( 1 3, 3 4 ) Selected Answers SECTION P.2. Quick Review P.1. Exercises P.1. Quick Review P.2. Exercises P.2 SELECTED ANSWERS 895

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1 44_Demana_SEANS_pp89-04 /4/06 8:00 AM Page 89 SELECTED ANSWERS 89 Selected Answers Quick Review P.. {,,, 4,, 6}. {,, }. (a) 87.7 (b) ; ,,,, 4,, 6 Eercises P (terminating)..6 (repeating). ; All real numbers less than or equal to ; All real numbers less than 7 9. ; All real numbers less than 0.., or. 7., 9.,., 4. The real numbers greater than 4 and less than or equal to 9. The real numbers greater than or equal to, or the real numbers which are at least 7. The real numbers greater than 9. 4; endpoints and 4; bounded; half-open. ; endpoint ; unbounded; open. 9 or [9, ); Bill s age or [.099,.99]; dollars per gallon of gasoline 7. a ab 9. a d (a) Associative propert of multiplication (b) Commutative propert of multiplication (c) Addition inverse propert (d) Addition identit propert (e) Distributive propert of multiplication over addition ,870,000,000, (a) Because a m 0, a m a 0 a m 0 a m implies that a 0. (b) Because a m 0, a m a m a m m a 0 implies that a m a m. 67. False. For eample, the additive inverse of is, which is positive. 69. E 7. B 7. 0,,,, 4,, ,, 4,,,, 0,,,, 4,, 6 SECTION P Quick Review P Distance: A B C D Eercises P.. A(, 0), B(, 4), C(, ), D(0, ). (a) First quadrant (b) On the -ais, between quadrants I and II (c) Second quadrant (d) Third quadrant Perimeter ; Area 0.. Perimeter ; Area. 0.6., 6 7. (, 4 )

2 44_Demana_SEANS_pp89-04 /4/06 8:00 AM Page SELECTED ANSWERS 9... [99, 00] b [0, 0] [99, 00] b [0, 0] [99, 00] b [0, 0]. (a) about $8,000 (b) about $77, The three sides have lengths,, and. Since two sides have the same length, the triangle is isosceles. 9. (a) (no answer) (b) (89) 4. ( ) ( ) 4. ( ) ( 4) 9 4. center: (, ); radius: center: (0, 0); radius: c d. 7; 6. Midpoint is, 7. Distances from this point to vertices are equal to or 9. True. l ength of AM length of AB because M is the midpoint of AB. B similar triangles,l ength o length f AM of AC l ength of AM l ength of AB, so M is the midpoint of AC. 6. C 6. E 6. If the legs have lengths a and b, and the hpotenuse is c units long, then without loss of generalit, we can assume the vertices are (0, 0), (a, 0), and (0, b). Then the midpoint of the hpotenuse is 0 a, b 0 a, b. The distance to the other vertices is a b a b 4 4 c c. 67. Qa, b 69. Qa, b SECTION P. Quick Review P Eercises P.. (a) and (c). (b). es 7. no 9. no. 8. t z t 9. (a) The figure shows that is a solution of the equation 6 0. (b) The figure shows that is a solution of the equation (a). (b) and (c) z , 4,, Multipl both sides of the first equation b. 6. (a) no (b) es 6. False. 6 because 6 lies to the left of on the number line. 6. E 67. A 69. (a) (no answer) (b) (no answer) (c) 800/80 799/800 (d) 0/0 0/0 (e) If our calculator returns 0 when ou enter 4, ou can conclude that the value stored in is not a solution of the inequalit b b 7. F ha 9 C SECTION P.4

3 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 897 SELECTED ANSWERS 897 Eploration. The graphs of m b and m c have the same slope but different -intercepts.. [ 4.7, 4.7] b [.,.] m= [ 4.7, 4.7] b [.,.] m= [ 4.7, 4.7] b [.,.] m=4 [ 4.7, 4.7] b [.,.] m= In each case, the two lines appear to be at right angles to one another. Quick Review P Eercises P ( ). 4 ( ) (a): the slope is., compared to in (b).. 4; [, 0] b [ 0, 60] 49. m 8 4, so asphalt shingles are acceptable. [, ] b [ 0, 80]. (a) (b) $7. trillion (c) $9. trillion (d). 0; 7 7. Ymin 0, Yma 0, Yscl 9. Ymin 0/, Yma 0/, Yscl / 4. (a) (b) 7 4. (a) (b) 7 4. (a) 87.; 4,000 (b) 9.7 ears (c) 87.t 4,000 74,000; t 0.04 (d) ears 47.,000 ft [99, 00] b [, 0]. (a) 7 7 (b) [0, ] b [000, 7000] [0, ] b [000, 7000] (c) The ear 006 is represented b 6. So the value of for 6 is 67 million, a little larger than the U.S. Census Bureau estimate of 6 million.

4 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS b ; a 6 9. (a) No; perpendicular lines have slopes with opposite signs. (b) No; perpendicular lines have slopes with opposite signs. 6. False. The slope of a vertical line is undefined. For eample, the vertical line through (, ) and (, 6) would have slope (6 )/( ) /0, which is undefined. 6. A 6. E 6. False. The slope of a vertical line is undefined. For eample, the vertical line through (, ) and (, 6) would have slope (6 )/( ) /0, which is undefined. 67. (a) (b) (c) (d) a is the -intercept and b is the -intercept when c. [, ] b [, ] [, ] b [, ] [, ] b [, ] (e) (f) When c, a is the opposite of the -intercept and b is the opposite of the -intercept. [ 0, 0] b [ 0, 0] [ 0, 0] b [ 0, 0] [ 0, 0] b [ 0, 0] a is half the -intercept and b is half the -intercept when c. 69. As in the diagram, we can choose one point to be the origin, and another to be on the -ais. The midpoints of the sides, starting from the origin and working around counterclockwise in the diagram, are then A a,0, B a b c,, C d b, c e, and D d, e. c e The opposite sides are therefore parallel, since the slopes of the four lines connecting those points are: m AB ; mbc ; b d a c e m CD ; mda. 7. A has coordinates b d a b, c, while B is b c a,, so the line c containing A and B is the horizontal line, and the distance from A to B is a b b a. SECTION P. Eploration.. B this method, we have zeros at 0.79 and... The answers in parts,, and 4 are the same ;.0707 [, 4] b [, 0] [, 4] b [, 0] [, 4] b [, 0] Quick Review P ( ) 7. ( )( ( ) ( ) ) 9. ( ) ( )

