Math 2412 General Review for Precalculus Last Updated 12/15/2015

Transcription

1 Math 4 General Review for Precalculus Last Updated /5/05 Give the equation of the horizontal asymptote, if any, of the function.. h(x) = 9x - 9x - 8 6x - 7x f(x) = x + 4 x. h(x) = x - x - 9 9x + 5. g(x) = x + 9 x - 49 Give the equation of the oblique asymptote, if any, of the function. 4. f(x) = x + x - 5 x - 4 x 9. f(x) = x f(x) = x + x + 5x - x. + 6x f(x) = -0x + x + 5x + 8-5x - Graph the function. x 7. f(x) = (x - )(x + ) 0. f(x) = x + x - x - x - 6

2 Solve the equation.. log (4 + x) - log (x - 5) = log 4. log (x + 4) + log (x - ) = 4. x + x - 6 = 0 Solve the equation. Express irrational answers in exact form and as a decimal rounded to decimal places. x 4. = 5 - x Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution (x - ) = 6. The first recorded population of a particular country was 5 million, and the population was recorded as 9 million 8 years later. The exponential growth function A =5e kt describes the population of this country t years since the first recording. Use the fact that 8 years later the population increased by 4 million to find k to three decimal places. 7. Conservationists tagged 70 black-nosed rabbits in a national forest in 009. In 00, they tagged 40 black-nosed rabbits in the same range. If the rabbit population follows the exponential law, how many rabbits will be in the range 9 years from 009? 8. A fossilized leaf contains % of its normal amount of carbon 4. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon The amount of a certain drug in the bloodstream is modeled by the function y = y0 e t, where y0 is the amount of the drug injected (in milligrams) and t is the elapsed time (in hours). Suppose that 0 milligrams are injected at 0:00 A.M. If a second injection is to be administered when there is milligram of the drug present in the bloodstream, approximately when should the next dose be given? Express your answer to the nearest quarter hour. 0. A cup of coffee is heated to 94 and is then allowed to cool in a room whose air temperature is 7. After minutes, the temperature of the cup of coffee is 40. Find the time needed for the coffee to cool to a temperature of 0. Assume the cooling follows Newton's Law of Cooling: U = T + (Uo - T)e kt. (Round your answer to one decimal place.). In a town whose population is 000, a disease creates an epidemic. The number of people, N, infected t days after the disease has begun is given by the function 000 N(t) =. Find the number of +. e t infected people after 0 days. Use a calculator to solve the equation on the interval 0 <. Round the answer to two decimal places.. csc = 5. 5 tan - 4 = 0 Solve the equation on the interval 0 <. 4. sin + sin = 0 5. cos - cos + = 0 6. tan + sec = 7. sin - cos = 0 8. cos - sin = + sin

3 Use a graphing utility to solve the equation on the interval 0 x < 60. Express the solution(s) rounded to one decimal place. 9. cos x + cos x = Simplify the expression. cos 0. + tan + sin Establish the identity.. sec u + tan u = cos u - sin u. sin x - cos x + sin x + cos x = csc x Use the information given about the angle, 0, to find the exact value of the indicated trigonometric function.. tan = 0, < < Find sin( ). 4. csc = - 5, tan > 0 Find cos( ). 4. tan 75 The polar coordinates of a point are given. Find the rectangular coordinates of the point. 4., The rectangular coordinates of a point are given. Find polar coordinates for the point. 44. (-, -) The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, ). 45. x + y - 4x = 0 The letters r and represent polar coordinates. Write the equation using rectangular coordinates (x, y) r = + cos Identify and graph the polar equation. 47. r = - cos 5. tan = 7 4, < < Find tan( ). Find the exact value of the expression. 6. cos sin - + sin sin = 4, 0 < < Find sin. 48. r = 4 sin( ) 8. sin = 4, tan > 0 Find cos. 9. tan =, < < Find tan. Use the Half-angle Formulas to find the exact value of the trigonometric function. 40. sin cos 75

