Chapter 2 Rocket Launch: AREA BETWEEN CURVES

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1 ANSWERS Mathematics (Mathematical Analysis) page 1 Chapter Rocket Launch: AREA BETWEEN CURVES RL-. a) 1,.,.; $8, $1, $18, $0, $, $6, $ b) x; 6(x ) + 0 RL-. a), 16, 9,, 1, 0; 1,,, 7, 9, 11 c) D = (-, ); R = [0, ); yes, a function (interval notation is not required in students answers) d) x = 0 RL-. a) D= (-, ); R=(0, ] and (, ) b) No; jump at x = 1. RL-. a) $0 d) ($ /hour) (hours) = $ RL-8. a) (x + )(x 1)b) x(x + )(x + ) c) (x + )(x + ) RL-9. a) 8 = 6 m b) 0 + = 7.76 m RL-10. a) x = 0 or b) x = or 9 RL-11. a) 1 7 b) 9 c) 9 d) RL-1. a) 98 = 7 b) y = -(x + ), y + = -(x ), y = -x + RL-1. a) x + 1x + 9 b) x ax bx + ab RL-1. a) Shift the first graph units to the left. RL-1. b) f(x) = 0, 0, -, -, 0; g(x) =,, 1, 0, RL-16. Shift f(x) down units to get h(x). RL-17. b) g(x) = f(x + ) RL-18. c) Shifts the graph to the right 1 unit d) h(x) = (x + 1) for x < ; (x 1) for x RL-19. a) f(x) = x- + 1 b) f(x) = x+ - 1 RL-0. y = (x ) (x ) + RL-1. b) Shifted to the left units c) f(x + ) RL-. a) (x + 8)(x 1) b) x(x + 8)(x 8) RL-. x = 1.7, y = CPM Educational Program v..1

2 ANSWERS Mathematics (Mathematical Analysis) page RL-. a) (x + 9y )(x + y)(x y) b) x ( + x ) RL-. a) x =.6, y = 9.06 b) x = 1.0, y = RL-6. f - 1 (x) = x+1 RL-7. a) 10 = b) cm c) RL-8., 9, 16, 1 RL-9. a) 10 b) 06 c) 1. d) 6 e) RL-0. He did not use integer values for p. RL-1.,,61,and 7 RL-. a) (, 9) and (-, ) d) The two graphs do not intersect, so there is no point of intersection. RL-. a) We need to take the square root of a negative number c) i RL-7. 1 ± i RL-8. a) 0 b) 0 c) 1080 RL Â RL-0. k RL = 0 RL-. a) -18 i b) 1 ± i c) + i 6 RL-. 1, 1.6,.,.8,., and RL-. a) - + i b) 1 10i RL-. a) i b) -1 i 001 CPM Educational Program v..1

3 ANSWERS Mathematics (Mathematical Analysis) page RL-6. a) Initializes the sum to 0 b) In step, change 1 to 7; in step, change to 0 c) The loop goes on forever d) In step, change k to 10 k. RL-8. a) b) Expressions: 16, ; Sum: 1, 0,, c) 1, d) 0, RL-9. c) Since we are using Y1, we need the independent variable X d) Change the test value of k from to 10 e) 9 RL-0. Change Y1 to 10 x. RL-1..9 RL-. a) none; just different variables b) none; the sum is just expressed differently. RL-. b) He walks for 1 hour; c) miles hr hr = miles d) miles RL-. i) 7, ii) 18., iii) d a) -1 b) -1 c) yes RL-. 1 RL-6. a) 7ib) i c) -16 d) -7i RL-7. a),.7,.,., and b) 1., RL-8. a) (x 1)(x + 1) b) (x 1) (x ) c) (x 1) (x + 1) RL-9. a) 18 b) 1 6 c) 6 RL-60. a) 7 b) 1 i c) i d) - + 6i RL-61. ± i RL-6. c) RL-6. b) miles/hr hr = miles c) miles RL RL-6. y = 0.x + or y = 0.(x ) or y. = 0.(x 1) 001 CPM Educational Program v..1

4 ANSWERS Mathematics (Mathematical Analysis) page RL-66. Â 0.k + RL-67. a) b) c) 17 d) e) i f) a bi RL-68. a) 9 ± 9 i b) i c) i RL-70. a) y = 00x 7; constant rate of change b) E(x) = for 0 < x 0.; 00x 7 for 0. < x 0.7; 7 for 0.7 < x RL-7. Change Y1 and the start and end of the index. RL-7. a) i b) 7i c) + i RL-7. g(x) = + 0.x a) Â 0.i + b) 17 RL-7. a) / b) / c) 7 / d) 7 / e) 7 / f) 7/ RL-76. x = 0, -, i=1 RL-77. a) -i b) 1-1 i c) 1 + i RL-78. a) 81 b) c) - d) i e) -i f) RL-79. d) Answer will likely be between 7.7 and RL-80. Answers should range from. to.9. RL-81. Students should add the results of problems RL-79 and RL-80. RL-8. a) c) Â 0.k +. d) Â 0.k +. RL-8. b) 0.k +. Â RL RL-8. c) It tells the index to start at B d) Change X 100 to X E. RL-86. a) 81 b) CPM Educational Program v..1