5 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 899 SELECTED ANSWERS 899 Eercises P.. 4 or. 0. or.. or or or or or or or intercept: ; -intercept: 7. -intercepts:, 0, ; -intercept: [, ] b [, ] [, ] b [, ] [, ] b [, ] [, ] b [, ]. 0; ; t 6 or t 0 4. or 6 4. or 4. (a) 4 (the one that begins on the -ais) and (b) 4 (c) The -coordinates of the intersections in the first picture are the same as the -coordinates where the second graph crosses the -ais. 47. or 49. or. 4.6 or 0.44 or...4 or.9 7. (a) There must be distinct real zeros, because b 4ac 0 implies that b 4ac are distinct real numbers. (b) There must be real zero, because b 4ac 0 implies that b b 4ac 0, so the root must be. (c) There must be no real zeros, because b a 4ac 0 implies that b 4ac are not real numbers d wide; 0 d long ft 6. False. Notice that () 8, so could also be. 6. B 67. E 69. (a) c (b) c 4 (c) c (d) c (e) There is no other possible number of solutions of this equation. For an c, the solution involves solving two quadratic equations, each of which can have 0,, or solutions. 7.., or approimatel and 4.0 SECTION P.6 Quick Review P a d Eercises P.6. 8 i. 4i. ( )i 7. i i. 4i. 48 4i. 0i 7. 4i 9. i.,.,. i / /i. / 4/i 7. / 7/i 9. 7/ /i 4. i i 4. False. An comple number bi has this propert E 49. A. (a) i; ; i; ; i; ; i; (b) i; ; i; ; i; ; i; (c) (d) (no answer). (a bi) (a bi) bi, real part is zero. (abi)(cdi) (acbd)(adbc)i (ac bd) (ad bc)i and (abi)(cdi) (a bi) (c di) (ac bd) (ad bc)i are equal 7. (i) i(i) 0 but (i) i(i) 0. Because the coefficient of in i 0 is not a real number, the comple conjugate, i, ofi, need not be a solution.

6 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS SECTION P.7 Quick Review P.7.. or. 7. z z Eercises P ( )( 4). (, 9] [, ). (, ) , (, ] [7, ) [7, ]. (, ),. (, ),. [, 0] [, ) , , 4,. (,.4] [0.08, ).,,. No solution 7. [.08, 0.7] [.9, ) 9..,. (a) 0 (b) 0 (c) 0 (d) ( )( ) 0 (e) ( )( 4) 0 (f) ( 4) 0. (a) t 4 sec up; t sec down (b) when t is in the interval 4, (c) When t is in the interval (0, 4] or [, 6). Reveals the boundaries of the solution set 7. (a) in. 4 in. (b) When is in the interval (, ]. 9. no more than $00, True. The absolute value of an real number is alwas greater than or equal to zero. 4. D 4. D 47. (.69, 4.) (0.6,.9) CHAPTER P REVIEW EXERCISES. Endpoints 0 and ; bounded.. v ,000,000,000. (a) (b) (c) (d) (e) ; 4.. ; ; 4 7. ( 0) ( 0), or 4 9. Center: (, 4); radius:. (a) , , 0 (b) (0) (80) a 7; b (a) (b) (c) 07.6, which is ver close to 08. (d) 4 [0, ] b [00, ] [0, ] b [00, ] ; or 4. 7 or ,. i or or or (6, ] (, ] 6. (, ], 6. (, 0.7) (.7, ) 67. (,.8] [0.4,.] 69. (, 7) (, ) 7. (, ) 7. i i i 8. (a) t 8 sec up; t sec down (b) When t is in the interval (0, 8] or [, 0) (approimatel). (c) When t is in the interval [8, ] (approimatel). 8. (a) When w is in the interval (0, 8.]. (b) When w is in the interval (.9, ) (approimatel).

7 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 90 SELECTED ANSWERS 90 SECTION. Eploration $4.6 Eploration. Percentages must be 00. A statistician might look for adverse economic factors in 990, especiall those that would affect people near or below the povert line. Quick Review.. ( 4)( 4). (9 )(9 ). (4h 9)(h )(h ) 7. ( 4)( ) 9. ( )( ) Eercises.. (d)(q). (a)(p). (e)(l) 7. (g)(t) 9. (i)(m). (a) Increasing, ecept for a slight drop from 999 to 004. (b) 974 to 979. Women (), Men (). Women: 0.8., men: , 69.9% 9. (a) and (b) L. square stones..t [, ] b [, 9] The lower line shows the minimum salaries, since the are lower than the average salaries. 7. Year. There is a clear drop in the average salar right after the 994 strike. 9.. ; 4..; [ 0, 0] b [ 0, 0] [ 0, 0] b [ 0, 0] [, ] b [ 0, 0] [ 4, 4] b [ 0, 0] or (a) $46.94 (b) 0 mi 49. (a) ( 00 ) /00 00/00 for all 0. (b) (c) Yes (d) For values of close to 0, 00 is so small that the calculator is unable to distinguish it from zero. It returns a value of 0 /00 0 rather than. [0, ] b [0, ]. (a) or. or. (b). Let n be an integer. n n n(n ), which is either the product of two odd integers or the product of two even integers. The product of two odd integers is odd. The product of two even integers is a multiple of 4, since each even integer in the product contributes a factor of to the product. Therefore, n n is either odd or a multiple of 4.. False; a product is zero if at least one factor is zero. 7. C 9. B 6. (a) March (b) $0 (c) June, after three months of poor performance (d) About $000 (e) After reaching a low in June, the stock climbed back to a price near $40 b December. LaToa s shares had gained $000 b that point. (f) An graph that decreases steadil from March to December would favor Ahmad s strateg over LaToa s.

8 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 6. (a) Subscribers Monthl Bills (b) The linear model for subscribers as a function of ears after 990 is [7, ] b [0, 00] [7, ] b [, ] (c) The fit is ver good: (d) The monthl bill scatter plot has an obviousl curved shape that could be modeled more effectivel b a function with a curved graph. Some possibilities: quadratic (parabola), logarithm, sine, power (e.g., square root), logistic. (We will learn about these curves later in the book.) [7, ] b [0, 00] (e) Subscribers Monthl Bills (f) Cellular phone technolog was still emerging in 99, so the growth rate was not as fast, eplaining the lower slope on the subscriber scatter plot. The new technolog was also more epensive before competition drove prices down, eplaining the anomal on the monthl bill scatter plot. [4, ] b [0, 00] [4, ] b [0, 60] SECTION. Eploration. From left to right, the tables are (c) constant, (b) decreasing, and (a) increasing.. positive; negative; 0 Quick Review , Eercises.. Function. Not a function; has two values for each positive value of.. Yes 7. No 9. (, ). [, ] b [, ] [ 0, 0] b [ 0, 0]

9 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 90 SELECTED ANSWERS (, 0] 9. (, )[0, ) [ 0, 0] b [, ] [, ] b [, ]. Yes, non-removable. Yes, non-removable [ 0, 0] b [ 0, 0] [ 0, 0] b [, ]. Local maima at (, 4) and (, ), local minimum at (, ). The function increases on (, ], decreases on [, ], increases on [, ], and decreases on [, ). 7. (, ) and (, ) are neither, (, ) is a local maimum, and (, ) is a local minimum. The function increases on (, ], decreases on [, ], and increases on [, ). 9. Decreasing on (, ];. Decreasing on (, ]; increasing on [, ) constant on [, ]; increasing on [, ) [ 0, 0] b [, 8] [ 0, 0] b [0, 0]. Increasing on (, ]; decreasing on [, ). Bounded 7. Bounded below 9. Bounded [ 4, 6] b [, ] 4. f has a local minimum of.7 at 0.. It has no maimum. [, ] b [0, 6]