4 Graph the polar equation. 49. r = - cos Use the vectors in the figure below to graph the following vector. 56. u + z Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary i Write the complex number in rectangular form cos + i sin 6 6 Find zw or z as specified. Leave your answer in polar w form. 5. z = + i w = - i Find zw. 5. z = 8 cos + i sin The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. 57. P = (-6, ); Q = (4, -4) Find the quantity if v = 5i - 7j and w = i + j. 58. v + w Find the unit vector having the same direction as v. 59. v = -i - 5j w = cos 6 + i sin 6 Find z w. Write the expression in the standard form a + bi. 54. ( + i)0 Find all the complex roots. Leave your answers in polar form with the argument in degrees. 55. The complex fourth roots of -6 Write the vector v in the form ai + bj, given its magnitude v and the angle it makes with the positive x-axis. 60. v = 5, = Two forces, F of magnitude 60 newtons (N) and F of magnitude 70 newtons, act on an object at angles of 40 and 0 (respectively) with the positive x-axis. Find the direction and magnitude of the resultant force; that is, find F + F. Round the direction and magnitude to two decimal places. 4

5 6. An audio speaker that weighs 50 pounds hangs from the ceiling of a restaurant from two cables as shown in the figure. To two decimal places, what is the tension in the two cables? Find the vertex, focus, and directrix of the parabola. Graph the equation. 69. y = -6x Find the angle between v and w. Round your answer to one decimal place, if necessary. 6. v = 8i + 6j, w = 4i + 9j Find the vertex, focus, and directrix of the parabola. Graph the equation. 70. x - x = y An airplane has an air speed of 550 miles per hour bearing N0 W. The wind velocity is 50 miles per hour in the direction N0 E. To the nearest tenth, what is the ground speed of the plane? What is its direction? State whether the vectors are parallel, orthogonal, or neither. 65. v = 4i - j, w = 4i + j Decompose v into two vectors v and v, where v is parallel to w and v is orthogonal to w. 66. v = i - 5j, w = -i + j 67. An SUV weighing 4900 pounds is parked on a street which has an incline of 0. Find the force required to keep the SUV from rolling down the hill and the force of the SUV perpendicular to the hill. Round the forces to the nearest hundredth. Round your answer to the nearest tenth. 68. Find the work done by a force of 4 pounds acting in the direction of 44 to the horizontal in moving an object 4 feet from (0, 0) to (4, 0). 7. An experimental model for a suspension bridge is built in the shape of a parabolic arch. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower. The towers are both 4 inches tall and stand 40 inches apart. Find the vertical distance from the roadway to the cable at a point on the road 6 inches from the lowest point of the cable. Find the center, foci, and vertices of the ellipse. 7. x + 5y - 4x + 0y + 07 = 0 5

6 7. A bridge is built in the shape of a semielliptical arch. It has a span of 0 feet. The height of the arch 7 feet from the center is to be feet. Find the height of the arch at its center. Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola. 74. x - 4y + 8x + 6y - 4 = Two recording devices are set 4000 feet apart, with the device at point A to the west of the device at point B. At a point on a line between the devices, 00 feet from point B, a small amount of explosive is detonated. The recording devices record the time the sound reaches each one. How far directly north of site B should a second explosion be done so that the measured time difference recorded by the devices is the same as that for the first detonation? 80. Ron throws a ball straight up with an initial speed of 40 feet per second from a height of 7 feet. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height? What is the maximum height of the ball? 8. A baseball player hit a baseball with an initial speed of 70 feet per second at an angle of 40 to the horizontal. The ball was hit at a height of feet off the ground. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height? What is the distance the ball traveled? Find the parametric equations that define the curve shown. 8. Convert the polar equation to a rectangular equation r = 4-4 cos 77. r = 4 + cos 78. r = + sin Graph the curve whose parametric equations are given. 79. x = t, y = t + ; - t 8. Find parametric equations for an object that moves along the ellipse x 9 + y = with the 4 motion described. The motion begins at (0, ), is clockwise, and requires seconds for a complete revolution. Solve the system of equations by substitution. 84. x + 7y = - x + y = 4 6

