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l parts are adequately supported. Problems like these can be modeled and solved by using vectors. See Exercise 50 in Section 10.6. 616 Chapter Outline 10.1 The Law of Cosines 10.2 The Law of Sines 10.3 The Complex Plane and Polar Form for Complex Numbers 10.4 DeMoivre’s Theorem and nth Roots of Complex Numbers 10.5 Vec... |
far is the boat from its starting point? turn and goes for another 45° 3 6 Start 26. A plane flies in a straight line at 400 miles per 15° hour for 1 hour and 12 minutes. It makes a turn and flies at 375 miles per hour for 2 hours and 27 minutes. How far is it from its starting point? 27. The side of a hill makes an a... |
e Law of Sines to find b. c sin C 12 a sin A 5 sin 37° sin A sin A 5 sin 37° 12 sin A 0.2508 A 14.5° or A 180° 14.5° 165.5° 630 Chapter 10 Trigonometric Applications C 37°, Because that is impossible, then the sum of angles A, B, and C would be is the only possible 14.5° A 165.5° If and greater than 180°. measure of an... |
south of airport A. How far has the plane traveled? 267°. Finally, 53. Charlie is afraid of water; he can’t swim and refuses to get in a boat. However, he must measure the width of a river for his geography 14.6 m 19 m 50° Section 10.3 The Complex Plane and Polar Form for Complex Numbers 637 56. A triangular lot has s... |
i sin 7p 12 b i sin p 8 b 12 cos a 3p 8 i sin 3p 8 b 7 2 a cos p 4 i sin p 4 b 35. 12 cos a 11p 12 i sin 11p 12 b 36. 37. 8 cos a 5p 18 cos 4 a p 9 6 cos a 7p 20 4 cos a p 10 i sin i sin i sin i sin 5p 18 b p 9 b 7p 20 b p 10b Section 10.3 The Complex Plane and Polar Form for Complex Numbers 643 254 38. cos a 26 cos a... |
h positive integer n, there are n distinct nth roots of unity, which have the following form. cos 2kp n i sin 2kp n , for k 0, 1, 2, p , n 1. Section 10.4 DeMoivre’s Theorem and nth Roots of Complex Numbers 649 Example 5 Find Roots of Unity Find the cube roots of unity. Solution Apply the formula for roots of unity wit... |
which is also called its norm. H Magnitude The magnitude (or norm) of the vector 2a2 b2. v 7 7 v a, b I H is CAUTION Example 2 Find Components and Magnitude of a Vector The order in which the coordinates of the initial point and terminal point are subtracted to a, b obtain I H cant. For the points x2, y22 Q P and : x1,... |
16i 22j 1 2 ■ Direction Angles ai bj v H a, b I If u determined by the standard position angle between terminal side is v, as shown in Figure 10.6-1. is a vector, then the direction of v is completely whose 360°, and 0° b v y θ x 〈a, b〉 a Figure 10.6-1 Section 10.6 Applications of Vectors in the Plane 663 The angle def... |
nd air speed of the plane. 47. A plane is flying in the direction 200° with an air u speed of 500 miles per hour. Its course and and 450 miles per hour, ground speed are respectively. What is the direction and speed of the wind? 210° 48. A river flows from east to west. A swimmer on the south bank wants to swim to a po... |
v2 b a v 26 20 a b 1 4i 2j 26 5 i 13 5 j 2 projuv v u u2 b a u 26 73 a b 1 8i 3j 208 73 i 78 73 j 2 ■ The projection vectors from Example 5 are shown in Figure 10.6.A-6. y projuv v projvu u x Figure 10.6.A-6 676 Chapter 10 Trigonometric Applications Projections and Components Recall from Section 10.6 that 1 v v is a u... |
. . . . 657 Section 10.6 Unit vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Linear combination of i and j . . . . . . . . . . . . . . . 662 Direction angle of a vector . . . . . . . . . . . . . . . . . 663 Section 10.6.A Dot product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 An... |
re 10.C-1 Euler’s formula can be used to define complex powers of e, that is, xiy e x e cos y i sin y The equation terms of a real power of e and the cosine and sine of a real number. 2 1 defines a complex power of e in ez e xiy e cos y i sin y iy e e 1 2 x x ■ xiy. e Example 3 Complex Power of e Find the exact value a... |
