Unformatted text preview: CHAPTER 7 INVEHSE FUNCTIONS
63. y = ﬂat) : ail/(1+1) A. D : {ac l 9: 7E —1} = (—oo,—1)U(—1, 00) B. No Litintercept; yintercept : f(0) = 6’1 C. No symmetry D. lim e'l/(’”+1) = 1 since #1/(w + 1) —> 0, so y = 1 is 2:42:00 alIA. lim+e_1/(1+1) = 0 since —1/(a: + 1) —> —oo, lim e‘l/(“U : 00 since —1/(m + 1) —> 00, so
3—4—1 (v——>—1‘ m : —l is aVA. E. f’(w) : (”Win/(3: + 1)2 :> f'(ac) > 0 for all :1: except 1, so f is increasing on (goo, —1) and (—1, 30). F. No extreme values H.
l/(a;+1) l/(cc+1) ‘ 71/(z+1) 2 G.f”()='e—_(T+e (32):_e (2+1)
(x+1) (90+1) (90+1) r> f”(3:) >0 4:) 223+1 <0 4:} :1: < —%,sofisCUon (—00, —l) and (—1,—%), andCD on (—%, 00). f has an IP at (—%, 5‘2). . y = f(a:) 2 ea” + e72” A. D = R B. y—intercept: f(0) = 2;
no zc—intercept C. No symmetry D. No asymptotes E. f/(m) = 35“  2e41, so f’(~) > 0 c) 3e“ > 2e”ac [multiply by 62"] 41> 65m > g (i) 53c > lug 4i) an > ging m —0.081. Similarly, f’(a') < 0 (i) is < élng. f is decreasing on (—00, § 111 g) and increasing on (% 111%,00). F. Local minimum value f (g In g) = (§)3/5 + (3772/5 m 1.96; no local maximum. G. f"(1:) = 963“” + 46’2”, so f”(ac) > 0 for all :5, and f is CU on (—007 00). No 1P .f(z):e$3’“D —>0as:c—> —oo,and 1'8
f(ac) 7, 00 as m —+ 00. From the graph, it appears that f has a local minimum of about f(0.58) : 0.68, and alocal '
maximum of about f(——0.58) = 1.47. To ﬁnd the exact values, we calculate f/(m) : (3m2 , 1) 6‘13””, which is 0 when 341:2 — 1 : 0 c) z = i—kg’. The negative root corresponds to the local maximum f (—ﬁ) : 6‘“ ﬂ? _ (‘1’ ‘5) : eZﬁ/g, and the positive root corresponds to the local minimum f (f) = 60/ £93 " (1/ V5) : e‘2ﬁ/9. To estimate the inﬂection points, we calculate and graph f”(m) = % [(3332 — 1)em3"m] = (3:52 — l)e”3 ‘1 (3932 — 1) + ez34(6$) = e“3 ‘m (93:4 — 6x2 + 6x + 1). From the graph, it appears that f”(:1:) changes sign (and thus f has inﬂection points) at :1: z —0.15 and
as w 7 1.09. From the graph of f , we see that these xvalues correspond to inﬂection points at about (—0.15, 1.15)
and (409,032). ...
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 Spring '06
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