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Chapter 3 80 - 230 CHAPTER 3 DIFFERENTIATION RULES 11 f(x...

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Unformatted text preview: 230 CHAPTER 3 DIFFERENTIATION RULES 11. f(x) = \3/—_1-ac = (1 —:r)1/3 :> my) : —g(1—e)r2/3.se 2 f(0) : land f'(0) = —%. Thus. f($)%f(0)+fl(0)(me0):1—ém.Weneed \ I 3/1 - a: — 0.1 <1—%;I: < 13/1 — as + 0.1. which is true when n L -2 —1.204 < m < 0.706. 12. flat) 2 tanm => f’ (:r) 2 SCC2 1:. so f(0) = 0 and f’(0) = 1. 1 Thus. f(m) % f(0) -I— f'(0)(ac , 0) : 0+ 1(1: # 0) : 2:. We need tang: — 0.1 < w < tans: + 0.1. which is true when —0.63 < :13 < 0.63. 1 —4 f’(:1:) : ;4(1 + 2e)*5(2) : £55 so f(0) : land f’(0) = —8. Thus. flat) z f(0) + f'(0)(x — 0) : 1 + (43w; f 0) = 1 — 89:. We need 1/(1 + 23:)4 — 0.1 < 1 — 89: < 1/(1 + 2x)4 + 0.1. which is true 70.08 when 70.045 < a: < 0.055. 14. f(m) : 6“” => f’(a:) : e$.so f(0) : 1 and f'(0) : 1. Thus. f(1:) % f(0) + f'(0)(:c 7 0) : 1 + 1(;v — 0) : 1 +m. We need e35 e 0.1 < 1 + :r < eI + 0.1. which is true when —0.483 < m < 0.416. 15. Ify : f(:c). then the differential dy is equal to f'(0v) dm. y = :04 + 555 => dy : (4303 + 5) dm. 16. y : cosmv => dy : —sin7r;c - TFdIII : #Trsinnmdm 17_ y:g;1nm :> dy: <w~$+lnav1> dm:(1+ln:c)da: 18.y:1/1+t2 :. dy:%(1+t2)_1/2(2t)dt: 1+t2dt u+1 (u—1)(1)—(u+1)(1) ‘2 g . 7 ————d 19. y— u- 1 dy (11.? DZ du (u~ 1)2 u 20. y z (1 + 2W4 : dy : ;4(1 + 2W5 ~2dr : —8(1 + 2r)’5 (17‘ 21. (a)y:$2+2m :> dy: (2x+2)d:z: (b) When a: z 3 and due : g. dy : [2(3) + 21%): 4. 22. (a) y : (BI/4 => dy : iem/4dm (b) When in : 0 and dm : 0.1. dy : (§e°)(0.1): 0.025. ...
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