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Chapter 4 148

# Chapter 4 148 - 410 CHAPTER 4 APPLICATIONS OF...

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Unformatted text preview: 410 CHAPTER 4 APPLICATIONS OF DIFFERENTIATlON 38. f(x) = sinmcos2 :1: => f’(:c) = cos3 x — 2 sin2 1: C053: => f"(m) = -7sinx cos2 x + 2 sin3 a: l 2.5 0.4 l A A l m 'A *1 ‘25 —().4 From the graphs of f’ and f”. it appears that f is increasing on (0. 0.62). (1.577 2.53), (3.76. 4.71) and (5.67, 27r) and decreasing on (0.62157). (2.53., 3.76) and (4.71. 5.67); f has local maxima of about f(0.62) : f(2.53) = 0.38 and f(4.71) : 0 and local minima of about f(1.57) = 0 and f(3.76) : f(5.67) : —0.38; f is CU on (1.08. 2.06). (3.14, 4.22) and (5.20, 271') and CD on (0.1.08). (2.06. 3.14) and (4.22. 5.20); and f has inﬂection points at about (0. 0). (1.08, 0.20). (2.06, 0.20). (3.147 0). (4.22, 40.20). (5.20, —0.20) and (271'. 0). 39. 1 From the ra h. we estimate the oints of inﬂection to be about 3 p P (\$0820.22). f(:B) : e—l/z2 :> f’(;c) : 2334364032 :> III/(3U) = 2 [m’3(2m_3)e’1/z2 + (fl/\$2 (73\$_4)] : 22—661”? (2 — 33:2). —5 5 0 This is 0 when 2 7 3x2 = 0 (I) a: : ::\/g. so the inﬂection points are (z: %.e_3/2). 1 1 1 40' (a) 1.1 (b) “3”) — ﬁ—/ .ILIEOW — m - 5’ . 1 1 - \$1me(\$):_1+1:2~ 1' f()* 1 —0 1 ‘10—_— 10 11m f(m) : __ : 1 v0.1 wHO- 1 +0 (c) From the graph of f. estimates for the 1P are (—0.4. 0.9) and (0.4.0.08). 61/1 [tel/”(2:5 i 1) + 2x + 1] \$4(el/1 + 1)3 (e) From the graph. we see that f” changes sign at a: : i0.417 (d) f"(I) — — (ac : 0 is not in the domain of f). IP are approximately (0.417. 0.083) and (70.417, 0.917). ...
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