<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/fdd3c887-ff5c-440e-ac61-9d4e18aef435/bg1.png"><div class="t m0 x0 h2 y1 ff1 fs0 fc0 sc0 ls36 ws0">01. <span class="ff2 ws1">Considere a função <span class="ff3 ls0">f</span>de\u2026<span class="_0 blank"></span>nida em <span class="ff4 ls1">R<span class="ff5 fs1 ls2 v1">2</span></span><span class="ws2">p or<span class="_1 blank"> </span><span class="ff3 ls3">f</span><span class="ff6 ws3">(<span class="ff3 ws4">x;<span class="_2 blank"> </span>y </span><span class="ws5">) = 2</span><span class="ff3">x<span class="ff5 fs1 ls4 v1">2</span><span class="ls5">y<span class="ff7 ls6">\ue000</span></span><span class="ws6">xy <span class="ff5 fs1 ls7 v1">2</span></span></span><span class="ls6">+</span><span class="ff3 ws7">xy </span></span>.</span></span></div><div class="t m0 x1 h2 y2 ff2 fs0 fc0 sc0 ls36 ws8">(a) Determine e classi\u2026<span class="_0 blank"></span>que os pontos críticos de <span class="ff3 ls8">f</span><span class="ws9">em <span class="ff4 ls1">R<span class="ff5 fs1 ls9 v1">2</span></span><span class="ff6">;</span></span></div><div class="t m0 x1 h2 y3 ff2 fs0 fc0 sc0 ls36 ws1">(b) Mostre que <span class="ff3 ls8">f</span>não tem máximo nem mínimo absolutos em <span class="ff4 lsa">R<span class="ff5 fs1 ls4 v1">2</span></span><span class="ff3">:</span></div><div class="t m0 x0 h3 y4 ff1 fs0 fc0 sc0 ls36 ws3">Solução:</div><div class="t m0 x1 h4 y5 ff2 fs0 fc0 sc0 ls36 ws8">(a) Os p<span class="_3 blank"> </span>on<span class="_4 blank"></span>tos críticos de <span class="ff3 ls0">f</span><span class="ws1">são as soluções do sistema <span class="ff3 ws3">f<span class="ff8 fs1 lsb v2">x</span><span class="ff6 wsa">= 0;<span class="_5 blank"> </span></span>f<span class="ff8 fs1 lsc v2">y</span><span class="ff6 wsa">= 0</span><span class="lsd">:</span></span><span class="wsb">T<span class="_6 blank"></span>emos,<span class="_1 blank"> </span>p ortan<span class="_4 blank"></span>to:</span></span></div><div class="t m0 x2 h5 y6 ff9 fs0 fc0 sc0 ls36">8</div><div class="t m0 x2 h5 y7 ff9 fs0 fc0 sc0 ls36"><</div><div class="t m0 x2 h5 y8 ff9 fs0 fc0 sc0 ls36">:</div><div class="t m0 x3 h2 y9 ff6 fs0 fc0 sc0 ls36 ws3">4<span class="ff3 wsc">xy <span class="ff7 ls6">\ue000</span><span class="lse">y<span class="ff5 fs1 ls7 v1">2</span></span></span><span class="ls6">+<span class="ff3 lsf">y</span></span><span class="wsa">= 0<span class="_7 blank"> </span><span class="ff2">(I)</span></span></div><div class="t m0 x3 h2 ya ff6 fs0 fc0 sc0 ls36 ws3">2<span class="ff3">x<span class="ff5 fs1 ls7 v1">2</span><span class="ff7 ls6">\ue000</span></span>2<span class="ff3 wsc">xy </span><span class="ls6">+<span class="ff3 ls10">x</span></span><span class="wsa">= 0<span class="_8 blank"> </span><span class="ff2 ws2">(I I)</span></span></div><div class="t m0 x0 h4 yb ff2 fs0 fc0 sc0 ls36 wsd">De (I) obtemos <span class="ff3 ls11">y</span><span class="ff6 ws3">(4<span class="ff3 ls12">x<span class="ff7 ls6">\ue000</span><span class="ls5">y</span></span><span class="wse">+ 1)<span class="_9 blank"> </span>=<span class="_9 blank"> </span>0<span class="_1 blank"> </span></span></span><span class="wsf">e daí segue que <span class="ff3 ls13">y</span><span class="ff6 ws10">= 0<span class="_1 blank"> </span></span><span class="ws11">ou <span class="ff3 ls13">y</span><span class="ff6 ws10">= 4<span class="ff3 ls14">x</span><span class="ws12">+ 1</span></span><span class="ws13">.