10 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 4. Local minimum: 4.09 at 0.8. Local maimum:.9 at 0.8. [, ] b [ 0, 0] 4. Local maimum: 9.6 at.0. Local minimum: 0 at 0 and 0 at 4. [, ] b [0, 80] 47. Even 49. Even. Even. Odd. ; 7. ; [ 0, 0] b [ 0, 0] [ 8, ] b [ 0, 0] 9. ; ; 6. 0; [ 0, 0] b [ 0, 0] [ 4, 6] b [, ] 6. (b) 6. (a) 67. (a) f() crosses the horizontal (b) g() crosses the horizontal (c) h() intersects the horizontal asmptote at (0, 0). asmptote at (0, 0). asmptote at (0, 0). [ 0, 0] b [ 0, 0] [ 0, 0] b [, ] [, ] b [, ]

11 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 90 SELECTED ANSWERS (a) The vertical asmptote is 0, and 7. True; this is the definition of the graph of a function. this function is undefined at 0 7. B 7. C (because a denominator can t be zero). (b) [ 0, 0] b [ 0, 0] Add the point (0, 0). (c) Yes 77. (a) k (b) 0; but the discriminant of is negative (), so the graph never crosses the -ais on the interval (0, ). (c) k (d) 0; but the discriminant of is negative (), so the graph never crosses the -ais on the interval (, 0). [, ] b [, ] 79. (a) (b) (c) (d) (e) Answers var 8. (a) (b) (c) (d) Answers var 8. (a). It is in the range. (b). It is not in the range. (c) h() is not bounded above. (d). It is in the range. (e). It is in the range. 8. Since f is odd, f() f() for all. In particular, f(0) f(0). This is equivalent to saing that f(0) f(0), and the onl number which equals its opposite is 0. Therefore f(0) 0, which means the graph must pass through the origin. SECTION. Eploration. f() /, f() ln. f() /, f() e, f() /( e ). No. There is a removable discontinut at 0. Quick Review Eercises.. (e). (j). (i) 7. (k) 9. (d). (l). E. 8. E. 7, 8 7. E., 4, 6, 0,, 9.,, /, sin., /,. /, e, /(+ e ). /, sin, cos, /( e ) 7.,, /, sin 9. Domain: all reals; Range: [, ). Domain: (6, ); Range: all reals. Domain: all reals; Range: all integers. (a) Increasing on [0, ) (b) Neither (c) Minimum value of 0 at 0 (d) Square root function, shifted 0 units right 7. (a) Increasing on (, ) (b) Neither (c) None (d) Logistic function, stretched verticall b a factor of 9. (a) Increasing on [0, ); decreasing on (, 0] (b) Even (c) Minimum of 0 at 0 (d) Absolute value function, shifted 0 units down 4. (a) Increasing on [, ); decreasing on (, ] (b) Neither (c) Minimum of 0 at (d) Absolute value function, shifted units right 4.,

12 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS No points of discontinuit No points of discontinuit 49.. No points of discontinuit 0. (a). (a) g() [, ] b [, ] f() [, ] b [, ] (b) f() g() (b) The fact that ln(e ) shows that the natural logarithm function takes on arbitraril large values. In particular, it takes on the value L when e L. 7. Domain: all real numbers; Range: all integers; Continuit: There is a discontinuit at each integer value of ; Increasing/decreasing behavior: constant on intervals of the form [k, k ), where k is an integer; Smmetr: none; Boundedness: not bounded; Local etrema: ever non-integer is both a local minimum and local maimum; Horizontal asmptotres: none; Vertical asmptotes: none; End behavior: int() as and int() as. 9. True; the asmptotes are 0 and. 6. D 6. E 6. (a) even (b) even (c) odd 67. (a) Pepperoni count ought to be proportional to the area of the pizza, which is proportional to the square of the radius (b) 0.7 (c) Yes, ver well. (d) The fact that the pepperoni count fits the epected quadratic model so perfectl suggests that the pizzeria uses such a chart. If repeated observations produced the same results, there would be little doubt. 69. (a) f() /, f() e, f() ln, f() cos, f() /( e ) (b) f() (c) f() e (d) f() ln (e) The odd functions:,,/, sin

13 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 907 SELECTED ANSWERS 907 SECTION.4 Eploration f g f g 4 ( ) ( ) 0.6 ln(e ) ln sin cos sin sin cos Quick Review.4. (, ) (, ). (, ]. [, ) 7. (, ) 9. (, ) Eercises.4. ( f g)() ; ( f g)() ; ( fg)() ( )( ). There are no restrictions on an of the domains, so all three domains are (, ).. ( f g)() sin ; ( f g)() sin ; ( fg)() sin. Domain in each case is [0, ).. ( f/g)() ; 0 and 0, so the domain is [, 0) (0, ). (g/f )() ; 0, so the domain is (, ). 7. ( f/g)() / ; 0, so ; the domain is (, ). ( g/f )() / ; 0 and 0; the domain is [, 0) (0, ]. 9.. ; 6. 8;. f(g()) ; (, ); g( f()) ; (, ) [0, ] b [0, ] 7. f(g()) ; [, ); g( f()) ; (, ] [, ) 9. f(g()) ; [, ]; g( f()) ; 4 [, ]. f(g()) ; (, 0) (0, ); g( f()) ; (, 0) (0, ). One possibilit: f() and g(). One possibilit: f() and g() 7. One possibilit: f() and g() 9. One possibilit: f() cos and g()

14 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS. V 4 r 4 (48 0.0t) ; 77,74.6 in.. t.6 sec. (, ) 7. and 9. and 4. and 4. and or and 4. False; is not in the domain of (f/g)() if g() C 49. E. f g D e ln (0, ) ( ) [, ) ( ) (, ] ( ) 0 (, ). (a) g() 0 (b) g() (c) g(). 4 0 SECTION. Eploration [ 9.4, 9.4] b [ 6., 6.]. T starts at 4, at the point (8, ). It stops at T, at the point (8, ). 6 points are computed.. The graph is less smooth because the plotted points are further apart.. The grapher skips directl from the point (0, ) to the point (0, ), corresponding to the T-values T and T 0. The two points are connected b a straight line, hidden b the Y-ais. 7. Leave everthing else the same, but change Tmin back to 4 and Tma to. Quick Review ,, and 0 Eercises.. (6, 9). (, ). (a) (6, 0), (4, 7), (, 4), (0, ), (, ), (4, ), (6, 8) (b). ; It is a function. (c) [, ] b [, ]

15 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 909 SELECTED ANSWERS (a) (9, ), (4, 4), (, ), (0, ), (, ), (4, 0), (9, ) (b) ( ) ; It is not a function. (c) [, ] b [, ] 9. (a) No (b) Yes. (a) Yes (b) Yes. f (), (, ). f (), (, ) (, ) 7. f (), 0 9. f (), (, ). f (), (, ). One-to-one. One-to-one 7. f(g()) [ ] ( ) ; g( f()) [( ) ] () 9. f(g()) [( ) / ] ( ) ; g( f()) [( ) ] / ( ) /. f(g()) ( )( ) ( ) ; g( f()). (a) 08 euros (b). This converts euros () to dollars (). (c) $ e and ln are inverses. If we restrict the domain of the function to the interval [0, ), then the restricted function and are inverses True. All the ordered pairs swap domain and range values. 4. E 4. C 4. (Answers ma var.) (a) If the graph of f is unbroken, its reflection in the line will be also. (b) Both f and its inverse must be one-to-one in order to be inverse functions. (c) Since f is odd, (, ) is on the graph whenever (, ) is. This implies that (, ) is on the graph of f whenever (, ) is. That implies that f is odd. (d) Let f(). Since the ratio of to is positive, the ratio of to is positive. An ratio of to on the graph of f is the same as some ratio of to on the graph of f, hence positive. This implies that f is increasing. 47. (a) 0.7 (b) 4 ( ). It converts scaled scores to raw scores. 49. (a) No (b) No (c) 4; es. When k, the scaling function is linear. Opinions will var as to which is the best value of k.

