BC Calculus – FDW 3rd Edition – Section 4.5 Even Answers Quick Review 4.5 x f(x) g(x) 2. 1 cos x ( x 21)sin x 4. -.322 6. y 2ex e 1 8. 0.7 -1.457 -1.7 10. ( x 1) 0.8 -1.688 -1.8 0.9 -1.871 -1.9 Section 4.5 1.0 -2.000 -2.0 2.(a) L( x ) 45 x 95 (b) error < 10-3 1.1 -2.069 -2.1 4.(a) L( x) x (b) error < 10-2 1.2 -2.072 -2.2 -3 6.(a) L( x ) x 2 (b) error < 10 1.3 -2.003 -2.3 8.(a) 1.2, 1.002100 1.2 101 (b) 1.003, 3 1.009 1.003 105 10. Method 1/ Use a b a n na n 1b1 ... then n (a) 4 3x 4 1 3 (b) 2 x 2 1 3 1 2 1 3 23 4 3x 1 2 2 12 2 12 (c) 1 21 x 3 13 23 1 2 1 1 3 4 3 1 x2 21 x 13 2 1 2 1 42 1 2 2 x 1 43 34 4 3 1 4x 1 2 x 2 2 1 x4 1 23 21 x 1 623 x Method 2/ Factor out (divide out) to make the first term (or ‘a’) equal to 1, then use: 1 + nx A problem like (c) is already to go! For (a) and (b) factoring out 41/3 or 21/2 may be tricky! 1 1 1 2 1 2 (a) 4 3x 3 4 3 1 34x (b) 2 x 2 2 1 x2 2 1 3 1 2 (c) No factoring here! 1 4 3 1 13 34x 4 3 1 4x 1 2 1 2 1 12 1 2 x 2 3 1 x2 2 x2 4 1 23 21 x 1 623 x So? Well… if our binomial is of the form (1 + b)n vs (a + b)n, then method 2 is best! 1 x 27 3 y 2.963 12. y 27 1 x 81 9 y 8.949 14. y 18 16. x 1.452627, 1.164035 (Why six decimal places? Why?!?) 18. 1.189207 20.(a) dy 22.(a) dy 2 2 x 2 1 x2 2 1 2 x 2 1 1 x 2 2 dx (b) dy .024 dx (b) dy .2 24.(a) dy csc 1 3x cot 1 3x dx (b) dy .205525 26.(a) dy dx (b) dy .0375 2 x y 2 x BC Calculus – Section 4.5 Even Answers (continued) 28. dy 5e5 x 5x 4 dx 30. dy 8 x ln8 8 x 7 dx 32.(a) .231 (b) .2 (c) .031 34.(a) .04060401 (b) .04 (c) .00060401 36. A 8 a dr 4 cm2 38. S 12a dx 6 cm2 40. A 2 r dh .1 r cm2 42. dV 4 r 2dr 4 82 .3 241in2 44. A s 4 3 dA 2 s4 3 ds 23 20 .5 8.7 cm2 46.(a) .08 .2513 (b) 2% 48. 3% 50.(a) within .5% (b) within 5% 52. Measure the height to within 1/3 % of ‘h’. 54. A 4 r 2 , we have dA 8 r dr 8 r 161 2r which is the surface area error in in2. 2 1 3 56.(a) dT L2 g 2 dg (b) T decreases and the clock speeds up. dT & dg have opposite signs (c) dg .9765, so g 979.0235 58. False. Recall the product rule: d(uv) = udv + vdu 60. A 62. A 64. If x1 h, then f '( x1 ) 1 1 2h 2 and x2 h 66.(a) b0 f ( a ), b1 f '( a ), b2 1 h 2 11 2h 2 f ''( a ) 2 h . If x1 h, x2 h . (b) 1 x x 2 (c) They appear identical. (d) 1 ( x 1) ( x 1)2 , the same 2 (e) 1 2x x8 , they appear the same (f) f: 1 + x, g: 1 – (x – 1), h: 1 2x 68. Formulas for differentials? Just multiply the derivative formulae by ‘dx’. 70. The error is zero at x = a. The error is negligible when compared with (x – a).