) The shape of the room and two parabolic dishes carry the quietest sound from one focus to the other. The width of the ellipse is 13 feet 6 inches and the length of the ellipse is 47 feet 4 inches. Assume that the ellipse is centered at the origin. Find its equation, sketch its graph, and locate the foci. Solution Bec... |
are asymptotic to the dashed lines. x ± a ± 2 y (0, 3) y = 3 2 x y y = − x3 2 y = x3 2 − x2 4 y2 9 = 1 (–2, 0) (2, 0) (– 13, 0) (–2, 0) (2, 0) ( 13, 0) x x (0, –3) y = − x3 2 Figure 11.2-2 Locate the foci by using the formula c 222 33 14 9 113 3.6 c 2a2 b2 with a 2 and b 3. Therefore, the foci are the right in Figure ... |
e 11.3-2 2 1 distance from distance from x, y 1 x, y 1 x 0 to to 2 2 2 0, p 2 1 0, p 2 1 y p 2 1 2 distance from 1 distance from 1 2 2 2 x x x, y x, y to y p x, p to 710 Chapter 11 Analytic Geometry Square both sides of the equation and simplify. x 0 1 2 y˛ x 2py p2 02 y˛ x˛ 2 4py 1 y p 2 2 2 1 2 2 2py p2 standard form... |
x2 9 y 2 36 1 shifted 5 units to the right Section 11.4 Translations and Rotations of Conics 717 and 4 units downward, as shown in Figure 11.4-1. • The center of the ellipse is 5, 4 . 2 1 • The major axis lies on the vertical line • The minor axis is on the horizontal line y 4. x 5. y 8 4 −8 −4 0 4 8 x −4 −8 Figure 11.... |
etermine its exact location by radio, as illustrated in the following example. Example 10 LORAN Application Three LORAN radio transmitters Q, P, and R are located 200 miles apart along a straight line and simultaneously transmit signals at regular intervals. These signals travel at a speed of 980 feet per microsecond, ... |
e shows that 1 2 u b 2 2 1 u b cos u cos b sin u sin b r cos b x r cos r˛1 cos u 2 1 u cos u v sin u y r sin u b r sin b 2 1 2 sin u A similar argument with sine leads to the following result. 1 and the addition identity for 2 The Rotation Equations If the xy coordinate axes are rotated through an angle to produce the ... |
in Figure 11.5-6, with r positive. Since r is the distance the distance formula shows that from (0, 0) to x, y , 1 2 r 2x2 y2 Also, by the definitions of the trigonometric functions in the coordinate plane, cos u x r sin u y r tan u y x These equations can be used to obtain the relationship between polar and rectangula... |
are constants such that r a sin u b cos u that the graph of Hint: Multiply both sides by r and convert to rectangular coordinates. ab 0 , show is a circle. x, y 50. Critical Thinking Prove that the coordinate r 6 0. conversion formulas are valid when P has coordinates and verify that the point Q with rectangular coordi... |
a 3 , p 2 b and 10 7 , 3p 2 b . a ■ Graphing Exploration Find a viewing window that shows a complete graph of the hyperbola in Example 3. Example 4 Polar Equations of Conic Sections (6, π) (3, 0) Find a polar equation of the ellipse with one focus at (0, 0) and vertices (3, 0) and 6, p . 1 2 Figure 11.6-8 Solution Bec... |
elation, and sketch the graph. x y˛ 2, write a set of parametric equations to Then write the parametric equations of the following successive transformations of the parent relation, and sketch each graph. 758 Chapter 11 Analytic Geometry y 4 2 0 −2 −4 a. a horizontal stretch by a factor of 5 b. then a horizontal shift ... |
ing one equation for t and substituting the result into the other equation. In Exercises 28–30, find a parameterization of the given curve. Confirm your answer by graphing. 28. line segment from Exercise 27. 14, 5 1 to 1 2 5, 14 2 Hint: See 2 (within one degree) needed so that the ball travels at least 150 feet. 764 Ch... |
eads to the following conclusion. Hyperbolas with center at (c, d) have the following parameterizations. Equation (x c)2 a2 (y d)2 b2 1 (y d)2 a2 (x c)2 b2 1 Parameterization x a sec t c y b tan t d (0 t 2P) x b tan t c y a sec t d (0 t 2P) Example 3 Parameterization of a Conic Identify the conic section whose equation... |
