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Chapter 4 85

# Chapter 4 85 - SECTION 4.6 GHAPHING WITH CALCULUS AND...

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Unformatted text preview: SECTION 4.6 GHAPHING WITH CALCULUS AND CALCULATORS 347 23. (a) M) 2 mm (b) Recall that ab = 61’1““. \$133+ 1111/7” = \$133+ (ea/1“”. As m —» 0+. 2 11173: —> —00. so :vl/C” = 60/1) I” —> 0. This indicates that there is - a hole at (0, 0). As a: —> 00. we have the indeterminate form 000. 0 8 lim 171/”: = lim e<1/m)lw.but lim h1_;z: g lim 1/_av : 0. so 7] lim III/'1 2 e0 : 1. This indicates that y : 1 is 21 HA. I—'OO (c) Estimated maximum: (2.72. 1.45). No estimated minimum. We use logarithmic differentiation to ﬁnd any 1 I 1 1 1 criticalnumbers.y—w1/I lny—wlnm '* %—E-;+(ln\$)(i?> => , 1/x 1—1111: , , y—m 2 0 Inn: 1 5 \$ e.F0r0<x<e.y>Oandform>e.y<0.so x f (e) : 61/8 is a local maximum value. This point is approximately (2.7183, 1.4447). which agrees with our estimate. (d) 0.1 From the graph. we see that f”(:c) : 0 at m z 0.58 and w m 4.37. Since f” changes sign at these values, they are m—coordinates of inﬂection points. 24. (a) f(:c) : (sin :c)5i” is continuous where sinm > 0. that is. 1-2 on intervals of the form (27m, (2n + 1)7r). so we have " graphed f on (0, 7r). . . . . -r (b) y : (51113:)5‘” => lny : srnw 1115mm. so 0 I . . . . . ln sinm H cotx lim lny : 11m smajlnsmx : 11m : 11m _ : lim (~ sin :c) : 0 1Ho+ 1—)0+ m—>O+ C50 :3 3—.o+ * csc m cot a: x—>0+ 2 lim y = 60 : 1. wHO+ (c) It appears that we have a local maximum at (1.57. 1) and local minima at (0.38. 0.69) and (2.76.069). y : (sinwwnx => lny 2 sinx lnsinm => I . cosm . . , _. y; : (Sln\$)(sin\$> —|— (lnsmm) cosx : cosm(1+lns1nat) => 3/ : (Sin\$)”mm(cosm)(1+lnsinx). y’ : 0 => cosa: : Oorlnsinw : —1 => m2 : g or sinw : 671. On (0,7r).sina: : 6’1 :> 3:1 2 sin—1(e’1) and \$3 : 7r 1 sin—1(e’1). Approximating these points gives us ...
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