) 19. cos 1 ( 3/2) 23. sin 1 ( 1/2) Section Verify the following identities. 1. sin θ sec θ = tan θ 3. cot u sec u sin u = 1 sin( x) 5. cos( x) = tan x 7. sin α = tan α cot α csc α 9. cot u + 1 = (csc u)(cos u + sin u) sin2 t cos t cos x sin x sin x cos x = csc x sec x + cos t = sec t 15. cos x 1 sin 2 x = sec x 17. (1 cos u)(1 + cos u) = sin 2 u 19. cos 2 x sin 2 x = 1 2 sin 2 x 6

7 21. (sec t + 1)(sec t 1) = tan 2 t 23. csc 2 x cot 2 x = (sin x cos x)2 sin x 39. cos θ + sin θ = cos y 1 cos y = = 2 cos x cot θ+1 csc θ sin2 y (1 cos y) tan 2 x sin 2 x = tan 2 x sin 2 x 45. csc θ cot θ+tan θ = cos θ cos A sec A 1 1+cos A = sec A sin 4 w cos 4 w = 1 2 cos 2 w 59. (sec x tan x) 2 = 1 sin x 1+sin x sin v cos v = cos v 1 sin v Section Verify each identity. 11. cot ( π 2 x) = tan x 13. csc ( π 2 x) = sec x 33. cos 2x = cos 2 x sin 2 x 35. cot(x + y) = 37. tan 2x = 2 tan x 1 tan 2 x cot x cot y 1 cot x+cot y 39. sin(v+u) cot u+cot v sin(v u) = cot u cot v 41. cot x tan y = cos(x+y) sin x cos y 43. tan(x y) = Section Verify each identity. cot y cot x cot x cot y (sin x + cos x) 2 = 1 + sin 2x 17. sin 2 x = 1 2 (1 cos 2x) cos 2x = tan x sin 2x 7

8 21. sin 2 x 2 = 1 cos x cot θ 2 = sin θ 1 cos θ 25. cos 2u = 1 tan2 u 1+tan 2 u csc 2x = 1+tan2 x tan x Section Find exact solutions over the indicated intervals, x a real number, θ in degrees sin x + 1 = 0, 0 x < 2π 3. 2 sin x + 1 = 0, all real x 5. tan x + 3 = 0, 0 x < π 7. tan x + 3 = 0, all real x 9. 2 cos θ 3 = 0, 0 θ < sin 2 θ + sin 2θ = 0, all θ 23. tan x = 2 sin x, 0 x < 2π cos 2 θ + 3 sin θ = 0, 0 θ < cos 2θ + cos θ = 0, 0 θ < sin 2 (x/2) 3 sin(x/2) + 1 = 0, 0 x 2π Section Solve each triangle. 1. α = 73, β = 28, c = 42 feet 3. α = 122, γ = 18, b = 12 kilometers 5. β = 112, γ = 19, c = 23 yards 7. α = 52, γ = 47, a = 13 centimeters Section Solve each triangle. 3. α = 71.2, b = 5.32 yards, c = 5.03 yards 5. γ = , a = 5.73 millimeters, b = 10.2 millimeters 9. a = 4 meters, b = 10.2 meters, c = 9.05 meters 11. a = 6 kilometers, b = 5.3 kilometers, c = 5.52 kilometers 8

9 Section Convert the polar coordinates to rectangular coordinates to three decimal places. 17. (6, π/6) 19. ( 2, 7π/8) 21. ( 4.233, 2.084) Convert the rectangular coordinates to polar coordinates with θ in degree measure, 180 < θ 180, and r ( 8, 0) Answers 25. ( 5, 5) 27. (9.79, 5.13) Section (A) 76 flies (B) 570 flies 65. (A) $4, (B) $12, $9,841 Section (A) 0 decibel (B) 120 decibels decibels more

10 Section / /e ± (1 + 89) 33. 1, e 2, e e e , Approximately 5 years 10

11 Section (A) a (B) 1/b (C) a/b (D) 1/a (E) b/a (F) b / Not defined Not defined 21. Quadrant II or III 23. Quadrant I or II 25. Quadrant II or IV 75. sin x = 3/2, tan x = 3, cot x = 1/ 3, csc x = 2/ 3, sec x = cos x = 1/ 2, tan x = 1, cot x = 1, csc x = 2, sec x = cot x = 1/ 3, sin x = 3/2, cos x = 1/2, csc x = 2/ 3, sec x = 2 11

