MATH 1 Final Exam Review Precalculus Mathematics for Calculus, 7 th ed., Stewart, et al 5.1 1. Mark the point determined by 6 on the unit circle. 5.3. Sketch a graph of y sin( x) by hand. 5.3 3. Find the amplitude, period, and horizontal shift of the graph of ycos3x. 6 5.3, 5.. Graph each function by hand: a. y cos(3x) b. ysin3x c. ytan x 3 5.3 5. Write the equation of the graph in the form y asin k( x b). y x 1 5.5 6. Evaluate sin 1. 3 5.5, 6. 7. Evaluate exactly. a. sin 1 sin. 1 b. cossin. 3 6.1 8. Find one positive and one negative angle coterminal with: a. 37 b. 3. 7 6.1 9. Convert 3 a. radians to degree measure. b. 196 to radian measure. Round to the nearest hundredth. 6.1 10. The roller on a treadmill has a diameter of 3 inches. How many revolutions per minute must it make in order for Josh to run at a speed of 10 miles per hour? (1 mile = 5,80 ft)
6. 11. If c represents the length of the hypotenuse, solve the right triangle with a 6 and B 71. 8 6. 1. If sec and is an acute angle, find sin exactly. 3 6. 13. You are a tour guide lost in the Rockies and have just come to a gorge. You estimate the angle of elevation of the top of the useful part of a tree that is 5 feet away from you to be 70. You also estimate that the gorge is 50 feet wide and want to allow an extra 6 feet on each end for safety. Will the useable part of the tree be long enough to bridge the gorge if you cut the tree? (Note: the tree just happens to be standing six feet from the edge of the gorge.) 6. 1. You are estimating the height of a lighthouse surrounded by water. When you first observe it from the shoreline, the angle of elevation seems to be about 16. After walking inland 00 feet in a direct line with the first observation point, the angle of elevation is about 10. Find the height of the lighthouse if the ground is level. 6.3 15. Write cot in terms of sin given that is in Quadrant II. 6., 7. 16. Simplify so that the answer contains no trig functions. 1 b 1 a 1 a. tan cos b. sin tan cos 7 3 6.5 17. Solve the triangle in which a. a, B 7, C 10 b. a 10, B 7, b 6 6.5 18. A pilot is about to land on the 8500 foot runway in Los Angeles. The angles of depression from the plane to the ends of the runway are 16 and 17.. Find the distance the plane must fly before it touches down. 6.6 19. Solve the triangle in which a 0, b 7, c 16. 6.6 0. The distance between two planes is calculated by using LORAN to find the position of each plane relative to the tracking station. If one plane is 30 miles from the station, a second is 7 miles, and the angle between the two distances is 37, find how far apart the planes are. 7.1 1. Simplify cos. 1 sin 7.1. Verify the trig identity: sec x tan x 1 tan x. 7. 3. Write an equivalent expression using only one trig function; evaluate exactly. sin(61 )cos(7 ) cos(61 )sin(7 )
7.. Find cos(105 ) exactly. (Hint: use an addition or subtraction formula) 7. 3 5. Given that sin A ; A and tan B ; B, find cos( A B) 5 3 exactly. 7.3 1 3 6. If cos ;, find a. sin( ) b. cos 3 7. 7. Algebraically find all exact solutions in radians. a. 3sec 0 b. 7.5 8. Find all real solutions. sin sin 1 c. cos x x 0. sin cos 1 0 8.1 9. Convert the point 3, from polar to rectangular coordinates. 3 8.1 30. Convert the point (, ) from rectangular to polar coordinates. 5 8.1 31. a. Graph the point 3, in a polar coordinate system. 6 5 b. State polar coordinates for the point 3, using a positive value of r. 6 8.1 3. Convert r 6sin to a rectangular equation. 8. 33. Convert x ( y 5) 5 to a polar equation. 8. 3. Graph r 5sin(3 ). 8. 35. a. Graph the curve x cos t, y sin t, 0 t. b. Find a rectangular-coordinate equation by eliminating the parameter. 8. 36. A ball is thrown with an initial speed of 9 feet per second from a height of 7 feet at an angle of 50 to the horizontal. You may assume that the ground is level. a. Find the height of the ball after seconds. b. Determine the horizontal distance the ball travels. Use the parametric equations x v cos t, y h v sint 16t 9.1 37. Given u = 1,, v =, 3, find u 3v. 0 0 9.1 38. Find the direction of the vector 3,. 9.1 39. The speed of an airplane is 00 mi/hr relative to the air. The wind is blowing due west with a speed of 35 mi/hr. In what direction should the airplane head in order to arrive at a point due south of its location? 9. 0. Find the angle between 3, and 7, 1.
