Chapter 3 63 - SECTION 3.8 DERIVATIVES 0F LOGARITHMIC...

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Unformatted text preview: SECTION 3.8 DERIVATIVES 0F LOGARITHMIC FUNCTIONS 213 31. y : f(a:) : lnlnm => f’(:v) = —1—<1) => f’(e) = 1 so an equation ofthe tangent line at (6,0) is lnw $ 6 1 1 y—0——(m e),ory~ a; Lora: ey~e e e 3 / 1 2 I 12 ' ' ‘ 32. y 2 1n(z — 7) :> y : 3 7 - 32: => y (2) : fl = 12, so an equation ofa tangent line at (2, 0) 18 m _ _ y—0=12(m—2) or y=12m—24. 33. f(w) = sinw + lna: => f’(a:) : cosm +1/IL‘. This is reasonable, because the graph shows that f increases when f’ is positive, and f’($) = 0 when f has a horizontal tangent. Ina: w(1/m)—lnm 1—lna: 34. — — l— h . y x _> y 9:2 $2 1— 0 1— 1 y’(1) = 12 = 1 and y’(e) = 62 = 0 => equations of tangent lines are y — 0 =1(x — 1) org 2 a: — 1 and y—l/e=0(a:—e) Cry: 1/6. 35. y=(2$+1)5(934—3)6 —s lny—ln<(2m I 1)5(ac4 3)6) 1 1 1 1n =51n2$+1 +6111 4—3 => _ '25. . . y ( ) (13 ) yy 2$+12+6 254—3 10 24 3 3 41:3 2 2zc+1 $4—3 2m+1 304—3 [The answer could be simplified to y’ : 2(2m + 1)4(:c4 — 3)5 (299v4 + 12m3 — 15), but this is unnecessary] 2 36.y=\/Ee1 (x2+1)10 :> lnyzln\/E-I—Ine“”2-I—ln(:zc2—I—1)10 => 1ny=§lnm+$2+101n(m2+1) 1 1 1 1 => —y'=---+2$+10- ~2m :> y'=\/a—cez2(m2+1)10(i+2x+ 20x) 3; 2 :1: 3724—1 sin2 :0 tan4 ac 31y: ($2+1)2 => lny=1n(sinQa:tan‘lzL')—111(:t:2+1)2 => lnyzln(sin:tc)2—I—ln(tain;z:)4—ln(:icz+1)2 => lny:21n|sina:|+4ln]tana:|—2ln(m2+1) => 1 ,_ . l 1 2 1 yy— sinx coszc+4-tan$~sec x—2-m2+1-2x => ($2 + 1)2 tancc _ m2 + 1 $2+1 3:2—1 1 y yl=4$2+1~l a? _ a: _14x2+1 ~2m _ 41324—1 952—1 2 z2+1 $2~1 ‘2 332—1 954—1 —1—:c4 932—1 39.y::L'$ :> lny=ln$I => lnyzmlnx => y’/y=m(1/w)+(lnx)~1 => y’:y(1+lnx) :> y’2m2(1+ln:c) , _ sin2mtan4cc (2cota:+ 4sec233 43: > 38.3]: 4 => 111g: filn(at2+1) — filn(m2—1) :> ...
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