Precalculus A. Semester Exam Review

Transcription

1 Precalculus A MCPS

2 The semester A eaminatin fr Precalculus cnsists f tw parts. Part 1 is selected respnse n which a calculatr will NOT be allwed. Part is shrt answer n which a calculatr will be allwed. Pages with the smbl indicate that a student shuld be prepared t cmplete items like these with r withut a calculatr. The frmulas belw are prvided in the eaminatin bklet. Trignmetric Identities sin cs 1 sec 1 tan csc 1 ct sin sincs cssin cs cscs sinsin cs cs sin cs 1 1 sin sin sin cs tan tan tan 1 tan tan tan tan 1 tan 1 cs sin 1 cs cs 1 cs sin tan sin 1 cs Triangle Frmulas Law f Sines: sin A sin B sin C Law f Csines: a b c c a b abcsc Area f a Triangle: 1 sin ab C Arc Length s r ( in radians) s r ( in degrees) 360 MCPS

3 PART 1 NO CALCULATOR SECTION 1. Sketch the graph f the piece-wise functin f. Lk at the graph f the piece-wise functin belw., if 0 1, if 0 Which f the fllwing functins is represented b the graph? A B f f,if 0,if 0,if 0,if 0 f C, if 0, if 0 f D,if 0,if 0 MCPS

4 3. Lk at the graph f the piece-wise functin belw. O Which tpe f discntinuit des the graph have at the fllwing -values? a. 3 b. 1 c Let f c 7,if 5, if 5 What is the value f c that will make f cntinuus at 5? 5. Let g 4 c, if 11 5, if 11 What is the value f c that will make g cntinuus at 11? MCPS

5 6. Which f the fllwing is true abut the functin f 4? 3 A The functin is cntinuus fr all real numbers. B The functin is discntinuus at 3nl. C The functin is discntinuus at 4nl. D The functin is discntinuus at 3and Lk at the graph f a functin belw. Des the graph represent an dd functin, an even functin, r a functin that is neither dd nr even? MCPS

6 8. Lk at the graph f a functin belw. Des the graph represent an dd functin, an even functin, r a functin that is neither dd nr even? 9. Determine whether each functin belw is dd, even, r neither dd nr even. 3 a. f sin b. g 4 c. rcs 4 MCPS

7 10. If f 3, which f the fllwing statements is NOT true? A B C The graph f f is smmetric with respect t the -ais. f is an even functin. The range f f is all real numbers. D As, f. 11. Lk at the graph f the functin belw. O a. What is the dmain f this functin? b. What is the range f this functin? MCPS

8 1 1. Fr each functin belw, find a frmula fr f dmain f f 1. and state an restrictins n the a. f b. 3 f True r False. a. The functin g f 5 represents a vertical stretch f the graph f f b a factr f 5, fllwed b a vertical translatin dwn units. b. The functin graph f f. 1 g f represents a vertical and hrizntal shrinking f the Match the transfrmatins that wuld create the graph f g frm the graph f f. g 3 f g f 3 A Stretch the graph f f hrizntall B Stretch the graph f f verticall 1 g f 3 C Shrink the graph f f hrizntall g f 1 D Shrink the graph f f verticall 3 MCPS

9 Fr items 15 and 16, use the graphs f f and g belw. f g 15. Which f the fllwing represents the relatinship between f and g? A gf 3 1 B g f 3 C g f 3 D 1 g f Sketch the graph f f. 17. Write the definitins f the si circular functins f an angle in standard psitin, passing thrugh the pint,, with r. 18. If 4 sin with cs 0, what are the values f ther five trignmetric functins? 5 MCPS

10 19. Fr each f the fllwing, state the quadrant in which the terminal side f lies. a. sin 0, tan 0 b. cs 0, tan 0 c. sec 0, csc 0 0. Cnvert t radian measure. Leave ur answer in terms f. a. 40 b On the unit circle belw the crdinates f pint A are 1, 0 and the crdinates f pint B are 0.8,0.6. Find the value f the fllwing. a. sin B b. cs c. tan A. Sketch the graphs f the si circular functins n the interval. 3. What are the perids f the si circular functins? 4. What is the perid f tan 8? 5. What is the value f b such that csb 6. What is the value f c such that cscc has a perid f 3? has a perid f 10? MCPS

11 7. Determine the eact value f the fllwing. a. sin 6 b. 5 cs 4 c. 5 tan 3 d. 3 sin cs f. tan e. g. 7 tan 4 h. 4 cs 3 i. 11 sin 6 j. sin 4 k. 5 cs 6 l. 7 tan 6 m. 5 sec 4 n. 5 ct 6. 4 csc 3 p. sec 8. Cmplete the table fr the inverse circular functins. Dmain 1 Sin Cs Tan 1 1 Range 9. Identif the functins represented b the graphs belw. a. 3 1 O 1 3 b. c MCPS

12 30. Determine the eact value f the fllwing. a. Sin 1 1 b. Cs c. Tan 3 d. Sin 1 3 e. Cs f. Tan g. Sin 1 h. Cs 0 1 i. Tan 0 j. cs Sin k. sin Tan 1 l. tan Cs 1 1 m. sin Csc n. 1 1 tan Sin Cs cs Write a sinusidal equatin fr each f the fllwing graphs. a b. c MCPS