cos t cos 8t and y 8 sin t sin 8t, 0 t 2p Section 11.7 In Exercises 78–81, sketch the graph of the curve whose parametric equations are given, and by eliminating the parameter, find an equation in x and y whose graph contains the given curve. 78. x 3 cos t, y 5 sin t, 0 t 2p 79. x cos t, y 2 sin2 t, 0 t 2p 80. x et, y ... |
ally Solving systems of equations by graphing often gives approximate solutions, while algebraic methods produce exact solutions. Furthermore, algebraic methods are often as easy to use as graphical methods. Two common algebraic methods are substitution and elimination. Substitution Method Solving Systems with Substitu... |
electric heat? of solar heat? c. In what year will the total cost of the two heating systems be the same? Which is the cheapest system before that time? Section 12.1 Solving Systems of Equations 789 38. One parcel of land is worth $100,000 now and is increasing in value at the rate of $3000 per year. A second parcel i... |
ced row echelon form matrices • Solve applications by using matrices • Solve application s using systems. It is often convenient to use an array of numbers, called a matrix, as a method to represent a system of equations. For example, the system is written in matrix form as x 2y 2 2x 6y 2 1 2 a 2 6 2 2b In this shortha... |
x y 2z 0 7x y 3z 2 x y 5z 6 3x 3y z 10 x 3y 2z 5 20. 3x y z 6 x 2y z 0 In Exercises 21–35, solve the system by any method. 21. 22. 23. 25. 27. 29. 31. 32. 11x 10y 9z 5 x 2y 3z 1 3x 2y z 1 x 2y 3z 4w 8 2x 4y z 2w 3 5x 4y z 2w 3 x y 3 5x y 3 9x 4y 1 x 4y 13z 4 x 2y 3z 2 3x 5y 4z 2 4x y 3z 7 x y 2z 3 3x 2y z 4 x y z 0 3x ... |
j The following diagram shows how the dimensions of the product matrix are related to the dimensions of the factor matrices: > m n matrix > 1 2 > > n p matrix 1 2 > m p matrix > 1 2 CAUTION Before finding the entries of a product matrix, check the dimensions of the factor matrices to make sure that the product is defin... |
of the system’s constants. Thus, the system can be solved AX B by solving the corresponding matrix equation. Solving the equation means finding the entries of the matrix X, which are the variables of the system. Section 12.4 Matrix Methods for Square Systems 815 Identity Matrices and Inverse Matrices To solve a matrix ... |
nits of mix T each day. Horses receive 8.1 units of mix R, 2.9 units of mix S, and 5.1 units of mix T each day. Bears receive 1.3 units of mix R, 1.3 units of mix S, and 2.3 units of mix T each day. If 16,000 units of mix R, 28,000 units of mix S, and 44,000 units of mix T are available each day, how many of each type ... |
inequality true, then the solution includes the region on the side of the line containing the test point. • If the coordinates of the test point make the inequality false, then the solution includes the region on the side of the line that does not contain the test point. The test-point method can be used to solve any ... |
. . . . . . . . . . . . . . . 779 Solution of a system. . . . . . . . . . . . . . . . . . . . . . 780 Number of solutions of a system . . . . . . . . . . . . 782 Substitution method. . . . . . . . . . . . . . . . . . . . . . 782 Elimination method . . . . . . . . . . . . . . . . . . . . . . 783 Inconsistent system . .... |
the x 4, 2 2x 2 –10 10 form x 3 6x x 2 2 10x 8 A x 4 Bx C 2 2x 2 x . –10 Figure 12.C-4 1 2 1 x Multiplying both sides of the equation by the common denominator, x 4 , and collecting like terms yields x 4 2 Cx 4Bx 4C 2A 4C 2 2x 2 2 2 2x 2 x 2 A x 1 Ax 2 2Ax 2A Bx 2A 4B C Bx C A B 1 x Because there is no sponding coeffi... |
e of retirement for all people in the U.S. d. the heights of all adult women in the U.S. Solution a. The last four digits of a phone number are assigned randomly, so all digits have about the same frequency. The distribution is uniform. b. In a typical corporation, most employees earn relatively low salaries while a fe... |