<span class="_a blank"> </span>Se <span class="ff3 ls15">y</span><span class="ff6 ws10">= 0</span></span></span>, en<span class="_4 blank"></span>tão de (I<span class="_3 blank"> </span>I)</span></div><div class="t m0 x0 h2 yc ff2 fs0 fc0 sc0 ls36 ws14">resulta <span class="ff6 ws3">2<span class="ff3">x<span class="ff5 fs1 ls16 v1">2</span></span><span class="ls17">+<span class="ff3 ls10">x</span></span><span class="wsa">= 0</span></span><span class="ws15">, o que nos dá <span class="ff3 ls10">x</span><span class="ff6 wsa">= 0<span class="_1 blank"> </span></span><span class="ws16">ou <span class="ff3 ls10">x<span class="ff6 ls18">=</span></span><span class="ff7 ws3">\ue000<span class="ff6 ls19">1</span><span class="ff3">=<span class="ff6 ls19">2</span></span></span></span>.<span class="_b blank"> </span>Com isso encon<span class="_4 blank"></span>tramos os p<span class="_3 blank"> </span>on<span class="_4 blank"></span>tos críticos:</span></div><div class="t m0 x0 h4 yd ff3 fs0 fc0 sc0 ls1a">P<span class="ff5 fs1 ls1b v2">1</span><span class="ff6 ls36 ws3">(0</span><span class="ls1c">;<span class="ff6 ls36 ws17">0) </span><span class="ls36 ws18">; P<span class="ff5 fs1 ls1d v2">2</span><span class="ff6 ws3">(<span class="ff7">\ue000</span><span class="ls19">1</span><span class="ff3">=</span><span class="ls19">2</span></span><span class="ls1e">;</span><span class="ff6 ws17">0) </span><span class="ls1f">:</span><span class="ff2 ws19">Se, p<span class="_3 blank"> </span>or outro lado,<span class="_9 blank"> </span></span><span class="ls20">y</span><span class="ff6 wsa">= 4</span><span class="ls21">x</span><span class="ff6 ws1a">+ 1</span><span class="ls1f">;</span><span class="ff2 ws1b">lev<span class="_4 blank"></span>amos esse v<span class="_6 blank"></span>alor em (I<span class="_3 blank"> </span>I) e encontramos:</span></span></span></div><div class="t m0 x4 h6 ye ff6 fs0 fc0 sc0 ls36 ws3">2<span class="ff3">x<span class="ff5 fs1 ls7 v3">2</span><span class="ff7 ls6">\ue000</span></span>2<span class="ff3 ls22">x</span>(4<span class="ff3 ls12">x</span><span class="wse">+ 1) + <span class="ff3 ls10">x</span><span class="ws1c">= 0 <span class="ff7 ws1d">(<span class="_c blank"></span>) <span class="ff6 ws3">6<span class="ff3">x<span class="ff5 fs1 ls7 v3">2</span></span><span class="ls6">+<span class="ff3 ls10">x</span></span><span class="ws1c">= 0 </span></span>(<span class="_c blank"></span>) <span class="ff3 ls10">x</span><span class="ff6 wsa">= 0<span class="_1 blank"> </span><span class="ff2 ws1e">ou <span class="ff3 ls10">x</span></span><span class="ls23">=</span></span><span class="ws3">\ue000<span class="ff6 ls19">1</span><span class="ff3">=<span class="ff6 ls19">6</span>:</span></span></span></span></span></div><div class="t m0 x0 h4 yf ff2 fs0 fc0 sc0 ls36 ws1f">Assim, determinamos os p<span class="_3 blank"> </span>ontos críticos:<span class="_d blank"> </span><span class="ff3 ls1a">P<span