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projection of u on v, determined by constructing a segment from the terminal point of a vector u perpendicular to another vector v at a point Q on the vector, where point O is the initial point of both vectors, denoted projv u. (p. 674) Pythagorean identities sin2 t cos2 t 1 (p. 456) the identity and the identities derived from it Pythagorean Theorem In a right triangle with legs a and b and hypotenuse c, a2 b2 c2. (p. 421) Q (p. 455) Quotient Law of Logarithms For all positive b, v, and w, and b 1, logba v wb logb v logb w. (p. 366, 373) R radian measure The radian measure of an angle in standard position is the length of the arc along the unit circle from the point (1, 0) on the initial side to the point P where the terminal side intersects the unit circle. (p. 436) radical equations as square roots, cube roots, etc.) of expressions that contain variables (p. 111) equations that contain roots (such radioactive decay decay in the amount of a radioactive substance that can be modeled by the function P x f 2 stance, and h is the half-life 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 of an organic object by using the amount of carbon-14 remaining in the object (p. 352) Glossary 1047 random sample a sample in which all members of the population and all groups of members of a given size have an equal chance of being in the sample (p. 843) random variable a function that assigns a number to each element in the sample space of an experiment (p. 869) range (of a data set) imum and minimum data values in a data set (p. 859) the difference between the max- range (of a function) the ordered pairs of a relation (p. 6, 142, 447) the set of second numbers in rational exponent A rational exponent is a rational number with a nonzero denominator
; for any positive real number c and rational number with positive denominator, (p. 330) t k c 1 k ct 2 1 1 c k t 2 1 or c ct 2k c t. B A t k k 2k t rational function a function whose rule is the quotient of two polynomials, defined only for input values that produce a nonzero denominator (p. 278) rational number the set of real numbers that can be expressed as a fraction of an integer numerator and an integer denominator, where the denominator is not equal to zero (p. 4) Rational Zero Test If a rational number written in r s, lowest terms, is a zero of the polynomial function anxn p a1x a0, x f 2 1 an, p, an are integers with a0, factor of the constant term, (p. 251) leading coefficient, where the coefficients 0 0, and s is a factor of the and a1 a0 an. then r is a rational zeros numbers (p. 250) zeros of a function that are rational rationalizing (a denominator) writing equivalent fractions with no radicals in the denominator (p. 332) real axis where each real number point (a, 0) (p. 638) the horizontal axis in the complex plane a 0i corresponds to the real numbers rational numbers and irrational numbers (p. 3) the set of numbers that consists of real solutions numbers (p. 88) solutions of an equation that are real real zeros 0 f x 1 2 solutions of an equation of the form that are real numbers (p. 245) reciprocal identities metric functions and their reciprocals (p. 455) identities that relate trigono- rectangular coordinate system See Cartesian coordinate system. recursive form of a geometric sequence In a geo run1 for some nonzero conmetric sequence un, un6 5 1048 Glossary stant r and all n 2. (p. 59) recursive form of a sequence a method of defining a sequence when given the first term and the procedure for determining each term by using the preceding term (p. 15) recursive form of an arithmetic sequence In an un6 un arithmetic sequence, 5 n 2. stant d and all (p. 22) un1 for some con- d reduced row-echelon form This form of an augmented matrix satisfies the following conditions: all rows consisting entirely of zeros (if any) are at the bottom; the
first nonzero entry in each nonzero row is a 1 (called leading 1); any column containing a leading 1 has zeros in all other entries; and each leading 1 appears to the right of leading 1s in any preceding row. (p. 797) reference angle the positive acute angle formed by the terminal side of x-axis (p. 449) in standard position and the u reflections The graph of reflected across the x-axis, and the graph of 1 is the graph of f reflected across the y-axis. (p. 177) is the graph of relation a correspondence between two sets; a set of ordered pairs (p. 6) 2, x is divided Remainder Theorem If a polynomial by then the remainder is x c f 1 (p. 244) 1 residual a measure of the error between an actual data value and the corresponding value given by a model; point contained by the model (p. 44) where (x, r) is a data point and (x, y) is a r y, c f. 2 1 2 Richter scale a logarithmic scale used to measure the magnitude of an earthquake (p. 368) right angle an angle with a degree measure of (p. 414) 90° solutions of an equation of the form roots and equal to the zeros of f (p. 83) f x 1 2 0 roots of unity the n distinct nth roots of 1 (the solutions of ) (p. 648) zn 1 rotation equations a coordinate plane to the corresponding point in the plane after a rotation (p. 729) equations that relate a point in rule of the function See function rule. S sample a subset of the population studied in a statistical experiment, whose information is used to draw conclusions about the population (p. 843) sample space the set of all possible outcomes in an experiment (p. 864) scalar a real number, often denoted by k, used in scalar multiplication (p. 655) scalar multiplication (with vectors) an operation in which a scalar k is multiplied by a vector v to produce another vector, denoted by kv (p. 655) scalar multiplication with matrices multiplication of each entry in a matrix by a real number (p. 805) scatter plot a graphical display of statistical data plotted as points on a coordinate plane to show the relationship between two quantities (p. 5) 0 (p. 673) Schwartz Inequality For any
vectors u and v, u v u v. 0 secant line (of a function) the straight line determined by two points on the graph of a function; the slope of the secant line joining points b, f 1 rate of change of the function from a to b (p. 218) on the graph of a function, equal to the average and a, f 22 22 b a 1 1 1 u secant ratio For a given acute angle in a right triu u angle, the secant of is written as sec and is equal to the reciprocal of the cosine ratio of the given angle. (p. 416) second a unit of degree measure equal to 1 60 of a minute, or 1 3600 of a degree (p. 414) second quartile See quartiles. second-degree equation See quadratic equation. self-similar the property possessed by fractals that every subdivision of the fractal has a structure similar to the structure of the whole (p. 305) sequence an ordered list of numbers (p. 13) sequence notation a customary method of denoting a sequence or terms of a sequence in abbreviated form: u1, u3, p, (p. 14) u2, un series the sum of the terms of a sequence (p. 76) sides (of an angle) that form an angle (p. 413) the two rays, segments, or lines Sigma notation See summation notation. simple harmonic motion motion that can be described by a function of the form a sin or x f 1 bt c 2 d a cos bt c d x f (p. 549) 1 2 1 2 1 2 simple interest interest that is generally used when a loan or a bank balance is less than 1 year; calculated where I is the simple interest, P is the prinby cipal, r is the annual interest, and t is time in years (p. 100) I Prt, sine ratio For a given acute angle gle, the sine of ratio of the opposite side length to the length of the hypotenuse. (p. 416) in a right trianis written as sin and is equal to the u u u sinusoid the wave shape of the graph of a sine or cosine function (p. 510, 548) sinusoidal function A function whose graph is the shape of a sinusoid and can be expressed in the form f x f or 2 1 2p b 2 is the amplitude, 2
where is the period, a cos 2 c b a sin bt c bt c d, d x a 1 1 1 0 0 is the phase shift, and d is the vertical shift. (p. 548) skewed distribution a type of distribution in which the right or left side of its display indicate frequencies that are much greater than those of the other side (p. 846) slant asymptote a nonvertical and nonhorizontal line that the graph of a function approaches as large (p. 286) x 0 0 gets slope of a line The value of the ratio y1 x1 are points contained by the y2 x2 ¢y ¢x, where line and x1, y12 x1 1 x2. and x2, y22 (p. 32) 1 slope-intercept form a linear equation in the form y mx b, sents the y-intercept (p. 33) where m represents the slope and b repre- solution of a system of equations a set of values that satisfy all the equations in the system (p. 780) solution of an equation the value(s) of the variable(s) that make the equation true (p. 30) solution to an inequality (in two variables) the region in the coordinate plane consisting of all points whose coordinates satisfy the inequality (p. 827) solving a triangle finding the lengths of all three sides and the measures of all three angles in a triangle when only some of these measures are known (p. 424) sound waves periodic air pressure waves created by vibrations (p. 558) special angles angles of degree measure 60° (p. 418) 30°, 45°, or square root of squares For every real number c, 2c2 (p. 109) c. 0 0 square system a system of equations that has the same number of equations as variables (p. 814) standard deviation a measure of variability that describes the average distance of data values from the mean, given by the square root of the variance (p. 857) standard equation of a hyperbola For any point (h, k) in the plane and positive real numbers a and b, Glossary 1049 2 1 x h a2 2 1 2 y k b2 2 1 or 1 2 x h b2 2 1 2 y k a2 2 1. (p. 701, 720) standard equation of a parabola For any point in the plane and non
zero real number p, h, k 2 1 x h. 1 720 or 2 1 2 2 2 1 1 (p. 710, standard equation of an ellipse For any point in the plane and real numbers a and b with x h a2 x h b2 y k b2 y k a2 h, k a 7 b 7 0, 1. 1 1 1 or p. 693, 720) standard form (of a line) used to display the equation of a line without fractions, a linear equation in the form to zero (p. 39) where A and B are not both equal Ax By C, standard normal curve a normal curve with a mean of 0 and a standard deviation of 1 (p. 890) standard notation (of triangles) a method of labeling triangles in which each vertex is labeled with a capital letter to denote the angle at that vertex, and the length of the side opposite that vertex is denoted by the same letter in lower case (p. 617) standard position (of an angle) an angle in the coordinate plane with its vertex at the origin and its initial side on the positive x-axis (p. 434) standard viewing window the window or screen of a graphics calculator that displays 10 y 10, graphics 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 stems, on the left and the remaining digits of the data values, called leaves, on the right; commonly used to display small data sets (p. 847) step function a function, such as the greatest-integer function, whose graph consists of horizontal line segments resembling steps (p. 157) sum function For any functions sum function is the new function f (p. 191 and x 21 2, their x g 1 2 sum of an infinite geometric series The sum S a1 1 r vergent geometric series with common ratio r such that r is a con- r3a1 r2a1 ra1 where p (p. 77) 6 1. a1, 0 0 1050 Glossary sum-to-product sin x ± sin y or cos x ± cos y (p. 599) trigonometric identities involving summation notation a
customary method of denoting the sum of terms by using the Greek letter Sigma c2 p cm ) as follows: (p. 25) c1 c3 ck © ( m a k1 supplementary angle identity For any acute angle u, (p. 628) sin u sin 180° u. 1 2 symmetric distribution a type of distribution in which the right and left sides of its display indicate frequencies that are mirror images of each other (p. 846) synthetic division an abbreviated notation for performing polynomial division when the divisor is a first-degree polynomial (p. 241) system of equations a set of two or more equations in two or more variables (p. 779) T tangent line (of a function) a line that touches the graph of a function at exactly one point; the slope of the tangent line to a curve at a point, equal to the instantaneous rate of change of the function at that point (p. 235) u tangent ratio For a given acute angle u angle, the tangent of is written as tan and is equal to the ratio of the opposite side length to the adjacent side length. (p. 416) in a right tri- u term (of a sequence) a number in a sequence (p. 13) terminal point (of a vector) that extends from point P to point Q (p. 653) the point Q in a vector terminal side the final position of a ray that is rotated around its vertex (p. 433) third quartile See quartiles. transformation form of a quadratic function a x h a 2 k, quadratic function in the form 2 1 a 0 where a, h, and k are real numbers and (p. 164) x f 2 1 Triangle Sum Theorem The sum of the measures of 180°. the angles in a triangle is (p. 421) trigonometric form of a complex number See polar form of a complex number. trigonometric functions of a real variable a function whose rule is a trigonometric ratio in the coordinate plane with domain values in radian measure (p. 445) trigonometric ratios (in a triangle) combinations of side length ratios of a right triangle (p. 415) the six possible trigonometric ratios (in the coordinate plane) six trigonometric ratios defined in terms of a triangle determined by the coordinates of a point on the terminal side of an angle in standard position and the origin (p. 444)
the two-stage path (of a network) a path in a directed network from one vertex to another with exactly one intermediate vertex (p. 810) U uniform distribution a type of distribution in which all of the data values have approximately the same frequency; its display is level (p. 846) unit circle the circle of radius 1 centered at the origin of the coordinate plane (p. 435) unit vector a vector with a length of 1 (p. 661) upper bound (for the real zeros of a polynomial function) the number s such that all the real zeros of a polynomial function f(x) are between r and s, where r and s are real numbers and (p. 255) r 6 s V variability the spread of a data set (p. 857) variance a measure of variability given by the average of squared deviations (p. 858) vector a quantity that involves both magnitude and direction; represented geometrically by a directed line segment or arrow and denoted by using its endpoints, or by a boldface lowercase letter, such as u such as (p. 653)! PQ, u. I a, b I H (p. 657) u (p. 658) H vector addition If a c, b d u v H and v c, d I H, then and v a, b I c, d H, I vector subtraction If u v a c, b d H. I vertex (of an angle) two rays, segments, or lines that form an angle (p. 413) the common endpoint of the vertex of a parabola the intersection of the axis of a parabola and the parabola; the midpoint of the segment from the focus of the parabola to the directrix (p. 709) vertical asymptotes a vertical line that the graph of a function approaches but never touches or crosses because it is not defined there (p. 284, 950) the graph of vertical compression For any positive number y c f c 6 1, compressed vertically, toward the x-axis, by a factor of c. (p. 179) is the graph of f x 1 2 vertical line a line that has an undefined slope and x c, an equation of the form real number (p. 37) where c is a constant y f vertical shifts For any positive number c, the graph is the graph of f shifted upward c units,
x of 1 c is the graph of f shifted and the graph of downward c units. (p. 174) See also sinusoidal function. y f c x 1 2 2 vertical stretch For any positive number graph of cally, away from the x-axis, by a factor of c. (p. 179) the is the graph of f stretched verti, vertical line test A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once. (p. 151) vertices of a hyperbola the points where the line through the foci intercepts the hyperbola (p. 701) vertices of a network the points that are connected in a network (p. 809) vertices of an ellipse the points where the line through the foci intercepts the ellipse (p. 692) W whole numbers natural numbers and zero: 0, 1, 2, the set of numbers that consists of p (p. 3) work The work W done by a constant force F as its point of application moves along the vector d is W F d. (p. 678) X then x-axis often the name of the horizontal axis of a coordinate plane with the positive direction to the right and the negative direction to the left (p. 5) x-axis symmetry A graph is symmetric with respect to the x-axis if whenever x, y x, y is on the graph, then is also on it. (p. 185) 1 x-coordinate usually the first real number in an ordered pair (p. 5) 2 2 1 the x-coordinate of a point where a graph x-intercept crosses the x-axis; the x-intercepts of the graph of f, equal to the zeros of f and the solutions of (p. 83) 0 x f 1 2 x-intercept form of a quadratic function a quadx s a f where ratic function in the form 1 a 0 a, x, s, and t are real numbers and x t, 2 21 (p. 164) x 2 1 Glossary 1051 x-intercept method a method of solving an equation of the form finding the x-intercepts (p. 84) by graphing y f 0 and -axis nate system with positive direction upward (p. 790) the vertical axis in a three-dimensional coordi- zero polynomial the