16 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS SECTION.6 Eploration. The raise or lower the parabola along the -ais.. Yes Eploration. Graph C. Points with positive -coordinates remain unchanged, while points with negative -coordinates are reflected across the -ais.. Graph F. The graph will be a reflection across the -ais of graph C. Eploration. The. and the stretch the graph verticall; the 0. and the 0. shrink the graph verticall. [ 4.7, 4.7] b [.,.] Quick Review.6. ( ). ( 6). ( /) Eercises.6. Vertical translation down units. Horizontal translation left 4 units. Horizontal translation to the right 00 units 7. Horizontal translation to the right unit, and Vertical translation up units 9. Reflection across -ais. Reflection across -ais. Verticall stretch b. Horizontall stretch b 0., or verticall shrink b Translate right 6 units to get g 9. Translate left 4 units, and reflect across the -ais to get g.. 0 h f g 6 6 f 6 h g 6. f() 7. f() 9. (a) (b). (a) f() ( 8) (b) f() 8(). Let f be an odd function; that is, f() f() for all in the domain of f. To reflect the graph of f() across the -ais, we make the tranformation f(). But f() f() for all in the domain of f, so this transformation results in f(). That is eactl the translation that reflects the graph of f across the -ais, so the two reflections ield the same graph.

17 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS (a) 8 (b) 7 4. (a) 4 (b) 9 4. Starting with,translate right units, verticall stretch b, and translate down 4 units. 4. Starting with, horizontall shrink b and translate down 4 units. 47. ( 4) Reflections have more effect on points that are farther awa from the line of reflection. Translations affect the distance of points from the aes, and hence change the effect of the reflections. 7. First verticall stretch b 9, then translate up units. 9. False; it is translated left. 6. C 6. A 6. (a) (b) Change the -value b multipling b the conversion rate from dollars to en, a number 6 that changes according to international market conditions. This results in a vertical 4 stretch b the conversion rate. Price (dollars) Month 67. (a) The original graph is on the top; (b) The original graph is on the top; (c) the graph of f() is on the bottom. the graph of f( ) is on the bottom. [, ] b [ 0, 0] [, ] b [ 0, 0] (d) [, ] b [ 0, 0] [, ] b [ 0, 0]

18 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 9 SELECTED ANSWERS SECTION.7 Eploration. n = ; d = 0 n = 4; d = n = ; d = n = 6; d = 9 n = 7; d = 4 n = 8; d = 0 n = 9; d = 7 n = 0; d =,, 9, 4, 0, 7. Linear: r 0.978; power: r 0.990; quadratic: R ; cubic: R ; quartic: R. Since the quadratic curve fits the points perfectl, there is nothing to be gained b adding a cubic term or a quartic term. The coefficients of these terms in the regressions are zero. Quick Review.7. h (A/b). h V/(r ). r Eercises.7 V 4 7. h A r A r r r A 9. P A( r/n) nt ( r/n) nt ( )() Let C be the total cost and n be the number of items produced; C 4,00.7n.. Let R be the revenue and n be the number of items sold: R.7n.. V r 7. A a /4 9. A 4r. 4 60; 4; ,4;,00. 8 t, so t. hr () 9.8, 0.7(7) 0.; The $ shirt is a better bargain, because the sale price is cheaper. 9..9%. (a) (00 ) 0.(00) (b) Use 7.4 gallons of the 0% solution and about 4.86 gal of the 4% solution.. (a) V (0 )(8 ) (b) (0,) (c) Appro..06 in. b.06 in.. 6 in. 7. Appro..6 in. 9. Appro..4 mph 4. True; the correlation coefficient is close to if there is a good fit. 4. C 4. B 47. (a) C 00,000 0 (b) R 0 (c) 000 pairs of shoes (d) The point of intersection corresponds to the break-even point, where C R. 49. (a) u(),000 (b) s(),000 (c) R u () 6 (d) 4 R s () 79 (e) (f) You should recommend stringing the rackets; fewer strung rackets need to be sold to begin making a profit (since the intersection of and 4 occurs for smaller than the intersection of and ). [0, 0,000] b [0, 00,000]

19 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS 9. (a) (b) List L {., 06., 0., 96.6, 9.0, 87., 8., 79.8, 7.0, 7.7, 68, 64., 6., 8.,.9,.0, 0.8, 47.9, 4., 4.} (c) It fits the data etremel well. [0, ] b [00, 00] CHAPTER REVIEW EXERCISES. d. i. b 7. g 9. a. (a) All reals (b) All reals. (a) All reals (b) [0, ). (a) All reals (b) [8, ) 7. (a) All reals ecept 0 and (b) All reals ecept 0 9. Continuous. (a) Vertical asmptotes at 0 and (b) 0. (a) none (b) 7 and 7. (, ) 7. (, ), (, ), (, ) 9. Not bounded. Bounded above. (a) none (b) 7, at. (a), at 0 (b) none 7. Even 9. Neither 4. ( )/ 4. / [, ] b [, ] [, ] b [, ] 49.. [, ] b [, ] [, ] b [, ]. ( f g)() 4; (, ] [, ). ( f g)() ( 4); [0, ) 7. lim 9. s / 6. 00h t/(0) 6. (a) (b) The regression line is (c),948 (thousands of barrels) [4, ] b [940, 700] [4, ] b [940, 700] 67. (a) h r (b) r r (c) [0, ] (d) (e).7 in. h r r [0, ] b [0, 0] h = r

20 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS Chapter Project , e [, ] b [ 00, 600] SECTION. Eploration. $000 per ear. $0,000; $8,000 Quick Review ( ) 7 (, 4) (, ) Eercises.. Not a polnomial function because of the eponent. Polnomial of degree with leading coefficient. Not a polnomial function because of the radical 7. f() f() (, 4) ( 4, 6) (, ) (, ) 0

21 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS 9. f(). (a). (b) 7. (e) (0, ) (, 0) 9. Translate the graph of units right and the result units down Translate the graph of units left, verticall shrink the resulting graph b a factor of, and translate that graph units down Verte: (, ); ais:. Verte: (, 7); ais: 7. Verte: 6, 7 ; ais: 6 ; f() Verte: (4, 9); ais: 4; f() ( 4) 9. Verte:, ; ais: ; g(). f() ( ) ; Verte: (, );. f() ( 8) 74; ais: ; opens upward; Verte: (8, 74); ais: 8; opens does not intersect -ais downward; intersects -ais at about 6.60 and 0.60, or (8 74 ) [ 4, 6] b [0, 0] [ 0, ] b [ 00, 00]