f the fall semester data. 31. Compare the shape of the two data sets. Which type of distribution do you think best describes these data sets? What might explain the differences in these data sets? 32. Create a histogram with a class interval of 10 for the data below. 68, 84, 59, 72, 62, 76, 61, 63, 68, 56, 70, 79, 54, ... |
the square root of the variance to find the standard deviation: s 27.44 2.73 Population versus Sample If data is taken from a sample instead of the instead of n when averentire population, it is common to divide by aging the squared deviations. The result is called the sample standard deviation and is denoted by s. For... |
et affected if a constant k is added to each value? 54. Critical Thinking How is the standard deviation of a data set affected if a constant k is added to each value? 55. Critical Thinking How is the mean of a data set affected if each value is multiplied by a constant k? 56. Critical Thinking How is the standard devia... |
hich is a reasonable estimate of the expected value. Average sum of two number cubes 7.3 6.98 7.26 7.005 6.978 Number of trials 100 200 300 400 500 Figure 13.3-3 870 Chapter 13 Statistics and Probability To calculate the expected value of a random variable from a probability distribution, multiply each value by its pro... |
f the probability of an event is P E 2 1 number of trials with an outcome in E n Example 1 Experimental Estimate of Probability An experiment consists of throwing a dart at the target in Figure 13.4-1. Suppose the experiment is repeated 200 times, with the following results: red yellow blue 43 86 71 Write a probability... |
he number of permutations can be written using factorials as 26 25 24 26 25 24 23 22 p 3 2 1 23 22 p 3 2 1 26! 23! In the case where order is not important, the number of combinations can be written as 26 25 24 3 2 1 26! 3 2 1 1 23! 2 26! 3! 23! Section 13.4 Determining Probabilities 881 Permutation and combination for... |
r outcome is preferred. Section 13.4.A Excursion: Binomial Experiments 885 The essential elements in a binomial experiment are given below. A set of n trials is called a binomial experiment if the following are true. 1. The trials are independent. 2. Each trial has only 2 possible outcomes, which may be designated as s... |
hree normal curves, with m and s labeled. y 1 0.8 0.6 0.4 0..5 x −4 −2 0 2 4 6 8 10 Figure 13.5-1 The normal curve with a mean of 0 and standard deviation of 1 is called the standard normal curve. The equation of the standard normal curve is y 1 12p x 2 e 2 y 0.4 0.2 −4 −3 −2 −1 0 1 2 3 4 Figure 13.5-2 x The standard n... |
ook two national standardized tests while applying for college. On the first test, m 32 and s 6. the second, on which test did he do better? m 475 and If he scored 630 on the first test and 45 on and on the second test, s 75, 17. Four students took a national standardized test for which the mean was 500 and the standar... |
bability that the spinner lands on white 3 times. 28. Suppose the experiment is repeated 25 times, with the following results: WRW, RRW, WWW, WWW, WWW, WWW, WWR, RWW, WWW, WRW, WWW, WWR, WWW, RWW, WWW, RWR, WWW, WRW, WWW, RWW, WWW, WWW, WRW, WWW, WWR Write a probability distribution of the random variable, based on the... |
mathematical problems involve the behavior of a function at a particular value: What is the value of the function f when x c? x 1 2 rather than at The underlying idea of limit, however, is the behavior of the function near x c You have dealt with limits informally in previous chapters, but this section will discuss li... |
12. lim xS2 x5 32 x3 8 13. lim xS3 x2 x 6 x2 2x 3 14. lim xS1 x2 1 x2 x 2 15. lim xS1 x3 1 x2 1 17. lim xS0 tan x x x3 19. lim xS0 x tan x x sin x 21. lim xS3 2x 23 x 3 23. lim xS0 x ln 0 x 0 16. lim xS2 x2 5x 6 x2 x 6 18. lim xS0 tan x x sin x 20. lim xS0 x sin 2x x sin 2x 22. lim xS25 2x 5 x 25 24. lim xS0 e2x 1 x 2... |