constant polynomial 0 (p. 240) y-axis often the name of the vertical axis of a coordinate plane with the positive direction up and the negative direction down (p. 5) Zero Product Property If a product of real numbers ab 0, is zero, then at least one of the factors is zero; if (p. 89) then b 0. a 0 or y-axis symmetry A graph is symmetric with respect x, y to the y-axis if whenever x, y is on the graph, then is also on it. (p. 184) 1 y-coordinate usually the second real number in an ordered pair (p. 5) 2 2 1 zeros of a function inputs of the function that produce an output of zero (p. 83) z-value a value that gives the number of standard deviations that a data value in a normal distribution is located above the mean (p. 894) 1052 Glossary Acknowledgments Photo Credits: page 2, Ron Behrmann/International Stock Photography; page 80, Orion Press/Natural Selection; page 140, Peter Van Steen/HRW Photo; page 326, Laurence Parent; page 412, Todd Gipstein/National Geographic Society Image Collection; page 472, Peter Van Steen/HRW Photo/Courtesy Jim Reese at KVET, Austin, Tx; page 522, Coco McCoy/Rainbow/PictureQuest; page 570, Telegraph Color Library/FPG International; page 616, Scott Barrow/International Stock Photography; page 690, Stone; page 778, Tom McHugh/Photo Researchers, Inc.; page 842, SuperStock; page 908, Tom Sanders/Photri/The Stock Market Illustration Credits: Abbreviations used: (t) top, (c) center, (b) bottom, (l) left, (r) right, (bkgd) background All work, unless otherwise noted, contributed by Holt, Rinehart and Winston. Chapter Four: page 306 (tc), Pronk&Associates; Chapter Six: page 470 (cl), NETS; page 471 (br), NETS. Chapter Eleven: page 713 (b), NETS. Chapter Thirteen: page 882 (br), NETS. Acknowledgments 1053 Answers to Selected Exercises 2.5, 0 1 H ; 3, 1 2 ; I D 1 3, 1 1.5, 3 1 5
. 2 P 1 4, 2 1 b. y (–5, 4) ; 2 (4, 1) x 2 (3, –2) (–2, –3) Chapter 1 1. Section 1.1, page 10 3, 3 ; 2 1 F ; 0, 2 1 2 6, 3 B 1 0, 0 A E 1.5, 3 G ; 3. P 1 2 1 ; 2 2, 0 C ; 2 1 2 7. y 500 400 300 200 100 2 x 0 987654321 10 9. a. About $0.94 in 1987 and $1.19 in 1995. b. About 26.6% c. In the first third of 1985 and from 1989 onward. 11. a. Quadrant IV b. Quadrants III or IV 13. a. y (–2, 3) (–5, –4) (3, 2) x (4, –1) 1054 Answers to Selected Exercises c. They are mirror images of each other, with the x-axis being the mirror. In other words, they lie on the same vertical line, on opposite sides of the x-axis, the same distance from the axis. 15. Yes. Each input produces only one output. 17. No. The value 5 produces two outputs. 19. (500, 0); (1509, 0); (3754, 35.08); (6783, 119.15); (12500, 405); (55342, 2547.10) 21. Each input (income) yields only one output (tax). 23. Postage is a function of weight since each weight determines one and only one postage amount. But weight is not a function of postage since a given postage amount may apply to several different weights. For instance, all letters under 1 oz use just one first-class stamp. 25. Domain: all real numbers between 3 inclusive; range: all real numbers between 3, inclusive and 3, 4 and 27. 2 is 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 un un1 5 30 0 10 0 4 2 4 3 11 2 11 3 25 2 25 3 53 2 53 3 109 400; 0.8un1; un u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 u0 9. 11. 13. 15. For 2 rays: 6; un u4 1; u2 un1 for 3 rays: n 1 for 3; u3 n 3 for 4 rays: 325 0 25 17. 19. 21. 23. u25 300 12, u1 seats un un1 2 for 2 n 35 ; u30 70 for n 1 ; 6500, for n 1 ; 1.06un1 0.75un1 u0 u10 u0 u8 u0 u20 30,000 un, $53,725.43 35,000 un, 26,901 students 50,000 un, 41,216 u35 ; 4000 0.9un1 40,250. for n 1 ; 25. a. The items listed are the first ten primes. Every other number less than 29 can be factored into a product of smaller integers. b. 59, 61, 67, 71 27. 4, 9, 25, 49, 121 29. 3, 7, 13, 19, 23 31. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 33. n 1: 5 1 n 2: 5 1 n 3: 5 1 n 4: 5 1 n 5: 5 1 n 6: 5 1 n 7: 5 1 n 8: 5 1 n 9: 5 1 n 10 13 21 34 12 2 9 32 3 16 42 4 49 72 5 121 112 6 324 182 7 841 292 8 2209 472 9 5776 762 10 15,129 1232 2 2 2 55 1 2 1 2 Section 1.3, page 29 1. 13; un 2n 3 15 0 0 10 u6 104.858 cm Answers to Selected Exercises 1055 3. 5; un n 4 15 4 6 0 2 5. 8; un n 2 21 2 15 Section 1.4, page 40 1. a. C b. B c. B d. D
3. Slope, 2; y-intercept, b 5 5. Slope, 3 7 ; y-intercept, b 11 7 7. Slope, 5 2 15. y 9. Slope, 4 11. t 22 13. t 12 5 10 L P(1, C) x (0, 0) (1, 0) Slope of 10 9. 224 17. 30 3 2 1 3 n 1 2 4 2; 0 5 7. 45 11. 13. 87 15. 21 4 3 2n 19. un1 un arithmetic with 1 2 d 2 21. un un1 5 3n 2 d 3 2 c 2n 2 d 2 2n 4 7n 10 3n 2 15 2 arithmetic with 23. 25. 27. 1 un1 un arithmetic with 14; un 25; un u5 u5 29. u5 0; un 5 3 n 1 1 2 2 3 2 ; 49; 17. 23. y x 2 y 7 3 x 34 9 19. y x 8 21. y x 5 25. Perpendicular 27. Parallel 29. Parallel 31. Perpendicular 33. Yes. The slopes are,, 2 5 and 9 8 perpendicular result in a right triangle. y 3 2 y 3x 7 5 2 39. 37. x. 35. Two sides y x 5 43. k 11 3 41. y x 2 45. un un u1 1 2 1 4n 6 n 1 d 2 n 1 4 2 or 47. The common difference is y 4x 6 6. d 6 2 or y 6x 13 un un 2 21 n 1 u1 1 n 1 7 1 6n 13 u1 1 2 1 8n 10 un The slope is n 1 d 2 n 1 8 2 or m 8 un y 8x 10 and the y-intercept is b 10. 51. Both have slope and different y-intercepts. A B y 449x 9287 x 9; y $13,328 1 x 24; y $20,063 53. a. b. 1990 2 2005 1 2 31. 710 33. 156 2 3 35. 2550 37. 20,100 39. $77,500 in tenth year; $437,500 over ten years 41. 428 43. 23.25, 22.5, 21.75, 21, 20.25, 19.5, 18.75 1056 Answers to Selected Exercises 57. a. b. 55
. $375,000; $60,000 y 5x 150 x 5, y 125 pounds x 7, y 185 pounds 50x 110,000 22x 110,000 59. a. c ft ; 2 1 6 ft 2 r 72x b. x 2 d. x 5000 1 61. a. 63. a. y 8.50x 50,000 x 10 b. $11, $9.50, $9 per hat b. x 30 15. No High School Diploma 12.31x 238 y2 1. a. Section 1.5, page 53 x 5 y 3 4 4 Sum of squares 3 Model B still has least error. Slope 1.0564054 y 1.0564054x 21.077892 b. 3. a. b. c. Line described in b predicts a higher number of workers. 5. negative correlation 7. very little correlation 9. a. 0.09, 0.17, 1.22, 3.13, 5.14, 8.26 b. not linear; 35 0 0 60 11. a. 4.6, 8.3, 14.3, 23.5, 37.2, 56.9, 84.3, 121.4, 170.7, 234.2 b. not linear; 1000 0 0 110 13. a. 446.9, 405.2, 515.8, 785.3, 298, 413 b. linear; positive correlation 7100 0 0 1000 0 0 High School Graduate 15.17x 354 y4 1000 0 0 Some College y1 20.74x 392 1000 0 0 College Graduate 34.86x 543 y3 1000 0 0 12 12 12 12 10 y 0.0292x 4.0149 y 0.4078x 16.8494 17. a. b. c. The income of the rich is increasing faster than the income of the poor is decreasing. d. The income gap will increase. Answers to Selected Exercises 1057 19. a. y 4x 82 b. The amount of federal money in loans is increasing and the amount in grants and workstudy is decreasing. 13. u6 5 16 ; un 1 1 n1 5 2 2n2 15. 315 32 19. un un1 21. un un1 n 1 a 2b n1 1 a 2b 5n2 2 n1 51 2 17. 381 1 2 ; geometric with r 1 2 5
; geometric with r 5 20 23. u5 1; un 1 1 n164 2 4n2 1 n1 1 2 4n5 25. u5 1 16 ; un 1 4n3 27. u5 8 25 ; un 2n2 5n3 29. 254 31. 4921 19,683 33. 665 8 100 0 0 Loan data Grant/work-study data c. 1983 40 21. a. 35. a. Since for all n, the ratio r is un1 un 1.71 1 1.71 1.191n1 1.191n 1 2 2 1.71 the sequence is geometric. b. $217.47 1.191n 1 1.71 21 1.191n 1.191 1 2 2 1.191, 39. 37. 23.75 ft 31 a n1 41. $1898.44 2n1 1 231 1 2 231 1 1 2 cents $21,474,836.47 1 1 43. un log u1 log log u1 u1rn1 u1rn1. log un 2 rn1 n 1 log 1 2 2n1 r 2. 5 and the sum of the preceding k 1 2 1 2n1 1 2k1 1 2 45. The sequence is 2k1, kth term is terms is the k1 a n1 2k1 1. log r and 6 2 th partial sum of the sequence, So for any k, the 47. 37 payments Chapter 1 Review, page 67 is an irrational number. 1. 3. 13 e 2.718 is an irrational number. 5. 0 is a whole number. 7. 1121 is a natural number. 9. 5 is a natural number. 11. Answers may vary; for example, numbers that are not rational. p and 12 are real 13. This is not a function; the input 2 has more than one output. 15. This is a function; for each input there is exactly one output. 17. all real numbers greater than or equal to 2 and not equal to 3 −1 0 17 y 1.27x 23.8318 b. c. The model gives a negative median time for approval in 2009, and it will not be useful for data extrapolation. y 0.08586x 22.62286 14.72 23. a. b. for 1992, a terrific prediction. c. This model may not remain valid for any future dates. The rate of improvement seems
66 98.92 69.17 29.75 The finite differences show that the data is not linear. 20.79 20.10 0.69 34.89 25.23 9.66 69.17 48.55 20.62 93. Second method is better. Answers to Selected Exercises 1059 Chapter 1 can do calculus, page 79 1. 1 4. 2 3 2. Diverges 5. 500 0.6 833 1 3 7. 4 212 8. Diverges 10. 13. 2 9 8428 99 11. 14. 37 99 10,702 4995 3. 3 7 6. 5.7058 9. 1 2 12. 15. 597 110 18,564 4995 16. Because there are infinitely many terms that are getting larger and larger, the sum cannot converge to a finite number. 17. The sum approaches 1 3. 0 0 30 −1 Chapter 2 Section 2.1, page 87 1. 3 3. 3 5. 2 7. 11. 15. 19. 21. 25. 29. x 2.426 x 1.475, x 1.379, x 2.115, x 2.102 x 0.951 x 2.390 9. 13. 17. 1.237 1.603 0.254, 1.861 23. 27. 33. x 7.033 35. 31. x 2 3 39. x 13 41. 2004 1.164 x 1.750, x 1.453, x 0, x 1.601, 0.507, 1.329 1.192 x 1.752 x 0, x 0.651, x 2.207 1.151 37. x 1 12 43. 1999 3. 1. 5. or 3 Section 2.2, page 95 x 3 or 5 y 1 2 u 1 or 4 3 x ± 3 x ± 140 ± 6.325 x2 9; x2 40; 11. 13. 15. 9. 7. x 2 or 7 t 2 or 1 4 or 4 3 x 1 4 1060 Answers to Selected Exercises 17. 19. 21. 23. 25. 27. x2 4; x ± 2 3x2 12; 5s2 30; 25x2 4 0; s ± 16 ± 2.449 s2 6; x2 4 25 w2 28 3 ; x ± 2 5 3w2 8 20; w ± ; 28 3 A ± 3.055 x 1 ± 113 1 15 2 w A
B or A 1 15 2 B 29. x 2 ± 13 31. x 3 ± 12 33. No real number solutions 35. 39. ± 12 x 1 2 4 ± 16 5 u 47. x 3 or 6 51. x 5 or 3 2 37. x 2 ± 13 2 41. 2 43. 2 45. 1 49. x 1 ± 12 2 53. No real number solutions 55. No real number solutions 57. 61. 65. x 1.824 or 0.470 y ± 1 or ± 16 y ± 2 or ± 1 12 59. 63. 67. x 13.79 x ± 17 x ± 1 15 69. k 10 or 10 71. k 16 73. k 4 Section 2.3, page 105 1. The two numbers: x and y; their sum is 15: the difference of their squares is 5: x y 15; x2 y2 5. 3. English Language Length of rectangle Width of rectangle Perimeter is 45 Area is 112.5 Mathematical Language x y x y x y 45 or 2x 2y 45 xy 112.5 5. Let x be the old salary. Then the raise is 8% of x. Hence, old salary raise $1600 1600 8% of x 2 1 x 0.08x 1600. x 7. The circle has radius pr2 p 82 64p. r 16 2 8, so its area is Let x be the amount by which the radius is to be reduced. Then p and the new area is 1 48p less than the original area, that is, 2 64p 48p, p 2 16p. p or equivalently, 8 x 8 x 8 x 2, 2 1 1 2 2 r 8 x which must be 9. $366.67 at 12% and $733.33 at 6% 11. 2 2 3 qt 13. 65 mph 15. 34.75 and 48 17. Approximately 1.75 ft 19. Red Riding Hood, 54 mph; wolf, 48 mph 21. a. 6.3 sec b. 4.9 sec 23. a. Approximately 4.4 sec 25. 23 cm by 24 cm by 25 cm b. After 50 sec r 4.658 27. 29. x 2.234 31. 2.2 by 4.4 by 4 ft high Section 2.4, page 116 12 y 2, 3w 2 0 −1 6 8 0 3. −6 −5 −4 −3 −2 −1 1098765
43210 Since 3w 6 or 10, dividing by 3, w 2, 3 5. 0 2x Since 2x 1 or 9, dividing by 2, x 1 2, 9 2. 3x 7. 0 2 1 2 0 5 or 0 3x 2 5 0 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 Since 3x 7 or 3, dividing by 9. x 6 or 3 15. x 3 2 3, x 7 3 1., 13. x 2 11. x 3 2 17. x 5 or 1 or 3 or 1 19. x 1 or 4 or 5 133 2 or 5 133 2 21. For any real number x, the distance between 2x2 cannot be a negative number. Therefore, 12 has no real number solution. 3x and 2x2 3x 0 23. Let x 0 Joan’s ideal body weight. The difference Therefore, 0 x 120 0.05x or x 120 0.05x 0 0.05x. x 120 0.95x 120 x 126.3 1.05x 120 x 114.3 To the nearest pound, Joan’s ideal weight is either 114 pounds or 126 pounds. 25. If the true wind 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 1 63. x 1 43. x ± 0.73 or 2.59 or x 3 51. or 1.40 x 1 or 7 57. x 7 41. 45. 49. 55. 59. or 4 x 1 2 x 1.17 x 6 x 0.457 x 3 ± 141 4 65. u x2 1 K2 B 67. b a2 A2 1 B 69. a. I x x2 1024 1 3 2 2 Section 2.5, page 124 b. 22.63 ft 1. 5. x 0 −2 3. −3 −2 −1 0 1 32 9. −4 −2 0 1 11. [5, 8] 17. q, a 3 2
T 23. 1, q 1 2 29. q, a 4 7b 35. 5, q 3 2 13. 19. 3, 14 2, q 2 2 1 1 25. (2, 4) 31. 37., q 7 b 17 S x 6 b c a 15. 21. 27. 33. 8, q 2 q, 8 5 T 3 a 3, 1, 5 2b 1 8b S S between Joan’s actual weight and her ideal weight is x 120. 39. c 6 x 6 a c 41. 1 x 3 Answers to Selected Exercises 1061 43. x 9 121 2 x 1 133 2 or or x 9 121 2 x 1 133 2 45. 47. 49. 51. 53. or x 1 or or 2.26 x 0.76 1 6 x 6 2 x 3.51 or 55. 0.5 6 x 6 0.84 57. x 6 1 3 or x 7 2 59. 2 6 x 6 1 or 1 6 x 6 3 61. x 7 1 63. x 9 2 or x 7 3 65. 67. 3 6 x 6 1 or 17 6 x 6 17 x 5 or x 7 5.34 69. x 6 3 or 1 2 6 x 6 5 71. x 7 1.43 73. x 3.79 or x 0.79 75. Approximately 8.608 cents per kwh 81. 77. More than $12,500 1 6 x 6 19 10 6 x 6 35 and 83. y 20 x 85. 1 t 4 79. Between $4000 and $5400 87. 2 6 t 6 2.25 Section 2.5.A, page 131 1. 4 3 x 0 5. x 6 2 or x 7 1 9. x 6 53 40 or x 7 43 40 3. 7. 11. 6 x 6 11 6 7 6 x 11 20 x 7 2 13. x 6 5 or 5 6 x 6 4 3 or x 7 6 5x 15 5x 4 0 0 1 0 0 1 11 5x 4 2 6 E. 0 11 0. Thus, 2 Chapter 2 Review, page 135 1. 5. x 2.7644 x 3.2843 9. No real solutions 3. 7. x 3.2678 x 1.6511 11. z 3 ± 2111 5 13. x 3 or 3 or 12 or 12 15. 2 17. gold, 3 11 19. 2 2 9 hrs 25. 25 oz
; silver, 8 11 oz 21. 9.6 ft 23. 4 ft 27. b 1 2 2 1 29. x 1 2 or 11 2 31. 33. 0 4 3x 1 0 3x 1 4 or 3x 1 4 3x 3 3x 5 x 5 3 x 1 2x2 x 2 0 Squaring both sides, x2 x 2 0 x 1 0 2 x 2, 1 x 2 21 1 Both of these check in the original equation. x2 6x 8 x 1 0 35. Set the numerator equal to 0. or x 1 4 or x 5 4 1 x 2 x2 6x 8 0 x 4 0 2 x 2, 4 21 1 7 6 x 6 3 37. 15. 17. 19. 21. 23. 25. 1 6 x 6 13 x 1 or x 7 16 or 1 x 0 13 6 x 6 1 x 6 16 x 2 or 0 6 x 6 2 3 1.43 6 x 6 1.24 or 2 6 x 6 8 3 or 27. x 6 0.89 or x 7 1.56 29. x 2 or x 14 3 31. 1.13 6 x 6 1.35 or 1.35 6 x 6 1.67 then multiplying both sides by 5 33. If 0 x 3 0 shows that. 0 x 3 But 5 x 3 2 0 0 1 0 1062 Answers to Selected Exercises Neither solution causes the denominator to be 0. x 5 15 39. No solutions 2 41. a. 8, q 1 2 b. 1 q, 5 4 43. 7 4 S, q b 45. 47. 51. 55. 57. 59. 61. b 49. e 53. x 7 or x 7 4, q and 1 a 3 0 x 1 q, 2 2 or 1 x 1 4, 5 8b S x 6 213 or x 6 1 113 6 3 6 x 6 213 x 7 1 113 or 6 y 17 x 4 3 or y 13 or x 0 Chapter 2 can do calculus, page 139 1. Numerical Method a. Each base must be greater than 0 and less than 10 yards. The nr in the chart indicates that no rectangle can be formed with a base length of 5 yards or more because the opposite bases of a rectangle are the same length, and, since 5 5 10, make the sides of a rectangle. there would be no wire left to b. length 1 yd 1.5 yd height 4 yd 3
.5 yd 2 yd 3 yd 2.5 yd 2.5 yd 3 yd 2 yd 3.5 yd 1.5 yd 4 yd 1 yd area 4 yd2 5.25 yd2 6 yd2 6.25 yd2 6 yd2 5.25 yd2 4 yd2 2.25 yd2 4.5 yd 5 yd 5.5 yd 0.5 yd nr — nr — 6 nr b. x y 1 2 3 4 5 5.92 5.66 5.20 4.47 3.32 area 5.92 11.32 15.6 17.88 16.6 — 6 c. 20 c. 8 0 0 d. The maximum area 6.25 yd2 appears to occur when the base length is 2.5 yd. Analytical and Graphical Method 2l 2w 10 2w 10 2l 5l l2 A l2 5l 2 8 0 0 6 Using the maximum finder on a graphing calculator indicates that the maximum area of 6.25 occurs at approximately 2.5 yd. yd2 2. Numerical Method a. The base must be greater than 0 and less than 6 units. The nr in the table indicates that no rectangle can be formed with a base of 6 or more units because exist if x 7 6. y 236 x2 would not 0 0 7 d. A maximum area 17.88 square units appears to occur when x is 4. Analytical and Graphical Method y 236 x2 A x y A x 236 x2 20 0 0 7 Using the maximum finder on a graphing calculator indicates that the maximum area of 18 square units occurs when x is approximately 4.24. 3. Numerical Method a. The base must be greater than 0 and less than 4 units. The nr in the table indicates that no rectangle with a base of 4 or more units if x 4. because y 0 Answers to Selected Exercises 1063 c. 3 0 0 b. x y 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1.75 1.5 1.25 1 0.75 0.5 0.25 4.0 nr area 0.875 1.5 1.875 2.0 1.875 1.5 0.875 — 11. y is not a function of
x. 13. 23 1 2.73 15. 322 3 22 1 1.69 17. 4 19. 34 3 21. 59 12 a k 2 2 1 a k 2 23. 25. 1 1 2 x 2 27. 8 33. t2 1 2 1 2 6 4x x2 1 2 x 29. 1 35. 1 2 x 2 1 s2 2s s 1 31. 1 37. 3 2 4 39. 2x h 1 41. 1 2x h 2x 43. All real numbers 45. All real numbers d. A maximum area of 2 square units appears to occur at x 2. Analytical and Graphing Method 4x x2 2 3 0 0 4 Using the maximum finder on a graphing calculator indicates that the maximum area is 2 when x is approximately 2. Chapter 3 Section 3.1, page 148 47. All nonnegative real numbers 49. All nonzero real numbers 51. All real numbers 53. All real numbers except 2 and 3 55. [6, 12] 57. a. the greatest integer less than or 4 2 1 ; 0 0 f 3 equal to 0 is 0. f f f 1.6 2 1 1 1 1 4 2.3 3 1.6 3 2 2.3 5 2p b. c. 4 5 2p d. e. The domain of f is all real numbers. 5.76 3 1.283 0 2 4 4 3 3 2 f f 59. a. 1 0.69 c. 1 e. The domain of f is all real 0 2 2.3 b. d. f f 1 1 2 5 f f 61. a. c. e. The domain of f is all real numbers. 1.6 2 5 2p 0 2 2.3 7.6 b. d.920 1.6 2 5 2p 2 numbers 20. 3.4 5.566 63. a. A pr2 65. V 4x3 b. A 1 4 ˛pd2 1 x C 67. a. 5.75x 45,000 x x 7 0. 69. a. Let t be the number of hours since he started b. The domain is 2 1. The set of inputs is the number of hours you work 3t in each pay period. The set of outputs is the amounts of your paychecks. The function rule is the amount of pay for each hour worked multiplied by the number of hours worked. 3. The set of
inputs is temperatures of gas. The set of outputs is pressures of gas. The function rule is the formula P k T. 5. y is a function of x. 7. y is not a function of x. 9. y is a function of x. 1064 Answers to Selected Exercises d t 2 1 2.25 5 t 3 4b a 8.5 8.5 3 b. The domain is t such that t 2..5 2.5 6 t 4 0 t 4. of annual income and of tax. 71. Let x f 1 2 f x 1 2 x amount amount 0 0.02 1 80 0.05 u The domain is x 0. x 2000 2 x 6000 1 2 if x 6 2000 if 2000 x 6000 if x 7 6000 Section 3.2, page 160 1. 5. The domain is 7. The domain is 6, 9 7, 8.. 4 4 3 3 9. This is not a function; for example, there are three output values for an input value of 4. 11. This is not a function; for example, there are three output values for an input value of 2. 13. This is a function. 1 and 15. Increasing on 6, 2.5 17. Constant on 1, 1. 1 1 2 2.5, 0 2 0, 1.7 2 q, 1 1 4 1 2 19. Increasing on q, 5.8 1 and 5.8, 0.5 0.5, q 1 2 21. Increasing on (0, 0.867) and 1 and (0.867, 2.883) q, 0 on 1 2 2 1 2 and 1 1.7, 4 2 ; decreasing on and 1, q 3 2 ; decreasing on ; decreasing on 2.883, q ; decreasing 2 23. Minimum at x 0.57735; maximum at x 0.57735 25. Minimum at 27. Minimum at x 0.7633; maximum at x 0.7633 29. a. maximum at 50x x2 x 1; x 1 A x b. To maximize area each side should be 25 in. long. 1 2 31. a. SA x 1 2 2x2 3468 x b. Base is approximately 9.5354 in. 9.5354 in.; height is same (that is, 9.5354 in.). 33. 3.1 −4.7 4.7 Therefore, the points of inflection are approximately at
