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Chapter 4 31 - SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE...

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Unformatted text preview: SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 293 (c) No minimum or maximum (6) (a) f”(m) = (1w ( m2+1)2 2 2 1/2 13 1 _ (a: + ) (m2+1)1/2 _ (m2+1)—m2 _ $2+1 _ (x2+1)3/z 2721—)3/2 >0.sofisCUoan.NoIP a: +1 48. (a) 11m xtanm : ooand li1r1/2+:ctanm : 00, sex 2 g andcc : 7% are VA. 33"” 2— z—>—7r (b) f(ac) : actanm. 7% < ac < % f'($) =mseczm+tanm > 0 (e) (i) 0 < :c < g. so f increases on (0, g) and decreases on (—Tr (1)1 51 (c) f(0) : 0 is a local minimum value. (d) f”(m) :2sec2x+2:c tanx seczw > 0f0r—g < x < §.sofis CUon (—§,g). NoIP 49. fix) : 1n(1 — In x) is defined when ac > 0 (so that lnm is defined) and 1 i Ina: > 0 [so that 1n(1 7 1n ac) is defined]. The second condition is equivalent to 1 > lna: 4:) :r < 6, so f has domain (0, e) . (a) As at —> 0+7 lnm —» —oo., so 1 — lnap~> 00 and flat) —> 00. As a: —> e’, lnrc ——> 1‘, so 1 — 1113: —> 0+ and f (av) —> —00. Thus. at = 0 and a: : e are vertical asymptotes. There is no horizontal asymptote. b l/ —— — — — _ __ . . ~ t . . ‘ . ' ( ) (CC) 1 1 ( > (1 l ) < 0 on (0, 6) Thus l lS decreasmg on IIS domaln (0 6) (c) f’(m) 95 0 on (0. e) . so f has no local maximum or minimum value. — [1(1 — lncc)]/ : :c(~1/m) + (1 — lnac) [2:(1 —111$)]2 1122(1 — 11131:)2 (d) f"($) — 11130 $20 — lnaic)2 SOfH($)>0 4:} Incc<0 <:> 0<$<1.Thus,fisCUon (0.1) and CD on (1, e) . There is an inflection point at (17 0) . I 50. f(m) : 1 :em has domain R. (a)1lirr:of(x) = Ill—{130$ — $1330 87ng 1 _ 0:1 _ 1. soy _ 1 isaHA. 1213100flm): mlir_n00 1::x : r00 : 0. so y 2 0 is a HA. No VA. (b) f/(m) = W : fl > O for all :10. Thus, f is increasing on R. ...
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