less than or equal to a given number x. See Section 3.1. Use a calculator as an aid in analyzing these problems. 30. For x h 1 exists. 2 31. For x g 1 exists. 2 x 3 4 x 4 3 , find lim xS2 h 1 x 2 , if the limit x x 3 4 , find lim xS2 g 1 x 2 , if the limit 3 x 4 2 x x 3 4 , find lim xS3 r 1 x 2 , if the limit 32. For r... |
no matter what positive number lim xS5 you specify in measuring how close you want f(x) to be to 12, you must be able to find how close x must be to 5 in order to guarantee that f(x) is that close to 12. [3] 2 1 e x d should be to 12 will be denoted by the Greek letter Hereafter, the small positive number you specify i... |
c. c, f ˛1 1 22 (c, f(c)) y c a. y c. Section 14.4 Continuity 937 (c, f(c)) c y b. y x x x x c c Figure 14.4-2 d. Figure 14.4-2 shows that a function is discontinuous, that is, not continx c. uous, at if the graph has a break, gap, hole, or jump when x c Calculators and Discontinuity When a calculator uses “connected”... |
ncil from the paper. As suggested in Figure 14.4-7, there is no way that this can be done unless the graph crosses the horizontal line y k, to the point a, b 3 b 1 6 k 6 f where f a, f 22 22 Section 14.4 Continuity 945 (b, f(b)) (c, f(c)) = (c, k) y f(b) k f(a) (a, f(a)) x a c b Figure 14.4-7 The first coordinate of th... |
written xS f(x) L. lim x 6 a x 7 a ” with “ Limits as x approaches negative infinity are defined analogously by replacing “ ”, “increasing” with “decreasing”, and “larger and larger positive” with “smaller and smaller negative” in the preceding definition. These definitions are informal because such phrases as “arbitr... |
ind lim xSq 4x 3x 2x 2x 1 . Section 14.5 Limits Involving Infinity 959 48. Critical Thinking Let f x 1 2 be a nonzero polynomial x g with leading coefficient a, and let nonzero polynomial with leading coefficient c. Prove that be a 2 1 a. if degree f b. if degree f x 1 2 x 2 1 6 degree g x 1 2 , then lim xSq 0. degree ... |
easing function of time: 2 To find the total distance traveled by a moving object over the time individe the interval into n equally spaced times terval a 6 t 6 b, 1 t v f t0 a, t1, p , tn b, where each time interval is ¢t b a n in duration. ti2 , The velocity at the beginning or end of each time interval is given by s... |
3y3 2 2xy3 b3 21 45y2 3 2 b2 3b 2 21 2 2 1 5d 2 4 2d 3d 1 2 1 3 c4d5c 2x 2 2 1 2 2y 2 1 3y2 2 3 3x 2 2y3 3 1 2 3x 2y4 0 2 In Exercises 39–42, express the given number as a power of 2. 39. 41. 1 1 64 2 2 24 16 2 3 2 40. 3 1 8b a 42 16 b In Exercises 43–60, simplify and write the given expression without negative expone... |
y learning to recognize multiplication patterns that appear frequently. Here are the most common ones. Quadratic Factoring Patterns Difference of Squares Perfect Squares u2 v2 (u v)(u v) u2 2uv v2 (u v)2 u2 2uv v2 (u v)2 Example 2 a. b. c. 2, 3y x2 9y2 x2 9y2 y2 7 y2 36r2 64s2 can be written x 3y x 3y 21 2 17 B A A 2 8... |
50. u3 v3 u2 v2 u2 uv v2 u v 51. c d 2 2 1 c2 d2 cd 52 53. 1 x2 1 x 1 y2 55. 3 6 y 1 1 y 1 1 3x 5 6x2 1 4y 1 y 1 x2 57. 59 54. 56. 58. 60. 33. 35. 37. 39. 40. 41. 42. 3x 9 2x 8x2 x2 9 36. 4x 16 3x 15 2x 10 x 4 5y 25 3 y2 y2 25 38. 6x 12 6x 8x2 x 2 u u 1 u2 1 u2 t2 t 6 t2 6t 9 t2 4t 5 t2 25 2u2 uv v2 4u2 4uv v2 8u2 6uv ... |
lies on the positive x-axis. 47. Show that the midpoint M of the hypotenuse of a right triangle is equidistant from the vertices of the triangle. Hint: Place the triangle in the first quadrant of the plane, with right angle at the origin so that the situation looks like the figure. In Exercises 29–36, find the equation... |
since 8 5 13 2 . Thus, the ninth term is 13 8 b a x5y8. Section B.1 The Binomial Theorem 999 Example 5 Find the ninth term of the expansion of 2x2 a 13 41y 16 b . Solution We shall use the Binomial Theorem with n 13 and with 2x2 in place of x and in place of y. The remarks on the previous page show that the 41y 16 nint... |
hat is, we assume that n k 1, that is, that We begin with the induction hypothesis:* of this inequality by 2 yields: 2k 7 k. Multiplying both sides 3 2 2k 7 2k 2k1 7 2k. Since k is a positive integer, we know that of the inequality we have k 1, k 1. Adding k to each side k k k 1 2k k 1. Combining this result with inequ... |