and 0.6, 0.6 1. 2 1 0.6, 0.6 10 2 37. a. 10 10 10 b. This function is increasing over the interval and decreasing over the interval c. There is a local minimum at the point (1, 0). d. This function is concave up over the interval q, 1. 2 1 1, q 2 1 1 q, q. 2 e. There is no point of inflection. 39. a. 10 10 10 10 b. This function is increasing over the intervals and decreasing over the 2, q q, 0 and 2 1 1 interval (0, 2). 2 c. There is a local maximum at the point (0, 2) and a local minimum at the point 2, 2. 2 1 d. This function is concave upward over the interval interval 1, q 2 q, 1 1 1. e. There is a point of inflection at (1, 0). 2 and concave downward over the −3.1 41. y This function is concave up over the interval q, 0. and concave down over the interval Therefore, the point of inflection is at (0, 0). 2 1 35. 2.5 −2.5 2.5 −2.5 0, q 1 2 4 3 2 1 −1 −2 −3 −4 x 1 2 The function is concave up over the approximate intervals and down over the approximate interval and concave 0.6, 0.6 q, 0.6 0.6, q. 2 1 2 1 1 2 Answers to Selected Exercises 1065 43. y 51. y 2 x 6 0 55. When 0 x 2, x is positive and 6 4 2 −5 −3 1 x 45. a. y b 2 x 2 x 0 47. a. y 2 (−4, 0) (4, 0) x (0, −2) 49. a. y b (0, 5) (10, 5) x (5, 0) 1066 Answers to Selected Exercises y = [-x] x 1 1 53. y y = 2[x] x 1 1 negative, so 0 x Therefore and x 2 0 0 x 1 0 x 2. when 0 0 x 2 is x 2. 2 1 x 2 2 57. Domain: all real numbers x such that x 2 or x 2; range: all nonnegative real numbers 59. Domain: all real numbers;
range: all real numbers 61. Many correct answers, including 6 4 2 −2 −1 2 4 −2 −4 −6 63. Entire graph: 75 69. 10 2 10 near the origin: 10 2 10 65. Entire graph: 60 16 near the origin: 2 62 2 32 5 2 2 10 10 10 10 x t, y t4 3t3 t2 71. 10 10 10 x t4 3t2 5, y t Section 3.3, page 170 3. (1, 2), downward 5. 1. (5, 2), upward y-intercept 3 y-intercept 5 9. The x-intercepts are 7. The parabola opens upward. The parabola opens downward. x 2 and x 3. The parabola opens upward. x 3 4 11. The x-intercepts are and x 1 2 ˛. The 13. Vertex is parabola opens upward. 3, 4 1 y-intercept 14. The x-intercepts are or x 1.586, 4.414.. 2 x 3 22, x 3 22 4 67. 20 10 45 5 20 5 10 10 Answers to Selected Exercises 1067 15. Vertex is 1, 4 y-intercept 5 there are no x-intercepts.. 2 1 10 25. 27. 29. 31 10 10 33x2 4x 13 2x 1 x 7 21 1 x 1 3x 1 21 x 1 2 2 2 1 2 x 5 2b a 2x2 1 49 2 2 2 10 17. Vertex is (4, 14) y-intercept 2. The x-intercepts are at approximately 0.258 and 7.742. 15 5 10 19. Vertex is 1, 4. 1 y-intercept 3. The x-intercepts are 2 10 1 and 3. 10 10 10 35. 39. f 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 1 x3 x 4 3 11. 5 3 10 13. 10 10 10 21. Vertex is 0.5, 24.5 24. 1. 2 y-intercept is The x-intercepts are 3 and 4. 10 10 10 10 23 x, h h x 2 1 23 x 1 x 2 5 15. 5 5 30 2x2 14x 20 2 Answers to Selected Exercises 5 g x 1 2 x 4 3 1, g x 1 2 x 4 3 x 23. g 1 1068 17. 10 35. 3 10 10 5 5 10 19 21. 5 5 5 5 h x 2 1 23. g 25 3x ˛, h 1 2x 3 2 1 2 0 x 3 0 27. g x 1 2 1.5 1 x 3 2 2 29. Shift the graph 4 units to the right and 1 unit upward; reflect the graph across the x-axis, and stretch it vertically by a factor of 3. 31. Reflect the graph across the y-axis, shift it 2 units to the left, stretch vertically by a factor of 4, and shift 3 units downward. 33. Compress the graph horizontally by a factor of 1, 1.3 units to the right, and shift 0.4 units 3.23 shift it upward. 3 g x 2 1 1 4 23 x 3 1, g 23 x x 1 2 37. 10 10 10 10, y x 39. 10 9.4 9.4 10 f x 1 2 3 2 x 4, f 1 x x 1 2 41. 10 10 10 10 ˛, g x 1 2 x3 Answers to Selected Exercises 1069 43. y 51. y 45. 47. 49. y y y g x 1 2 3f 53. y 55 57. 5.1 −4.7 4.7 −1.1 21 x2 2x 2 1070 Answers to Selected Exercises 59. 3.1 37. − 4.7 4.7 −3.1 321 x2 g x 2 1 61. 6.2 −9.4 9.4 −6. 63. a. Shifts upward by 28 units b. The graph is stretched vertically by a factor of 1.00012. Section 3.4.A, page 189 1. Symmetric with respect to the y-axis 3. The graph does not have symmetry with respect to the x-axis, y-axis, or
origin. However, it is symmetric with respect to the point (0, 2). 5. y 23 x x2 7. Yes 13. No Symmetric with respect to the origin. 9. Yes 15. Yes 11. Yes 17. Yes 19. Origin 21. Origin 23. y-axis 25. Odd 27. Even 29. Even 31. Even 33. Neither 35. 39. Many correct graphs, including the one shown here: (−7, f(−7)) (−5, f(−5)) −7 −5 −3 3 1 −1 −2 (−3, f(−3)) −4 (−2, f(−2)) (4, f(4)) (1, f(1)) 1 3 5 (6, f(6)) 7 1 2 2 1 x, y x, y on the graph implies that 41. Suppose the graph is symmetric to the x-axis and is 2 x, y the y-axis. If (x, y) is on the graph, then 1 on the graph by x-axis symmetry. Hence, is on the graph by y-axis symmetry. Therefore, x, y is on the 1 graph, so the graph is symmetric with respect to the origin. Next suppose that the graph is symmetric to the y-axis and the origin. If on the graph, then axis symmetry. Hence, the graph by origin symmetry. Therefore, the graph implies that is on the graph, so the graph is symmetric with respect to the x-axis. The proof of the third case is similar to that of the second case. is on the graph by y- x 1 1 x, y 1 x, y x, y 2 2 x, y, y x, y is on on is Section 3.5, page 196 1. 3. 5. f g 21 1 f g 21 1 g f 21 1 real numbers x x x 2 2 2 x3 3x 2; x3 3x 2; x3 3x 2 ; domain for each is all 1 x x 21 x2 2x 5; 2 x2 2x 5; f g 1 1 g f x domain for each is all real numbers except 0 x 21 x2 2x 5 1 x f g 21 x 2 1 1 2 ; 21 fg f gb1 1 a x 2 x 2 3x4 2x3; 3x 2 x3 ˛; a g f b1 x 2 x
3 3x 2 Answers to Selected Exercises 1071 7. fg 1 2 1 f g b1 21 x 1 x2 1 21 b1 x 2 a 2 21 21 x 1 2 2x2 1 2x 1 2x2 1 2x 1 B x2 1 x 1 2x 1 x 1 x2 1 B 2x 1 2x2 1 1 x 1 B 9. Domain of fg: all real numbers except 2; domain of f g : all real numbers except 2 11. Domain of fg: all real numbers; domain of f g : all real numbers except 3 4 10 −10 1 4 1 1 −3 x g f x 2 21 x f f x 2 21 1 1 39. 41. 43 25; f g 1 f g 1; 4 x 2 21 and 1 g f x 3 15. 1 g f 2; is all real numbers. 17. 30 x2 3; 21 x 1 2 2 domain of g f x 2 21 1 1x ; domain of f g f x 1 2 45. 0.5x2 5.5x2 5 0 13. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37 and 2 21 g f x ; 1 1x 1 0, q 2 f f is 1 x6;. ff ff x x 2 2 21 21 1 1 x2 ; 1 f g g f x 21 x 21 2 2 f a g f g g f 21 21 x7 1 1 x B A x9 x 2 x x 2 21 f f 21 x 2 9 9x 2 b 1 x 2 13 x 2 2 3 23 2 2 where 1 A where A x 21 x 21, 9x 2 9 x 2 2 1 13 and and x2 2, B 7x3 10x 17, x 13 x 1 2 x 2 21, where A x 1 2 3x2 5x 7, 49 By the definition of absolute 0 f 1 47. a. x x 3, g f 2 g f 21 1 value. When if if 0 x f 1 6 0 x f 1 the graph of f 2 2 g f 1 2 the graph of f. When below the x-axis, but the graph of reflection of f across the x-axis; therefore, the graph will be above the x-axis. is the 6 0, f x b 1 2 is the same as the graph of f is 1072 Answers to Selected Exercises y1
x3 3, y2 51. 53. 55. 2x 1, y2 y1 21 15,600 19.5n n gives the unit price as a function of n, the number of telephones produced. 57. V 256pt3 3 ; 17,157.28 cm3 59. s 10t 3 61. One such function is f x 1 x ˛. x 1 2 Section 3.5.A, page 203 1. 2, 2.16, 2.3328, 2.5194, 2.7210, 2.9387, 3.1737, 3.4276 3. 0.2, 0.64, 0.9216, 0.2890, 0.8219, 0.5854, 0.9708, 0.1133 5. 0.5, 0.375, 0.3223, 0.2888, 0.2647, 0.2462, 0.2312, 0.2189 7. approaches infinity 9. converges to 0 11. converges to 0 13. The fixed points are 3 and 2. 15. The fixed points are 0, 1, and 2 28 2 17. The fixed points are 1. and 2 28 2 ˛. 3. y 1 1 x 3 sample points on the inverse: 8, 6 1 2 2, 2 1 0, 3, 2 1, 2 5. 10 −10 10 −10 t t3 3t2 2 t3 3t2 2 t x1 y1 x2 y2 7. 10 19. Any real number greater than or equal to 0 is a fixed point. Any negative number is an eventually fixed point. −10 10 21. a. b. x 0.5; x 1 0 1 2 f x x terms of orbit: 0.5, x 1 terms of orbit: 1.5, 0.5, 0.5,... ; 0 x 0.5; is an eventually fixed point. f and 0.5, 0.5,... ; 0.5 is a fixed point. and 0.5 x f 0 2 1 0,... ; 0 is a periodic point. x 1; periodic point. x 1 terms of orbit: 0, 1, 0, 1,... ; 1 is a terms of orbit: 1, 0, 1, and x 0; x 1 and 10 t t4 3t2 t
4 3t2 t x1 y1 x2 y2 x 11. x 5y2 4, y ± x 4 5 B Section 3.6, page 212 1. y 4 2 3 6 1 f(y) 1 2 3 4 5 9. g 13. g 17. g 21 23. No 3 5 x B 2 1 x 3 B 15. g x 1 2 19. x x2 7 4, 1 x 0 1 2y2 1 ˛, y ± B 2 1 x 2x 5x 1 1 x 25. Yes 27. Yes 29. No Answers to Selected Exercises 1073 31. 10 −5 5 49. 1 g f 25 x x 21 25 x5 x A B 25 x A −10 3 51. 33. 35. −10 10 −3 5 − and g f 1 x 2 21 B f f x 1 22 1 x 2f 2 1 x 3f 2 1 2x 1 2 1 3 1 2x 1 1 2 2 1 3x 2 2 2 1 3x 2 1 2 3x 2 2 3x x5 21 2x 1 3x 2 d 2x 1 3x 2 d 7x 3x 2 7 3x 2 c x 53. Let y f x 1 2 mx b. Since m 0, we can solve for x and obtain Hence, the rule of the x y b m. inverse function g is g 1 and we have 21 x. f and b a 1 b a 2 1. g 2 f x 1 22 1 mx 22 1 b x x 2 21 x b m b a mx b 2 Slope a b b a y x 1. b. The line has slope 1, is Length PR 2 c. 55. a. has slope 1 and by (a), line PQ Since the product of their slopes the lines are perpendicular 2a2 2ac c2 b2 2bc c2 2a2 b2 2c2 2ac 2bc; 2 1 2c2 2bc b2 c2 2ac a2 2a2 b2 2c2 2ac 2bc is the Since the two lengths are the same, perpendicular bisector of segment PQ. Length RQ 2 d. 93 1 3 ft/sec 3. a. 0.709 gal/in. b. 2.036 gal/in. 5. a. 250 ties/mo c. 500 ties/mo e. g. ties/mo ties/mo 188
1500 55.5 9. 7. a. 2 2x h 17. b. 11. 92.5 1 b. 438 ties/mo d. 563 ties/mo f. h. ties/mo ties/mo 750 375 c. 462.5 13. 1.5858 19. 2t h 8000 15. 1 21. 2pr ph −5 37. One restricted function is x 39. One restricted function is that x h ; 2 1 2 h inverse function x 1 2 with x. x 0 0 x g 2 1 x2 x 0 (so x h 1 1x. with 2 x 0; with Another restricted inverse function x 0; inverse function function is g h 1 1x. x x 2 x g 2 1 x2 1 2 41. One restricted function is h 1 x 0; inverse function g x 2 1 x2 6 2 x 2 12x 6. with 43. One restricted function is f x 0; inverse function x 1 2 1 x2 1 with 21 21 f g x 2 x 2 45. 47. 1 1 1 f x 2 21 f 1 g x g 22 1 x f 22 and and 2 1 2 1 1 x x x 2 21 a g f 1 b g 1 x 1b a 1074 Answers to Selected Exercises. 1. a. 14 ft/sec b. 54 ft/sec c. 112 ft/sec Section 3.7, page 220 23. a. Average rate of change is 7979.9, which b. The graph represents a function of x because it means that water is leaving the tank at a rate of 7979.9 gal/min. 7979.99 gal/min. 6.5p 7980 6.1p gal/min. 6.2p d. c. c. 6p b. 25. a. b. e. It’s the same. 27. a. C, 62.5 ft/sec; D, 75 ft/sec t 9.8 sec b. Approximately c. The average speed of car D from t 4 to t 4 to t 10 sec is the slope of the secant line joining the (approximate) points (4, 100) and (10, 600), namely, 600 100 10 4 83.33 ft/sec. The average speed of car C is the slope of the secant line joining the (approximate) points (4, 475) and 800 475 10 4 54.
17 ft/sec. (10, 800), namely, 29. a. From day 0 until any day up to day 94, the average growth rate is positive. b. From day 0 to day 95 c. 27, meaning that the population is decreasing at a rate of 27 chipmunks per day d. 20, 10, and 0 chipmunks per day Chapter 3 Review, page 226 11 3 7 2t 9 1 2b 2 2 7 5 3 3 x3 4 2 2 x 2 x 2b a 1 5 2b 7 2x 2h passes the vertical line test. 21. y x 23. −6 −4 1 −1 6 4 2 −2 −2 −4 −6 4 y x = t2 − 4 y = 2t + 1 (−3 ≤ t ≤ 3) x 2 4 6 25 20 5. All real numbers 2 except for 3. 7. a. For 20 miles, it costs $150. For 30 miles, it costs $202.50 b. 39 miles −10 10 1 2 1 1, f 1 2b a 1, f a 3 2b 2 −10 9. f 0 0, f 2 1 3, 3.5 11. 3 15. No local maxima; minimum at 13. 4 3 x 0.5; 0.5, q increasing on ; 2 This function is concave up for all values of x; there are no points of inflection. decreasing on 1 1 q, 0.5 17. Maximum at 1 minimum at q, 5.0704 x 5.0704; x 0.2630. Increasing on and (0.2630, ); decreasing on (5.0704, 0.2630) This function is concave up on the interval 8 a b 3 q, 8 3b and concave down on the interval There is a point of inflection at, q. 2 a x 8 3. 19. a. The graph does not pass the vertical line test; therefore, it does not represent a function of x. Vertex is (4, 1). The y-intercept is 17. There are no x-intercepts. x2 2x 7 x f 1 2 2 27. 5 −15 10 −15 1, 6 Vertex is There are no x-intercepts.. 1 2 The y-intercept is 7. Answers to Selected Exercises 1075 29. 10 −10 10 −10 Vertex is 1 The
y-intercept is The x-intercepts are 0.35, 4.2025 2 4.08. 2.4. and 1.7. 39. Compress the graph of g toward the x-axis by a factor of 0.25, then shift the graph vertically 2 units upward. 41. Shift the graph of g horizontally 7 units to the right; then stretch it away from the x-axis by a factor of 3; then reflect it across the x-axis; finally, shift the graph 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. 1 −10 55. 4x 57. 51. Odd c. 2 1 x3 3 35. parent function: g x 1 2 x 0 0 −10 10 −10 10 37. parent function: f 1 10 x2 x 2 59. 82 27 x 3, 2 6 x 6 3 f 61. For and x 2 the same. For g f is the graph of of f across the x-axis. the graphs of f and, the graph of g f are, a reflection of the graph f x 63. x2, g 65. 2, 1, 0, 1, 0, 1, 0, 1 x 2 1 1 2 2x 1 67. y Inverse f x −10 10 −10 69. 71. 1076 Answers to Selected Exercises 73. x 25 y3 1, f x 1 y2, y ± 21 x x 1 2 x 2y 1, f 1 x 2 1 1 23 x5 1 x 2 1 75. The graph of f passes the horizontal line test and hence has an inverse function. It is easy to verify either geometrically [by reflecting the graph of f across the line f calculating function. ] or algebraically [by ] that f is its own inverse y x x 22 f 1 1 y 3 f(x) = −3 −3 1 x 3 x 2. 0.625 seconds The instantaneous velocity of the ball is 0 when the ball reaches its maximum height. Thus, the
maximum height of the ball will be 81.25 feet. 3. s t 1 4. 14 2 16t2 300; 5. 96 0.111111 feet per second 6. 3 7. 2a 8. instantaneous rate of t 4: tangent line at change 16 y 16t 76 ; equation of 100 −2 8 −100 77. 79 21 21 0.25 4 3 g f x 2 21 1.5 x 1.5 1.5 x 6 x 6 6 x 7x x 3 1 1 2 1 4x 6 2 0.25x 1.5 2x 1 x 3 b a 2x 1 x 3 b 2x 1 2 1 2 x 3 2x 3x 1 x 2 b a 3x 1 x 2 b 3x 1 3 1 2 x 2 3x 1 3 1 x 2 x 2 7 1 7x 7x x 3 7 x 3 7x 21 1 89. a. For example, from 3 to 1 b. For example, from 1 to 2 c. For example, from 6 to 8 d. Both intervals are portions of the same line, so their slopes are the same. 91. a. $290/ton b. $230/ton c. $212/ton Chapter 3 can do calculus, page 237 1. t s 2 1 44 16t2 20t 75 feet per second change 25.132741 9. instantaneous rate of r 1, for each change of 1 unit in the When radius, the surface area of the sphere will increase by approximately 25.132741 square units. 10. instantaneous rate of change 100 dollars per phone When sold, the profit increases by approximately $100. for every additional phone x 1000, Chapter 4 Section 4.1, page 248 1. Polynomial of degree 3; leading coefficient 1; constant term 1 3. Polynomial of degree 3; leading coefficient 1; constant term 1 5. Polynomial of degree 2; leading coefficient 1; constant term 3 7. Not a polynomial 9 3x3 2x2 4x 1; 9 8 1 7 8 33 0 3 2 9 11 5 6 1 2x3 x2 3x 11; 3 25 2 quotient remainder 25 13. 7 5 5 0 35 3 245 4 1,694 6 11,830 242 35 1,690 5x3 35x2 242x 1690; 11,836 quotient remainder 11,836 Answers to Selected Exercises 1077 1 3 81.
a. 83. 6 b. 5 8 85. 3 87. 2x h 11. quotient remainder 7 3 2 3 2 9. No 11. Degree 3, no; degree 4, no; degree 5, yes 15. 2 1 1 quotient remainder 17. Quotient 19. Quotient 21. Quotient 23. No 29. 222, 1 30 41. No 35. 4 8 4 6 2 4 x3 4x2 4x 6; 2 8 6 7 12 19 19 3x3 3x2 5x 11; x2 2x 6; 5x2 5x 5; remainder remainder 0 remainder 12 7x 7 25. Yes 31. 2 27. 0, 2 33. 6 37. 170,802 39. 5,935,832 43. No 45. Yes 1 1 x 21 1 f 49. 51. 47. 2x 7 x 3 21 x 2 3x 5 2 21 2x 1 2 21 x 1 x 4 x 3 x 3 21 2 x5 3x4 5x3 15x2 4x 12 x x 1 2 x5 5x4 5x3 5x2 6x 55. Many correct answers, including 53. 21 21 21 21 21 21 21 57. Many correct answers, including 21 17 100 1 21 x 5 21 1 2 x 8 x 2 59. 61. f x 2 1 k 1 k 1 63. x4 x2 1, 65. If x c were a factor of then c would be a solution of c4 c2 1. satisfy is impossible. Hence, x4 x2 1 0, c4 0 But x c that is, c would c2 0, and is not a factor. so that 67. a. Many possible answers, including: if n 3 and then since c 1, x3 1 x 1 x 1 b. Since n is odd solution of is a factor of 1 1 is not a solution of c 2 xn cn 0. xn cn and hence c x by the Factor Theorem. is not a factor of x3 1 0. c is a x c n cn Thus, 2 2 1 1 69. k 5 71. d 5 Section 4.2, page 258 1. x ± 1 or 3 3. x ± 1 or 5 5. x 4, 0, 1 or 1 2 7. x 3 or 2 9. 13. x 2 x 2 2x2 1 x 1 x 2 2 21
5; 19. Lower 17. 21 2 1 1 2 x2 3 11. 15. x 5, x2 3 x3 2, or 3 x 2 21 2 1 1 upper 2 2 21. Lower 7; upper 3 23. x 1, 2, or 1 2 25. x 1, 1 2, or 1 3 1078 Answers to Selected Exercises 27. x 2 or 5 ± 137 2 31. x 1, 5, or ± 13 29. 33. x 1 2 x 1 3 or ± 12 or ± 13 or 1.8393 35. x 2.2470 or 0.5550 or 0.8019 or 50 37. a. The only possible rational zeros of 1 2 x are x2 2 f zero of f(x) and irrational. 23 rational zeros are 13 ± 1 is a zero of ± 3. or b. ± 2 ± 1 or 12 ± 1 (why?). But ± 2. Hence, or 12 12 is a is x2 3 ± 1 whose only possible ± 3 (why?). But or 39. a. 8.6378 people per 100,000 b. 1995 41. 2 by 2 in. c. 1991 43. a. 6° /day at the beginning; b. Day 2.0330 and day 10.7069 c. Day 5.0768 and day 9.6126 d. Day 7.6813 6.6435° /day at the end Section 4.3, page 269 1. Yes 3. Yes 5. No 7. Degree 3, yes; degree 4, no; degree 5, yes 13. The graphs have the same shape in the window but and 1000 y 5000 40 x 40 with don’t look identical. 2 2 zero of even multiplicity. and 1 15. 17. is a zero of odd multiplicity, as are 1 and 3 are zeros of odd multiplicity; 2 is a 19. (e) 21. (f) 23. (c) 25. The graph in the standard viewing window does not rise at the far right as does the graph of the highest degree term so it is not complete. x3, 27. The graph in the standard viewing window does not rise at the far left and far right as does the graph of the highest degree term not complete 20 y 40 60 y 320 35 y 20 0.005x4, so it is and and and 33. 31. 29. 33 x 2 right half: 35.
