1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6

Transcription

1 A Answers Important Note about Precision of Answers: In many of te problems in tis book you are required to read information from a grap and to calculate wit tat information. You sould take reasonable care to read te graps as accurately as you can (a small straigtedge is elpful), but even skilled and careful people make sligtly different readings of te same grap. Tat is simply one of te drawbacks of grapical information. Wen answers are given to grapical problems, te answers sould be viewed as te best approximations we could make, and tey usually include te word approximately or te symbol meaning approximately equal to. Your answers sould be close to te given answers, but you sould not be concerned if tey differ a little. (Yes tose are vague terms, but it is all we can say wen dealing wit grapical information.) Section approx. 1, 0, (a) min 0 min 8 min 6 min cooling; 5 min cooling (c) 5.5 min cooling; 10 min cooling (d) Wen t 6 min. 5. We estimate tat te area is approximately (very approximately) 9 cm Metod 1: Measure te diameter of te can, fill it alf full of water, measure te eigt of te water and calculate te volume. Submerge te bulb, measure te eigt of te water again, and calculate te new volume. Te volume of te bulb is te difference of te two calculated volumes. Metod 2: Fill te can wit water and weig it. Submerge te bulb (displacing a volume of water equal to te volume of te bulb), remove te bulb, and weig te can again. By subtracting, find te weigt of te displaced water and use te fact tat te density of water is 1 gram per 1 cubic centimeter to determine te volume of te bulb. Section (a) (c) 0 (d) 2 (e) undefined 3. (a) (c) x + 2 (if x 2) (d) 4 + (if 0) (e) a + x (if a x) (a) t 5: ; t 10: ; t 20: any t > 0: t 50 3t (c) Decreasing, since te numerator remains constant at 5000 wile te denominator increases. 7. Te restaurant is 4 blocks sout and 2 blocks east. Te distance is blocks. 9. (a) feet (c) tan(θ) so θ 1.37 ( 78.5 ) 11. (a) Te equation of te line troug P (2, 3) and Q (8, 11) is y (x 2) 6y 8x 2. Substituting x 2a + 8(1 a) 8 6a and y 3a + 11(1 a) 11 8a into te equation for te line, we get: 6(11 8a) 8(8 6a) 66 48a a 2 for all a, so te point wit x 2a + 8(1 a) and y 3a + 11(1 a) is on te line troug P and

2 A2 answers Q for any a. Furtermore, 2 8 6a 8 for 0 a 1, so te point in question must be on te line segment PQ. dist(p, Q) , wile: dist(p, R) (8 6a 2) 2 + (11 8a 3) 2 (6 6a) 2 + (8 8a) (1 a) (1 a) 2 100(1 a) a 1 a dist(p, Q) 13. (a) m 1 m 2 (1)( 1) 1 (c) Because 20 units of x-values are pysically wider on te screen tan 20 units of y-values. (d) Set te window so tat: (xmax xmin) 1.7(ymax ymin) 15. (a) y 5 3(x 2) or y 3x 1 y 2 2(x 3) or y 8 2x (c) y (x 1) or y 1 2 x (a) y (x 2) or y 3 2 x + 2 y (x + 1) or y 3 2 x (c) x Distance between te centers (a) (c) 0 (tey intersect) (d) (e) Find dist(p, C) (x ) 2 + (y k) 2 and compare te value to r: inside te circle if dist(p, C) < r P is on te circle if dist(p, C) r outside te circle if dist(p, C) > r 23. A point P (x, y) lies on te circle if and only if its distance from C (, k) is r: dist(p, C) r. So P is on te circle if and only if: if and only if: (x ) 2 + (y k) 2 r (x ) 2 + (y k) 2 r (a) 5 12 undefined (vertical line) (c) 12 5 (d) 0 (orizontal line) 27. (a) (c) (by inspection) 3 units, wic occurs at te point (5, 3) 29. (a) If B 0, we can solve for y: y A B x + C B, so te slope is m A B. Te required slope is A B (te negative reciprocal of A B ) and te y-intercept is 0, so te equation is y A B x or Bx Ay 0. (c) Solve te equations Ax + By C and Bx Ay 0 simultaneously to get: x AC A 2 + B 2 and y BC A 2 + B 2 (d) Te distance from tis point to te origin is: AC BC ( A 2 + B 2 )2 + ( A 2 + B 2 )2 Section A: a, B: c, C: d, D:b 3. A: b, B: c, C: d, D: a 5. (a) C A (c) B A 2 C 2 (A 2 + B 2 ) 2 + B 2 C 2 (A 2 + B 2 ) 2 (A 2 + B 2 )C 2 (A 2 + B 2 ) 2 C 2 A 2 + B 2 C A 2 + B 2