5 ANSWERS Mathematics (Mathematical Analysis) page RL-87. The equation needs to be entered and the starting and ending values need to be adjusted; 10 Ï RL-88. b) g(x) = x + for x < Ì ; k(x)= Ó + for x Ï Ì Ó ( x - ) for x < for x RL-90. a) -1 and - b) Just one at x = - c) x = - ± i; it does not cross the x-axis, hence the complex roots. RL-91. x = ±, y = + RL-9. c = -, b = RL-9. a) x = b) x =, y = RL-9. 1±i RL-9. a)  k b)  k c)  k RL-96. a) (x + 6)(x )b) (x y)(b + 7) RL-97. a) (1, 1), (, ), (, 9) b) height c) 1,,, ; 1 d) The height of the rectangle is the function value at the left endpoint of each interval, e) 1(1) + 1() + 1(9), f) 1 RL-98. a) height b) 1,,, ; 1 c) They start from the right endpoint of the interval d) 1() + 1(9) + 1(16) = 9; uses right-endpoint values of the interval. 100 RL-99. Some possibilities are averaging the two values or using more rectangles. RL-100. a) 0. b) 1,.,, 6., 9, 1. c) They are the y-values of the left-endpoints [ ] d) width, f) S 6 = 0.( 0.k + 1)  RL b) 0., 1,., c) 0.( ) 19 RL-10. k  or  0.k Other answers are acceptable, as well. k= CPM Educational Program v..1

6 ANSWERS Mathematics (Mathematical Analysis) page 6 RL x = -x + 1 means 1 = -x + x so x x + 1 = 0. This has complex roots, therefore the graphs do not intersect; 1±i RL-10. no a) - ± i b) 7 ± i c) ± i RL-10. p RL ; square root 16 and then cube the result. RL-107. a) x - 10 RL-108. a) a b b) c b c) a c RL-109. a) x π - b) x c) x π ±1 d) all reals RL-110. g(x) = 1 x- - 1 RL-111. a) gain of about 0 million RL-11. It counted as negative b) 1. RL-11. a). b). c) -. d) There is more area below the x-axis than above. RL-11. a) 9 b) shifts graph up units; 1; additional by rectangle c) RL-11. a) no change b) add c) nothing; add a rectangle width x k units RL-116. a) added by 1 rectangle b) 8.66 c) a RL-117. a) width of the rectangles b) 1.79 c) RL-118. a) 0. b),.,.8,.,.6, 6.0 c) upper = 0.8, lower = d) use more rectangles RL-119. x for x -, x + 7 for x < - RL-11. right:., left: 19. RL-1. f(x) = x if x 0, x if x > 0 RL-1. a) b a b) b c c) c a 001 CPM Educational Program v..1

7 ANSWERS Mathematics (Mathematical Analysis) page 7 RL-1. a) 8 b) a + 16a + c) -7, 1 d) g 1 (x) = - ± x ) Ï RL-1. b) g(x) = - x if x 1 Ì Ó x + ifx> 1 RL-16. a) 1 x - x- 1 b) 9x 11 c) x 0x + RL-18. a) 90 miles b) Area under the curve = the distance traveled; i.e. hr mi hr = mi = m RL-19. a) 0. b).0,.,.8,.,.6, and.0e) k RL-10. a) 0.  1. 0.k+ RL-11. a) left = 0.( ); right = 0.( ) RL-1. W + 1 b) left endpoints =  0. ( + 0.k) +,  right endpoints = 0. ( + 0.k) + RL-1. a) upper =.01, lower =.81 b) 0. - (0.k + 1) + 6 RL-1. ª RL-1. Verify that + i is a solution to x 10x = -9. RL-16. (1 + i, + i), (1 i, i) RL-17. A = 6  RL-18. A n- 1 is larger RL-19. a) x = B±C A+1 b) 0, ±0., ±0.i 001 CPM Educational Program v..1