Left half: 250,000; 90 x 120 37. 39. Overall: y-axis: and 3 x 3 0.1 x 0.2 20 y 30 50,000 y and and 2 x 3 15,000 y 5000 20 y 20; and near 4.997 y 5.001 and 41. a. The graph of a cubic polynomial (degree 3) has at most 3 1 2 local extrema. When is large, x 0 0 ax3, the graph resembles the graph of that is, one end shoots upward and the other end downward. If the graph had only one local extremum, both ends of the graph would go in the same direction (both up or down). Thus, the graph of a cubic polynomial has either two local extrema or none. b. These are the only possible shapes for a graph that has 0 or 2 local extrema, 1 point of inflection, is large. and resembles the graph of when x ax3 0 2, 0, 4, and 6 0 43. a. Odd b. Positive c. 45. (d) 47. y (0, 4) 4 2 −2 (−1, 0) −4 −3 −2 f(x) = x3 − 3x2 + 4 x 1 2 3 4 (2, 0) 49. y (2, 4) 4 2 h(x) = 0.25x4 − 2x3 + 4x2 x −4 −3 −2 −1 (0, 0) −2 1 2 3 4 5 (4, 0) 53. y (−1.30, 2.58) (−0.34, 0.77) (−1.69, 0) 4 2 f(x) = x5 − 3x3 + x + 1 (0.34, 1.23) (1.51, 0) x −4 −3 −2 −1 −2 1 2 3 4 (1.30, −0.58) d. 5 55. (−0.71, 174.27) y 175 150 125 100 75 50 (−3, 0) (−2.75, 0) −5 −4 −3 −2 −1 1 2 3 (−2.88, −2.34) h(x) = 8x4 + 22.8x3 − 50.6x2 − 94.8x + 138.6 (1.4, 0) (1.5
, 0) 1 2 x (1.45, −0.37) 57. a. The solutions are zeros of 2 1 x and 3.99 y 4.01 4 0.01x3 0.06x2 0.12x 0.08. g This polynomial has degree 3 and hence has at most 3 zeros. 1 x 3 b. c. Suppose f(x) has degree n. If the graph of f(x) y k had a horizontal segment lying on the line k x for some constant k, then the equation would have infinitely many solutions (why?). x has degree n (why?) But the polynomial 2 and thus has at most n roots. Hence the equation f the graph cannot have a horizontal segment. has at most n solutions, which means 51. y 25 −8 −7 −6 −5 −4 −3 −2 −1 −25 (−.12, −44.73) −50 (6.72, 0 10 g(x) = 3x3 − 18.5x2 − 4.5x − 45 −75 −100 −125 −150 −175 (4.23, −167.99) Answers to Selected Exercises 1079 59. a. The general shape of the graph should be as b. shown here. The graph should cross the x-axis at the points specified, and bounce off the axis at 2. y 0.5179820180x2 20.88711289 80°; 9 A.M.: 69°; c. Noon: 2 P.M.: 14.65684316x 9. a. 50,000 1 0 83° 20 b. On the TI-83, only 3 of the roots are shown. 2 is skipped over entirely. b. Quartic c. y 1.595348011x4 58.04379735x3 630.033381x2 2131.441153x 36466.9811 d. $42,545.95 e. According to this model, income will drop steeply after 2002. 11. a. 24,000 0 0 22 b. y 0.084189248x4 0.528069153x3 66.26642628x2 397.2751554x 3965.686061 c. 1996: $19,606; The estimate is lower. Section 4.4, page 290 1. All real numbers except 3
. All real numbers except 5. All real numbers except 5 2 3 15 12, 7. Vertical asymptotes x 1 x 0; 9. Hole at vertical asymptote 11. Hole at x 2; vertical asymptotes 13. any window with and 3 15 12 and 1, and x 6 x 1 x 2 115 x 110 31 x 35 any window with any window with 40 x 42 y 3; y 1; y 5 2 ; 15. 17. 19. Asymptote: y x; window: 14 x 14 and 15 y 15 21. Asymptote: y x2 x; window: 15 x 6 and 40 y 240 c. Again, d. Try the windows x 2 is missed. 20 x 3, 5,000,000 y 1,000,000. 3 x 2, part 5 x 11, 5000 y 60,000. 5000 y 5000 100,000 y 100,000 1 x 5, Then try For the third and lastly Section 4.3.A, page 276 1. Cubic 3. Quadratic 5. a. y 0.634335011x3 11.79490831x2 51.88599279x 4900.867065 b. 1987: 4898.0 per 100,000; 1995: 4635.6 per 100,000 c. 3138.2 d. Answers may vary. 7. a. 100 5 0 20 1080 Answers to Selected Exercises 23. y 29. y −6 −4 −2 2 −2 x 2 2 −1 −2 1 2 4 x vertical asymptote horizontal asymptote x 5 y 0 25. y 2 −2 −3 −2 31. x 2 vertical asymptote horizontal asymptote x 2.5 y 0 27. y 6 4 2 −1 −2 2 x 33. vertical asymptote horizontal asymptote y 3 x 1 vertical asymptote horizontal asymptote x 3 y 1 y 6 4 2 −3 −2 −1 1 2 3 x −2 −4 −6 −8 −10 vertical asymptotes horizontal asymptote x 11 −2 −5 −3 x 1 2 3 vertical asymptotes horizontal asymptote y 0 x 2, x 1 Answers to Selected Exercises 1081 35. y 431 −2 −3 x 1 2 3 vertical asymptotes horizontal asympt
ote y 0 x 5, x 1 37. y x −1 −2 −3 −4 vertical asymptote horizontal asymptote y 4 x 0 39. y 4 2 −2 1 3 5 −2 −4 vertical asymptotes horizontal asymptote x 1, x 5 y 1 y 41. 4 2 −2 −4 −6 −4 −2 vertical asymptotes x 3 hole at horizontal asymptote y 0 x 4, x 5 1082 Answers to Selected Exercises vertical asymptote x 2 x 3 1 −2 1 3 4 −3 oblique asymptote y x 1 45. y 28 24 20 16 12 8 4 x −4 −2 −4 −8 −12 y = 2x + 7 x 1 32 4 vertical asymptote oblique asymptote x 5 2 y 2x 7 47. y x 2 4 6 8 y = x2 + x + 1 −2 12 10 8 6 4 1 −2 −4 −6 x 1 2 3 vertical 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 x 2. 4.7 x 4.7 55. Overall: x 1; hole at 0.65 x 0.75 and and 2 y 2; there is a 73. a. to see the vertical asymptote, use and 3 y 3. 57. For vertical asymptotes and x-intercepts: 4.7 x 4.7 and close to the horizontal asymptote: and 8 y 8; 2 y 3 to see graph get 40 x 35 and 3 x 15 2 y 2; and hidden 0.02 y 0.01 61. 63. 59. Overall: and 4.7 x 4.7 x 4: area near 15.5 x 8.5 4.7 x 4.7 and 13 x 7 area
near the origin: 0.02 y 0.02 65. Overall: 16 y 8 12 y 8 20 y 20; and 2.5 x 1 and hidden b. 20 e. x p 1 2 4x 10 x 3 f. Shift the graph of f(x) horizontally s units (to 0 0 s 6 0 ); stretch (or r to the right if (away from s 7 0; r the left if shrink) the graph by a factor of 7 1, the x-axis if 0 6 1 0 6 r ); also if the x-axis; then shift vertically downward if if r ts 0 t 7 0; t 6 0 ). 0 0 0 t 0 tx 2 1 x s 0 toward the x-axis if r 6 0, reflect the graph in units (upward 0 g. 69. a. b. c.2 1 4.002 change 1 9.3 1 9.003 0.2381; 1 4.02 0.2488; 0.2499; instantaneous rate of 0.25 1 4 0.1075; 1 9.03 0.1107; 0.1111; instantaneous rate of change 1 9 0.1111 p d. They are the same. 71. a. y x 1 x 2 b. y x 1 x 2 c. Graph (a) has a vertical asymptote at graph (b) has a vertical asymptote at and x 2 x 2. 2x2 1.25 100 a 4x 1000 x2 b 2 0.06x2 3 C x 2 100 1 1 50 x x 16.85 in. 20 x 50 x x c 1 2 b. 75. a. b. between 25 gallons and 100 gallons c. x 50 gallons c a x x 1 x 2 1 2 40,000 2.60x x 77. a. 67. b. Stretch the graph of f(x) away from the x-axis by a factor of 2. c. The graph of h(x) is the graph of f(x) shifted vertically 4 units upward; the graph of k(x) is the graph of f(x) shifted horizontally 3 units to the right; the graph of t(x) is the graph of f(x) shifted horizontally 2 units to the left. d. Shift the graph of f(x) horizontally 3 units to the right, stretch vertically by a factor of 2, then shift vertically 4 units upward. c. 79. a
. y 2.60; $2.60. v 50u u 50 b. v 50 0 0 100,000 the average cost can never be below Answers to Selected Exercises 1083 c. 100 Section 4.5.A, page 306 15. 4 17. i 19. i 21. i Section 4.6, page 313 1. 3. 5. 1 1 f 3 0.39.4521 0 0 ; 2 2 2 02 0.4521. 0.5 i; 0.3; f 2 1 0.4521 2 0.5 0.5i; f 2 0 2 1 d 1.5207 0.25 1.5i ; 1.2 0.5i; 0.01 0.7i; f 2 0 2 1.6899 0.514i d 1.7663. The seventh iteration is more than 2 units from the origin. 9. The thirteenth iteration is more than 2 units from the origin. 11. The eighth iteration is more than 2 units from the origin. 13. i is the Mandelbrot set. 15. 1 is not in the Mandelbrot set. 17. The cycle approaches approximately The number 0.2 0.6i is in the 0.275 0.387i. Mandelbrot set. 1. g(x) is not a factor of f(x). 3. g(x) is not a factor of f(x). 5. g(x) is not a factor of f(x). 7. x 0 (multiplicity 54); x 4 5 (multiplicity 1) x p (multiplicity 14); (multiplicity 15); (multiplicity 13) 9. 11. 13. x 0 x p 1 x 1 2i 15. x 3 or 17. 19. 21 15 f x 1 2 A i b 21 1 2i; or x 1 2i 215 i or 3 215 x 1 a 3 3 313 2 x 3 2 3 2 x 3 2a 1 13i i b a 1 3 i; 215 3 x 1 2i 2 215 3 x 1 3 313 2 x 3 2 3 or 2 313 2 1 13i; i ba i; i or x 2 1 or i or x 1 or x 1 13i i; 2A 1 BA or x i x 1 21 15 or x 25 21 12i or x 15 or BA BA 2 x i 21 12i; x 22i x 1 13i
B 23. Many correct answers, including 21 25. Many correct answers, including 21 1 2 f 27 2x 21 x 4 1 2 x 3 3 2 1 21 2 x 12i B BA 313 2 i b 50 0 35,000 d. If the object is close, a small change in u leads to a large change in v. However, when u is large, a small change in u leads to nearly no change in v, so that u may change substantially while the object stays in focus. 3. 2 10i 12 13 a 2 10 11i b 7. 11. 2i 25. 31. 1 3 i 10 17 11 17 i 37. 6i 27. 33. 12 41 7 10 15 41 11 10 i i 41. 45. 49. 4i 215 322 A 41 i i B Section 4.5, page 300 1. 8 2i 43. 11i 2i 1 2 1 13i 21 20i 2 29 i 4 41 i 5 29 5 41 113 170 214i 41 170 i A i B 2 3 2 522 A 12 ± 13, y 3 2 17 2 ± 2 5. 9. 13. 23. 29. 35. 39. 47. 51. 53. 57. 61. 65. 67. 69. 25 2210 i 55. 59. 63. i B x 2, y 2 x 1 3 x 1 4 ± ± 114 3 131 4 i i 1 x 2, 1 13i, 1 13i x 1, 1, i, i 71. z a bi, 73. If with a, b real numbers, then 2 1 z a bi a bi and hence, 2bi. If z z 2bi 0. z z, Conversely, if then b 0. which implies that is Hence, a bi z z 2 1 b 0 real, then z z. Therefore, 0 z z 2bi, z a is real. 75. 1 z a a2 b2R Q b a2 b2R i Q 1084 Answers to Selected Exercises 2 7. 9. 11. x 44 7 1 4 ± 13 3 2 1 2, x f x 2 x 2 AA A 3 213 BB A 1 13. x A 3 213 BB 15. a. There are no rational zeros b. An irrational zero lies between 3 2. and 0. A third is and 1 Another lies between between 1 and 2. 17. 3 19. d 21. When x4 4x3 15 synthetically, the
last row, 1 alternating signs. Therefore, for the real zeros. 1 is divided by 5 20, has is a lower bound x 1 5 5 1 23. rational zeros: and 4; irrational zero: 1.328 is a zero of multiplicity 2; 4 is a zero of is a zero of multiplicity 1; 3 is a 3 multiplicity 1; zero of multiplicity 1 3 2 i 25. z a bi c di 2 z w. and 27. Answers may vary. 29. i, iv, and v are false. 29. 31. 33. 35. 37. 39. 41. 43 x2 2x 5 2 1 1 21 21 21 x2 4x 5 x2 2x 2 x2 4x 5 x 2 x 3 x2 2x 5 x 4 1 x4 3x3 3x2 6x 6 2x3 2x2 2x 2 x2 6x 10 2 x2 2x 2 21 2 1 1 2 2 45. Many correct answers, including 1 i 47. Many correct answers, including x2 49. x f 2 1 3, 1 2 1, 2i, 2i 57. a. Since 53. x3 5x2 13 i, 1 2 2 2 1 7 2i 13 2 i x 3 6i 1 2 i, i, 1, 2 51. Hence 2 1 2 ac bd zw 2 ac bd ad bc 2 w c di, z w ad bc i. z w w c di, a c 1 b. Since zw and ac bd i, i, 55. b d 1 2 b d i. 2 a bi 2 i, i, 2 Since 1 2 z w ad bc i, 2 i. Since 2 a bi Hence 1 1 1 1 1 z a bi c di 21 zw z w. 2 (definition of f(z)) 59. a. 1 f z 1 2 2 1 2 az3 bz2 cz d az3 bz2 cz d a z3 b z2 c z d az3 bz2 cz d az3 bz2 cz d f z 1 b. Since f Hence 0, z we have 2 1 is a zero of f(x). z 2 (Exercise 57(a)) (Exercise 57(b)) r r for r real) ( (Exercise 57(b)) (definition of where gi1 g31 gk 1 61. If f(z) is a po
lynomial with real coefficients, then g11 degree g11 p degree g21 is a polynomial with real coefficients f(z) can be factored as each and degree 1 or 2. The rules of polynomial multiplication show that the degree of f(z) is the degree sum: gk1 z g31 z. degree 2, then this last sum is an even number. But f(z) has odd degree, so this can’t occur. Therefore, at least one of the is a first-degree polynomial and hence must have a real zero. This zero is also a zero of f(z). degree gi1 z g21 2 If all of the have gi 1 z z z 2 2 2 2 2 Chapter 4 Review, page 317 1. (a), (c), (e), (f) 5. 0 17 10 7 16 14 2 1 5 other factor: x5 3x4 2x3 5x2 7x 2 0 4 4 b. 31. Use 10 x 20 and 2 x 2 overall graph and for behavior around the origin. 10,000 y 500 and 10 y 5 for the 33. Use 0.2 x 2 and 20 y 40; 1 x 0.2, 20 y 20 35. Use 2 x 3 and 1, 0, 5 y 5. a local maximum at (0, 3), There is an x-intercept at 1 2 and a local minimum at. a, 49 4 27b 3 5 y 5. x 0.618, 1.618 and There are and a local 37. Use 3 x 3 x-intercepts at minimum at x 0.909. 39. a. 120,000 0 0 25 y 0.04x4 29,552.18; 0.19x3 62.06x2 $79,223 in 2007 and $127,317 in 2015 3.1 y 3.1. 1615.35x There is a and x 2 and a horizontal 41. Use 4.7 x 4.7 vertical asymptote at asymptote at y 1. Answers to Selected Exercises 1085 43. Use 4.7 x 4.7 3.1 y 3.1. vertical asymptote at a hole at the x-axis is the horizontal asymptote. x 1, and There is a, and x 1 Chapter 5 Section 5.1, page 334 5. 0.09 11. 81
3 ± 131i 10 51. 55. 57. 59. 2 A 3 1 23i or x 2 A 3 x 2 or i, i, 2, 1 or i or i or 1 23i 61. Many correct answers, including x4 2x3 2x2 x f 1 2 63. a fixed orbit of one point: 65. 67. 69. 1 1 1 21 x 1 x 1 21 x2 1 21 x 2 x 2 21 x2 1 x 3 ; 2 x2 1 1 ; 2 x i ; 2 1 21., 3 a 5 x 1 21 x 1 x i 4 5b x 2 21 x 2 21 x i 1 21 21. x 3 2 x i 21 x i 2 21 x i. 2 21 Chapter 4 can do calculus, page 325 b. 1. a. 1, 4 1 2. a. (1, 0) 2 1 3. a. approximately b. 1 2 and (2, 4) 1, 4 2, 0 or (1, 0) 3.785 cm 6.980 cm 2 or c. (3, 20) c. 1 3, 16 2 b. 8.560 cm 1.365 cm V 126.49 cm3 approximately 6.324 cm and height is approximately 3.163 cm. when side length is 4. a. approximately 4.427 inches by 4.427 inches. b. The largest volume occurs when x 10 3. 5. a. r 4.09977 b. r 1.996; 37.566 in2 6. a. Approximately 206 units are produced. b. The minimum value of about 577 dollars per unit occurs when about 269 units are produced. 7. r 1.769, h 58 pr 2 5.8996 8. The maximum area of about 220.18 square feet occurs when x is about 9.31 feet. 9. 4 sq. units 10. 5 x2 1.5. The point that is closest to (0, 1) has the exact value of 7 2, 3 2 b aB are okay., but approximations 1086 Answers to Selected Exercises 99. 1.5 19. g x 1 2 2x5 10 −3 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. Shift the graph of h vertically 5 units downward. 3. Stretch the graph of h vertically by a factor of 3. 31. Neither 35. When x is large, 5. Shift the graph of h horizontally 2 units to the left, then vertically 5 units downward. 37. 4 33. Odd x 0, e so 1 e1 e ex e 39. 1 x ex 0 ex. 3 e3 e 1 2 8.84 7. Shift the graph of h vertically 4 units upward. 9. Compress the graph of h vertically by a factor of 1 4. 41. xh 51 2 2 5x 2 h 43. 1 exh e ex e x 1 2 45. The x-axis is a horizontal asymptote; local 11. Reflect the graph of h across the y-axis, then shift maximum at (1.443,0.531). horizontally 2 units to the right. 13. Reflect the graph of h across the y-axis, stretch horizontally by a factor of 1 0.15 6 2 3, then stretch vertically by a factor of 4. x 15. f x 1 2 5 2b a 10 0 10 0 −5 17. x g 1 2 3 x 2 −5 5 5 47. No asymptotes; local minimum at (3, 0.0078). 49. No asymptotes; no extrema. 51. a. About 520 in 15 days; about 1559 in 25 days b. in 29.3 days 53. a. 1980: 74.06; 2000: 76.34 b. 1930 55. a. 100,000 now; 83,527 in 2 months; 58,275 in 6 months b. No. The graph continues to decrease toward zero. 57. a. The current population is 10, and in 5 years it will be about 149. b. After about 9.55 years. 59. a. Not entirely x b. The graph of f81 appears to coincide with the graph of g(x) on most calculator screens; when 2.4 x 2.4, the maximum error is at most 0.01. 2 c.
Not at the right side of the viewing window; x f121 2 Section 5.3, page 353 1. Annually: $1469.33; quarterly: $1485.95; monthly: $1489.85; weekly: $1491.37 3. $585.83 5. $610.40 7. $639.76 Answers to Selected Exercises 1087 9. $563.75 11. $582.02 49. y 13. About $3325.29 15. About $3359.59 17. About $6351.16 19. About $568.59 21. Fund C 23. $385.18 25. About $1,162,003.14 27. $4000 29. About 5.00% 31. About 5.92% 33. a. About 9 years; about 9 years; about 9 years b. Doubling time is not dependent on the amount invested, but on the rate at which it is invested. 35. About 9.9 years 37. a. About 12.6% 39. a. b. 12.6%; about 12.7%; about 12.7% 3x1 c. No; yes f b. 3 18 6 3x or 41. a. g x 43. a. 2 x 1 E 1 b. $7966 2 100.4 5550 1.014 2 1 1.0368 1 x x 2 b. 115.38 million c. In the sixth year 45. About 256; about 654 b. 11.036 b. $0.86; $0.74 c. 16.242 47. a. 6.705 49. a. 51. a. x 2 1 1 f x 0.97 c. About 75 years 20 t 140 t f 2 0.5 A 2 1 B b. About 11.892 mg; about 3.299 mg c. About 325 days 53. About 5566 years old Section 5.4, page 361 3. 9. 1. 4 7. 13. 17. 21. 102.8751 750 ezw x2 2y log 3 0.4771 ln 5.5527 12 7 5. 103 1000 4.6052 0.01 e 2.5 e1.0986 3 15. 11. log 0.01 2 ln 25.79 3.25 ln w 2 r 19. 23. 25. 243 27. 15 29. 33.
x y 35. x2 37. 1 41. They are exactly the same. 1 2 1, q 31. 931 2 39. q, 0 1 2 f(x) = log(x − 31 −2 51. y h(x) = −2 log 1 −2 x x 53. 55. 57. 0 x 9.4 asymptote at 10 x 10 0 x 20 and 6 y 6 and x 1 ) (vertical and 3 y 3 59. 0.5493 63. a. ln 1 3 h h 2 ln 3 6 y 3 61. 0.2386 b. h 2.2 65. a. About: 17.67, 11.90, 9.01, 6.12, 4.19, 3.22, 2.25 b. The rule of thumb is that the number of years it takes for your money to double at interest rate r% is 72 divided by r. 67. a. 77 69. a. 9.9 days n 30 error of 0.00001 when 71. b. 66; 59 b. About 6986 0.7 x 0.7. gives an approximation with a maximum 43. Stretch the graph of g away from the x-axis by a Section 5.5, page 369 factor of 2. domain: all positive reals; range: all reals 45. Shift the graph of g horizontally 4 units to the right. domain: all reals 7 4; 47. Shift the graph of g horizontally 3 units to the left, then shift it vertically 4 units downward. domain: all reals 7 3; range: all reals range: all reals 1. 103 3. About 3.63 5. About 0.9030 0.2219 9. About log 13. e 1 2 3 ln 1 u 2v 1 2 x 3 1 2 19. 25. log 2 20xy 1 u 1 6 v 2 3 17. 23. 0.1461 7. About x2y3 7 x ln ln 1 11. 15. 2 2 1 2u 5v 21. 27. a. For all x 7 0 1088 Answers to Selected Exercises logarithms on page 364, b. According to the fourth property of natural eln x x for every x 7 0. 29. False; the right side is not defined when x 6 0, but the left side is. 31. True by the Power Law 33.