3 A3 7. (a) f (1) 4, g(1) is undefined, H(1) 1 (c) f (3x) (3x) x 2 + 3, g(3x) 3x 5 (for x 5 3 ), H(3x) 3x 3x 2 (d) f (x + ) (x + ) x 2 + 2x , g(x + ) x + 5, H(x + ) x+ x (a) m 2 m 2x (c) If x 1.3, ten m ; if x 1.1, ten m ; if x 1.002, ten m f (a+) f (a) 2a + 2 (if 0). If a 1:. If a 2: 2 +. If a 3: 4 +. If a x: 2x + 2. g(a+) g(a) a+ a. If a 1: If a 2: If a 3: If a x: x+ x. 13. (a) Approx. 250 miles, 375 miles. Approx. 200 miles/our. (c) By flying along a circular arc about 375 miles from te airport (or by landing at anoter airport). 15. (a) Largest: x 2; smallest: x 4. (c) Largest: at x 5; smallest at x Te pat of te slide is a straigt line tangent to te grap of te pat at te point of fall: 21. Approximate values: 23. On your own. Section 0.4 x f (x) g(x) (a) 18, 2.2 If T 11 C, WCI if 0 v v v if 6.5 < v if v > g(0) 3, g(1) 1, g(2) 2, g(3) 3, g(4) 1, g(5) 1. 3 x if x < 1 g(x) x if 1 x 3 1 if x > 3 5. (a) f ( f (1)) 1, f (g(2)) 2, f (g(0)) 2, f (g(1)) 3 g( f (2)) 0, g( f (3)) 1, g(g(0)) 0, g( f (0)) 0 7. (a) (c) f ((3)) 3, f ((4)) 2, (g(0)) 0, (g(1)) 1 x f (x) g(x) (x) f (g(1)) 1, f ((1)) 3, ( f (1)) 3, f ( f (2)) 3, g(g(3.5)) 3 (c) 19. (a) s(1) 2, s(3) 4 3, s(4) 5 4 s(x) x+1 x

4 A4 answers 9. If L(d) represents location on day d: England if d Monday or Tuesday France if d Wednesday L(d) Germany if d Tursday or Friday Italy if d Saturday 11. Assuming te left portion is part of a parabola: { x f (x) 2 if x < 2 x 1 if x > (a) B(1) 1 f (1) , B(2) , 13 1x B(3) 3 1. B(x) x f (x) x 1 (if x > 0) (a) f (g(x)) 6x A, g( f (x)) g(3x + 2) 6x A. If f (g(x)) g( f (x)), ten A 1. f (g(x)) 3Bx 1, g( f (x)) 3Bx + 2B 1. If f (g(x)) g( f (x)), ten B Grap of f (x) x x : 21. f (x) sin(x) works. Te value of A in f (x) A sin(x) determines te relative lengts of te long and sort parts of te pattern. 23. (a) g(1) 1, g(2) 1, g(3) 0, g(4) starting wit x 1, 2, 10 or any value 27. f (1) f (0.5) 1.25, f (1.25) 1.025, f (1.025) , f ( ) ,... f (4) 2.125, f (2.125) , f ( ) , f ( ) , (a) f (2) , f ( 14 3 ) , f ( 50 9 ) , f ( ) f (4) , f ( 16 3 ) , f ( 52 9 ) , f ( ) f (6) 6. c 6 (c) Solve c g(c) 3 c + A to get 3c c + 3A 2c 3A c 3A 2 is a fixed point of g. 31. On your own. Section (a) x 2, 4 x 2, 1, 0, 1, 2, 3, 4, 5 (c) x 2, 1, 1, 3 3. (a) all x (all real numbers) x > 3 2 (c) all x 5. (a) x 2, 3, 3 no values of x (c) x 0 7. (a) If x 2 and x 3, ten x 2 + x 6 0. True. If an object does not ave 3 sides, ten it is not a triangle. True. 9. (a) If your car does not get at least 24 miles per gallon, ten it is not tuned properly. If you cannot ave dessert, ten you did not eat your vegetables. 11. (a) If you will not vote for me, ten you do not love your country. If not only outlaws ave guns, ten guns are not outlawed. (poor Englis) If someone legally as a gun, ten guns are not illegal. 13. (a) Bot f (x) and g(x) are not positive. x is not positive. (x 0) (c) 8 is not a prime number. 15. (a) For some numbers a and b, a + b a + b. Some snake is not poisonous. (c) Some dog can cb trees. 17. If x is an integer, ten 2x is an even integer. True. Converse: If 2x is an even integer, ten x is an integer. True. (It is not likely tat tese were te statements you tougt of; tere are lots of oter examples.)