8 ANSWERS Mathematics (Mathematical Analysis) page 8 RL-10. a) 1 b) 1 c) 8 ( 10) 7 RL-11. A = 1 RL-1. a) 8 6i RL-1. b) 80 RL-1. a). square units b) i c) It s the units! b) It is too high. RL-1. b) 16 to 10 RL-16. b) 1 miles RL-17. b) not as far RL-18. a) 7. c) It s the units. c) about 66.7 miles b) It is too high. RL-19. a) 8. square units b) 6. square units c) same RL-10. a) 0 miles b) Vertical distances are halved c) Draw the picture again or use the fact that the first car always travels twice as fast, so it goes twice as far. RL-11. a) 0. b) 0.6 c) 0.08 d) E- B N RL-1. a) f(1) b) f(1.6) c) f(10) d) f(9.6) e) f(b), f(b + W), f(e), f(e W) RL RL-1. i) ii) iii) 0 iv) 1 RL-1. b) Each graph must intersect at least once. RL-16. y = x 1- x RL-17. c- 7 a +b RL-18. a) 1000 b) 1 9 c) 9 d) 9 a RL CPM Educational Program v..1

9 ANSWERS Mathematics (Mathematical Analysis) page 9 RL-160. a) (E B)/NÆ W c) The height of the Xth rectangle d) Adds the area of the Xth rectangle e) We are incrementing by the width of the sub-interval, not by one RL-161. a) BÆ X and X (E W) b) ª.7 c) They will get closer to each other RL-16. b) 80 miles c) height RL-16. a) t b) One base is 0, and the other is f(x) or 0t + 0 c) 10t + 0t Ï t for 0 t 6 RL-16. a) J(t) = Ì Ó (t - 6) for t > 6 Ï 6t for 0 t 8 b) C(t) = Ì Ó 8 + 1(t - 8) for t > 8 c) 7.6 hours, d) 7. < t < 9 RL-166. a) x = ± i, therefore it has no real roots and cannot cross the x-axis b) We are looking for: one repeated linear factor gives one real root, two different linear factors give two real roots, the quadratic that cannot be factored with real coefficients gives two non-real roots c) (x + i)(x i) RL-167. a) Three real linear factors (one repeated), therefore real (1 single, 1 double) and 0 non-real roots b) One linear and one quadratic factor, therefore 1 real and complex (non-real) roots c) Four linear factors therefore real, 0 non-real roots d) Two linear and one quadratic factor, real and complex (non-real) roots. RL-168. x = ± RL-169. b) x + y =, y 0 c) p RL-170. She needs a 9%. RL-171. (-, -), (, 6) RL i RL-17. a) 1716 ft (left endpoint), 1991 ft (right), 181. ft (trapezoidal) b) y =.x is a pretty good fit c) ft d) 1 ft/sec, 806 ft/sec 001 CPM Educational Program v..1

10 ANSWERS Mathematics (Mathematical Analysis) page 10 RL-17. ª.1 to.16 depending on left or right rectangles RL-177. a) g(x) = x + c) Area under g(x) is twice as large. RL RL-180. a) repeat 1, i, -1, -i, etc. b) 1, i, -1, -i RL-181. a) 1 b) i c) -1 RL-18. He needs an 86%. RL-18. Since the denominator on the first fraction is smaller and both fractions have the same numerator, the first fraction is larger than the second. RL-18. a) a(a + 8) b) (x + )(x + 10) c) (y 1)(y + 7) RL-186. a) b) - c) - d) - RL-187. n = pk u RL-189. a) 0.( ) b) 0.( ) c) 0.( ) RL-19. a) A(x 7, x 7) b) x dx RL-19. lower: left-endpoint rectangles, upper: trapezoids RL-19. b) 0, 60, 90, 10, 10, 160 d) d(t) = 60t for t, (t ) for t > RL-19. a) sub-intervals of width 0. b) 1 x c) 0. 0.k+1 Â 8 Ú d) left RL-196. a) Ú [- (x - 1) + ]dx c).1 d) lower 1 e) more sub-intervals RL-198. a) The sum gives a lower bound b) The sum gives an upper bound. 001 CPM Educational Program v..1

11 ANSWERS Mathematics (Mathematical Analysis) page 11 RL-199. a) c) f(x) dx and h(x) dx RL-00. a) f -1 (x) = x 6 + b) f -1 (x) = c) x =.108 and d) x = log8 Ú - RL-01. a) b) 0. Â 0.k+ c) A( x, x ) Ú - x- RL-0. a) The roots are complex numbers b) y = (x ) + or x 6x + 1 c) x = ± i RL-0. a) V = 60 cm, A = 1 cm b) V = 88 cm, A = 88 cm RL-0. a) = 16 b) 6 - = 1 6 c) 10 = 1000 RL-06. a) b) c) 8 Interlude Introduction to Logarithms IL-1. a) b) x = y ] x y 6 1/ [ -1 ] [ 1 ] 1 [ ] [ none ] 0 [ none ] 1/ [ - ] 1/ IL-. a) = 6 b) y = log (x) IL-. a) b) c) IL-. a) = 8; b) 6 = 196; c) 7 - = 1 9 d) 91/ = IL-. a) x = log 7 y; b) x = y c) y = log 11 x d) K = log W (B) 001 CPM Educational Program v..1