9. 1. A horse pulls a wagon horizontally by exerting a force of 0 pounds on the handle. If the handle makes an angle of 0 with the horizontal, find the work done in moving the wagon 300 feet. 10.1. Solve the system. x 3y 3 3x y 1 10.1 3. A chemist needs 100 ml of a 7% solution of sulfuric acid, but only has 10% and 0% solutions available. Find how much of each should be used. 10.3. Michigan State University and the University of Michigan scored a total of 50 points during their 015 football game. There were 1 scoring plays: touchdowns, extra point kicks, and field goals at 6, 1, and 3 points respectively. There were twice as many touchdowns as field goals. Find the number of each scoring play for the game. 10.3 5. Solve each system. If the system is dependent, state the complete solution. a. x y 3z 1 5x y z 5 3x 3y z 1 b. 3x y z x y z 1 7x y 5 c. 3x y z x y z 1 7x y 11.1 6. A parabolic reflector for a telescope is six inches in diameter. It is ground so that its vertex is 0.15 inch below the rim (on the concave side). a. Insert an xy coordinate system with vertex at the origin and find the equation of the parabola. b. How far is the focus from the vertex? 11. 7. Find the vertices, foci, and eccentricity of the ellipse with equation x 9y 36. 11. 8. Write the equation of an ellipse centered at the origin containing the point (,0) and with one focus at (0,3). 11. 9. Find the vertex, focus, and directrix of the parabola x 6x y 1 0. Draw the graph. 11. 50. Find the center, vertices, foci, and asymptotes of 5x y 00x 8y 96 0. Draw the graph. 1.1 51. List the first four terms and the 100 th n term of the sequence an. n 1 3 5 6 1.1 5. Find a formula for the nth term of the sequence 1,,,,,... 5 10 17 6 1.1 53. a. Find the fourth partial sum of the sequence 3,1, 7,8,75, b. Evaluate the sum 3k. k 1
1.1 5. Write sigma notation for the series 3 5 7 9 11. 3 5 6 1.1 55. Find the first five terms of the sequence where a1 3, an 1 an 1 for n 1. 1. 56. Find the st 31 term of the sequence 8, 5,,1,,... 1. 57. Find a 3 a for the arithmetic sequence which has a 7 7 and a 0 159. 1. 58. Evaluate the sum 3 k 1 k 1 3 1. 59. Each row in an auditorium seats more people that the preceding row. There are 5 seats in the first row and 8 rows. Find the seating capacity of the auditorium. 1.3 60. Find the 10th term of the geometric sequence 81 1,18,7,,... 1.3 61. Find a formula for the nth (general) term of the sequence, 1, 6, 3, 1.3 6. Find the sum. 1 k 1 k 1 8 1.3 63. Determine whether the infinite geometric series divergent. If it is convergent, find its sum. k 1 8 is convergent or k 1 1.3 6. You have a job offer with a salary of $50,000 for the first year. If you accept the job and get a.5% raise every year for the next 30 years, what are your total earnings for the entire 30 years? 1.6 65. Expand and simplify: a. b. 1 x 5 x 3) using the binomial theorem ( using Pascal s triangle. x 1.6 66. Find the tenth term in the binomial expansion of x. 1.6 67. The probability that a softball player hitting.50 will get a hit 3 times in the next eight at bats is the term in which.5 has an exponent of 3 in the binomial expansion of 8 (0.5 0.75). Find that probability. 17