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7.2_49 - m[3 CHAPTER INVEFISE FUNCTIONS 49 y = ﬂat =...

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Unformatted text preview: m [3 CHAPTER? INVEFISE FUNCTIONS 49. y = ﬂat) = ln(sin;t') A. D={:siniR|sin:t'>0} U (2mr.(2n+1)1r) n=—oo = -- - U (—471'. -37r) U {—211'. -rr} U (0. 11') U (2w.31r) U - - - B. No y-intercepl: z-intercepts: f(x} = 0 4: ln(sin:c) = 0 4: sin: = e" = 1 a a: = 2m- + g for each integer n. C. f is periodic with period 271-. D. lim f(x) = —oo and lim f(:i:} = —oo. so t—o{2mr}+ :-[(2n+l)1r]‘ the lines I = mr are VAs for all integers n. E. foe) = 99—“ = cot 1:. so f’{:c) ) 0 when 2mr < a: < 2nrr +§ sin: for each integer n. and f'(-::) < 0 when 21m + 3;— < .1: < (21: + 1)rr.Thus. f is increasing on (2nrr.2mr + 3—) and decreasing on (21m + g, (21: + Dir} for each integer n. F. Local maximum values f(2mr + 325) = 0. no local minimum. G. f”(at) = — csc’x < 0. so I is CD on (2mr. (2n + 1}rr) for each integer o. No I? 51. y = ﬂz) = ln{1 + 2:2) A. D = R B. Both intercepts are 0. C. ﬂ—I) = ﬁx). so the curve is symmetric ' ' 2 _ .0 _ 21' about the y-axis. D. .521... lnfl + a: ) — 00. no asymptotes. E. f (z) — 1 + 1:9 a: > 0. 501' is increasing on (0.00) and decreasingon (—00.0). H. F. f(0) = 0 is a local and absolute minimum. 2(1+m’) — 2::(2z) = 2(1— 1-“) (1+3?)2 (1 +3.3)2 lz] < 1.501” is CU on(—1.1}.CDon (—oo. —1) and (Loo). IP {1.1:12)and(—1.1n2). G. f”(:t:) = )0 41b I. We use the CA5 to calculate f’(z) = M and 21: + xsinx 22:35in +4sinx — cos“: + \$2 + 5 2:20:05“: — 45in: - 5) f"(-’I=} = . From the graphs. it seems that f’ > 0 {and so f is increasing) on approximately the intervals (0.2.7). (4.5.8.2) and {10.9, 14.3). It seems that 1'" changes sign {indicating inﬂection points) at a: w 3.8. 5.7. 10.0 and 12.0. Looking back at the graph of ﬁr) = 111(22: + rsin I). this implies that the inﬂection points have approximate coordinates(3.8,1.7). (5.7.2.1). (10.0. 2.7). and (12.0. 2.9). ssy=(2..~.+1)‘(a:‘—3)-‘ => my=tn((2m+1)‘(z‘-3)‘) =5 “ I um‘Li ...
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