22 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 7. f() ; Verte:, ; ais: ; opens upward; does not intersect -ais; verticall stretched b [.7, ] b [,.] 9. ( ) 4. ( ) 4. ( ) 4 Strong positive 47. Weak positive 49. (a) (b) Strong positive $940 [, 4] b [0, 0]. (a) The slope tells us that hourl compensation for production workers increases about 4 /r. (b) About $.70. (a) [0, 00] b [0, 000] is one possibilit. (b) either 07, units or 7,66 units 7.. ft 9. (a) R() (6, )( ) (b) (c) 90 cents per can; $6,00 [0, ] b [0,000, 7,000] 6. (a) About ft above the field (b) About 6.4 sec (c) About 7 ft/sec downward 6. (a) h 6t 80t 0 (b) 90 ft,. sec [0, ] b [ 0, 00]

23 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 97 SELECTED ANSWERS (a) (c) On average, the children gained 0.68 pounds per month. (d) [, 4] b [0, 40] (b) (e) 9.4 lb [, 4] b [0, 40] 69. The Identit Function f() Domain: (, ); Range: (, ); Continuous; Increasing for all ; Smmetric about the origin; Not bounded; No local etrema; No horizontal or vertical asmptotes; End behavior: f(), lim lim f() 7. False. The initial value is f(0) 7. E 7. B 77. (a) (i), (iii), and (v) (b) (i), (iii), (iv), (v), and (vi) (c) (ii) is not a function. 8. (a) The two solutions are b ac b 4 and b ac b 4 b ; their sum is a a a b a. b (b 4ac) c (b) The product of the two solutions given above is. 4a a 8. a b, (a b) 4 8. Suppose f() m b with m and b constants and m 0. Let and be real numbers with. Then the average rate of change of f is f( ) f( ) [ 4.7, 4.7] b [.,.] (m b) (m b) m ( ) m, a nonzero constant. On the other hand, suppose m and are constants and m 0. Let be a real number with and let f be a function defined on all real numbers such that f( ) f( ) m. Then f() f( ) m( ) and f() m (f( ) m ). Notice that the epression f( ) m is a constant; call it b. Then f( ) m b; so, f( ) m b and f() m b for all. Thus f is a linear function. SECTION. Eploration. The pairs (0, 0), (, ) and (, ) are common to all three graphs. [.,.] b [.,.] [, ] b [, ] [ 0, 0] b [ 00, 00]

24 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS Quick Review... d. q 4 7. / / Eercises.. power, constant. not a power function. power, constant c 7. power, constant g 9. power, constant k. degree 0, coefficient 4. degree 7, coefficient 6. degree, coefficient 4 7. A ks 9. I V/R. E mc. The weight w of an object varies directl with its mass m, with the constant of variation g.. The refractive inde n of a medium is inversel proportional to v, the velocit of light in the medium, with constant of variation c, the constant velocit of light in free space. 7. power 4, constant ; Domain: (, ); Range: [0, ); Continuous; Decreasing on (, 0). Increasing on (0, ); Even. Smmetric with respect to -ais; Bounded below, but not above; Local minimum at 0; Asmptotes: none; End Behavior: lim 4, lim 4 [, ] b [, 49] 9. power 4, constant ; Domain: [0, ); Range: [0, ); Continuous; Increasing on [0, ); Bounded below; Neither even nor odd; Local minimum at (0, 0); Asmptotes: none; End Behavior: lim 4 [, 99] b [, 4]. shrink verticall b ; f is even.. stretch verticall b. and reflect over the -ais; f is odd. [, ] b [, 9] [, ] b [ 0, 0]

25 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 99 SELECTED ANSWERS 99. shrink verticall b ; f is even. 7. (g) 9. (d) 4. (h) 4 [, ] b [, 49] 4. k, a 4. f is increasing in Quadrant I. f is undefined for k, a 4. f is decreasing in Quadrant IV. f is even. 47. k, a. f is decreasing in Quadrant I. f is odd , power, constant 8.. L m/sec. (a) (b) r.04 w 0.97 (c) (d) Approimatel 7.67 beats/min, which is ver close to Clark s observed value [, 7] b [0, 40] [, 7] b [0, 40] 7. (a) (b) ; es (c) (d) Approimatel.76 m W and m W, respectivel [0.8,.] b [ 0., 9.] [0.8,.] b [0., 9.] 9. False, because f() () / / f(). The graph of f is smmetric about the origin. 6. E 6. B

26 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 6. (a) The graphs of f() and h() [0, ] b [0, ] [0, ] b [0, ] [, ] b [, ] f g h k Domain Range Continuous es es es es Increasing (, 0) (, 0) Decreasing (, 0), (0, ) (0, ) (, 0), (0, ) (0, ) Smmetr w.r.t. origin w.r.t. -ais w.r.t. origin w.r.t. -ais Bounded not below not below Etrema none none none none Asmptotes -ais, -ais -ais, -ais -ais, -ais -ais, -ais End Behavior lim f() 0 lim g() 0 lim h() 0 lim k() 0 are similar and appear in the st and rd quadrants onl. The graphs of g() and k() 4 are similar and appear in the st and nd quadrants onl. The pair (, ) is common to all four functions. (b) The graphs of f() / and h() /4 [0, ] b [0, ] [0, ] b [0, ] f g h k Domain [0, ) (, ) [0, ) (, ) Range 0 (, ) 0 (, ) Continuous es es es es Increasing [0, ) (, ) [0, ) (, ) Decreasing Smmetr none w.r.t. origin none w.r.t. origin Bounded below not below not Etrema min at (0, 0) none min at (0, 0) none Asmptotes none none none none End behavior lim f() lim g() lim g() lim h() lim k() lim [, ] b [, ] k() are similar and appear in the st quadrant onl. The graphs of g() / and k() / are similar and appear in the st and rd quadrants onl. The pairs (0, 0), (, ) are common to all four functions. 67. T a.. Squaring both sides shows that approimatel T a. 69. If f() is even, g() f() f() g(). If f() is odd, g() f() (f()) f() g(). If g() f(), then f() g() and f() g(). So b the reasoning used above, if g() is even, so is f(), and if g() is odd, so is f(). 7. (a) The force F acting on an object varies jointl as the mass m of the object and the acceleration a of the object. (b) The kinetic energ KE of an object varies jointl as the mass m of the object and the square of the velocit v of the object. (c) The force of gravit F acting on two objects varies jointl as their masses m and m and inversel as the square of the distance r between their centers, with the constant of variation G, the universal gravitational constant.