34. 2n 6 n! for all n 4 35. Let n be a positive integer. Suppose that there are three pegs and on one of them n rings are stacked, with each ring being smaller in diameter than the one below it (see the figure). We want to transfer the stack of rings to another peg according to these rules: (i) Only one ring may be mo... |
ing example outlines the procedure for graphing an equation. Example 1 Using Technology to Graph a Function Use technology to graph the relation 2u3 8u 2v 4 0. Solution Step 1 Choose a preliminary viewing window. If it is unknown where the graph lies in the plane, start with a viewing . This window setting is window wi... |
en, or it may require graphing if the calculator display is misleading. Calculator Exploration 1. Tick Marks a. Set Xscl 1 so that adjacent tick marks on the x-axis are one unit apart. Find the largest range of x values such that the tick marks on the x-axis are clearly distinguishable and appear to be equally spaced. ... |
on that represents the type of graph desired and press ENTER. • Enter the names of the lists that contain the x-data and the y-data. Choose the graph type and Mark symbol. Option GS.D. represents a scatter plot. • GRAPH Casio 9850 STAT EXE GPH SET • Select the graph number: GPH1, GPH2, or GPH3. • Highlight Graph Type a... |
the angle through which the point rotates over time (p. 440) approach infinity Output values of a function that get larger and larger without bound as input values increase are said to approach infinity. (p. 201) arc an unbroken part of a circle (p. 434) arc length the length of an arc, which is equal to the radius tim... |
jacent side length to the length of the hypotenuse. (p. 416) cotangent ratio For a given acute angle u triangle, the cotangent of equal to the reciprocal of the tangent ratio of the given angle. (p. 416) is written as cot and is u in a right u coterminal angles angles formed by different rotations that have the same in... |
type of relation in which each member of the domain corresponds to one and only one member of the range (p. 7) A function consists of a set of inputs called the domain, a rule by which each input determines one and only one output, and a set of outputs called the range. (p. 142) function notation There is a customary m... |
2 2bc cos A, standard notation, b2 a2 c2 2ac cos B, (p. 617) and c2 a2 b2 2ab cos C. Law of Sines For any triangle ABC in standard nota- tion, a sin A b sin B c sin C . (p. 625) Laws of Exponents For any nonnegative real numbers c and d and any rational numbers r and s, crcs crs, c 0 cr dr 2 2 1 r 1 c cr cr cs d 0 r cr... |
tion origin the point of intersection of the axes in a coordinate system (p. 5) origin symmetry A graph is symmetric with respect x, y to the origin if whenever x, y is also on it. (p. 187) 1 orthogonal vectors perpendicular vectors; vectors u and v such that is on the graph, then u v 0 (p. 673) 2 2 1 outlier from the ... |
of the substance (p. 352) x h, corresponds to when the decay began, where P is the initial amount of the sub- 0.5 2 1 x 0 1 the four regions into which a coordinate quadrants plane is divided by its axes, usually indicated by Roman numerals I, II, III, and IV (p. 5) radiocarbon dating a process of determining the age ... |
cs calculators (p. 84) 10 x 10 listed in the ZOOM menu of most and standardize (data) normally distributed in order to match the standard normal curve (p. 893) to adjust the scale of data that is stem plot a display of quantitative data in a tabular format consisting of the initial digit(s) of the data values, called s... |
output of 1 2 ; 0 of 5 2 ; and 3 of 5 2 . 2; 3 of 0; 2 of 1; 1 of 2.5; and 0 29. 1 of is output of 1.5. 31. 1 is output of 2; 1; 1 of 0; 1 of 3 1 2 2, 3 (approximately) of ; and 1.5 of 1. 33. a. 3, 4 4 3 d. 0.5 b. 3 e. 1 4 2 c. f. 1 Section 1.2, page 19 1. 20 0 3. 20 0 5. u1 6 and un un1 2 10 0 −10 10 10 10 7. u1 6 and... |