13 3. Determine the equatin that best describes a sine curve with amplitude 3, perid f 6, and a phase shift f t the right. 33. Fr each equatin belw, state the amplitude, perid, phase shift, vertical translatin, and an reflectins f the sinusid relative t the basic functin f sin r g cs Sketch the graph, marking the - and -aes apprpriatel. 6 a. h sin 3 5 b. h5cs 1 c. h sin Simplif each epressin belw as a single functin f a single angle. D nt evaluate. a. sin17 cs17 b. 3 3 cs cs sin sin c. sin 7 cs3 cs 7sin 3 d. tan11 tan 5 1 tan11 tan5 e. 1 sin 9 f. cs8cs5 sin 8sin 5 g. 1 cs 3 h sin cs cs sin If 1 ct A and 0 A, determine the fllwing. 5 a. sin A b. A cs A c. sin A d. cs 36. Slve the fllwing equatins ver the interval a. sin 0 b. 3cs 4 5cs Slve the fllwing equatins n the interval 0. a. tan 1 0 b. sin 3sin 1 0 MCPS

14 38. Cmplete the fllwing chart. Radius Angle (radians) Arc length 6 inches feet 6 10 meters 30 meters 39. Prve the fllwing identities. a. sin ct cs b. sin cs 1 sin csc c. sin 1 ct d. sin cs sec csc cs sin e. sin sin sin cs f. sin sin tan tan MCPS

15 PART CALCULATOR SECTION A calculatr ma be used n items 40 thrugh 54. Make sure that ur calculatr is in the apprpriate mde (radian r degree) fr each item. Unless therwise specified, answers shuld be crrect t three places after the decimal pint. 40. A ball n a string is swinging back and frth frm a ceiling, as shwn in the figure belw. d Let d represent the distance that the center f the ball is frm the wall at time t. Assume that the distance varies sinusidall with time. When t 0 secnds, the ball is farthest frm the wall, d 160 cm. When t 3 secnds, the ball is clsest t the wall, d 0 cm. When t 6 secnds, the ball is farthest frm the wall, d 160 cm. a. Sketch a graph f the distance as a functin f time. b. Write a trignmetric equatin fr the distance as a functin f time. c. What is the distance f the ball frm the wall at t 5 secnds? d. What is the value f t the first time the ball is 40 cm frm the wall? Yur answer shuld be crrect t three places after the decimal pint. MCPS

16 41. Sara is riding a Ferris wheel. Her sister Kari starts a stpwatch and recrds sme data. Let h represent Sara s height abve the grund at time t. Kari ntices that Sara is at the highest pint, 80 feet abve the grund, when t 3 secnds. When t 7 secnds, Sara is at the lwest pint, 0 feet abve the grund. Assume that the height varies sinusidall with time. a. Write a trignmetric equatin fr the height f Sara abve the grund as a functin f time. b. What will Sara s height be at t 11.5 secnds? Yur answer shuld be crrect t three places after the decimal pint. c. Determine the first tw times, t 0, when Sara s height is 70 feet. Yur answer shuld be crrect t three places after the decimal pint. 4. At Ocean Tide Dck, the first lw tide f the da ccurs at midnight, when the depth f the water is meters, and the first high tide ccurs at 6:30 a.m., with a depth f 8 meters. Assume that the depth f the water varies sinusidall with time. a. Sketch and label a graph shwing the depth f the water as a functin f the number f hurs after midnight. b. Determine a trignmetric functin that mdels the graph. c. Suppse a ship requires at least three meters f water depth is planning t dck after midnight. Determine the earliest pssible time that the ship can dck. 43. Slve the fllwing equatins fr, where a. 3cs 9 7 b. 3sin 7sin Hw man triangles ABC are pssible if A0, b40, and a 10? 45. Given ABC, where 41 A, B58, and c 19.7cm, determine the measure f side b. Yur answer shuld be crrect t three places after the decimal pint. 46. In ABC, a 9, b 1, c 16. What is the measure f B? Yur answer shuld be crrect t the nearest tenth f a degree. 47. Determine the remaining measurements f a triangle with A58, a11.4, b 1.8. Yur answers (sides and angles) shuld be crrect t the nearest tenth. MCPS

17 48. Frm a pint 00 feet frm its base, the angle f elevatin frm the grund t the tp f a lighthuse is 55 degrees. Hw tall is the lighthuse? Yur answer shuld be crrect t three places after the decimal pint. 49. A truck is travelling dwn a muntain. A sign sas that the degree f incline is 7 degrees. After the truck has travelled 1 mile (580 feet), hw man feet in elevatin has the truck fallen? Yur answer shuld be crrect t three places after the decimal pint. 50. The wner f a garage shwn belw plans t install a trim alng the rf. The lengths required are in bld. Hw man feet f trim shuld be purchased? Yur answer shuld be crrect t three places after the decimal pint feet An airplane needs t take a detur arund a grup f thunderstrms, as shwn in the figure belw. Hw much farther des the plane have t travel due t the detur? Yur answer shuld be crrect tp three places after the decimal pint miles 5. Determine the area f triangle ABC if a4, b10, and mc 30. MCPS

18 53. A real estate appraiser wishes t find the value f the lt belw. 160 feet 6 50 feet a. Determine the length f the third side f the lt. b. Find the area f the lt. Yur answer shuld be crrect t three places after the decimal pint. c. An acre is 43,560 square feet. If land is valued at $56,000 per acre, what is the value f the lt? Yur answer shuld be crrect t the nearest cent. 54. Find the area f the quadrilateral belw. Yur answer shuld be crrect t three places after the decimal pint MCPS