27 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS 9 SECTION. Eploration. (a) ; (b) ; (c) ; (d) ;. (a) ; (b) ; (c) ; (d) ; Eploration Quick Review.. 4. ( )( ). ( )( ) 7. 0, 9. 6,,. Eercises.. Shift to the right b units,. Shift to the left b unit, stretch verticall b. -intercept: (0, 4) verticall shrink b,reflect over the 0 0 -ais, and then verticall shift up units. -intercept: 0,. Shift 4 to the left units, verticall stretch b, reflect over the -ais, and verticall shift down units. -intercept: (0, ) local maimum: (0.79,.9), zeros: 0and (c). (a). One possibilit:. One possibilit: [ 00, 00] b [ 000, 000] [ 0, 0] b [ 000, 000] [, ] b [ 8, ] lim f() ; lim f() lim [ 8, 0] b [ 0, 00] f() ; lim f() lim.., 7., [, ] b [ 4, 6] f() ; lim f() [, ] b [ 0, 0] lim f() ; lim f()

28 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 9 SELECTED ANSWERS 9. (a) There are zeros: the are.,, and... (c) There are zeros: approimatel 0.7 actuall, 0., and.. 4 and. and 7. 0,, and 9. Degree ; zeros: 0 (mult., graph crosses -ais), 4. Degree ; zeros: (mult., graph crosses -ais), (mult., graph is tangent) (mult., graph is tangent) [, ] b [ 0, 0] [, ] b [ 0, 0] [ 6, 4] b [ 00, 0].4, 0.74,.67.47,.46, , 0.4,, , 6, and 6.,,. f() 8 7. f() It follows from the Intermediate Value Theorem. 6. (a) (b) (c) (d) 6.9 ft (e) mph [0, 60] b [ 0, 0] [0, 60] b [ 0, 0] 6. (a) (b) 0.9 cm or True. Because f is continuous and f() and f(), the Intermediate Value Theorem assures us that the graph of f crosses the -ais between and. 7. C 7. B 77. The eact behavior near is hard to see. A zoomed-in view around the point (, 0) suggests that the graph just touches the -ais at 0 without actuall crossing it that is, (, 0) is a local maimum. One possible window is [0.9999,.000] b [ 0 7, 0 7 ]. [0, 0.8] b [0,.0] 79. A maimum and minimum are not visible in the standard window, but can be seen on the window [0., 0.4] b [.9,.]. 8. The graph of ( ) increases, then decreases, then increases; the graph of onl increases. Therefore, this graph can not be obtained from the graph of b the transformations studied in Chapter (translations, reflections, and stretching/shrinking). Since the right side includes onl these transformations, there can be no solution. 8. (a) Substituting, 7, we find that 7 ( ) 7, so Q is on line L, and also f() , so Q is on the graph of f(). (b) (c) The line L also crosses the graph of f() at (, ). [.8,.] b [6, 8]

29 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS (a) and D u D u u impl D u. 8 D u 8 implies D u u( 8). Combining these ields u u( 8), which D u 8 8 implies 8 8. (b) Equation (a) sas 8 8. So, Thus. (c) B the Pthagorean Theorem, 8 D 900 and D 400. Subtracting equal quantities ields 00. So, Thus, 00 ( 8) 64 ( 8), or This is equivalent to (d) Notice that 8 0. So, the solution we seek is.7, which ields.4 and D 6.. SECTION.4 Quick Review ( )( ) 7. 4( )( ) 9. ( )( )( ) Eercises.4. f() ( ) f() ;. f() ( f() 4)( ) ; 4 f( ) 7 9/. f() ( 4 )( ) 8; f( ) Yes. No. Yes 4. f() ( )( )( 7) f () ( 4)( )( ). ;,,, 6.,, 9 ;, 4. No zeros outside window 47. There are zeros not shown (appro..00 and.00) 49. Rational zero: ; irrational zeros:. Rational: ; irrational:. Rational: and 4; irrational:. Rational: and 4; irrational: none 7. $6.7; (b) is a zero of f () (c) ( )( 4 9) (d) One irrational zero is.04. (e) f() ( )(.04)( ) 6. False. ( ) is a factor if and onl if f() A 67. B 69. (d) 0.67 m 7. (a) Shown is one possible view, on the window [0, 600] b [0, 00]. (b) The maimum population, after 00 das, is 460 turkes. (c) P 0 when t. about das after release. (d) Answers will var. [0, 600] b [0, 00]

30 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 7. (a) 0 or positive zeros, negative zero (b) no positive zeros, or negative zeros (c) positive zero, no negative zeros (d) positive zero, negative zero 7. Answers will var, but should include a diagram of the snthetic division and a summar: ( ) (a) (b) 7,, and (c) There are no rational zeros. 79. (a) Approimate zeros:.6,.07, 0.90,.9. (b) f() g() (.6)(.07)( 0.90)(.9) (c) Graphicall: graph the original function and the approimate factorization on a variet of windows and observe their similarit. Numericall: Compute f(c) and g(c) for several values of c. SECTION. Eploration. f(i) (i) i(i) 4 0; f (i) (i) i(i) 0; no.. The Comple Conjugate Zeros Theorem does not necessaril hold true for a polnomial function with comple coefficients. Quick Review.. i. 7 4i. ( )( ) 7. 9 i 9.,, /, / Eercises.. 9; zeros: i; -intercepts: none ; zeros: (mult. ), i; -intercept: (b) 9. (d). comple zeros; none real. comple zeros; real. 4 comple zeros; real 7. Zeros:, 9 i; f() ( )( 9i)( Zeros:, i; f() ( )( )( i)( 4. Zeros: 7,, i; f() ( 7)( )( i)( i). Zeros:, i; f() ( )( )( i)( i). Zeros:, i; f() ( )( )( i)( i) 7. ( )( ) 9. ( )( ) 4. ( )( 4)( ) 4. h.776 ft 4. Yes, f() ( ) No, either the degree would be at least or some of the coefficients would be nonreal. 49. f() (a) D 0.080t 0.96t.6t.779 [, 9] b [0, ] (b) Sall walks toward the detector, turns and walks awa (or walks backward), then walks toward the detector again. (c) t.8 sec (D. m) and t.64 sec (D.6 m).

31 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS 9. False. If i is a zero, then i must also be a zero.. E 7. C 9. (a) Power Real Part Imaginar Part (b) ( i) 7 8 8i ( i) ( i) 9 6 6i ( i) 0 i 0 0 (c) Reconcile as needed. 6. f(i) i i(i) i(i) i i 0 6. Snthetic division shows that f(i) 0 (the remainder), and at the same time gives f() ( i) i h(), so f() ( i)( i). 6. 4, i, i SECTION.6 Eploration. g(). k() 4 [, 7] b [, ] [ 8, ] b [, ] Quick Review.6.,.. 7. ; 7 9. ; Eercises.6. Domain: all ; lim f(),lim f(). Domain: all, ; lim f(), lim f(), lim f(), lim. Translate right units. 7. Translate left units, reflect across -ais, Asmptotes:, 0 verticall stretch b 7, translate up units. Asmptotes:, 0 f() 6

32 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 9. Translate left 4 units, verticall stretch b, translate down units. Asmptotes: 4, Vertical asmptote: none; Horizontal asmptote: ; lim f() lim f(). Vertical asmptotes: 0, ; Horizontal asmptote: 0; lim 0 lim f(), lim f() lim f() 0. Intercepts: 0, and (, 0). No intercepts Asmptotes:,, Asmptotes:, 0, and 0, and 0 f(), lim 0 f(), lim f(), [ 4, 6] b [, ] [ 4.7, 4.7] b [ 0, 0] 7. Intercepts: (0, ), (.8, 0), and (0.78, 0); Asmptotes:, 9. Intercept: 0,, and Asmptotes:, 4 [, ] b [ 4, 6] [ 0, 0] b [ 0, 0]. (d); Xmin, Xma 8, Xscl, and Ymin, Yma, Yscl. (a); Xmin, Xma, Xscl, and Ymin, Yma 0, Yscl. (e); Xmin, Xma 8, Xscl, and Ymin, Yma, Yscl 7. Intercept: 0, ; asmptotes:,, 0; lim f(), lim f(), lim 6 f(), (/) lim (/) f() ; Domain:, ; Range:, (0, ); Continuit: all, ; Increasing: (, ),, 4, Decreasing: 4,,, ; Unbounded; [ 4.7, 4.7] b [.,.] Local Maimum at 4, 6 ; Horizontal asmptote: 0; Vertical asmptotes:, ; End behavior: lim f() lim f() 0