False; the graph of the left side differs from the graph of the right side. 35. Answers may vary: log 3 log 2 1.585 and 3 2b log a b e 0.1761 thus log 3 log 2 log 3 2b a 39. A 3, B 2 41. 2 37. 43. Approximately 2.54 45. 20 decibels 4 69. Horizontal shift of units to the right, then 3 compress horizontally by a factor of 1 3. Domain: all real numbers Range: all real numbers 7 4 3 71. Compress the graph vertically by a factor of 1 3, then a horizontal translation of 1 unit to the right, then a vertical translation of 7 units upward. Domain: all real numbers Range: all real numbers 7 1 47. Approximately 66 decibels 49. 100 times 73. True 75. True 77. False 79. 397398 51. a. 1.2553 b. 3.9518 c. log x ln x ln 10 Section 5.5.A, page 376 log 0.01 2 log r 7k 5. 1. 9. log3a 1 9b 2 3. 7. log 23 10 1 3 log7 5,764,801 8 11. 104 10,000 13. 102.8751 750 15. 53 125 19. 25. 29. 10zw x2 2y 1 2 x 21. 243 27. 6 0 log4x f x 1 2 Not defined 17. 2 1 2 4 23. 2x2 y2 1 0 2 0.5 4 1 31. x log6x h x 1 2 1 36 2 1 6 1 1 0 216 3 33. 35. 37. x 0 2 log7x f x 1 2 Not defined x 3 log21 h x 2 1 b 3 2.75 6 x 3 2 39. b 20 27 49 1 7 2 1 3 1 1 6 4 29 15 x2y3 z6 41. 5 43. 3 45. 4 47. log 49. log x2 3x 1 2 55. ln 1 x 1 2 x 2 b 2 a 51. log21 5c 2 53. log4a 1 49c2b 57. log21 x 2 59. ln 1 e2 2e 1 2 61. 3.3219 63. 0.8271 65. 1.1115 67. 1.6199 81. 85. 87. 89. logbu logau logab 83
. log10u 2 log100u logbx 1 2 b3 2v logbA g f x 2 1 is false. x 2 1 logbv 3 logb2v logbb3 ; hence B only when x b32v. x 0.123, so the statement y 6 4 2 (−0.3679, 0.3195) −6 −4 −2 −2 −4 h(x) = x log x2 Hole at (0, 0) x 2 4 6 (0.3679, −0.3195) Section 5.6, page 386 1. x 4 3. x 1 9 5. x 1 2 or 3 7. x 2 or 1 2 9. x ln 5 ln 3 1.465 2.7095 11. 13. 15. x ln 3 ln 1.5 x ln 3 5 ln 5 ln 5 2 ln 3 x ln 2 ln 3 3 ln 2 ln 3 1.825 0.1276 17. x 1 ln 5 2 2 0.805 19. x 1 ln 3.5 1.4 2 0.895 5 2.1b 2 ln x a ln 3 x ln 2 0.693 21. 25. 1.579 23. x 0 or 1 or x ln 3 1.099 Answers to Selected Exercises 1089 510 20 20 20 0.792 b. 500 x ln 3 ln 4 41. x 3 47. x 5 −50 0 15. a. 2 27. 29. 31. 35. 43. x ln 3 1.099 x ln 2 ln 4 x ln A 1 2 t 2t2 1 or then x 5 x 9 ln u ln v, 37. 5 137 2 x 33. If 49. x ± 210001 u v so x 6 B eln u eln v, 39. x 9 1 x 45. 51. e 1 2 e 1 e 1 A 53. Approximately 3689 years old 55. Approximately 950.35 years ago 57. Approximately 444,000,000 years 59. Approximately 10.413 years 61. Approximately 9.853 days 63. Approximately 6.99% 65. a. Approximately 22.5 years b. Approximately 22.1 years 67. $3197.05 71. a. About 1.3601% k 21.459 73. a. 69. 79
.36 years b. In the year 2027 b. t 0.182 75. a. There are 20 bacteria at the beginning and 2500 three hours later. b. ln 2 ln 5 0.43 77. a. At the outbreak: 200 people; after 3 weeks: about 2718 people b. In about 6.09 weeks k 0.229, c 83.3 b. 12.43 weeks 79. a. Section 5.7, page 396 1. Cubic, exponential, logistic 3. Exponential, quadratic, cubic 5. Exponential, logarithmic, quadratic, cubic 7. Quadratic, cubic 9. Quadratic, cubic 11. Ratios: 5.07, 5.06, 5.06, 5.08, 5.05; exponential is appropriate 13. a. For large values of x the term close to zero so the quantity is slightly larger than 1, which means 1 0.0216x 56.33e 1 56.33e is 0.0216x 0 0 b. 10 0 0 c. 1330 0 0 17. 19. ln x, ln y 5 1 26 appears the most linear. Power model ln x, ln y and 5 1 Power or logarithmic model ln x, y 5 1 26 26 are both nearly linear. 21. a. 105,000 2 442.1 1 56.33e close to) 442.1. 0.0216x is always less than (but very 0 92,000 8 1090 Answers to Selected Exercises b. 105,000 25. a. 1800 f(x) g(x) 8 0 92,000 c. 0 0 22 b. y 7.05x2 78.34x 398.73 d. The logarithmic model predicts continued but slowing growth while the logistic predicts a cap of about 102,520. Therefore, the logarithmic model seems the better one for the long haul. 23. a. 80 0 0 b. 5 0 0 c. Exponential. y 152.22 0.97x 1 175 0 0 110 110 110 2 c. y 6413.2 1 107.2e d. 2325.01, 2419.97 e. The quadratic model will give an ever 0.1815x increasing number of kids, and the rate of increase will continue to increase. Before too terribly long
the number of kids home schooled by the quadratic model will exceed the number of kids in the world. The logistic model, on the other hand, gives us a maximum that can never be exceeded. y 17.5945 13.4239 ln x 27. a. b. 77.4 years c. 2012 29. a. 85 0 0 b. y 10.48 1.16x 1 2 13 Answers to Selected Exercises 1091 c.-d. 25. x f 1 2 56,000 1.065 1 x 2 Worldwide shipments (thousands) Predicted number shipments (thousands) 14.7 15.1 16.7 18.1 21.3 23.7 27 32.4 38.9 47.9 60.2 70.9 84.3 12.2 14.1 16.4 19 22 25.5 29.6 34.4 39.9 46.2 53.6 62.2 72.2 Worldwide shipments ratio (current to previous) 1.03 1.11 1.08 1.18 1.11 1.14 1.2 1.20 1.23 1.26 1.18 1.19 Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 e. An exponential model may not be appropriate. Chapter 5 Review, page 403 1. c2 3. 10 3 b a 42 5 5. 1 2 v u 1 2 7. c2d4 2 9. 2 22x 2h 1 22x 1 11. Reflection across the x-axis, stretch vertically by a factor of 2 13. Reflection across the y-axis, stretch horizontally by a factor of 2 15. Vertical translation of 4 units upward 17. 3 x 3 and 0 y 2 19. a. 62,000 63,000 64,000 S 60,000 1000 33,708 35,730 37,874 t 1 b. c. Compunote is the best choice d. Calcuplay will be paying more this time, but 1.06 S 30,000 2 1 1 2 t1 your total earnings will be more from Compunote 21. a. About $1341.68 b. $541.68 23. a. About $2357.90 b. After about 32.65 years 1092 Answers to Selected Exercises 29. 27. About 3.75 grams ln 756 6.628 log 756 2.8785 et rs 37
. 33. r2 1 1 2 31. u v ln e7.118 1234 39. Undefined 35. 41. Reflection across the y-axis, horizontal translation of 4 units to the right; Domain: all real numbers 6 4; Range: all 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 ´ 2 1 1, q 2 1 59. 2 65. x 1 2 69. x 2 73. About 1.64 mg 61. (c) 57. 63. 67. 71. 75. Approximately 12 years 77. $452.89 81. a. 11° F b. 30 79. 7.6 5 0 50 c. The points (x, ln (y)) are approximately linear. d. e. y 22.42 10.27° F 0.967x 2 1 Chapter 5 can do calculus, page 411 1. y x 1; 5 −1 5 −5 25 −1 31 −5 4. y e2 x 2 1 2 e2 10 −5 5. y x 3 −5 −1 10 −1 6. y e x 1 2 1 e 2 10 −1 −5 5 5 5 5 5 7. y e2 x 2 2 1 e2 2 10 −1 −5 8 10 −1 10 −1 −5 9. y x 1 ln 3 1 2 −5 10. y 3 ln 3 1 21 3 x 1 2 10 −5 −1 11. y 9 ln 3 1 21 9 x 2 2 10 −5 −1 5 5 5 5 5 Answers to Selected Exercises 1093 41. The area of the triangle is a h. The altitude h 1 2 forms a right triangle with side b as the hypotenuse and side h opposite sin u b S h b sin u. h so u, Thus, the area of the 5 43. triangle is 1 2 A 4320.123 a h 1 2 a 1 Section 6.2, page 429 b sin u ab sin u. 2 1 2 A 33.246 45.
1. 7. 13. 17. 19. 22 3 12. y 1 9 a ln 3 b1 x 2 1 9 2 5 −1 −5 Chapter 6 Section 6.1, page 419 1. 5. 9. 11. 13. 15. 19. 23. 27. 47.26° 23°9¿36– sin u 3. 15.4125° 4°12¿27– ˛, tan u 7., cos u 3 2 11 3 sin u ˛, sec u ˛, tan u 211 211 3 ˛, csc u A cot u 3 22 7 ˛, cos u 2 27 27 ˛, sec u 2 ˛, csc u m ˛, tan u h d d ˛, csc u m A cot u 2 23 m ˛, cos u d h ˛, sec u m sin u h cot u d h 11 2 A 23 2 7 3 A sin 32° 0.5299 sec 47° 1.4663 u 45° 17. 21. 25. tan 6° 0.1051 u 30° u 60° 1 2 29. 3 8 31. 16 7 33. False; sin 50° 0.7660 2 sin 25° 0.8452 35. True; cos 28° 1 37. False; 2 2 0.7796 1 sin 28° 2 1 2 0.7796 39. tan 75° 3.7321 tan 30° tan 45° 1.5774 u 1° 0.1° 0.01° 0.0001° sin 0° 0; of 0°, cos u 0.9998 0.999998 0.99999998 0.9999999998 sin u 0.0175 0.00175 0.000175 0.0000175 the right triangle definitions do not apply. cos 0° 1 Since no triangle has an angle 1094 Answers to Selected Exercises c 36 h 2522 2 c 423 3 3. c 36 5. c 8.4 9. h 300 11. h 5023 15. a 1023 3 a 10 cos 50° 6.4, A 40°, c 10 sin 50° 7.7 b 6 C 76°, sin 14° 21. C 25°, a 5 tan 65° 10.7, 23. C 18°, c 3.5 cos 72° 1.1 a 3.
5 sin 72° 3.3, 24.8, c 6 24.1 tan 14° b 5 cos 65° 11.8 25. About 48.59° 27. About 48.19° 29. 31. 33. 35. A 33.7°, C 56.3° A 44.4°, C 45.6° A 48.2°, C 41.8° A 60.8°, C 29.2° 37. a. b. 23.18 feet. 6.21 feet. 39. 460.2 ft 41. 8598.3 ft 43. No 45. Approximately 263.44 feet 47. 351.1 m 53. a. 56.7 ft 55. 173.2 mi 49. 10.1 ft b. 9.7 ft 51. 1.6 mi 57. 52.5 mph 59. 449.1 ft Section 6.3, page 441 1. 40°, 2p 9 radians 5. 10°, p 18 radians 9. 288°, 8p 5 radians 3. 20°, p 9 radians 7. 240°, 4p 3 radians 11. 36° 13. 18° 15. 135° 19. 75° 21. 972° 23. p 30 17. 4° 25. p 15 31. 39. 5p 4 3p 5 33. 31p 6 41. 7 2p 19. a. sin 10p 3 10p 3 tan 0.8660, cos 10p 3 0.5, 1.7321 b. Since the sine and cosine are both negative, the 5p 12 5p 3 9p 4, 17p 4 3p 4 29. 37. 3p 4, 15p 4, 25p 6, 7p 4, 13p 6 23p 6 27. 35. 43. 45. 47. 55. 11p 6, 4p 3 17 4 63. 8.75 69. 7p 77. 3 radians 49. 57. 7p 6 50 9 51. 41p 6 53. 8p cm 59. 2000 61. 5 65. 3490.66 mi 67. 942.48 mi 73. 42.5p 75. 2pk 4p 71. 171.9° 1 2 79. a. 400p radians per min b. 800p in. per min or 200p 3 ft per min 81. a. 5p radians per sec b. 6
.69 mph 83. 15.92 ft 85. approximately 8.6 miles Section 6.4, page 452 sin t 7 1., cos t 2 3. sin t 5. sin t 253 6 261 10 2103, cos t, cos t 253 5 261 23 2103, tan t 7 2, tan t 6 5, tan t 10 23 7. 9. sin t 1 25 sin t 4 5, cos t 2 25, tan t 4 3, cos t 3 5, tan t 1 2 21. a. terminal side is in the third quadrant. sin 9.5p 1, undefined cos 9.5p 0, tan 9.5p is b. Since the sine is 1 terminal side is on the negative y-axis. 0.2752, and the cosine is 0, the 17 cos 23. a. sin 1 tan 1 17 2 17 2 0.9614, 3.4939 1 2 b. Since the sine is positive and the cosine is negative, the terminal side is in the second quadrant. 17π 6 x 25. y reference angle = π 6 y 27. 1.75π x reference angle = π 4 11. sin 13p 6 1 2 ˛; cos 13p 6 23 2 13. sin 16p 0; cos 16p 1 29. y 15. a. sin 7p 5 7p 5 tan 0.9511, cos 7p 5 0.3090, 3.0777 b. Since the sine and cosine are both negative, the terminal side is in the third quadrant. 17. a. sin a 14p b 9 14p 9 b tan a 0.9848, cos 14p 9 b a 0.1736, 5.6713 b. Since the sine and cosine are both positive, the terminal side is in the first quadrant. – π 7 x reference angle = π 7 Answers to Selected Exercises 1095 31. sin 7p 3 b a 23 2, cos 7p 3 b a 1 2, tan 7p 3 b a 23 33. sin 11p a 4 b 22 2, cos 11p a 4 b 22 2, tan 11p a 4 b 1 35. sin 3p 2 b a 1, cos 3p 2 b a 0, tan 3p 2 b a sin u cos u is undefined. 37. sin 23p 6 b a 1 2, cos 23p 6
b a 23 2, tan 23p 6 b a 1 23 23 3 39. sin 19p 3 b a 23 2, cos 19p 3 b a 1 2, tan 19p 3 b a 23 41. sin 15p 4 b a 22 2, cos 15p 4 b a 22 2, tan 15p 4 b a 1 43. sin 5p 6 b a 1 2, cos 5p 6 b a 23 2, tan 5p 6 b a 1 23 23 3 45. 47. 49. 55. 57. 59. is undefined and tan u sin u 1, cos u 0, sin u 0, cos u 1, and tan u 0 22 2 1 23 A 51. 22 4, cos t 3 234 B, tan t 5 3 sin t 5 234 53. 23 2 sin t 1 25 sin t 3, cos t 2 25 cos t 1, 210 210, tan t 1 2 tan t 3, 9. 11. 13. 15. 19. cos t 0.9457, sec t 1.0574, sec t 3.7646, cot t 0.2755, sin t 0.1601, cot t 6.1668, sin2 t cos2 t 1 4 tan t 0.3438, csc t 3.0760 cos t 0.2656, csc t 1.0372 cos t 0.9871, sec t 1.0131 cot t 2.9089, sin t 0.9641, tan t 0.1622, 17. cos t 21. cos t 2 23. cos t sin t cos t 25. 0 27. even 2sin t 0 33. sin t 23 2 39. 45. 51. 3 5 221 5 32 22 2 29. even 35. sin t 23 2 41. 47. 3 4 2 5 53. 32 22 2 31. odd 37. 43. 49. 3 5 3 4 221 5 55. possible 61. csc t csc cot t cot 1 1 57. not possible sec t sec ; t ± 2p t ± p 2 2 59. not possible t ± 2p ; 2 1 Chapter 6 Review, page 464 1. 41.115° 3. (d) 9. 13. 4 7 C 34°, b 13.3, c 7.4 11. 5. 4 265 7. 265 7 C 50°, a 6.4, c 7.7 15. 225.9 ft 17
. The boat has moved about 95.3 feet. 19. 255° 21. p 5 23. ˛ 3p 4 25. 16p 3 27. 2 revolutions per minute 29. 37. 3 5 23 3 31. 0 39. 2 33. 41. ˛ 1 2 23 2 43. quadrants 2 and 3 45. 9 4 35. 23 61. r cos t, r sin t 1 2 63. Domain: all real numbers with q, 1 65. Domain: all real numbers with Range: 1, q ´ 2 1 1 2 Range: all real numbers Section 6.5, page 460 u a multiple of p u a multiple of p 47. sin t cos t cos t sin t tan2 t 49. e 51. ˛ 3 5 53. 1 55. b 1. cos t 3. 1 5. 1 7. csc2 t 1096 Answers to Selected Exercises Chapter 6 can do calculus, page 471 1. sin t cos t 2 t f 1 p 12 2 cos t 2 1 1 sin t 4. t f 1 p 12 p 144 max of 0.5 when x 0.7854 max of 0.5 when x 0.78538 p 144 sin t 2 cos t 2. 2 t f 1 p 12 5. a. b. 200 sin t cos t p radians; 4 width 1022 height 522 meters max of 1.1375 when x 0.2618 max of 1.1387 when x 0.28362 meters and p 144 3. 3 sin t sin p 2 a t b p 12 p 144 max of 2.2321 when x 0.5236 max of 2.236 when x 0.45815 max of 3.1566 when x 1.309 max of 3.1622 when x 1.2435 6. approximately 13.23 feet from the statue 7. a. road cost 10,000˛a b. approximately $117,321 10 1 tan tb 20,000˛a 1 sin tb Chapter 7 Section 7.1, page 483 1. 3. 5. 13π 2 −π ≤ x ≤ −2 ≤ y ≤ 2, −π ≤ x ≤ 3π, −4 ≤ y ≤ 4 −π ≤ x ≤ 4π, −2 ≤ y ≤ 2 Answers to Selected Exercises 1097 7. 11. 15. 17. p 2 3p 2 3p 2 ˛,
3p 3p 2, 5p 6 t 6 3p 2 4 p 2 7p 4 and 9. 1 13. 1 3p 2π 45. 47. 19. all values on the interval p, 2p 3 4 except 3p 2 21. t p 4 2np or t 3p 4 2np, where n is any −2π 2np or t 4p 3 2np, where n is any 2np or t 5p 3 2np, where n is any 49. d integer t 2p 3 integer t 4p 3 integer t p 6 integer t p 6 integer t 3p 4 integer t p 3 23. 25. 27. 29. 31. 33. 2np 2np or t 11p 6 2np, where n is any or t 5p 6 2np, where n is any 2np or t 5p 4 2np, where n is any np, where n is any integer 35. Reflect the graph of f across the horizontal axis. 1 domain: all real numbers; range: 1 g t 1 2 7 −7 1 −1 2π 2π 53. f 55. a. odd; b. even; c. odd; d. even; e. odd; 51. e t sin 2 1 t cos 1 t tan 2 1 t sin 2 1 t tan 2 1 2 sin t cos t tan t sin t tan t 57. 1.4 61. −2 59. 11 40 2 −40 37. Shift the graph of f vertically 5 units upward. domain: all real numbers except odd multiples of p 2 ; range: all real numbers a. 0 c. 15.4 yards e. When t 0.25, b. 0 d. 3.6 yards d is undefined. The beam is parallel to the wall at this time. 39. Stretch the graph of f away from the horizontal axis by a factor of 3. domain: all real numbers; range: 3 g 3 t 1 2 41. Stretch the graph of f away from the horizontal axis by a factor of 3, then shift the resulting graph vertically 2 units upward. domain: all real 1 g numbers; range: 43. Shift the graph of f vertically 3 units upward. 4 domain: all real numbers; range 1098 Answers to Selected Exercises Section 7.2, page 490 1. The graph of s t 3 sec t 2 is the graph of stretched vertically by a factor of 3 and 1 2 t sec t g shifted down 2
units. 2 1 3. The graph of csc t f t 2 1 t m 4 csc shifted 4 units up. t 1 2 1 2 is the graph of 5. The graph of p 1 2 t 2 1 sec t 1 is the graph of sec t g t 2 1 compressed vertically by a factor of 1 2 and shifted up 1 unit. 7. The graph of t sec t g shifted 8 units down. 2 1 t q sec 8 reflected across the vertical axis, and is the graph of t 1 2 2 1 9. amplitude: 0.3; period: 6p 11. amplitude: 1 2 ˛; period: 2p 3 9. The graph of v t p csc t is the graph of stretched vertically by a factor of p. 13. amplitude: 5; period: 20p 17 13. g t 2 1 1 4 sec t 5 17. g t 2 1 cot t 2 1 23. A 31. A 25. A 33 21. B 29 11. csc t 3 sec 15. g t 2 1 csc 19. D 27. B 35. − π 2 −2π 37. π 2 2π −1 4 −4 y sec t 39. Look at the graph of y t, draw in the line sec t intersect the graph of p 2 t p 2 and Q,. on page 488. If you it will pass through and obviously will not when But it will intersect each part of the graph that lies above the horizontal axis, to the right of t p 2 ; it will also intersect those parts that lie below the horizontal axis, to the left of p 2 The first coordinate of each of these infinitely. many intersection points will be a solution of sec t t. Section 7.3, page 498 1. amplitude: 1; period: 2p 3. amplitude: 1; period: 2p 3 5. amplitude: 4; period: 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. g
is the graph of f reflected across the y-axis; amplitude: none; period: p. 25. g is the graph of f horizontally compressed by a factor of 5 8 ˛; amplitude: 1; period: 5p 4 ˛. 27. g is the graph of f vertically stretched by a factor of 3; amplitude: 3; period: 2p. 29. g is the graph of f vertically compressed by a factor of 1 3 ˛; amplitude: none; period: p. 31. g is the graph of f vertically stretched by a factor 1 2 ˛; of 5 and horizontally compressed by a factor of amplitude: 5; period: p. 33. g is the graph of f reflected across the x-axis, vertically stretched by a factor of 2, and horizontally stretched by a factor of 5; amplitude: none; period: 5p. 35. g is the graph of f vertically compressed by a factor of factor of 2 5 1 8 ˛; and horizontally compressed by a amplitude: 2 5 ˛; period: p 4 ˛. 37. g is the graph of f vertically compressed by a and horizontally compressed by a 1 3 1 p ˛; factor of factor of 39. 5 0 −5 amplitude: none; period: 1 4π Answers to Selected Exercises 1099 11. amplitude: 7; period: 2p 7 ; phase shift: 1 49 ; vertical shift: 0 13. amplitude: 3; period: p; phase shift: 5p 8 ; vertical shift: 0 15. amplitude: 97; period: p 7 ; phase shift: 5 14 ; vertical shift: 0 17. amplitude: 1; period: 1; phase shift: 0; vertical shift: 7 19. amplitude: 3; period: 6; phase shift: 3 p ; vertical or f 3 sin t 2 1 8t 8p 5 b a a t p 5 b t 2p 3 b 3 t 4p 2 3 a 2 a 2 3 2 3 sin sin 9 b t 1.5 2 0.8pt 1.2p 1 0.5 sin 0.8p 0.5 sin 2 or 2 or 0.6 0.6 2 t 0 2 1 or f t 2 1 6 sin 6 5 t b a 1 shift: 5 21. f t 2 1 3 sin 8 23. 25. 27. 29 sin 5 2 5 2 sin sin a 0 or 10p 9 1 10p 2