class="ff5 fs1 ls1b v2">3</span></span><span class="ff6 ws3">(0<span class="ff3 ls1c">;</span><span class="ws20">1) </span></span><span class="ls24">e</span><span class="ff3 ws3">P<span class="ff5 fs1 ls1d v2">4</span><span class="ff6">(<span class="ff7">\ue000</span><span class="ls19">1</span></span>=<span class="ff6 ls19">6</span><span class="ls1e">;<span class="ff6 ls19">1</span></span>=<span class="ff6 ws21">3) </span><span class="ls25">:</span></span><span class="ws22">A classi\u2026<span class="_0 blank"></span>cação dos p<span class="_3 blank"> </span>on<span class="_4 blank"></span>tos</span></div><div class="t m0 x0 h4 y10 ff2 fs0 fc0 sc0 ls36 ws1">críticos dar-se-á p<span class="_3 blank"> </span>or meio da análise do sinal do determinante <span class="ffa ws23">hessiano </span>:</div><div class="t m0 x5 h7 y11 ff3 fs0 fc0 sc0 ls26">H<span class="ff6 ls36 ws3">(</span><span class="ls27">P<span class="ff6 ls36 ws24">) = <span class="ff9 v4">\ue00c</span></span></span></div><div class="t m0 x6 h5 y12 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x6 h5 y13 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x6 h5 y14 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x6 h5 y15 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x6 h5 y16 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x7 h4 y17 ff3 fs0 fc0 sc0 ls36 ws3">f<span class="ff8 fs1 ws25 v2">xx </span><span class="ff6">(</span><span class="ls28">P<span class="ff6 ls29">)</span></span>f<span class="ff8 fs1 ws26 v2">xy </span><span class="ff6">(</span><span class="ls27">P</span><span class="ff6">)</span></div><div class="t m0 x7 h4 y18 ff3 fs0 fc0 sc0 ls36 ws3">f<span class="ff8 fs1 ws27 v2">y x<span class="_e blank"> </span></span><span class="ff6">(</span><span class="ls27">P<span class="ff6 ls2a">)</span></span>f<span class="ff8 fs1 ws27 v2">y y<span class="_f blank"> </span></span><span class="ff6">(</span><span class="ls27">P</span><span class="ff6">)</span></div><div class="t m0 x8 h5 y19 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x8 h5 y12 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x8 h5 y13 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x8 h5 y14 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x8 h5 y15 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x8 h5 y16 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x0 h4 y1a ff2 fs0 fc0 sc0 ls36 ws1">calculado em cada p<span class="_3 blank"> </span>on<span class="_4 blank"></span>to crítico.<span class="_a blank"> </span>F<span class="_0 blank"></span>ormemos uma tab<span class="_10 blank"> </span>ela classi\u2026<span class="_0 blank"></span>catória.</div><div class="t m0 x9 h4 y1b ff3 fs0 fc0 sc0 ls36 ws3">f<span class="ff8 fs1 ws28 v2">xx </span>f<span class="ff8 fs1 ws29 v2">xy </span>f<span class="ff8 fs1 ws27 v2">y y<span class="_11 blank"> </span></span><span class="ls2b">H</span><span class="ff6">(</span><span class="ls27">P<span class="ff6 ls2c">)</span></span><span class="ff2">natureza</span></div><div class="t m0 xa h4 y1c ff3 fs0 fc0 sc0 ls1a">P<span class="ff5 fs1 ls1b v2">1</span><span class="ff6 ls36 ws3">(0</span><span class="ls1c">;<span class="ff6 ls36 ws2a">0) 