33 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 97 SELECTED ANSWERS Intercepts: 0,, (, 0); asmptotes:, 4, 0; lim h(), lim lim h() Domain:, 4; Range: (, ); 4 h(), lim h(), 4 Continuit: all, 4; Decreasing: (, ), (, 4), (4, ); No smmetr; Unbounded; No etrema; Horizontal asmptote: 0; Vertical asmptotes:, 4; End behavior: lim h() lim h() 0 4. Intercepts: (, 0), (, 0), 0, 9 ; asmptotes:,, ; lim f(), lim f(), lim f(), lim f() Domain:, ; Range: (, 0.60] (, ); Continuit: all, ; Increasing: (, ), (, 0.67); Decreasing: (0.67, ), (, ); No smmetr; Unbounded; Local maimum at (0.67, 0.60); Horizontal asmptote: ; Vertical asmptotes:, ; End behavior: lim f() lim 4. Intercepts: (, 0), (, 0), 0, ; asmptotes:, ; lim h(), lim [ 9.4, 9.4] b [, ] [.87,.87] b [.,.] [ 9.4, 9.4] b [, ] f() Domain:, Range: (, ); Continuit: all ; Increasing: (, ), (, ); No smmetr; Unbounded; No etrema; Horizontal asmptote: none; Vertical asmptote: ; Slant asmptote: ; End behavior: (a) (a) lim h(), lim h() h() [ 0, 0] b [ 0, 0] [ 0, 0] b [ 0, 60] (b) (b) [ 40, 40] b [ 40, 40] [ 0, 0] b [ 00, 00] (a) (b) [, ] b [ 00, 00] [ 0, 0] b [ 000, 000]

34 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS. Intercept: 0, 4 ; Domain: (, ); Range: [0.77, 4.7]; Continuit: (, ); Increasing: [0.4,.44]; Decreasing: (, 0.4], [.44, ); No smmetr; Bounded; Local ma at (.44, 4.7), local min at (0.4, 0.77); Horizontal asmptote: ; Vertical asmptote: none; End behavior: lim f() lim f(). Intercepts: (, 0), 0, ; Domain: ; Range: (, ); Continuit: ; Increasing: [0.84, 0.44], [.94, ); Decreasing: (, 0.84], [0.44, ), (,.94]; No smmetr; Not bounded; Local ma at (0.44, 0.86), local min at (0.84, 0.44) and (.94,.970); Horizontal asmptote: none; Vertical asmptote: ; End behavior: lim h() lim h() ; End-behavior asmptote: 4 [, ] b [, ] [ 0, 0] b [ 0, 0]. Intercepts: (.7, 0), (0, ); 7. Intercept: (0, ); 9. Intercepts: (, 0), 0, ; Domain: ; Range: (, ); Continuit: ; Increasing: [ 0.84, ),, ; Decreasing: (, 0.84]; No smmetr; Not bounded; Local min at (0.84, 0.90); Horizontal asmptote: none; Vertical asmptote: ; End behavior: lim f() lim f() ; End-behavior asmptote: 4 8 Asmptote: ; End-behavior asmptote: [, ] b [ 0, 0] Asmptote: ; End-behavior asmptote: [ 0, 0] b [ 00, 400] [, ] b [ 0, 0] 6. Intercepts: (.476, 0), (0, ); Asmptote: ; End-behavior asmptote: 4 [, ] b [, ] 6. False. is a rational function and has no vertical asmptotes. 6. E 67. D

35 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 99 SELECTED ANSWERS (a) No: the domain of f is (, ) (, ); the domain of g is all real numbers. (b) No: while it is not defined at, it does not tend toward on either side. (c) Most grapher viewing windows do not reveal that f is undefined at. (d) Almost but not quite, the are equal for all. 7. (a) The volume is f() k, where is pressure and k is a constant. f() is a quotient of polnomials and hence is rational, but f() k, so is a power function with constant of variation k and power. (b) If f() k a,where a is a negative integer, then the power function f is also a rational function. (c) 4. L 7. Horizontal asmptotes: and ; 7. Horizontal asmptotes: ; Intercepts: 0,,,0 ; Intercepts: 0, 4,,0 ; h() f() 0 4 [, ] b [, ] [ 0, 0] b [, ] 77. The graph of f is the graph m shifted horizontall dc units, stretched verticall b a factor of bc ad /c,reflected across the -ais if and onl if bc ad 0, and then shifted verticall b ac. SECTION.7 Quick Review LCD: 6; Eercises.7 7. LCD: ( )( ); ( ) ( ).. or 7. 4 or, the latter is etraneous. 7. or 9. or 4. or, the latter is etraneous.. or, the latter is etraneous.. or 0, the latter is etraneous. 7. or 0, both of these are etraneous (there are no real solutions). 9.. Both or No real solutions or 0.66 or.49. (a) The total amount of solution is ( ) ml; of this, the amount of acid is plus 60% of the original amount, or 0.6(). 7 (b) 0.8 (c) C() 0.8; 69. ml. (a) C() 000. (b) 476 hats per week (c) 60 hats per week. (a) P() 6 4 (b).49 (a square); P (a) S 000 (b) Either. cm and h 6.88 cm or.7 cm and h. cm. 9. (a) R() (b) 6. ohms 4. (a) D(t) 4. 7 t (b) t.74 h. 4. 7t

36 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS 4. (a) (b) About 98. billion dollars 4. False. An etraneous solution is a solution of the equation cleared of fractions that is not a solution of the original equation. [0, ] b [0, 0] 47. D 49. E. (a) f(),, 0 (b) 0, (c) f() { (d) undefined, or 0 The graph appears to be the horizontal line with holes at and 0... [ 4.7, 4.7] b [.,.] SECTION.8 Eploration (+)( )(+) (+)( )(+) (+)(+)(+) (+) (+)( )( ) (+) (+)(+)( ) (+) (+)(+)( ). (a) Negative Negative Positive. (a) Positive Negative Negative (b) 4 (b) [, ] b [ 000, 000] [, ] b [ 0, 0] Quick Review.8. lim f() ;lim f(). lim 9. (a),,, (b) ( )( )( ) Eercises.8 g(), lim g(). 7 ( )/ 7.. (a),, (b) or (c) or. (a) 7, 4, 6 (b) 7 or 4 6 or 6 (c) 7 4. (a) 8, (b) 8 or 8 7. (, ) (, ) 9. (, ) (, ). [, ] [, ). [, 0] [, ). (, /) (, ) 7. [., ) 9. (, ). (a) (, ) (b) (, ) (c) There are no solutions. (d) There are no solutions.. (a) 4 (b) (, ) (c) There are no solutions. (d) 4. (a) (b),4 (c) or 4 (d), or 4 7. (a) 0, (b) (c) 0 (d) 0 9. (a) (b),, (c) or