t 0.2 9 t 2p 9 b 10t p 2 b 31. a. f b. g t t 2 2 1 1 12 sin 12 cos 10t a 2t p 2 b a 8t p a 2 b 35. a. f t 2 1 1 2 sin 8t b. g t 2 1 1 2 cos 37. a. b. 39. a. f f f b sin 4 cos 2 sin 2 cos 3p 41. 12 −12 π 6 1 − π 6 43. 4 0 −4 45. d 47. b 49. f 51. 55. 59 sin t 2 sin pt 3 2 2 5 sin 5t 1 53. 57.8 sin 4t 3 2 sin 4t 2 61. local maximum of 1 at t 0 and t 2p 3 ; local minimum of 1 at t p 3. 63. there is a local maximum of 23 2 at t p; local Section 7.4, page 508 1. amplitude: 1; period: 2p; phase shift: 1; vertical shift: 0 3. amplitude: 5; period: p; phase shift: 0; vertical shift: 0 5. amplitude: 1; period: 2p; phase shift: p; vertical shift: 4 7. amplitude: 6; period: 2 3 ; phase shift: 1 3p ; vertical shift: 0 9. amplitude: 4; period: 2p 3 ; phase shift: p 18 ; vertical shift: 1 1100 Answers to Selected Exercises minimum of 1 at t 3p 2. 65. 1 900,000 33. a. f t 2 1 sin 2t b. g t 2 1 cos 41. y −2π −π 3 1 −1 −3 43. y −2π −π 1 −1 45. y 3 2 1 −1 −2 −3 47. 3 0 −1 2.6180; local t 5p 6 5.7596 minimum at 49. Local maximum at t 11p 6 t p 6 51. Local maxima at 0.5236, 2.6180, t 5p 6 t p 2 1.5708, π x 2π t 3p 2 t 7p 6 4.7124; 3.6652, local minima at t 11p 6 5.7596 53. The graph of f the graph of intercepting the y-axis at 1. sin2 t cos2 t 1, t 1 x 2 2 f 1 a
horizontal line is the same as 55. Not an identity 57. Possibly an identity 59. 61. A 3.8332, b 4, c 1.4572 A 5.3852, b 1, c 1.1903 63. All waves in the graph of g are of equal height, which is not the case with the graph of f. It cannot be constructed from a sine curve through translations, stretches, or contractions. x Section 7.4A, page 515 1. 3. 5. 7. 9. 11. 13. 15.2361 sin 5.3852 sin 5.1164 sin t 1.1072 2 4t 1.1903 3t 0.7442 f 1 0 t 2p and 5 y 5 10 t 10 and 10 y 10 0 t p 50 and 2 y 2 1 0 t 0.04 and 7 y 7 0 t 10 and 6 y 10 1 2 one period 2 2 one period one period 1 one period 2 2 1 17. To the left of the y-axis, the graph lies above the t-axis, which is a horizontal asymptote of the graph. To the right of the y-axis, the graph makes waves of amplitude 1, of shorter and shorter period. Window: 3 t 3.2 2 y 2 and 19. The graph is symmetric with respect to the y-axis and consists of waves along the t-axis, whose amplitude slowly increases as you move farther from the origin in either direction. Window: 30 t 30 and 6 y 6 21. There is a hole at point (0, 1). The graph is symmetric with respect to the y-axis and consists of waves along the t-axis whose amplitude rapidly decreases as you move farther from the origin in either direction. Window: 30 t 30 and 0.3 y 1 2π 23. The function is periodic with period The graph lies on or below the t-axis because the logarithmic function is negative for numbers p. Answers to Selected Exercises 1101 π 2π h(t) = 3 sin 2t +( π 2 ) π 2 π 3π 2 2π x 19. 6 0 0, cos t between 0 and 1 and is always between 0 and 1. The graph has vertical asymptotes when t ± p ± 7p 1 2 2 points and ln 0 is not defined). Window: 2p t 2p and periods) 3 y 1
cos t 0 ± 5p 2 ± 3p 2 at these (four,...,, Chapter 7 Review, page 517 1. (c) 3. 5. 7. 0 n2p, p 3 np, where n is an integer where n is an integer 9. The graph of g is the graph of f reflected across the horizontal axis and compressed horizontally 1 2 ˛. domain: all real numbers except −2π where n is an integer; range: all real by a factor of t p 4 numbers n p 2, 11. 4 −2π 2π −2π −2π 21. 23. even 25. 27. 29. 400 33. 37. 2π 2π 2π −2 8 −8 4 −4 13. −2π −4 6 −1 15. A 17. C 2π −3π 31. C 35. 2 cos 5t 2 b a 6π 8 sin f t 2 1 2pt 28p 5 b a 2 −2 1102 Answers to Selected Exercises 39. Not an identity 41. Possibly an identity 43. 45. 2 t f 1 0 t p 50 10.5588 sin 4t 0.4580 1 and 5 y 5 2 one period 2 1 Chapter 7 can do calculus, page 521 1. 2 6x 18x2 54x3 162x4.... ; 1 6 x 1 6 2. 3 6x 12x2 24x3 48x4.... ; 6 16 x 5 16 3. 2 2x 2x2 2x3 2x4.... ; 9 16 x 5 8 4. 3 6x 12x2 24x3 48x4.... ; 1 4 x 11 32 5. cos x; q x q 6. 1 x ; 0 x 2 Chapter 8 7. ex; q x q 1. 3. 5. Section 8.1, page 528 x 0.5275 kp x 0.4959 2kp 1.8877 2kp or x 0.1671 2kp 4.5453 2kp or x 1.2161 2kp x 2.4620 2kp x 0.5166 2kp 13. a. The graph of 11. 9. 7. f 2 shows that 1 to 2p or 1.6868 kp 1.2538 2kp or 2.6457 2kp or
1.5708 2kp or 4.7124 2kp or or 1.8256 2kp or 2.8867 2kp or or or x 5.0671 2kp 3.8212 2kp 5.6766 2kp sin x sin x 1 on the interval from 0 only when x p 2. Since sin x has period obtained by adding or subtracting integer all other solutions are 2p, from p 2, that is,, 2p multiples of 2p 5p p 2 2 13p 2 7p 2 p 2 p 2 p 2 3 2 2p 2p 2 1 1 2 2 2p 1 2, etc., and, p 2 3 9p, 2 2p 3p 2 11p 2 p 2 2p, etc. 2 1, b. Similarly, the graph shows that sin x 1 only when x 3p 2, so that all solutions are obtained by adding or subtracting integer multiples of 2 1 2 :, 2p 2p 2p 3 2 3p from 2 3p 2p 7p 2 2 15p 2 5p 2 3p 2 3p 2 3p 1 2 x 0.1193 u 82.83°, 262.83° u 210°, 270°, 330° u 60°, 120°, 240°, 300° u 120°, 240° 29. Sin x k cos x k k 7 1 or and k 6 1. or 3.0223 2 2p 1 2, etc., and, 11p 2 2p p 2 9p 2 3p 2 2p, etc. 2, 1 x 1.3734 or 4.5150 u 114.83°, 245.17° 3, 3p 2 17. 21. a 65.38° 31. a 30° have no solutions when 5. 0 7. p 6 13. 2p 3 0.8584 15. 0.3576 23. 2.2168 1. Section 8.2, page 536 p 4 p 3 11. 3. 9. p 2 p 4 1.2728 cos u 1 2 19. 0.7168 21. ; tan u 23 p 2 p 6 29. 37. 5p 6 4 5 31. 39. p 3 4 5 33. 41. p 3 5 12 43. cos 45. tan sin 1 v sin 1 v 1 1 2 2 47 21 v2 1 v 21 v2 1 y π
2 15. 19. 23. 25. 27. 33. 17. 25. 27. 35. f(x) = cos −1 (x + 1) f(x) = sin x x 2π −2 −1 2 0 −2 Answers to Selected Exercises 1103 49. y 2π 3 π 3 x −2π −π π 2π −2π 3 51. a. u sin 1 40 x b a 53. y csc x b. 9.2° 5 −5 π 2 y csc x is one-to-one and has an − π 2 The graph of inverse. y csc 1 x − π 2 5 −5 55. a. Let 1 b. Let Then cos 2 and cos u w 1 w, u cos 1 v. u cos 1 v cos cos tan u w 1 w, u tan 1 v tan u tan Let tan 2 with 0 u p. 1 cos u and cos 1 cos u v, Then cos u v. p 2 1 with tan u and tan 1 1 v. tan u v, Then tan u v. u p. 2 tan and 2 1 w u. Let Then 1 w u. 1 u tan 1 2 4 xb a 57. a. tan 1 2 xb a b. x 9.13 feet Section 8.3, page 545 1. x p 3 2kp or 2p 3 2kp 3. x p 3 kp 1104 Answers to Selected Exercises 5. 9. 13. 15. 19. 23. 27. 31. 35. 37. 41. 43. 45. 47. 49. 51. 55. x ± 5p 6 2kp 7. x p 6 2kp or 7p 6 2kp 11. 17. 14.18° 27.57° x 0.4836 2kp or 3.6252 2kp x ± 2.1700 2kp x 0.4101 kp x p 6 x p 9 2kp or kp 3 kp 2p 3 21. 29. 25. x 0.2327 kp x ± 1.9577 2kp x ± p 2 4kp x ± 0.7381 2kp 3 x 3.4814, 5.9433 33. x 2.2143 2kp x 3p 4 x p 4 7p 4 p 2,, 2.1588,
5.3004 x p 6,,, 39. 5p 4 3p 2 x 0.8481, 1.7682, 2.2935, 4.9098 x 0.8213, 2.3203 x 0.3649, 1.2059, 3.5065, 4.3475 x 1.0591, 2.8679, 4.2007, 6.0095 x p 7p 4 4 x p 4 5p 4 3p 4 53, 5p 6, 3p 2, 5p 4 kp 6 1.2682, 0.7446, 0.2210, 1.7918 5p 4 1 4 t tan 6 1. 3. 5 125 sin pt 5 b cos 20pt 216 sin2 a 1 6 sin pt 2 b a 7. h t 2 1 20pt 2 6 cos pt 2 b a 9. d t 2 1 10 sin pt 2 b a 11. a. 10 0 0 b. Roughly periodic; y 1.358 sin 0.4778x 0.569 1 20 7.636 2 57. not possible x p 6n 63. 2kp n, 59. 5p 6n 74.0° 61. No solution 16.0° or 2kp n π 2 Section 8.4, page 555 c. No, the unemployment total will only be predicted in the range 6.2781 to 8.9944. 2.376 y 2.9138 sin 0.400x 1.809 b. About 15.7, which is somewhat reasonable but 2 1 13. a. may not be the best model to use. c. 5 5. (graph of C-major chord: C E G ) 4 0 −4 0.03 −1 −1 25 y sin 1 262 2p x sin 1 2 330 2p x sin 1 2 392 2p x 2 Chapter 8 Review, page 564 The model is not a good fit in the second year. y 2.1663 sin 0.513x 1.051 1.71 d. 1 2, where k is an integer. 1. 3. 5. 7. x kp x 0.8419 2kp 4.1784 2kp or x 0.5236 2kp tan 3t 3 5, 3t tan 1 3 5b a kp, 2.2997 2kp or 5.2463 2kp 2
.6180 2kp or or or 4.7124 2kp 25 1 tan 3 5b a 3 kp 3 t. In the first 2 seconds the solutions are 0.1801, 1.2273. 5 −1 −1 e. About 12.2 This model provides a much better fit. Section 8.4.A, page 562 1. 1 0.01 0 −1 y sin 1 294 2px 2 3. 1 0 −1 y sin 1 440 2px 2 9. p 3 19. π 2 0 − π 2 21. 25. 27. 2 515 x p 6 x 4p 9 11. p 3 13. 0 15. 2 17. 3p 4 4 23. 75° or 255° 2kp or 2kp x 11p 6 x 5p 9 2kp 3 or 2kp 3 x 3.78509 2kp 31. x tan 1 5 2b a kp 1.19029 kp 33. x p 4 kp or 37. x 0.8959 2kp 3p 4 or kp 35. x ± p 3 kp 2.2457 2kp Answers to Selected 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 8.379 1 2 seconds, 31. 33. Chapter 8 can do calculus, page 569 1. 1 2. 0 3. does not exist 4. 1 5. 1 6. 1 7. does not exist 8. does not exist 9. 1 10. 1 11. 0.75 12. 8 7 13. does not exist 16. does not exist 14. 3 17. 6 19. does not exist 20. 0.5 22. does not exist 23. 3 25. lim xS0 sin bx sin cx b c 15. does not exist 18. 4 21. 5 24. 0 Chapter 9 Section 9.1, page 580 1. Possibly an identity 3
. Possibly an identity 5. b 7. e 9. tan x cos x sin x cos xb a cos x sin x 11. cos x sec x cos x 1 cos xb a 1 13. tan x csc x sin x cos xb a 1 sin xb a 1 cos x sec x 15. tan x sec x sin x cos x 1 cos x sin x 1 cos x 1 21 17. 1 cos x 19. Not an identity x 2 x 2 21. sin x cos x sin 1 cos 1 x 1 2 23. cot x cos 1 x sin 1 2 2 1 cos2x sin2x 2 tan x cos x sin x cot x 1106 Answers to Selected Exercises 27. 25. Not an identity sec2x csc2x tan2x cot2x cot x 1 sin2x cos x sin x 1 cos2x 2 2 2 3 3 2 tan x 1 29. 1 2 1 2 sin2x tan2x 1 tan2x 1 2 1 cot2x 2 1 sin x cos x cot x 1 1 1 tan x 2 2 4 2 2 4 1 sin2x sin2x cos2x 1 sec2x sin2x 1 2 sin2x 1 1 a tan2x cos2xb sin2x 1 cos2x 1 sin2x 1 2a 1 21 2 tan2x 1 1 cos2xb sin2x tan2x 2 1 tan2x 35. sec x csc x 1 cos x 1 sin x sin x cos x tan x sin2x sec2x 2 21 37. cos4x sin4x cos2x sin2x 39. Not an identity 1 41. sec x csc x sin x cos x cos2x sin2x cos2x sin2x 2 21 1 cos x 1 sin x tan x sin x cos x tan x tan x tan x 2 tan x 43. sec x csc x 1 tan x sin x cos x sin x cos x sin2x 1 csc x sin x 1 1 cos x 1 sin x cos x sin x sin x cos x sin x cos x sin x cos x cos x sin x sin x 1 2 45. 1 csc x sin x 1 sin x sin x sin x sin x 1 sin2x 1 sin x sin x a cos2x sec x tan x 1 cos xba sin x cos xb 47. Not an identity 49. Conjecture: cos x. Proof
: 1 cos x sin2x 1 cos x cos x cos2x 1 cos x 1 sin2x 1 cos x 1 sin2x 1 1 cos x 2 cos x cos x 1 cos x 1 1 cos x 2 cos x 51. Conjecture: tan x: Proof: sec x csc x sin x cos x 1 sin x sec x cos x csc x cot x 2 sin x 21 sin x csc x 2 cot x 2 cos x sec x 1 cos x 53. 55. 57. 59. 1 sin x cot x 2 cos x sin x 1 sin x sin x cot x 2 cos x tan x cot x 1 sin x sec x 1 sin x sec x 1 cot x 1 sin2x 1 sin x sec x 1 2 1 cos x cos x 1 cos x sin x tan x 1 sin x 1 sin x cos2x 1 sin x 1 1 2 2 1 cos x 1 cos x 1 2 1 sin x cos x 1 sin x cos x 1 sin x 1 sin x 1 sin x 1 cos2x 2 sec x tan x 1 sin2 x 1 1 sin x cos x sin x cos x cos x cos3x 1 sin x 2 sin x sin x cos x cot x cot x cos x cos2x 1 sin x cos x cos x cos x cos x cos x sin x 1 sin x 1 sin x 1 sin x cos x cos x sin x cos x cos2x 1 sin x 1 sin x cot x cos x cos x cot x 41. 1 1 cos x 1 sin x 1 sin2x 2 2 cos x sin x cos x cos x sin x b cos x a cot x 1, so tan x 61. csc x cot x 1 log10 1 cot x 2 log101 csc x cot x tan x 1 log10 a tan xb 1 log101 tan x 2 csc x cot x 2 csc x cot x 2 1 csc x cot x 1 1 2 csc2x cot2x csc x cot x log10 a 2 1 log101 1 csc x cot xb csc x cot x 2 ; 2 2 so, csc x cot x csc x cot x cot x cot y log10 1 log101 tan x tan y 1 tan y tan y tan x tan x tan y cos x sin y cos y sin x cos x sin y cos x sin
y 2 tan x tan x cot x 1 1 2 63. 65. tan y cot y 2 cos2x sin2y 1 1 1 1 1 1 cos y sin x 1 sin2x 2 cos y sin x 21 cos x sin y 1 cos2y 2 1 cos x sin y 21 cos2y sin2x cos y sin x cos y sin x cos y sin x cos x sin y cos y sin x cos x sin y 21 21 21 2 2 2 2 2 cos y sin x cos x sin y Section 9.2, page 587 1. 16 12 4 7. 2 23 9. 2 23 19. 13. cos x sin 2 4 12 6 25. 31. 0.993 15. sin x 21. cos x 216 13 10 2.34 27. 33. 16 12 4 1/cos x 2 sin x sin y 11. 17. 23. 29. 0.393 f 1 35. x h 2 h f x 1 2 cos 1 cos x cos h sin x sin h cos x x h 2 h cos x 2 1 h cos x cos h cos x h a cos x cos h 1 h 44 2 125 the third quadrant. x y sin 1 ; b 37. sin x sin h h sin h sin x a h b 44 117 x y ; is in tan 1 x y 2 39. cos x y 1 56 65 2 ; tan 1 x y 33 56 2 x y ; is in the cos x. third quadrant. u v w sin cos u sin v cos w cos u cos v sin w sin u sin v sin w sin u cos v cos w 2 1 43. Since Hence, 1 1 Q p 2 x, x R y p sin y sin 2 sin2x sin2y sin2x cos2x 1 sin x cos p cos x sin p x p 2 1 sin x 0 21 2 21 cos p cos x sin p sin x p x 1 1 sin x cos x 1 2 x p sin x sin x cos p cos x sin p 2 1 0 21 2 cos x sin x sin x cos x cos x 21 45. sin 47. cos 49. sin 51. By Exercises 49 and 50, tan sin x cos x tan x x p 2 1 sin 1 cos 1 x p 2 x p 2 cos x cos y 1 cos 2 4 cos x cos y sin x sin y sin x sin y 2 3 1 2 4 53. 55. 57. 2 1 1 1 2 cos cos
x y 2 sin x sin sin x sin y 1 2 1 x y 1 1 cos x cos y sin x sin y 1 2 cos x cos y 1 cos 2 cos2x cos2y sin2x sin2y x y cos 2 sin x cos y cos x sin x sin y cos y 2 1 1 1 sin x sin y 2 2 2 cos x cos y sin x sin y sin x cos y cot x tan y cos x cos y sin x sin y 2 3. 2 23 5. 2 23 59. Not an identity Answers to Selected Exercises 1107 61. sin sin 1 1 x y x y 2 2 sin x cos y cos x sin y sin x cos y cos x sin y 1 cos x cos y 1 cos x cos y sin x cos x sin x cos x sin y cos y sin y cos y tan x tan y tan x tan y 63. Not an identity 65. Not an identity 59. Section 9.2.A, page 592 1. 0.64 radians 3. 2.47 radians 7. 1.37 radians or 1.77 radians 11. 1.39 radians or 1.75 radians 5. 9. p 2 p 4 or 3p 4 45. cos x 47. sin 4y 49. 1 51. 53. sin 16x sin 2 1 cos4x sin4x cos2x sin2x cos 2x 8x 2 4 cos2x sin2x 1 3 2 sin 8x cos 8x cos2x sin2x 2 21 55. Not an identity 1 cos 2x sin 2x 57. 1 1 2 cos2x 1 2 sin x cos x 2 2 cos2x 2 sin x cos x 1 2x x cot x 2 sin x cos x cos x sin x sin 3x sin 2 1 2 sin x cos2x sin x 2 sin3x 1 sin2x 2 sin x sin x sin x 2 1 2 2 sin2x 1 2 sin2x 3 4 sin2x 2 cos x 2 sin x 2 sin3x 1 2 sin2x 1 2 1 1 2 sin 2x cos x cos 2x sin x sin x 61. Not an identity 63. csc2 x 2b a sin2 Section 9.3, page 600 22 12 2 1. 5. 2 23 9. 22 12 2 3. 7. 22 12 2 22 13 2 11. 22 1 65. 67. sin x sin 3x cos
x cos 3x tan x sin 4x sin 6x cos 4x cos 6x cot x 1 1 1 cos x x 2 2R Q 2 cos 2x sin 1 2 cos 2x cos 1 x 2 x 2 2 1 cos x sin x cos x 2 sin 5x cos 1 2 sin 5x sin x 2 x 1 cos x sin x 2 2 sin a 2 sin x y cos 2 R x y 2 sin cot x y 2 a b 69. sin x sin y cos x cos y x y cos 2 Q x y 2 Q 1 cos x sin x sin R 71. a. and part (a), tan x 2 73. R vt cos a a R v 2v sin a g R 2v2 sin a cos a b cos a g R v2 sin 2a g 1 1 cos x sin x 21 1 cos x 2 1 1 cos x sin2x 1 cos x sin x 2 1 cos2x sin x 1 cos x 1 2 b. By the half-angle identity proved in the text sin x 1 cos x 2 1 1 cos x sin x sin x 1 cos x 13. 17. 21. 23. 25. 27. 29. 1 2 1 2 sin 10x 1 2 cos 20x 1 2 sin 2x 15. 1 2 cos 6x 1 2 cos 2x cos 14x 19. 2 sin 4x cos x 2 sin 2x cos 7x sin 2x 120 169 sin 2x 24 25 sin 2x 24 25 115 8 sin 2x, tan 2x 120 119, tan 2x 24 7, cos 2x 119 169, cos 2x 7 25, tan 2x 24 7 115 7, cos 2x 7 25, cos 2x 7 8, tan 2x 31. sin 33. sin 35. sin 0.5477, cos x 2 0.8367, tan x 2 0.6547 1 110, cos x 2 15 2 215 B, tan, cos B 3 110 x 2 1 x 3 2 15 2 215, tan 29 415 25 2 x 2 x 2 x 2 x 2 37. sin 2x 0.96 39. cos 2x 0.28 41. sin x 2 0.3162 43. cos 3x 4 cos3x 3 cos x 1108 Answers to Selected Exercises 3. x 0, p, 2p, 7p 6, 11p 6 7. x 0, 2p 11. x 3p 8, 7p 8, 11
p 8, 15p 8 15. cos x sec x cos x 1 1 cos2 x cos x sin x cos x cos x sin2 x cos x sin x tan x sin x 75. 2 sin2 1 a 2 a sin2 1 2 1 2 1 cos cos a 1 cos u 2 cos u cos a a 1 cos 2 1 2 Q u R ¢ 2 b Section 9.4, page 608 1. no solution 5. 9. 13. 15. 19. 23. 27. 31. 35 3p 4 x 3p 4 x 5p 12 x p 3 x p 4 3p 2 3p 2 3p 4 3p 4,, 5p 4 5p 4,, 7p 4 7p 4,, 7p 4 7p 4, 13p 12, p, p, 3p 4, 5p 4 37. a. f x 1 2 2 sin b. 2 c. x p 6 39. a. f x 1 2 x 7p 4 b a 22 sin 22 b. c. x f 2 1 x 3p 4, p 12, 5p 12 17. 21. 25.,, 13p 12 x p 3 x p 3 x p 3 17p 12,,, 5p 3 5p 3 5p 3, p 29. x p, 0, p 33. x 2p 5, 6p 5, 2p, 0, 4p 3, 0, p, 2p, 7p 4 x p a 3 b 11. 13. 2 1 2 sin 2x 9. Not an identity sin x cos x sin2x 2 sin x cos x cos2x 2 sin x cos x sin2x cos2x 1 cos4 x sin4 x 1 tan4 x cos2 x sin2 x 1 tan2 x sec2 x cos2 x sin2 x 1 tan2 x 21 21 cos2 x sin2 x 1 sin2 x cos2 x b 1 cos2 x 1 1 1 cos2 x sin2 x 1 tan2 x 2 2 2 1 2 1 cos2 x sin2 x cos2 x sin2 x a cos4 x 2 cos4 x 17. (a) 19. 3 2 2 cos x y x y cos 1 1 cos x cos y sin x sin y cos2x cos2y sin2x sin2y 1 sin2y cos2x 1 cos2x cos2x sin2y sin2y cos2x sin2y cos2x sin
2y 1 cos2x sin2y 4 3 1 2 2 cos x cos y sin x sin y 4 21. a. 3 5 23. 120 169 b. 117 44 c. 44 125 25. 142 212 10 29. 1.23 radians 2 2 sin2 x tan x 27. 31. 33. 1 1 1 1 2 sin2 x tan x 2 sin2 x cos x sin x cos x 1 cos 2x tan x 2 sin2 x sin x cos x 2 sin x cos x 2 cos x 2 cos3 x 2 cos x 2 cos x sin2 x sin 2x sin x 1 1 2 1 cos2 x 2 sin x cos x 2 2 1 35. 120 169 37. Yes. cos x 0 0 1 2 sin 2x 2 sin x cos x 2 p 12b p 6 b d 1 2 a a cos c p 6 1 cos 1 13 2 R B 2 2 13 4 R 2 22 13 2 ; cos p 12b a Chapter 9 Review, page 611 39. cos 1. 5. 7. 3. sin4 x cot t 1 3 sin4t cos4t sin2t sin t 1 cos t sin t 1 3 1 1 cos2t 1 sin2t 2 4 1 sin t 1 cos t sin2t cos2t 1 1 2 1 1 sin t 21 2 sin2t 1 1 cos t 1 cos t 2 2 1 cos t sin2t 1 1 cos t 2 sin2t cos2t 2 2 1 cos t sin t Answers to Selected Exercises 1109 19. 334.9 km 23. 84.9° 27. 231.9 ft 21. 63.7 ft 25. 8.4 km 29. 154.5 ft 31. 4.7 cm and 9.0 cm 33. 33.44° 35. 978.7 mi 37. AB 24.27, B 56.8°, AC 21.23, C 73.0° 39. 16.99 m Section 10.2, page 634 BC 19.5, A 50.2°, 1. 3. 5. 7. C 110°, b 2.5, c 6.3 B 14°, b 2.2, c 6.8 A 88°, a 17.3, c 12.8 C 41.5°, b 9.7, c 10.9 11. 32.5 9. 7.3 13. 82.3 17. No solution 55.