7 Solve the system of equations by elimination x - y = - x + 4y = A retired couple has $00,000 to invest to obtain annual income. They want some of it invested in safe Certificates of Deposit yielding 6%. The rest they want to invest in AA bonds yielding % per year. How much should they invest in each to realize exactly $0,400 per year? 87. A movie theater charges $8.00 for adults and $5.00 for children. If there were 40 people altogether and the theater collected $7.00 at the end of the day, how many of them were adults? Solve the system of equations. 88. x + y + z = -0 x - 5y - z = 4x + y + z = The Family Arts Center charges $ for adults, $ for senior citizens, and $9 for children under for their live performances on Sunday afternoon. This past Sunday, the paid revenue was $0,408 for 7 tickets sold. There were 40 more children than adults. How many children attended? Solve the system of equations. If the system has no solution, say that it is inconsistent. 90. x - y + 4z = 5x + z = 0 -x + y - 4z = - Solve the system of equations. 9. x + 4y - z = x + 5y - z = 5 x + y - z = 9 Solve the problem using matrices. 9. Find real numbers a, b, and c such that the graph of the function y = ax + bx + c contains the points (-, -4), (, -), and (, -9). 9. Jenny receives $70 per year from three different investments totaling $0,000. One of the investments pays 6%, the second one pays 8%, and the third one pays 5%. If the money invested at 8% is $500 less than the amount invested at 5%, how much money has Jenny invested in the investment that pays 6%? 94. Determinants are used to show that three points lie on the same line (are collinear). If x y x y x y = 0, then the points (x, y), (x, y), and (x, y) are collinear. If the determinant does not equal 0, then the points are not collinear. Are the points (-7, 8), (0, -), and (-4, 7) collinear? Solve the system of equations using Cramer's Rule if it is applicable. If Cramer's Rule is not applicable, say so x - 8y - z = -75 x + y + 7z = 84 7x + y + z = x - y + z = -4 x + z =0 -x + y - z = 6 Compute the product Each matrix is nonsingular. Find the inverse of the matrix. Be sure to check your answer

8 Show that the matrix has no inverse Solve the system using the inverse matrix method. 00. x + 4y - 5z = -8 x + 5y + z = - x + y + z = 5 Write the partial fraction decomposition of the rational expression. x - 0. (x - 4)(x - ) 0. 8x + 7x + 6 (x + )(x + ) 06. A person at the top of a 600 foot tall building drops a yellow ball. The height of the yellow ball is given by the equation h = -6t where h is measured in feet and t is the number of seconds since the yellow ball was dropped. A second person, in the same building but on a lower floor that is 408 feet from the ground, drops a white ball seconds after the yellow ball was dropped. The height of the white ball is given by the equation h = -6(t - ) where h is measured in feet and t is the number of seconds since the yellow ball was dropped. Find the time that the balls are the same distance above the ground and find this distance. Graph the solution set of the system of inequalities or indicate that the system has no solution. 07. x - y -8 x + y 0. x + (x - )(x + x + ) 04. x + x (x + 5) 05. The area of a rectangular piece of cardboard shown is 76 square inches. The cardboard is used to make an open box by cutting a 4-inch square from each corner and turning up the sides. If the box is to have a volume of 6 cubic inches, find the dimensions of the cardboard that must be used. 08. x + y x - y 0 8

9 The sequence is defined recursively. Write the first four terms. 09. a =, a = 5; an = an- - an- Express the sum using summation notation Find the sum Solve.. Suppose you just received a job offer with a starting salary of $7,000 per year and a guaranteed raise of $500 per year. How many years will it be before you've made a total (or aggregate) salary of $,05,000? Find the sum.. 4 k = 5 k+ Determine whether the infinite geometric series converges or diverges. If it converges, find its sum k- k= Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n n = - n n(n + ) = n(n + )(n + ) Expand the expression using the Binomial Theorem. 7. (x + )5 Use the Binomial Theorem to find the indicated coefficient or term. 8. The coefficient of x in the expansion of (x + )5 9

10 Answer Key Testname: GENERAL PRECAL REVIEW. y =. no horizontal asymptotes. y = 0 4. y = x y = x - 6. no oblique asymptote

11 Answer Key Testname: GENERAL PRECAL REVIEW 0.. {8}. {4} ln. ln 4. ln ln ln 5 5. {0.95} ,840 rabbits 8. 6,45 9. :45 P.M minutes. 77 people. {0.4,.7}. 0.67, ,, 5. 0,, 5 6. {0} 7. 4, 4, 5 4, ,, 7 6, , 80.0, sec. sec u + tan u =. cos u + sin u cos u = + sin u cos u = + sin u cos u - sin u - sin u = - sin u cos u( - sin u) = sin x - cos x + sin x sin x[( + cos x)+( - cos x)] = = sin x + cos x ( - cos x)( + cos x) - cosx = sin x = csc x. sinx cos u cos u( - sin u) = cos u - sin u