5 A5 19. (a) False. If a 3, b 4, ten (a + b) , but a 2 + b False. If a 2, b 3, ten a > b, but a 2 4 < 9 b 2. (c) True. 21. (a) True. False. If f (x) x + 1 and g(x) x + 2, Ten f (x) g(x) x 2 + 3x + 2 is not a linear function. (c) True. 23. (a) If a and b are prime numbers, ten a + b is prime. False: take a 3 and b 5. If a and b are prime numbers, ten a + b is not prime. False: take a 2 and b 3. (c) If x is a prime number, ten x is odd. False: take x 2. (Tis is te only counterexample.) (d) If x is a prime number, ten x is even. False: take x 3 (or 5 or 7 or... ) 25. (a) If x is a solution of x + 5 9, ten x is odd. False: take x 4. If a 3-sided polygon as equal sides, ten it is a triangle. True. (We also ave non-equilateral triangles.) (c) If a person is a calculus student, ten tat person studies ard. False (unfortunately), but we won t mention names. (d) If x is a (real number) solution of x 2 5x + 6 0, ten x is even. False: take x 3. Section (a) m y 9 x 3. If x 2.97, m If x 3.001, m If x 3 +, m (3 + )2 9 (3 + ) Wen is close to 0, 6 + is close to (a) m y x 2. If x 1.99, m If x 2.004, m If x 2 + : [ (2 + ) 2 + (2 + ) 2 ] 4 m 5 + (2 + ) 2 Wen is very small, 5 + is very close to All of tese answers are approximate. Your answers sould be close to tese numbers. (a) average rate of temperature cange p.m. 9 a.m ours 4 our At 10 a.m., temperature was rising about 5 per our; at 7 p.m., its was rising about 10 /r (falling about 10 /r). 7. All of tese answers are approximate. 300 ft 0 ft ft (a) average velocity sec 0 sec sec 100 ft 200 ft ft average velocity 5 30 sec 10 sec sec (c) At t 10 seconds, velocity 30 feet per second (between 20 and 35 ft/sec); at t 20 seconds, velocity 1 feet per second; at t 30 seconds, velocity 40 feet per second. 9. (a) A(0) 0, A(1) 3, A(2) 6, A(2.5) 7.5, A(3) 9 Te area of te rectangle bounded below by te t-axis, above by te line y 3, on te left by te vertical line t 1 and on te rigt by te vertical line t 4. (c) Grap of y 3x. Section (a) 2 1 (c) DNE (does not exist) (d) 1 3. (a) 1 1 (c) 1 (d) 2 5. (a) (DNE) 7. (a) 0.54 (radian mode!) (c) (a) 0 0 (c) (a) 0 1 (c) DNE 13. Te one- and two-sided its agree at x 1, x 4 and x 5, but not at x 2: x 1 g(x) 1 x 2 g(x) 1 g(x) 2 x 4 g(x) 1 x (a) (a) x 1 + g(x) 1 x 2 + g(x) 4 g(x) 2 x 4 + g(x) 1 x 5 + g(x) 1 x 1 g(x) DNE x 2 g(x) 2 x 4 g(x) 1 x (a) A(0) 0, A(1) 2.25, A(2) 5, A(3) 8.25 A(x) 2x x2 (c) Te area of te trapezoid bounded below by te t-axis, above by te line y 1 2 t + 2, on te left by te vertical line t 1 and on te rigt by te vertical line t 3.