ind speed differs by 5 feet per second true wind speed from the measured speed, let x 5 and x 20 0 0 x 20 5 or x 20 5 x 25 x 15 or The true wind speed is between 15 and 25 feet per second. CL 0.0097 0 x 4 0.0497 33. (in practice) and x 2 31. 27. 29. 35. x 1 or 2 37. x 9 39. x ± 3 x 1 2 47. x 1.658 53. No solutions 61. ... |
x 2 1 b 0 37. 41. b 4, c 8 a 1 2 43. Minimum product is 4; numbers are 2 and 2 45. Two 50-ft sides and one 100-ft side 47. $3.50 49. 30 salespeople 51. 1 second; 22 ft 53. The maximum height of 35,156.25 feet is reached 46.875 seconds after the bullet is fired. Section 3.4, page 182 5. x2 h x 2 1 5 1. 7. f x 1 2 h x 2 ... |
vertically 2 units upward. 43. e 45. 36 31 2b x 4 25 4 x 1 2 1 1 33. parent function: 21 2 x 2 f 1 10 (transformation form) (x-intercept form) 1x 0 0 125 Note: the right endpoint of each segment is a part of the graph; the left endpoint is not a part of the graph. −10 10 47. x-axis, y-axis, origin 49. Even 1 53. a. b.... |
al asymptote parabolic asymptote y x2 x 1 x 1 49. y 36 30 24 18 12 6 y = x2 + 2x + 4 −5 −3 −1 1 3 5 x −9 y x2 2x 4 and 8 y 4; hidden 2 x 2 x 5: and 0.5 y 0.5; 15 x 3 and 4 y 4; there is a hole at vertical asymptote parabolic asymptote 5 x 4.4 51. Overall: x 2 53. area near origin: hidden area near 0.07 y 0.02 9.4 x 9.4... |
a. c. x x 3 −1.. d2 −1 14 101. a. g is the graph of f moved 3 units left b. h is the graph of f moved 2 units down c. k is the graph of f moved 3 units left, then 2 units down. Section 5.2, page 343 21. 23. 25. 27. 29. 1 2 x : B 20 x 2 : C; h 1 and and and and 2 x : D : A; k 1 0 y 1 10 y 10 0 y 1 0 y 10 2 2 xh 2 h 1. S... |
ll real numbers 43. Vertical stretch by a factor of 3, vertical translation of 5 units downward; Domain: all positive real numbers; Range: all real numbers 45. 3 47. 3 4 49. 2 ln x 51. ln 9y x2 b a 53. (c) 55. The domain consists of those values of x for which is positive; x x 1 dw uv x 3 ± 257 4 2 uc d x e1 x 101 q, 0... |
eriod: 2p 7. amplitude: none; period: p 2 15. amplitude: none; period: 4 17. a. 2 c. t 0 or 2 b. d. t 1 2 t 1 or 3 2 19. g is the graph of f horizontally compressed by a factor of 1 5 ˛; amplitude: 1; period: 2p 5 ˛. 21. g is the graph of f horizontally compressed by a factor of 1 8 ˛; amplitude: 1; period: p 4 ˛. 23. ... |
Exercises 1105 0.01 29. x 2.49809 2kp or 39. 9.06° or 80.94° 41. a. 19 feet b. 3 ft below water c. 20 seconds d. Answers may vary: g 11 cos t 2 1 p 10 a t b 8 t e. Answers may vary: p 11 sin h 1 10 t 4.418 20k where k is any integer. and f. a 2 8 t p 2 b t 15.582 20k 43. Using approximate values, y 0.006 cos 2094.768x... |
now EAB (Why?) then is b. 94.24 ft 55. 13.36 m 57. 5.8 gal 59. 11.18 sq units 61. No such triangle exists because the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, which is not the case here. cos p a 4 12 2 a 22 13 2 p 6 b cos p 4 cos 13 2 b 12 1 2b 2 a 16 12 4 x ... |
y2 8 x2 9 x2 4 x2 1 x2 1 1 11. 9. 1 or 8x2 y2 8 1 or 8x2 y2 8 3. 7. x2 4 y2 16 y2 1 x2 9 1 by the −10 10 −10 Because of limited resolution, this calculator-generated graph does not show that the top and bottom halves of the graph are connected. 13. 10 21. 10 −10 −10 23. −10 10 −10 −10 10 −10 foci are at A y ± b a x; ar... |
x 9, . 1 2 21. This is an ellipse with center at 0, 5 and 1 the points 1 3 25, 5 1 2 and 1 2 6, 5 3 25, 5 1 2 . 2 3, 5 2 , vertices at and foci at the points 1 2 3 4 5 6 −3 −4 −5 −6 −7 2 7. This is a hyperbola with foci at the points and vertices at the points graph on the window ± 3, 0 9 x 9, . 2 1 Shown is the 6 y 6... |
counties in each of 5 states. 5. In #1 the data is qualitative. In #2 the data is qualitative. In #3 the data is quantitative and discrete. In #4 the data is quantitative and continuous. 7. 200 cartons. 9. 2500 families. 11. Exercise Aerobics Kick boxing Tai chi Stationary bike 13. Relative frequency 40% 16% 16% 28% 50... |