12 Answer Key Testname: GENERAL PRECAL REVIEW , 44., r = 4 cos 46. y = 5-0x 47. cardioid

13 Answer Key Testname: GENERAL PRECAL REVIEW 48. rose with four petals (cos 60 + i sin 60 ) 5. - i 5. 4 cos + i sin 5. 8 cos + i sin (cos 45 + i sin 45 ), (cos 5 + i sin 5 ), (cos 5 + i sin 5 ), 6(cos 5 + i sin 5 ) v = 0i - 5j

14 Answer Key Testname: GENERAL PRECAL REVIEW 59. u = - i - 5 j 60. v = 5 i + 5 j 6. Direction: ; magnitude: 9.0 N 6. Tension in right cable: 5.90 lb; tension in left cable: 4.59 lb mph; N5.7 W 65. Neither 66. v = 5 i j, v = i j lb and lb ft-lb 69. vertex: (0, 0) focus: (-4, 0) directrix: x = vertex: (6, 5) focus: (6, 8) directrix: y = in. 7. (x - 6) 5 + (y + ) = center: (6, -); foci: (7.7, -), (4., -); vertices: (8., -), (.8, -) ft 4

15 Answer Key Testname: GENERAL PRECAL REVIEW 74. center at (-4, ) transverse axis is parallel to x-axis vertices at (-6, ) and (-, ) foci at (-4-5, ) and (-4 + 5, ) asymptotes of y - = - (x + 4) and y - = (x + 4) ft 76. y = 4x x + 6y + 4x - 44 = x + 8y + 4y - 44 = x = 0, y = -6t + 40t sec,.5 sec, feet 8. x = 0.t, y = -6t + 09.t sec,.46 sec, feet 8. x = t +, y = -t + 5; 0 t 8. x = sin ( t), y = cos ( t), 0 t 84. x =, y = -; (, -) 85. x =, y = 8; (, 8) 86. $40,000 at % and $60,000 at 6% adults 88. x = -4, y = -5, z = ; (-4, -5, ) children 90. inconsistent 9. x = -z - 5, and y = z +, where z is any real number or {(x, y, z) x = -z - 5, and y = z +, where z is any real number} 9. a = -, b = -, c = 9. $ Yes 95. x =, y = 9, z = 8; (, 9, 8) 96. not applicable 5

16 Answer Key Testname: GENERAL PRECAL REVIEW x = 5, y = -, z = ; (5, -, ) 0. x x x x (x + ) 5 x x + x + x x + -0x - 0 x (x + 5) in. by 46 in sec; 404 ft

17 Answer Key Testname: GENERAL PRECAL REVIEW a =, a = 5, a = -, a4 = k = k k , years Converges; 5. When n =, the left side of the statement is n = =, and the right side of the statement is - n = - = - =, so the statement is true when n =. Assume the statement is true for some natural number k. Then, k + k+ = - k + k+ = - k - = - k+. So the statement is true for k +. Conditions I and II are satisfied; by the Principle of Mathematical Induction, the statement is true for all natural numbers. 7

18 Answer Key Testname: GENERAL PRECAL REVIEW 6. First we show that the statement is true when n =. ( + )( + ) For n =, we get = =. This is a true statement and Condition I is satisfied. Next, we assume the statement holds for some k. That is, k(k + )(k + ) k(k + ) = is true for some positive integer k. We need to show that the statement holds for k +. That is, we need to show that (k + )(k + )(k + ) (k + )(k + ) =. So we assume that k(k + ) = both sides of the equation k(k + ) + (k + )(k + ) = = = = k(k + )(k + ) k(k + )(k + ) k(k + )(k + ) is true and add the next term, (k + )(k + ), to + (k + )(k + ) + (k + )(k + ) k(k + )(k + ) + (k + )(k + ) (k + )(k + )(k + ) Condition II is satisfied. As a result, the statement is true for all natural numbers n. 7. 4x5 + 80x x + 70x + 40x