37 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS 9 (d). (a) (b) 4, (c) 4 or 4 (d) f() is never negative.. (, ) (, ). [, ] 7. (, 4) (, ) 9. [, 0] [, ) 4. (0, ) (, ) 4. 4, 4. (0, ) 47. (, 0) (, ) 49. (, ) [, ). [, ). [, ) 7. in. 4 in in in. or 4.0 in. 6 in. 6. (a) S 000 (b). cm.7 cm,. cm h 6.88 cm (c) about 48.7 cm 6. (a) (b) After 0 6. False, because the factor 4 does not change sign at C 69. D 7. Vertical asmptotes:, ; -intercepts: (, 0), (, 0); -intercept: 0, 4 undefined undefined ( )( ) 0 ( )(+) ( )(+) 0 (+)(+) (+)(+) ( )( ) ( )( ) ( )(+) ( )(+) (+)(+) Negative Negative Positive Negative Positive B hand: 0 Grapher support: 0 0 [, ] b [, ] [0, 0] b [ 40, 40] (a) 9 4 f() 4. (b) If stas within the dashed vertical lines, f() will sta within the dashed horizontal lines. (c) f() The dashed lines would be closer when and a b a ab and ab b ; so, a b. CHAPTER REVIEW EXERCISES.. Starting from,translate right units and verticall stretch b (either order), then translate up 4 units. 0 [, ] b [, ] 6. Verte: (, ); ais: 7. Verte: (4, ); ais: 4 9. (/9)( ). ( )

38 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 9 SELECTED ANSWERS.. 7. S kr [ 0, 7] b [ 0, 0] [ 4, ] b [ 0, 0] 9. The force F needed varies directl with the distance from its resting position, with constant of variation k.. k 4, a, f is increasing in the first quadrant, f is odd.. k, a, f is increasing in the fourth quadrant, f is odd Yes 4 7.,,, 6,, ; and are zeros. 9. i 4. i 4. i 4. (c) 47. (b) 49. Rational: 0. Irrational:. No nonreal zeros.. Rational: none. Irrational: approimatel.4, 0.7,.77. No nonreal zeros.., i; f() ( )( i)( i).,,, and ; f() ( )( )( )( ) 7. f() ( )( ) 9. f() ( )( )( ) Translate right units and verticall stretch b (either order), then translate down unit.; Horizontal asmptote: ; vertical asmptote:. 69. Asmptotes:,, and 7. End behavior asmptote: 7;. Intercept: (0, ). Vertical asmptote:. Intercept: 0,. [, ] b [, ] [ 7, ] b [ 0, 0] 7. -intercept: 0,, -intercept: (., 0); Domain: ; Range: (, ); Continuit: all ; Decreasing: (, ), (, 0.8]; Increasing: [0.8, ); Unbounded; Local minimum: (0.8,.6); Vertical asmptote: ; End-behavior asmptote: ; f() lim f() lim [ 0, 0] b [ 0, 0] 7. or (, /) (, ) 79. [, ) (, )

39 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page 9 SELECTED ANSWERS 9 8., 8. Yes; at approimatel (a) V (0 )(70 ) in (b) Either 4.7 or 8.6 in. 87. (a) V 4 (40 ) (b) (c) The largest volume occurs when 70 (so it is actuall a sphere). This volume is 4 (70),46,7 ft. [0, 70] b [0,,00,000] 89. (a) (b) (c) Using linear regression: In 008; Using quadratic regression: In 00 [0, ] b [0, 0] [0, ] b [0, 0] 9. (a) P(), P(70) 600, P(00) 648 (b) (c) The deer population approaches (but never equals) (a) C() (b) about. ounces of distilled water (c) (a) S 4000/ (b) 0 ft b 0 ft b. ft or 7., giving approimate dimensions 7. b 7. b (c) 7. 0 (lower bound approimate), so must be between. and about Chapter Project Answers are based on the sample date shown in the table... The sign of a affects the direction the parabola opens. The magnitude of a affects the vertical stretch of the graph. Changes to h cause horizontal shifts to the graph, while changes to k cause vertical shifts [0,.6] b [ 0., ] SECTION. Eploration. (0, ) is in common; Domain: (, ); Range: (0, ); Continuous; Alwas increasing; Not smmetric; No local etrema; Bounded below b 0, which is also the onl asmptote; lim f(). lim f() 0 Eploration.. k 0.69 [ 4, 4] b [, 8] Quick Review / 7. /a

40 44_Demana_SEANS_pp89-04 /4/06 8:0 AM Page SELECTED ANSWERS Eercises.. Not eponential, a monomial function. Eponential function initial value of and base of. Not eponential, variable base / (/). /. Translate f() b units to the right. 7. Reflect f() 4 over the -ais. 9. Verticall stretch f() 0. b a factor of and then shift 4 units up.. Reflect f() e across the -ais and horizontall shrink b a factor of.. Reflect f() e across the -ais, horizontall shrink b a factor of, translate unit to the right and verticall stretch b a factor of.. Graph (a) is the onl graph shaped and positioned like the graph of b, b. 7. Graph (c) is the reflection of across the -ais. 9. Graph (b) is the graph of translated down units.. Eponential deca; lim f() 0, lim f(). Eponential deca; lim f() 0, lim f() since 4 ( ) ( ) [ 0, 0] b [, ] [, 0] b [, 0] [, ] b [, 8] -intercept: (0, 4) -intercept: (0, 4) Domain: (, ); Range: (0, ); Continuous; Horizontal asmptotes: Horizontal asmptotes: Alwas increasing; Not smmetric; 0, 0, 6 Bounded below b 0, which is also the onl asmptote; No local etrema; lim f() ; lim f() [, ] b [, 9] [, 4] b [, 7] Domain: (, ); Range: (0, ); Continuous; Domain: (, ); Range: (0, ); Continuous; Alwas increasing; Not smmetric; Alwas increasing; Smmetric about (0.69,.); Bounded below b 0, which is the onl asmptote; Bounded below b 0 and above b, both of No local etrema; lim f() ; lim f() 0 which are asmptotes; No local etrema; lim f() ; lim f() 0. In 006. Near the end of 00. In (a) 00 (b) False. If a 0 and 0 b, then f() a b is decreasing. 6. E 6. A 67. (a) f() decreases less rapidl as increases. (b) as increases, g() decreases ever more rapidl. 69. a 0, c 7. a 0 and b, or a 0 and 0 b 7. As, b, so a b c and a b 0; 6. (a) (b) [, ] b [, ] Domain: (, ); Range: e, ; Decreasing on (, ]: Increasing on [, ); Bounded below b e ; Local minimum at, e ; Asmptote: 0; lim f() ; lim f() 0 Domain: (, 0) (0, ); Range: (, e] (0, ); Increasing on (, ]; Decreasing on [, 0) (0, ); Not bounded; Local maimum at (, e); Asmptotes: 0, 0; lim [, ] b [ 7, ] g() 0; lim g()

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