12 Section π/ π/2 13. π/6, π/3, π/2, 2π/3, 5π/6, π 15. π/4, π/2, 3π/4, π , 120, 180, 240, 300, , 180, 270, 360 Section π/6 31. π/ / / /

13 Section b/c 3. c/b 5. b/a 7. cot(90 θ) 9. sin(90 θ) 11. cos(90 θ) 19. a = 3.28, b = b = 127, c = a = 31.85, c = Section π/2 3. π/3 7. π/ π/ π/6 23. π/6 Section Use sec θ = 1/ cos θ and tan θ = sin θ/ cos θ 3. Use cot u = cos u/ sin u and sec u = 1/ cos u 5. Use sin( x) = sin x, cos( x) = cos x, and tan x = sin x/ cos x 7. Use tan α = 1/ cot α and csc α = 1/ sin α 13

14 9. Use csc u = 1/ sin u and cot u = cos u/ sin u 11. Use csc x = 1/ sin x and sec x = 1/ cos x 13. Use sin 2 t + cos 2 t = 1 and sec t = 1/ cos t 15. Use sin 2 x + cos 2 x = 1 and sec x = 1/ cos x 17. Use sin 2 u + cos 2 u = Use sin 2 x + cos 2 x = Use sec t = 1/ cos t, tan t = sin t/ cos t, and sin 2 t + cos 2 t = Use csc x = 1/ sin x, cot x = cos x/ sin x, and sin 2 x + cos 2 x = Use sin 2 x + cos 2 x = Use cot θ = cos θ/ sin θ and csc θ = 1/ sin θ 41. Use sin 2 y + cos 2 y = Use sin 2 x + cos 2 x = Use cot θ = cos θ/ sin θ, tan θ = sin θ/ cos θ, sin 2 θ + cos 2 θ = 1, and csc θ = 1/ sin θ 51. Use sec A = 1/ cos A 53. Use sin 2 w + cos 2 w = Use sec x = 1/ cos x and tan x = sin x/ cos x 63. Use sin 2 v + cos 2 v = 1 Section Use cofunction identities for sine and cosine, and tan x = sin x/ cos x 13. Use sin(π/2 x) = cos x, and sec x = 1/ cos x 33. Use cos(x + y) = cos x cos y sin x sin y with y = x 35. Use double-angle identity for tangent, and cot x = 1/ tan x 37. Use sum identity for tangent 39. Use sum and difference identities for sine 14

15 41. Use cos(x + y) = cos x cos y sin x sin y, cot x = cos x/ sin x, and tan y = sin y/ cos y 43. Use tan x = 1/ cot x Section Use sin 2 x + cos 2 x = 1 and sin 2x = 2 sin x cos x 17. Use cos 2x = 1 2 sin 2 x 19. Use sin 2x = 2 sin x cos x and cos 2x = 1 2 sin 2 x 21. Use half-angle identity for sine 23. Use half-angle identity for tangent, and cot(θ/2) = 1/ tan(θ/2) 25. Use double-angle identity for tangent, and cot 2u = 1/ tan 2u 27. Use csc 2x = 1/ sin 2x, sin 2x = 2 sin x cos x, and tan x = sin x/ cos x Section π/6, 11π/6 3. 7π/6 + 2kπ, 11π/6 + 2kπ, k any integer 5. 2π/3 7. 2π/3 + kπ, k any integer 9. 30, k(180 ), k(180 ), k any integer 23. 0, 2π/3, π, 4π/ , , 180, π/3, π, 5π/3 15

16 Section γ = 79, a = 41 ft, b = 20 ft 3. β = 40, a = 16 km, c = 5.8 km 5. α = 49, a = 53 yd, b = 66 yd 7. β = 81, b = 16 cm, c = 12 cm Section a = 6.03 yd, β = 56.6, γ = c = 14.0 mm, α = , β = α = 23.0, β = 94.9, γ = α = 67.3, β = 54.6, γ = 58.1 Section (5.196, 3.000) 19. (1.848, 0.765) 21. (2.078, 3.688) 23. (8, 180 ) 25. (5 2, 135 ) 27. (11.05, 27.7 ) 16