r except 27. Continuous at every real number except x 3 x 0 and . 29. 31. 33. 1 2 2x f has a removable discontinuity at g x 2 1 x 4. 35. If and 1 for all 1 x x 0, then x g 1 x 6 0. 2 1 Thus for all for all g lim xS0 x 2 2 does not exist. Hence the definition of continuity cannot be satisfied, no matter what g(0) is. 1 ... |
3 1 a r rb n. For example, rows 2 and 3 of Pascal’s triangle 1b are 1 1 2 1 3 3 1 that is, 2 0b a 2 1b a 2 2b a a 3 1b a 3 2b a 3 3 0b 3b a The circled 3 is the sum of the two closest entries in the row above: 2 , 1b 2 3 0b a 1b a r 0. Similarly, in the general case, and n 2 verify that the two closest entries in the r... |
sine function, 529–534, 539 arctangent function, 534–536, 540 area circular puddle, 195 under a curve, 904–906 maximum, 138–139, 169–170, 468–471 quarter-circle, 905–906 Riemann sums, 964–967 triangle, 632–633, 682 z-values, 895–896 A absolute value complex numbers, 638–639 definitions, 107–108, 134 deviations, 858 dis... |
408, 584, 922–923 dreidels, 882 dynamical systems, 199 differential calculus, 76 directed networks, 809–811 direction angles, 662–664 directrix, 709–711, 720 discriminants, 93–94, 172, 723–724 distance absolute value and, 108, 128 applications, 101–102, 113–114 average rates of change, 214–220 between two moving object... |
inequalities (continued) compound, 118–120 equivalent, 119 interval notation, 118–119 linear, 119–120, 827 multiplying by negative numbers, 119 nonlinear, 121–122 quadratic and factorable, 122–123, 129–130 solving, 119–123 systems of, 826–832 test-point method, 826 infinite geometric series, 76–79, 520–521 infinite se... |
al estimates, 877–878 probability density functions, 871–872, 891 product functions, 192, 943 Product Law of Exponents, 365 Product Law of Logarithms, 365, 373–374 product-to-sum identities, 599 profit functions, 146, 170, 196, 216–217 projectile motion, 546, 602, 761–762 projections, 674–677, 681 proofs, identities, 5... |
oordinates, 790–791 TRACE feature, 176, 185 transformation form, 164–165, 168–169, 225 transformations amplitude, 497–498, 503–505, 516, 563 combined, 180, 503–505 conic sections, 716–725 horizontal shifts, 175–176 logarithmic functions, 359–360, 375–376 parameterization of, 757–759 parent functions, 172–173 phase shif... |
2 cos cos x cos y 2 sin cos sin a a cos a sin |
s as π 6 , and so on. If you and your friends carry , π , 3π 4 , π 2 , π 3 backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data. Example 1.5 Data Sample of Quantitative Discrete Data The data are the number of books students... |
the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member ... |
50th student is chosen until 75 students are included in the sample. c. A completely random method is used to select 75 students. Each undergraduate student in the fall semester has the same probability of being chosen at any stage of the sampling process. 24 CHAPTER 1 | SAMPLING AND DATA d. The freshman, sophomore, j... |
cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range. Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking th... |
g to Table 1.9, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample. A relative frequency is the ratio (fraction or proportion) of the number of times a value of the ... |
2006? c. What is the relative frequency of deaths that occurred in 2000 or before? d. What is the percentage of deaths that occurred in 2011? e. What is the cumulative relative frequency for 2006? Explain what this number tells you about the data. This content is available for free at http://textbookequity.org/introduc... |
tion that drove him to carry out acts of increasingly daring fraud, like a junkie seeking a bigger and better high.[2] The committee investigating Stapel concluded that he is guilty of several practices including: • creating datasets, which largely confirmed the prior expectations, • altering data in existing datasets,... |
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