12 ANSWERS Mathematics (Mathematical Analysis) page 1 e) B = W K f) P = ( 1 ) Q IL-6. a) b L = N b) log b b L = L c) b log b N = N IL-7 a) b) -: c) 1 d) x IL-8. f(x) D: all reals, R: y > 0, no zeroes; g(x) D: x > 0, R: all reals, zeroes: x = 1 IL-9. a) b) 9, c) x d) y = log x e) x = y f) yes IL-10. a) 1 b) 0 c) -1 IL-11. a) 81 b) c) 1 9 d) - e) ] f) 1 IL-1 a) 0 b) 7, LS 7 = 1 c) d) w = a) 1 b) - c) d) 0 e) 1, d = ]IL-1. f) - IL-1. a) (x + y)(x y) b) (z y)(z + y) c) (x + y)(x y) L-1. a) p 1 = p b) = 9 c) 6 0 = 1 IL-16. a) - b) 1 c) 6 d) IL.17 a) yes; b) no; c) log + log = log 6; d) log + log = log 8 e) log + log = log 1 f) Since log 10 = 1, log + log = log 10. The pattern is confirmed. g) log xy IL-18. a) Both sides =. True in all cases. b) Both sides =. c) Both sides = -. Il-19. No. Following a pattern is not a proof the pattern may break down with more examples. We haven't shown WHY this is always true. IL-0. a) log 6 log = log b) log 8 log = log c) log x y IL-. a) log b) log 8 c) log 16 d) log 1 e) log 8 f) log x n IL-7. a) log or log b) log + log 7 c) Impossible. Logs are being multiplied, not added. d) Impossible. Arguments are being added, not multiplied. (log 1 is correct, but no log law is used.) e) Impossible. Bases are different. f) log IL-8. a) log 6 b) log c) log MN d) log( P Q ) 001 CPM Educational Program v..1

13 ANSWERS Mathematics (Mathematical Analysis) page 1 IL-9. a) 1.87 b).1 c).1 IL-0. a) 10 (7/10) b) c = c) ( 10) 7 ] d) Taking the root first makes the numbers smaller. e) same f).01 g) h) She can calculate log = since 10 log =. IL IL-. b) short = x, long = x c) x ] IL-. a) = 17 b) 0. 1 Â 0.x+1.6 i=1 ; others are possible. IL-. a) y = 1- x, y = x b) (1, 0), (-, -1) IL-. a) log M + log N b) 1 ( logm - logn) c) log M log M ] IL-6. a) y = x b) y = x -1 or 1 x c) y = x IL years IL-9. a) 1.91 b) c) 1.1 IL-0 a) yes b) We do not know how to use a calculator to find log c).10 d) yes IL-1. IL-. a).18 b) use Y1 = log x log a) 1.7, 1.8, 1.9 would all be reasonable because = 9, and 8 is a bit less than 9. c) 1.89 d) 8 ª 1.89 IL-. a) t = log 0.6 log 0.9 b).88 hours or hours, 1 minutes IL-. a) 1 b) c) d) - e) impossible f) g) h) impossible i) IL-7. The hypotenuse is the longest side of a right triangle, but <. IL-8. (8, ) or (-8, -) IL-9. a) 0 x 1 since it is a probability. b) 1 x c) x + (1 x) or x x + 1 d) x IL-0. a).6 b) 18.8 IL-1. a) 1 b) 0 x ±x CPM Educational Program v..1

14 ANSWERS Mathematics (Mathematical Analysis) page 1 IL-. a) 1 b) geometric c) The increase is 00%, even though each dose is 00% of the previous dose. d) dose e) d = r or d = r±1 f) r = log +log d- log log IL-. a) 10 / = 1.6 b) M = log E.87 c).69 x 1016 joules d) 10 1.(8. 7.1) ª 6.1 times IL-., acidic;., acidic; 7, neutral; 7.0, alkaline; 9, alkaline IL-. a) b) c) 0 d) - e) -1 f) - 1 IL-6. a) b) - 6 c) -6 IL-7. Domain: - < x; Range: - < y < IL-8. a) x > 1 IL-9. a) - 11 b) x π 0 c) x π d) all reals b) ± c) d) ± 6 e) IL-60. a). b) -. c) 1.d) 7.6 IL-61. a) A = 1.19 cm, C = 6.8 cm b) A =.619 cm, C = 7.1 cm IL-6. A =.619 ft 001 CPM Educational Program v..1