2°, 124.8°, A1 A2 19. C1 C2 104.8°, 35.2°, 14.1; 8.4 c1 c2 B2 21. No solution 65.8°, 9.8°, 23. a1 10.3; 58.2°, B1 A1 A2 2.1 a2 C 72°, b 14.7, c 15.2 a 9.8, B 23.3°, C 81.7° A 18.6°, B 39.6°, C 121.9° c 13.9, A 60.1°, B 72.9° C 39.8°, A 77.7°, a 18.9 25. 27. 29. 31. 33. 15. 31.4 114.2°, 35. No solution 37. 6.5 39. About 7691 41. 135.5 m 43. 5.4° 45. 5 ft 47. 5.3° 49. 30.1 km 51. About 9642 ft then EBA 180° ABD. 53. a. Use the Law of Cosines in triangle ABD to find is ABD; Use the Law of Cosines in triangle ABC to find CAB; have two of the angles in triangle EAB and can easily find the third. Use these angles, side AB, and the Law of Sines to find AE. 180° CAB. You 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 kp 43. 41. 0.96 sin p p 6 4 16 12 4 sin p 6. So, or 22 13 12 16 2 45. x 2.0344 kp Chapter 9 can do calculus, page 615 1. a. 1; sin x is changing at a rate of 1 unit per increase in x when 12 2 ; b. x 0. sin x is increasing approximately 0.7071 per unit increase in x when x p 4. c. 0; sin x is not changing per
unit increase in x when x p 2 d. 13 2 ; sin x is increasing approximately 0.8660 per unit increase in x when x p 6. 2. sin x 3. a. 0; cos x is not changing per unit increase in x x 0. when 12 2 ; b. c. cos x is decreasing approximately 0.7071 units per unit increase in x when 1; cos x is decreasing 1 unit per unit increase x p 4 in x when x p 2. d. 1 2 ; cos x is decreasing unit per unit increase 1 2 in x when x p 6. Chapter 10 Section 10.1, page 622 1. 3. 5. 7. 9. 11. 13. 15. 17. a 4.2, B 125.0°, C 35.0° c 13.9, A 22.5°, B 39.5° a 24.4, B 18.4°, C 21.6° c 21.5, A 33.5°, B 67.9° A 120°, B 21.8°, C 38.2° A 24.1°, B 30.8°, C 125.1° A 38.8°, B 34.5°, C 106.7° A 34.1°, B 50.5°, C 95.4° 54.2° at vertex at vertex (0, 0); 1, 4 1 2 48.4° at vertex 5, 2 1 2 ; 77.4° 1110 Answers to Selected Exercises real i i 1 2 1 real 1 21. 23. 25. 27. 29. 31. Section 10.3, page 642 1.–7. i (2i)(3, − 5 2 i) real 3 + 2i (1 + i)(1 − i) − 8 3 − i5 3 9. 13 11. 23 15. Many correct answers, including 13. 12 z 2i, w i 17. 19. real 1 i 1 5 i 1 cos 0.9273 i sin 0.9273 2 cos 5.1072 i sin 5.1072 2 cos 1.1071 i sin 1.1071 5 1 13 1 25 1 218.5 2 cos 2.1910 i sin 2.1910 1 2p 2p 3 b 3 i sin 3 323i 2 33. 6 cos a 35. 42 cos a 7p 6 i sin 7p 6 b 2123
21i 37. 3 2 a cos p 4 i sin p 4 b 322 a 4 b 322 a 4 b i 39. 222 cos a 7p 12 i sin 7p 12 b real 41. cos p 2 i sin p 2 43. 12 cos a 2p 3 i sin 2p 3 b 45. 222 cos a 19p 12 i sin 19p 12 b 47. The polar form of i is 1 cos 90° i sin 90° 1. Hence, 2 by the Polar Multiplication Rule i sin zi r 1 u 90° cos u 90°. 1 1 1 2 You can think of z as lying on a circle with center at the origin and radius r. Then zi lies on the same circle (since it too is r units from the origin), but 90° direction). farther around the circle (in a counterclockwise 2 Answers to Selected Exercises 1111 y b d c b1 a x a 2 29. 24 2 a 1 2 or 24 2 i b 1 a 2 or i b lies on L since b d b 2 1 a c, b d lies on M since 2 49. a. d. f. 51. a. b. c. 2 and d c x c b b. a y d b a 1 a c sin u221 sin2u22 i sin u121 cos u1 cos u2 1 r21 r21 r11 r13 1 sin u1 cos u2 i 1 r13 cos 2 cos u2 cos2u2 cos u1 u1 1 1 a c c. 4 i sin u22 i sin u22 2 cos u2 r2 cos u2 sin u1 sin u22 cos u1 sin u22 4 i sin u22 u22 4 u1 1 Section 10.4, page 652 24323 2 1. i 3. 243 2 i 5. 64 7. 1 2 23 2 i 9. i 11. 1, 1, i, i i sin p 15b, 4 cos a 11p 15 i sin 11p 15 b, i sin 7p 5 b 13. 4 cos a 4 cos a 15. 3 cos a p 15 7p 5 p 48 3 cos a 49p 48 i sin 49p 48 b, 3 cos a 73p 48 i sin 73p 48 b 17. cos a p 5 i sin p 5 b, cos a 3p 5 i sin 3p 5 b, cos p i sin p 1, 2 cos a 7p 5 i
sin 7p 5 b, cos a 19. cos a cos a 9p 5 p 10 9p 10 i sin 9p 5 b i sin p 10b, cos a p 2 i sin p 2 b, i sin 9p 10 b, cos a 13p 10 i sin 13p 10 b, cos a 17p 10 i sin 17p 10 b 21. 24 2 cos a p 8 i sin p 8 b, 24 2 cos a 9p 8 i sin 9p 8 b 1 2 i or 23 2 1 2 i or 23 2 1 2 i or 23. x 23 2 1 2 23 2 i or i or i 23 2 23 2 a 24 2 23 2 23 2 1 2 i b or 24 2 1 2 0.2225 ± 0.9749i, i b a 31. 1, 0.6235 ± 0.7818i, 0.9010 ± 0.4339i ± 1, 35. 1, 0.7660 ± 0.6428i, ± i, 33. 0.7071 ± 0.7071i, 0.7071 ± 0.7071i 0.1736 ± 0.9848i, 37. 1 0.5 ± 0.8660i, 0.9397 ± 0.3420i x6 1 x5 x4 x3 x2 x 1 x 1 so 21 x5 x4 x3 x2 x 1 0 the solutions of the sixth roots of unity other than 1; namely, 23 2 i, 1 2 i, 1 2 23 2 23 2 1, 1 2 1 2 i,, 2 are 23 2 i. 39. 12 41. For each i, ui n vui2 Hence 1 vui and is a solution of the equation. If is an nth root of unity, so n vn 1 r n 1. u12 cos u i sin u vuj, ui2 vui vn 21 1 1 1 2 then multiplying both sides by 1 v shows that In other words, if ui Thus, the solution is not equal to vu1, p, vun uj, are all then uj. ui vuj. vui distinct. 7. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. H 5 4, ; I 6, 10 I H u v 18, 10 H I u v H 1 322 I 23 4 u v h ; 13
5, 2 5 i 9. h u v 8, H 3 ; I 3u 2v 3 422, 3u 2v u v 1 322 ; I 9 822, H 3 422, H 2 922 I u v ;, 13 i 9 4 h, 7 ; i, 24 i 2 h 3u 2v 17 2 u v 2130 v w 1022 2 0 w 2u 1 v 225 7 v 2 3 u3 u2 u4 v c, d 2 I H v r s rc, rd u1 v 1 r s 1v 1 6 1 2 3, 9 I H c, d H c, d 2H I c, d c, 9 I H 0 0, 2 r v sv d 2 1 I H1 I sc, sd I H v, 0v 0 c, d 0, 0 H H 2 2c2 d2 v b d b b d d 2 2a2 b2 u 0 I I 2 1 2 i sin p 48b, 3 cos a 25p 48 i sin 25p 48 b, Section 10.5, page 660 1. 325 3. 234 5. 6, 6 H I 25. 23 2 1 2 i or 23 2 27. x 3i or 323 2 3 2 i i or 1 2 or 323 2 i 3 2 i 33. a. 2 b. 2 1 1 1112 Answers to Selected Exercises c. The slope is which is the same as the slope d c, of the vector v. d. The slope of the line is which is the same as b a, the slope of the vector u. Section 10.6, page 667 1. 3. 5. 7. u v 3i, u v i 2j, 3u 2v i 5j u v i 4j, u v 7i 4j, 3u 2v 18i 12j u v 23 i 22 j, u v 23 i 22 j, 3u 2v 223 i 322 j i 2j 5 2 9. 7i 7j 523 2, 5 2 i h 7.5175, 2.7362 H v 11. 9i 18j 13. v 15. 19. 21. 23. 25. 27. 29. I 17. 10, 1023 H I 1.9284, 2.2981 H v v v 10, u 60° v 241 6.4031, u 51.34° v 425, u 296.6
° v 5213, u 213.7° i 8 7 31. j 2113 2113 1 210 i 3 210 j 33. Direction: 82.5°; magnitude: 9.52 lb 35. Direction: 37. u sin 1 a 18.4°; 894.8 1500 b 36.6° magnitude: 80.4 kg 39. Parallel to plane: 68.4 lb; perpendicular to plane: 187.9 lb 41. 1931.85 pounds 43. Ground speed: 401.1 mph; course: 154.3° 45. Ground speed: 448.7 mph; course: 174.2° 47. Air speed: 96.6 mph; direction: 326° 49. 0.8023 mph 4° 6°; 53. 1002 lb on lies on the straight line through (0, 0) and which has slope 2 a c b d 51. 1005 lb on u v a c through (a, b) and (c, d), which also has slope a c b d they actually have the same direction by considering the relative positions of (a, b), (c, d), points and and w are parallel. Verify that For instance, if a c, b d Similarly, w lies on the line u v u v So,... 1 2 upward to the right, then (a, b) lies to the right d 6 b, and above (c, d). Hence so that a c 7 0 which means that the and endpoint of w lies in the first quadrant, that is, w points upward to the right. b d 7 0, c 6 a and Section 10.6.A, page 679 1. 3. 5. u v 7, u u 25, v v 29 u v 6, u u 5, v v 9 u v 12, u u 13, v v 13 7. 6 9. 20 11. 13. 1.75065 radians 15. 2.1588 radians 17. radians 28 p 2 19. Orthogonal k 2 25. 21. Parallel k 22 27. 23. Neither I 29. 31. 33. 37. 39. ; h 12 17 projuv, 20 17 i ; projvu projuv 0, 0 I H compvu 22 213 v w 1 a, b H I u a, b 1H I H 2 c r, d s I H ac ar bd bs u v u w a, b I
°, b 86.9 B 81.8°, C 38.2°, c 2.5 B 98.2°, C 21.8°, c 1.5 a 41.6; C 75°, c 54.1 and 19. 147.4 21. 13.4 km 23. Joe is 217.9 m from the pole and Alice is 240 m from the pole. 25. a. 3617.65 ft b. 4018.71 ft c. 3642.19 ft 27. 10 31. 210 220 29. 37.95 33. The graph is a circle of radius 2 centered at the origin. 35. 2 cos a p 3 i sin p 3 b 39. 223 2i 37. 422 422i 41. 81 2 8123 2 i 43. 1, cos i sin p 3 4p i sin 3 p 3 4p 3, cos, cos 2p 3 5p 3 i sin i sin 2p 3 5p 3 1,, cos cos cos p 8 9p 8 11, 1 H 210 I, i sin p 8 9p i sin 8 cos 5p 8, cos i sin, 5p 8 i sin 13p 8 2229 49. 55. 522 2, 522 2 h i 1 25 i 2 25 j 59. Ground speed: 321.87 mph; course: 126.18° 63. 3 26 projvu v 2i u2 v2 0 the same magnitude since u and v have 71. 1750 lb 45. 47. 53. 57. 61. 67. 69. 13p 8 51. 11i 8j −6 ; 0.7071 0.7071i 4. cos p 4 b i sin a a cos p i sin p p 4 b ; 2.7183 2 e 5. 1 6. 1 7. e a 8. ep i sin p 3 b cos p 3 cos 1 i sin 1 1 ; 12.5030 19.4722i 2 ; 1.3591 2.3541i Chapter 11 1. x2 49 Section 11.1, page 698 y2 4 y2 49 1 1 x2 9 5. 9. 2x2 y2 12 13. Ellipse −6 15. Ellipse 4 −4 4 −4 3. x2 36 y2 16 1 7. x2 6y2 18 11. x2 16 y2 9 1 6 6 21. 7p 23 65. 0.70 rad
ians 17. 8p 19. 223p 23. approximately 1,507,964 sq. ft. x2 a2 y2 a2 25. If a b, then 1. Multiplying both sides a2 gives x2 y2 a2, by radius a with center at the origin. the equation of a circle of Chapter 10 can do calculus, page 689 1. 1 cos 2 3 1 1 ; 0.4161 0.9093i 2. i sin i sin 3. cos.9900 0.1411i 27. As b gets larger, the ellipse becomes more elongated horizontally. As b gets closer to 0, the graph becomes very close to being a vertical line. However, it will never be a vertical line, because b cannot have a value of 0. 1114 Answers to Selected Exercises 10 10 10 29. Approximately 226,335 mi and 251,401 mi 19. Hyperbola 31. of OC; b length c 2a2 b2, Pythagorean Theorem c length or a length c2 a2 b2 of OF; since a2 b2 c2; of CF. 5. 1. 1 Section 11.2, page 707 y2 36 y2 5 y2 8 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; are 222, 0 and 222, 0 ; asymptotes B y 4 26 A and x B y 4 26 x 15. 10 −10 10 −10 foci are at A y ± b a x; are 220, 0 B y 2x and A and 220, 0 y 2x 17. 2 ; asymptotes B 25. The two branches of the hyperbola are very “flat” when b is large. With very large b and a small viewing window, the hyperbola may look like two horizontal lines, but it isn’t because its asymptotes y ± 2 b �
�x ± 2 b close to, but not equal to, 0 when b is large). are not horizontal (their slopes, are, 27. 8000 −3 3 −200 3800 −2 213 and 0, foci are at 0, a y ± a b x; 6 b y 2 3 x a and y 2 3 are 213 6 b −8000 ; asymptotes 29. y x 2 x 31. The distance between the vertices is 2a. One point on the hyperbola is the vertex at (a, 0). This point is the distance from the closest focus and a c a distance c a from the other focus. The Answers to Selected Exercises 1115 1 c a c a differences of these distances is 2 2a. Therefore, by the definition of a hyperbola, when the difference of the distances from any point on the hyperbola to the two foci is constant, this constant difference is 2a; this is the distance between the two vertices. OV a, OP b; a2 b2 c2 c2 a2 b2, each focus to the center O. since in the right triangle and in the equation of a hyperbola PV c, which is the distance from 33. Section 11.3, page 714 1. 7. y 3x2 y x2 4 11. Parabola 3. y2 20x 9. x2 8y 5. y2 8x −10 13. Parabola −10 4 −4 2 10 10 −10 15. Focus: 1 12b 0, a ; directrix: y 1 12 17. Focus: (0, 1); directrix: y 1 19. x2 1 2 y 21. y2 4x 1 1 2 2 x 2 6 y 2 16 13 x 3 36 x 3 4 2 1 5. 7. 9. 13. 15 16 y 2 2 1 17. Ellipse 23. 9 11 19. Parabola 21. Hyperbola −5 10 −1 25. 9 27. −7 −5 29. 7 −1 9 −1 7 5 23. The point closest to the focus is the vertex at (0, 0). 25. y2 8x 27. y2 2x −2 13 2 1 1. Section 11.4, page 726 y 3 16 y 4 36 x 2 4 x 7 25 1 1 33 1116 Answers to Selected Exercises 5910aans_11
14-1147 9/21/05 2:03 PM Page 1117 31. 9 −18 18 5. 7. u 53.13°; x 3 5 u 36.87°; ; y 4 5 v; y 3 5 33. −15 10 −10 10 −10 35. Ellipse; 37. Hyperbola; 6 x 3 and 7 x 13 2 y 4 and 3 y 9 39. Parabola; 41. Ellipse; 1 x 8 1.5 x 1.5 and 3 y 3 1 y 1 and 43. Hyperbola; 15 x 15 and 10 y 10 45. Parabola; 19 x 2 and 1 y 13 47. Hyperbola; 15 x 15 and 15 y 15 49. Ellipse; 6 x 6 and 4 y 4 51. Parabola; x 5 49 53. 2 1 2 9 x 4 y 3 16 2 2 1 1 and 2 y 10 x 5 16 1 2 2 or 55. The asymptotes of 1 are x2 a2 1 with slopes y2 a2 and 1, Since 1. these lines are perpendicular. 9, 1 a 2 ± 1 2 234 59. b y ± x, 1 1 2 1 1 b 0 2 57. 61. 63. 65. 67 105 200.45 1 2 2 1 1 1 2 y 55 2 1 y2 5,759,600 x2 1,210,000 The exact location cannot be determined from the given information. measurement in feet 49 y ± a a x or Section 11.4A, page 733 1. u2 2 v2 2 1 3. u2 4 v2 1 v u 3 5 u 4 5 u2 v 9. a. b. c. E cos u 2 1 2 u A cos2 u B cos u sin u C sin2 u 1 B cos2 u 2A cos u sin u 2C cos u sin u 1 C cos2 u B cos u sin u uv B sin2 u 1 2 D cos u E sin u v2 A sin2 u 1 2 2 v F 0 D sin u B¿ B cos2 u 2A cos u sin u 2C cos u sin u B sin2 u 2 B since Since sin 2u 2 sin u cos u and cos 2u cos2 u sin2 u, B¿ 1 cot 2u A C then B B¿ B cot 2u we have 2 B cos 2u
B cos 2u 0. A C B cot 2u cos2 u sin2 u C A B¿ 1 1 2 2 and sin 2u B cos 2u sin 2u B cos 2u. C A sin u cos u is the coefficient of uv 1 2 d. If 11. a. From Exercise 9 (a) we have B¿ 2 4A¿C¿ 1 2 2 2 4 C cos2 u 2 1 2 1 1 2 B2 C A 1 C A cos u sin u 2cos2 u sin2 u 4B 2 A2 C2 B2 2 2 1 cos2 u sin2 u 2 cos2 u A cos2 u B cos u sin u 2 4 C cos2 u B cos u sin u A sin2 u C A cos u sin u B cos2 u 2A cos u sin u 2C cos u sin u 1 B sin2 u 2 C sin2 u 2 1 cos2 u sin2 u B 2 1 3 A cos2 u B cos u sin u C sin2 u 4 1 B cos u sin u A sin2 u 4 1 sin2 u 4 1 4AB B2 4BC 1 2 cos2 u sin2 u sin4 u 4 cos2 u sin2 u 2 cos2 u sin2 u cos4 u sin4 u 4AC 1 (everything else cancels) B2 4AC 2 1 B2 4AC 2 1 B2 4AC 6 0, B¿ 0, 4AC 2 cos2 u sin2 u 2 cos3 u sin u cos u sin3 u 1 cos3 u sin u cos u sin3 u 1 cos4 u 2 cos2 u sin2 u sin4 u cos2 u sin2 u 2 then also 4A¿C¿ 6 0 B¿ 2 1 and so cos4 u sin4 u 2 B2 4AC cos4 u 2 4A¿C¿ 6 0. A¿C¿ 7 0. By Since Exercise 10, the graph is an ellipse. The other two cases are proved in the same way. 522 2 23 2, 1 2 b 22 2 b, 15. a 1 1 b. If Section 11.5, page 743 13. a 1. P T 2, a 4, a 5, 2p, 1 1 2 5p 6 b, 1, a others 3. 5. 7. 13. a, U, Q p 4 b
3p 2 b 5, 2p, 1 2 1, 7p 6 b a 3, 2p 3 b 6, p 3 b a 5, 3p 2 1, a,, R 5, p, S 1 7p 6 b, 7, a 2 5p 3 b, V 7, 0 1 2 or 6, a 5, p, 1 11p 6 b, and others 2 1, 13p 6 b a,, and 6, p 6 b, 323 3 a 2 225, 1.1071 2 b A B 9. 23 2 a, 1 2 b a 231.25, 2.6779 11. B 15. A Answers to Selected Exercises 1117 25. θ = π 2 −1 −0.5 0.5 1 θ = 0 −0.5 −1 −1.5 −2 θ = 3π 2 27. 29. θ = π 2 1 0.5 −0.5 −1 θ = 0 θ = π −2 −1.5 −1 −0.5 θ = π 2 1 0.5 −1 −0.5 0.5 1 θ = 0 −0.5 −1 θ = 0 5 10 31. θ = π 2 θ = 0 0.5 0.5 −0.5 −0.5 θ = 3π 2 17. π 2 θ = 4 −4 2 −2 −2 −4 19 11 − π 3 −1 θ = 0 21. θ = π 2 3 2 1 −1 −2 −3 −2 −1 1 radian 1 2 θ = 0 23. θ = − 3π 2 −10 −5 7.5 5 2.5 −2.5 −5 −7.5 −10 θ = − π 2 1118 Answers to Selected Exercises 33. −2 −1 θ = π 2 0.5 −0.5 θ = 3π 2 θ = 0 1 2 35. θ = π 2 3 2 1 −1 −2 − = 3π 2 37. θ = π 2 1.2 1 0.8 0.6 0.4 0.2 −0.2 0.2 0.2 0.4 0.6 0.8 1 1.2 θ = 0 θ = 3π 2 �
� = π 2 0.75 39. −1 −0.75 θ = 3π 2 41. θ = π 2 2 1.5 1 0.5 −0.5 −1 −1.5 −2 0.2 0.4 0.6 0.8 1 θ = 0 θ = 3π 2 43 45. θ = π 2 0.8 0.6 0.4 0.2 −0.2 −0.2 0.2 0.4 θ = 0 θ = 3π 2 47. a. 3 −4 4 θ = 0 1 −3 Answers to Selected Exercises 1119 b. 3 19. a. 4 6 6 4 b. 215 4, 210 4, 22 4 c. The smaller the eccentricity, the closer the shape is to circular. 21. 10 5 (4, π) −10 −5 5 10 −5 −10 23. 10 5 (−2, 0) −10 −5 −5 −10 (, π2 ) 3 5 10 25. (10, )π 2 10 5 −5 −5 −10 −10 5 ( 10 7, 10 )3π 2 −4 c. 3 4 3 −3 3 3 49. r a sin u b cos u 1 r2 ar sin u br cos u 1 x2 y2 ay bx 1 x2 bx y2 ay 0 1 a2 b2 x2 bx b2 a 4 b 4 y a a 2b y2 ay a2 4 b a2 b2 4 x b 2b a circle with center b 2, a 2b a and radius 2a2 b2 2. 51. Using the Law of Cosines in the following d2 r2 s2 2rs cos so. diagram, d 2r2 s2 2rs cos r, θ) d r θ − β s (s, β) Section 11.6, page 752 1. (d) 3. (c) 5. (a) 7. Hyperbola, e 4 3 11. Ellipse, e 2 3 9. Parabola, e 1 13. 0.1 15. 25 17. 5 4 1120 Answers to Selected Exercises 10 5 (3, π) −5 −5 −10 −10 27. 29. 31. (15, 0) 5 10 15 10 5 −5 −5 −10
15 10 5 ( 3 2, )π 2 5 10 )3π 2 (−10, )π (2, 2 −10 −5 −5 5 10 33. r 6 1 cos u 37. r 3 1 2 cos u 41. r 3 1 sin u 45. r 2 1 2 cos u Section 11.7, page 763 35. r 39. r 16 5 3 sin u 8 1 4 cos u 43. 47. r 2 2 cos u r 3 107 1 cos u 2 1 and 4, 7 2, 5 25. Both give a straight line segment between Q P 1 equations in (a) move from P to Q, and the parametric equations in (b) move from Q to P. t x a c a The parametric 27. Solving t. 2 x a substituting in ˛, 2 ˛ 1 for t gives 1 2 then gives 1 or x a. 2 and This is a linear equation and therefore gives a straight line. You can check by substitution that (a, b) and (c, d) lie on this straight line; in fact, these points correspond to respectively. x 6 18t, y 12 22t t 1, 0 t 1 t 0 and 1 2 29. 31. Local minimum at 6, 2 33. Local maximum at (4, 5) 1 2 35. Solve the first equation for t x 140 cos 31° ˛. Substitute into the second to get y 140 sin 31° a which is the equation of a parabola. x 140 cos 31°b 16 2a 1 x 140 cos 31°b 2, 37. a. 120 0 0 250 88 cos 48° 88 sin 48° x 1 y 1 0 t 4.5 t 2 t 16t2 4 2 b. Yes v 8024 50 39. 41. a. 3 105.29 ft/sec 1. 3. 5. 7. 9. 11. 13. 15. 21. 5 x 6 and 2 y 2 3 x 4 and 2 y 3 0 x 14 and 15 y 0 2 x 20 and 11 y 11 12 x 12 and 12 y 12 2 x 20 and 20 y 4 25 x 22 and 25 y 26 y 2x 5 y 2x 7 17. 19. y ln x 0 0 350 t 2 t 16t2 2 110 cos 28° 110 sin 28° x 1 y 1 0 t 3.5 b. About 3.2 sec c. 41.
