Instructors' Manual to accompany CALCULUS WITH ANALYTIC GEOMETRY Containing Answers to even-numbered exercises Solutions of selected exercises, even- and odd-numbered Comments on selected exercises ( $> ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers CHAPTER 1 SECTION 2, page G 2. Suppose a fI a and b TI O. Then a ,= a • a-1b-1 l l (aa- )(bb- ) = 1 1 = 1. l -1 4. (a/b) (e/d) = (ab-1) (cd-I) = ab-led- and (ae)/(bd) (ae) (bd) = (ae) (b-ld-l ) = ab-led-l by Ex. 3. 6. ba + de = badd + bbed = (ad) Ibd)-l + (be) (bd)-l = (ad + be) (bd)-l -ad-b+d-be :lote that (ad)/(bd) = lad)(bd)-l = (ad)(b-ld-l ) = (ab-l)(dd-l) ab-l by Ex. 3. 8. a < a < b implies a < b-l < a-I, hence a < ab-1 = alb andab-l < aa-1 1. 2 2 2 2 2 2 10. Set a + ... + d e Then (a/e )2 + ... + (d/e )2 = (a + ... 2 4 2 4 2 + d )/e = e /e = 1/e . (More generally, the reciprocal of a sum of n squares is a sum of n squares.) 12. tal = ± a and Ibl = ± b with the same choice of sign, so a 2 lal + Ibl = ± (a + b) = la + bl. 14. laI = I(a - b) + bl 2 la - bl + Ibl so tal - Ibl < la - bl· Also - (IaI - IbI) = Ibl - lal2lb-al = la bl,-hence Iial - Ibll (I I - Ibl) 2 la - ,- a bl· 16. Ix - al > Ix - bl 18. Ixl > 3 20. Ix - 71 < 2 22. -1.0 < x < 10 24. 2.9999 < x < 3 or 3 < x< 3.0001 26. -6 < x < -2 CHAPTER 1 28. -5.005 < x < -5 or -5 < x < -4.995 30. -- < x < - 2 2 32. 1 < x < 13 34. x > ~- 36. 7 < x < 9 or 9 < x < 11 ss. - ! < , <! 12 40. x < 30 42. x > 44. 9 < x < 17 46. x < -5 or x > 3 48. x < -2 or -1 < x < 0 50. x < 0 or x > - 52. i, Ü 9' 11 54. Ifa=b = c = 0, then |a| -/- |ib| + |c| =0. Otherwise at least one of the non-negative numbers \a\ , \b\ , \c\ is positive and then |a| + |jb| + Ici > 0. -6 56. | (x + y) - 121 = | (x - 1) + (y - 5) | £ |x - 7| -f \y - 5| < 10 + 10"6 < 10"5. 58. Clearly |x| < 5 + 10 < 6. Now \xy - 35| * |x(y - 1) + 1(x - 5j | <_|x||y-7|+ 7|x - 5| < 6 x 10 6 + 7 x 10 6 = 1.3 x 10 5 -5 < 2 x 10 60. |x3 - 27| = |x2 + 3x + 9||x - 3| < |χ2 + 3x + 9| x 1θ"6. Since |x| < 4 we have |x + 3x + 9| <_ |x | +3|x| + 9 < 16 + 12 + 9 = 37 < 50. Hence |x - 27I < 50 x 10~6 = 5 x 10~5. SECTION 3, page 10 (-15,60)· + (-2,2)· + + (95,40)· (3,0) 20 H—I I I I H—I · I I ► -M—I—I—h H—h-1—I—I 20 · (0,-2) ZU (75,-10) + «(1,-3) 2 CHAPTER 1 (0.03, 12)· 50 + 10 (-0.02,35) H 1 h H -H 1 ► (-0.02, 5) 0.01 50+ t(0.00, -60) H 1 \- 1 1 1 ► 0.01 10. 12. (Ο,.ν) 1 + H h H h-^ (0,0) ·, 0) x (λ 14. 16. 1 + 1 + -+-H—H ■■♦ i * » H 1—H H t—H^J ► 18. 20. "I""1 I l i t t» H—|—I—I—\- H 1 1—I ► 1 1 1 1 3 CHAPTER 1 22. All points except (0, 0) 24. (1 ± /Î, 3), (1, 3 ± /2) 26. (± 1, 0), (0, /3) SECTION 4, page 15 2 4 2 /— 2. 1, x - x -/- 1, x + x + 1, x + Vx + 1, h(2x + h + 1) . 10. 0.01 + H—I—h I I I ► 0.01 14. CHAPTER 1 16. H 1 1 h -I 1 1 1 ► 18. all x, all y 20. all x, all y 22. x φ -2, y =)= 1 24. x < 1, y > 0 26. x < -, y >0 28. all x, y > /ΪΓ 30. x > -4, y > 0 32. |x| > 1, y >_ 0 34. x > -1, y> 0 36. 4x -/- 2, 4x + 4x - 3 , 4 32 38. x + 1, -x + x - x -/-x 40. [f + g] (χχ) = f (χλ) + g(x^) < f(x ) -f g(x ) = [f + g] (x) 7 42. -2x2 + 6x - 1, -4x + lOx - 4 44. -x + 2, -x 46. -x + 4, -x - 2 48. π(2χ + 5)2, 2T\X2 -h 5 50. fix; 52. x 54. 3, x + 1, etc. 56. 3[-(x0 + χλ)] -5 = |[(3x - 5; + (3x - 5)] 0 1 60. i/rx x ;2 = α/χ 2; η/χ 2; 58. no 0 1 0 χ SECTION 5, page 20 2. (0,-3) (-2,-7)< CHAPTER 1 8. (-20,41) + 2000 (200, 1750) 4-1000 r(100, 850) ■+■ H 1 h- 100 200 -10 10. 12. (-1,0.5) 3000 (100,2515) 2000 1000 '(50,1265) 4- (i,-o.i) -H 1 ► 50 100 t 14 16. kx (-1, 120) (100,4040) (1,-80) 22. 3 24. -1 26. CHAPTER 1 28. y = -x + 1 30. y = 2x + 1 ^ο 2 x 5 34. y = 0 32. y = 3^ + 3- 36. y = jx + j 38. y = -yU + 2) 40. y = -lOx - 16 42. - - , -3 AA 1 7 46. a, b 44. —2, 2 ' 48. 1/a, 1/2? 50. y = -3x + 17 52. 90° 54. 75° (or 105°); also tan Θ = ±(2 + /ï) SECTION 6, page 26 6. lt I II » A 10. I l I H 1—I—I ► (1,-2) (3,-7) 34. (-1,1) t 36. + 1 (_! Ü) ^ 2' 2 } -I 1 ► -1 -2 + -4+ -6 + 9