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Chapter 3 84

# Chapter 3 84 - 234 CHAPTER 3 DIFFERENTlATION RULES d(11 du...

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Unformatted text preview: 234 CHAPTER 3 DIFFERENTlATION RULES _ d (11) du d1} d (d)d(uv)—E(uv)d\$=(11%+0E>dm=uﬁdm+vd—:dm:udv+vdu d vdu ud'u vdud dud u u —— — — x—u— :c d 7 =_ _ 2 da: dun _ da‘ d _vdu-udv (e) (1}) dx (v)dm 112 dac_ 112 \$ T d (f) d(:c") = a (13")dm : nmn'l dx 48. (a) f(ac) = sinx => f’(a:) : cosm. so f(0) = Oand f’(0) : 1. Thus. ﬂan) av. f(0) + f’(0)(:c — 0) = 0 + 1(ac — 0) : x. (b) 1 y=1.02 sinx 0.36 y=X ‘ y=0,985inx y=l,023inx y = 0.98 sin x 0.36 70.33 Y:X y= 1.02 sinx #036 *033 We want to know the values of m for which y = m approximates y : sin a: with less than a 2% difference; that is. the values of m for which < 0.02 c» —0.02 < £1,313 < 0.02 4:) SlnfE 5v — sinm sin x 1.02sincc < m < 098\$in if sinm < 0 70.023in\$ < a: i sinm < 0.02 sinm if sinm > 0 0.98sinzr < a: < 1.02 sinm if sinzc > 0 42> —0.02 sina: > \$ # sins: > 0.02 sinx if sinm < 0 In the ﬁrst ﬁgure, we see that the graphs are very close to each other near a: : 0. Changing the viewing rectangle and using an intersect feature (see the second ﬁgure) we ﬁnd that y : a: intersects y : 1.02 sin 30 at w m 0.344. By symmetry. they also intersect at ac x —0.344 (see the third ﬁgure). Converting 0.344 radians to degrees. we get 0.344(Lfg) % 19.70 m 20°, which veriﬁes the statement. 49. (a) The graph shows that f’(1) : 2, so L(a:) : f(1) + f'(1)(m i 1): 5 + 2(90 — 1): 2x + 3. f(0.9) % L(0.9) = 4.8 and f(1.1) m L(1.1) : 5.2. (b) From the graph. we see that f '(rc) is positive and decreasing. This means that the slopes of the tangent lines are positive. but the tangents are becoming less steep. So the tangent lines lie above the curve. Thus. the estimates in part (a) are too large. ...
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