67 ft x2 y2 9 23. 16x2 9y2 144 Answers to Selected Exercises 1121 80° 60° 40° 20° 43. a. 200 0 0 b. c. 40° 200 45° 40° 0 0 An angle of distance. c. 6 0 0 12 The particles do not collide; they are closest when 8 t 1.13. d. 0 −1 2 350 350 45° seems to result in the longest d is smallest when t 1.1322. Section 11.7A, page 769 1. 3. 5. x 9 5 cos t, y 12 5 sin t x 2 cos t 2, y 2 sin t 3 x 210 cos t, y 6 sin t 0 t 2p 1 0 t 2p 2 2 1 0 t 2p 2 1 −10 7 −7 10 7. x 1 2 cos t, y 1 2 sin t 1 0 t 2p 2 −3 2 −2 3 45. a. Since Then 3 cos., y CT PQ 3 3 sin u u t 3p 2 u, TCQ 3p t 3p 2 2 x OT CQ 3t 3 cos u 3t t 3p And. 2 b t 3p 2 b cos t cos 1 3p 2 sin t. sin t sin 3p 2 Therefore,. b. a cos a 3 3 sin t 3p a 2 b cos t sin t 0 1 t 3p a 2 b t 3p 2 b 21 3t 3 cos t sin t. Sin 3 a cos t sin sin t 2 1 2 1 2 3p 2 21 3t 3 sin t sin t cos 3p 2 cos t. 21 1 3 3 sin 0 1 2 t 3p a 2 b cos t 1 2 21 3 3 cos t Therefore, 1 cos t 3 1. 2 47. a. 6 0 0 12 The particles do not collide. b. t 1.1 1122 Answers to Selected Exercises 9. x 210 cos t, y 6 tan t 0 t 2p 2 1 17. x 4 cos t 1, y 28 sin t 4 0 t 2p 1 2 −10 10 −10 10 −7 11. x 1 cos t, y 1 2 tan t 1 0 t 2p 2 19. x t any real number t 2 9 −1 7 9 −1 5 6 −5 −6 4 −4 9 −1 −5 13. x t2 4, y t 1 t any
real number 2 −5 10 −10 15 15. x 2 cos t 1, y 3 sin t 5 0 t 2p 2 1 21. x 2 t 2 1 2 2, y t 1 any real number t 2 7 −3 −2 13 23. x 4 tan t 1, y 5 cos t 3 1 0 t 2p 2 9 −18 18 10 −15 Answers to Selected Exercises 1123 1; 0, ± 25 y2 x2 1 4 points A Shown is the graph on the window 6 y 6. This is a hyperbola with foci at the 0, ± 2 1 9 x 9, and vertices at the points B 2. 25. x 1 cos t 3, y 2 tan t 2 0 t 2p 1 2 9. −10 10 −10 10 Chapter 11 Review, page 772 1. This is an ellipse with foci at the points and vertices at the points A graph on the window 0, ± 225 9 x 9, B. 0, ± 2 2 1 Shown is the 6 y 6. 11. y2 25x 13. 15. y2 16x 17. Focus: 5 14b, 0, a directrix: y2 16x y 5 14 19. Ellipse, foci: (1, 6), (1, 0), vertices: (1, 7), 1, 1 1 2 9 −3 9 −9 3. This is the same graph as in Exercise 2; the equations are equivalent. 5. This is an ellipse with foci at the y2 16 1; 0, ± 2 x2 12 points 1 Shown is the graph on the window 6 y 6. and vertices at the points 2 0, ± 4 9 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. ± 5, 0 1 1124 Answers to Selected Exercises 23. This is a hyperbola with center at (0,
0), vertices at 6, 0, foci at the points and asymptotes 252, 0 2 1 the points (6, 0) and 252, 0 A y ± 2 and 1 B 3 ˛x. 2 37. Hyperbola 39. Ellipse 41. 43. 45. 47. 9 x 9 and 6 y 6 15 x 10 and 10 y 20 6 x 6 and 4 y 4 23 x 1 2 2 23 u 1 2 2 u y v v 10 49. 45° 51. θ = π 2 −10 10 −10 25. This is a parabola with vertex at the point (3, 3), y 23 25 8 b 8 focus at the point and directrix 3, a,. 20 15 10 5 1 2 3 4 5 6 27. This is a parabola with vertex at the point 5 4 focus at the point and directrix, 1 1 x 3 4 b a, 1, 1 (2, 3π/4) (−3, −2π/3) θ = π θ = 0 53. θ = π 2 2π 3 θ = 0, 2. 5520 −10 1 −1 −2 −3 θ = 0 10 20 15 10 5 −5 −10 −15 −20 θ = − π 2 4, 6 2 29. Center: y2 4 1 31. 1 x 3 16 2 2 1 33. 35. 1 y 1 2b a y 1 2 1 x 3 2b 2 2 1 Answers to Selected Exercises 1125 57. θ = π 2 69. Ellipse −2 − ( 12, π 2 ) 59. 61. −1 −2 −3 −4 θ = π 2 −1 −0.5 0.2 −4 −4 −2 θ = 0 2 4 θ = π Pole θ = 0 3π 2 ) 4,( 3π 2 θ = 71. r 2 1 cos u 73. r 24 5 cos u 75. 77. 79. 35 x 32 and 2 y 16 15 x 15 and 10 y 10 y 2x2 2 y 2 x −1 1 1, 0 as t goes from 0 1 Then point retraces its path, moving from 2 Point moves from (1, 0) to p. to 1, 0 1 x 3 2y or y 1 to (1, 0) as t goes
from 2 2 ˛x 3 x 5 cos t 3, y 5 sin t 5, 0 t 2p 2 p to 2p. 83. numbers 74 and 75 11 −1 9 −9 63. 2 ˛, 323 3 2 b a 67. Hyperbola 65. Eccentricity 2 3 B 0.8165 81. 85. θ = π Pole (−2, π) (4, 0) θ = 0 1126 Answers to Selected Exercises 87. x cos t 2, y 2 sin t 3, 0 t 2p 95. x 212 tan t 2, y 2 sec t 3, 0 t 2p −9 6 −6 9 −9 9 3 −9 Note that the diagonal lines shown here are not part of the graph but are just about where the asymptotes should be. t 4 x 32 2 5, y t, 10 t 10 1 2 97. 1.5 −25 5 −5 5 89. x 1 2 cos t, y 1 3 sin t, 0 t 2p −1.5 1 −1 91. x sec t, y 1 36 tan t, 0 t 2p 0.1 −2 2 −1.5 93. x 7 cos t 2, y 8 sin t 5, 0 t 2p 15 −5 15 −15 Chapter 11 can do calculus, page 777 1. approximately 30 2. approximately 20 3. approximately 8 4. approximately 4 5. approximately 10 6. approximately 5 Chapter 12 Section 12.1, page 788 1. Yes 3. Yes 5. No 7. 11. 15. 5 x 11 5 ˛, y 7 2 ˛, s 5 2 x 28, y 22 r 5 9. 13. 17. 7 x 2 7 ˛, y 11 2 ˛, y x 2, y 1 x 3c c 2d 2 19. Inconsistent 21. x b, 23. x b, y 3b 4 2 y 3b 2 4 25. Inconsistent ˛, where b is any real number ˛, where b is any real number 27. x 6, y 2 29. x 0.185, y 0.624 31. 33. x 3, y 1, z 4 x 66 5 ˛, y 18 5 Answers to Selected Exercises 1127 35. c 3, d 1 2 y 960x 2000; y 114x 14,
000 37. a. Electric: solar: b. Electric: c. Costs same in fourteenth year; electric; solar $14,570 $6800; solar: 39. a. b. c. y 7.50x 5000 y 8.20x 130,000 ≈ (7143, 58564) 0 0 15,000 3. 50, 2, −3) y y Costs equal at approximately 7143 cases. d. The company should buy from the supplier any number of cases less than 7143 and produce their own beyond that quantity. 2 4 2 (3, 0, 0) 4 x 41. 140 adults, 60 children 43. $19,500 at 2% and $15,500 at 4% 45. 3 4 lb cashews and 2 1 4 lb peanuts 47. 60cc of 20% and 40cc of 45% 49. 80 bowls; 120 plates Section 12.1.A, page 794 1. z (1, 4, 5) 4 2 y 2 2 4 4 x 1128 Answers to Selected Exercises 7. z 4 2 (2, −3, −1) 2 4 x 9. y 2 4 x 6 intercepts: y 2 z 6 x 3 intercepts: y 9 4 z 3 2 19. intercepts: no y-intercept x 2 z 3 21. Each equation can be represented by a plane. Two planes are parallel, are coincident, or they intersect in a line. The system of equations either has no solution, an infinite number of solutions which lie on the plane, or an infinite number of solutions, all of which lie on a straight line. Section 12.2, page 801 x 3 intercepts: y 3 z 3 1 11. 13. 15. 17. x 4 intercepts: y 2 z 8 intercepts: no y-intercept x 4 z 12 3. 9. 5. 11. y 5, z 2, £ 2x 3y 1 4x 7y 2 4x 2y 4z 2w 1 4x 4y 4z 2w 3 4x 2y 5z 2w 2 x 3 2 x 2 t, any real number x 1, y 1, y 3 x 1 2 ˛, 4 ˛, x 14, y 6, 19. No solution 2t, z t, 2t, 21. 17. 15. 13. x t, w 0 z 4, w t, where t is where
t is any real 25. No solutions x t 2, for any real number t y t 1, y 0, z 0 number x 1, y 2 z t, x 0, x 1, x 3, x 3 4 ˛, 23. 27. 29. 31. 33. 35. y 1, z 3, y 1, z 2, w 2 w 5. y 10 3 ˛, z 5 2 Answers to Selected Exercises 1129 37. 10 quarters; 28 dimes; 14 nickels 39. $3000 from her friend; $6000 from the bank; $1000 from the insurance company 41. $15,000 in the mutual fund; $30,000 in bonds; $25,000 in food franchise 43. Three possible solutions: 18 bedroom, 13 living room, 0 whole house 16 bedroom, 8 living room, 2 whole house 14 bedroom, 3 living room, 4 whole house 45. Tom: 8 hours; George: 24 hours; Mario: 12 hours. 47. 2000 chairs; 1600 chests; 2500 tables 49. 20 model A; 15 model B; 10 model C 1. 2 A B Section 12.3, page 811 11 4 7 A C 4 3 10 ≥ 23. 25. 27. 29. 31. To ≥ £ ≥ £ ≥ £ ≥ From: 0 1 0 1 1 1 £ 3.75 6.5 14.75 The total cost to bake and decorate each giant ≥ £ cookie is $3.75; sheet cake: $6.50; 3-tiered cake: $14.75. ; a11 $91,000 18200 5080 represents the total 91000 25400 amount of tuition the college got from lecture; a22 ¢ the college got from lab; are not meaningful in the context of the problem. represents the total amount of tuition and $5080 a12 a21 ≤ 33. a Ma Mil Min SL C Ma Mil Min SL. • A2. • A A2 A3 12 11 11 11 11 A3, 8 8 8 8 8 µ 11 7 10 10 7 11 10 7 7 10 • 11 10 7 7 10 8 4 8 8 4 11 7 10 10 7 • This matrix represents the total number of flights that are direct, have one layover, or have two layovers between each pair of cities. µ 5. 2C 10 14 2 8 4 2 £ 12 2 4; 3 3; 3 2; £ 7. AB defined, 9
. AB defined, 11. AB defined, 13. 3 2 0 8 11 10 ¢ 17 17 3 33 19 5 19. § AB BA not defined ≥ BA defined, 2 2 BA not defined 1 3 2 1 6 5 15. £ ≥ ¥ ; BA 4 24 9 2 21. AB 1130 ¢ £ 8 2 2 ≤ 3 21 24 8 6 15 ; BA ¢ 19 10 ≤ 0 9 2 0 8 0 0 Answers to Selected Exercises ≥ £ ≥ 5910aans_1114-1147 9/21/05 2:03 PM Page 1131 Section 12.4, page 819 35. a. ; CI2 3 2 4 1 ; I2C 3 2 4 1 CI3 ¢ ; ≤¢ 2u 0 ≥ • 4u. I2 1 0 3. I3 ¢ 1 0 0 0 1 0 1 ≤ 0 I3C £ 5. 1 ≤ 1 1 9. 2x 1 £ 4x y 0 11. x 2y 2z 1 y 2z 0 3x 2y 2z 0 u 2v 2w 0 v 2w 1 3u 2v 2w 0 r 2s 2t 0 s 2t 0 3r 2s 2t 1 13. 2 3 2 1 1 2 ≥ 17. No inverse £ 15. No inverse 3 1 19. 2 4 1 1 10 8 5 1 2 6 4 3 7 The A matrix and X matrix are the same; no. The first system has an infinite number of ≤¢ ¢ ≤ ¢ x t, solutions of the form ¢ ≤ ≤ ≤¢ t 3 y 1 2 2 ≤ ¢ the ; second system has no solution. If a coefficient matrix does not have an inverse, then any system with those coefficients will have an infinite number of solutions or no solution. b 0, y x2 1 c 1; 37. 39. 41. 43. 45. a 1, a 0.5, x, y 2 0, 5 2 2, 1 2 4, 7 8, y 2 1 5, 1 1 2, 0 1 1, 3 2 1 2, 5 2 1 10, 4 1 a 1, b 1.5, y 0.5x2 1.5x 2. c 2; y ax 3 bx 2 cx d 5 d 1 8a 4b 2c d 7 64a 16b 4c d 3 512a 64b 8c d y ax 4 bx 3 cx 2 dx e
1 625a 125b 25c 5d e 0 16a 8b 4c 2d e 3 a b c d e 5 16a 8b 4c 2d e 4 10,000a 1000b 100c 10d e c 1 ; y e x 4e x 1 47. A 0, F, C 0, D 0, E 0, and F t, 2 b 4, B 1 12 where t is any real number. The equation is t 12 which reduces to xy t 0, xy 12; hyperbola l 10,128.2, 49. h 224.4, b 2339.7 Section 12.5, page 824 ≥ 1. x 3, 21. 23. 25. 27. x 3, y 1 x 1, y 0, z 3 x 8, y 16, z 5 x 0.5, y 2.1, z 6.7, w 2.8 £ 29. no solution 31. 33. x 2 t, y 3.5 2.5t, where t is any real number. x 1149 161 y 426 161,, z 1124 161, w 579 161 z 1 2t, 3. 5. 7. 9. 11. w t, x 1, y 1 or x, y 3 2 y 9 or y 3 2, or or x 221, B x 7, x 0, x 2, x 4, or or x 3, x 6, or x 4, x 4 or x 221, y 2 Answers to Selected Exercises 1131 x 1.3163, or y 2.4814 or x 0.3634, y 0.9578 15. 13. x 1.6237, y 1.0826 or x 1.9493, x 1.4184, or x 0.9519, 19. No solutions 17. y 8.1891 x 2.8073, y 0.4412 y 0.5986 y 0.8145 5. 10 x 10; 10 y 10 21. x 13 2105 x 13 2105 8 8 y y ˛, ˛, 3 2105 8 3 2105 8 or 7. 10 x 10; 10 y 10 x 3.1434, or y 7.7230 or x 2.8120, or x 0.9324, or y 1.4873 x 0.0480, or 25. 23. y 2.2596 y
19.3201 x 2.1407, x 4.8093, y 7.7374 or y 11.7195 x 3.8371, y 7.7796 x 1.4873, y 0.0480 y 1.4873 x 0.0480, or x 1.4873, y 0.0480 r 5 29. center (0, 0); 31. center (7.5, 12.5); r 12.75 27. 33. (440.2, 38205.5) and (1893.1, 81794.5). 37. 35. Two possible boxes: one is 2 by 2 by 4 m and the other is approximately 3.123 by 3.123 by 1.640 m. 4 and 12 15 and 12 8 15 inches 43. 12 ft by 17 ft y 6x 9 39. 1.6 and 2.6 47. 45. 41. Section 12.5.A, page 832 10 x 10; 1. 10 y 10 9. 10 x 10; 10 y 20 11. 10 x 10; 10 y 10 3. 10 x 10; 10 y 10 13. 0 x 10; 0 y 10 corner points: 0, 0, 1 2 1 0, 3 2 1132 Answers to Selected Exercises 15. 0 x 10; 0 y 10 Chapter 12 can do calculus, page 841 corner points: 0, 0, 1 2 1 0, 4 2, 0, 1 2, 2 a 20 7, 18 7 b 17. At (0, 6), the objective function has a value of 30. 19. At (0, 0), the objective function has a value of 0. 21. 24 roast beef sandwiches for a profit of $72 23. 3000 peach trees and 27,000 almond trees Chapter 12 Review Exercises, page 835 1. 5. 9. x 5, x 35, x t 1, 19 11. 37 and y 7 y 70, 3. x 0, 7. y 2 x 2, y 4, z 6 z 22 y 2t, for any real number t z t 13. (c) 15. 100 17. 2x 6y 16 2x 3y 7 19. 2x 3z 2 4x 3y 7z 1 8x 9y 10z 3 21. x 26 9, y 11 9 ; consistent 23. no solution; inconsistent t 37 11 y 3 11 x 1 11 25., t
10 11 z t,, for any real number t; consistent 2 3 4 1 9 ¢ 4 7 3 ≤ 29. Not defined 33 46 85,,, £ z 21 34 x 4, y 3, z 2 ≤ ¢ x 1 y 14 85 85 y 5x 2 2x 1 x 3, y 9 or x 1, y 1 x 1 27, y 1 27 x 1 27, y 1 27 x 1.692, y 3.136 or or x 1.812, y 2.717 47. maximum is 150 at (5, 0); minimum is 20 at (0, 2) 49. minimum of 97.5 pounds at (85, 12.5) 27. 31. 35. 37. 39. 41. 43. 45 3x 1 x 2 x 1 2. 2 2 5. 1 25 x 2 5 2x 1 3 5 x 4 2x 2 1 3 25 x 3 3x 1 x 2 4 5 ˛x 1 x 2 2x 2 3 5 3 x 1 x 3 3. 4. 6. 7. Chapter 13 Section 13.1, page 851 1. The population is the entire student body; the sample is 50 students from each grade level. 3. The population is the total number of American families; the sample is 50 families in each of 10 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% 40% 30% 20% 10% Aerobics Kick boxing Tai chi Stationary bike Answers to Selected Exercises 1133 Frequency 6 8 5 4 1 1 Relative frequency 24% 32% 20% 16% 4% 4% 15. Color red blue purple green yellow orange 17. 40% 30% 20% 10% red blue purple green yellow orange 19. symmetric 21. skewed left 23. uniform 25 27 10 0 0 Key: 2 Key: 5 0 0 29. The number 90 is an outlier since it is quite a distance from the rest of the data. 31. The shape of the fall semester data is basically symmetric; the summer semester data is skewed right. It may be that in the summer enrolled students live

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