0<span class="_12 blank"> </span>1 0<span class="_13 blank"> </span><span class="ff7 ws3">\ue000</span><span class="ls2d">1</span><span class="ff2 ws3">sela</span></span></span></div><div class="t m0 xb h8 y1d ff3 fs0 fc0 sc0 ls1a">P<span class="ff5 fs1 ls4 v2">2</span><span class="ff6 ls36 ws3">(<span class="ff7 ls2e">\ue000</span><span class="ff5 fs1 v3">1</span></span></div><div class="t m0 xc h9 y1e ff5 fs1 fc0 sc0 ls2f">2<span class="ff3 fs0 ls1e v1">;<span class="ff6 ls36 ws2b">0) 0 <span class="ff7 ws3">\ue000</span><span class="ws2c">1 1<span class="_13 blank"> </span><span class="ff7 ws3">\ue000</span><span class="ls2d">1</span><span class="ff2 ws3">sela</span></span></span></span></div><div class="t m0 xa h4 y1f ff3 fs0 fc0 sc0 ls1a">P<span class="ff5 fs1 ls1b v2">3</span><span class="ff6 ls36 ws3">(0</span><span class="ls1c">;<span class="ff6 ls36 ws2d">1)<span class="_14 blank"> </span>4 <span class="ff7 ws3">\ue000</span><span class="ws2c">1 0<span class="_13 blank"> </span><span class="ff7 ws3">\ue000</span><span class="ls2d">1</span><span class="ff2 ws3">sela</span></span></span></span></div><div class="t m0 xb ha y20 ff3 fs0 fc0 sc0 ls1a">P<span class="ff5 fs1 ls4 v2">4</span><span class="ff6 ls36 ws3">(<span class="ff7 ls30">\ue000</span><span class="ff5 fs1 v3">1</span></span></div><div class="t m0 xc hb y21 ff5 fs1 fc0 sc0 ls2f">6<span class="ff3 fs0 ls31 v1">;</span><span class="ls36 v5">1</span></div><div class="t m0 xd hb y21 ff5 fs1 fc0 sc0 ls32">3<span class="ff6 fs0 ls33 v1">)</span><span class="ls36 v5">4</span></div><div class="t m0 xe hb y21 ff5 fs1 fc0 sc0 ls34">3<span class="ff7 fs0 ls2e v1">\ue000</span><span class="ls36 v5">1</span></div><div class="t m0 x7 hc y21 ff5 fs1 fc0 sc0 ls36">3</div><div class="t m0 xf hc y22 ff5 fs1 fc0 sc0 ls36">1</div><div class="t m0 xf hc y21 ff5 fs1 fc0 sc0 ls36">3</div><div class="t m0 x10 hc y22 ff5 fs1 fc0 sc0 ls36">1</div><div class="t m0 x10 h9 y21 ff5 fs1 fc0 sc0 ls35">3<span class="ff2 fs0 ls36 ws2 v1">min.<span class="_b blank"> </span>lo cal</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y23 w1 hd" alt="" src="https://files.passeidireto.com/fdd3c887-ff5c-440e-ac61-9d4e18aef435/bg2.png"><div class="t m0 x1 h4 y24 ff2 fs0 fc0 sc0 ls36 ws2f">(b) P<span class="_4 blank"></span>ara deduzir que <span class="ff3 ls37">f</span>não p<span class="_3 blank"> </span>ossui extremos absolutos, analisemos o comp<span class="_3 blank"> </span>ortamento de <span class="ff3 ls38">f</span>ao</div><div class="t m0 x0 h4 y25 ff2 fs0 fc0 sc0 ls36 ws1">longo da reta <span class="ff3 ls20">y<span class="ff6 ls23">=</span><span class="ls36 ws3">x</span></span><span class="ws30">. T<span class="_6 blank"></span>emos:</span></div><div class="t m0 x11 h4 y26 ff6 fs0 fc0 sc0 ls36">lim</div><div class="t m0 x12 he y27 ff8 fs1 fc0 sc0 ls39">x<span class="ffb ls36 ws31">!\ue0061 <span class="ff3 fs0 ls3 v6">f<span class="ff6 ls36 ws3">(<span class="ff3 ws32">x; x</span><span class="wsa">) =<span class="_15 blank"> </span>lim</span></span></span></span></div><div class="t m0 x13 hf y27 ff8 fs1 fc0 sc0 ls39">x<span class="ffb ls36 ws31">!\ue0061 <span class="ff9 fs0 ls3a v7">\ue000</span><span class="ff3 fs0 ws3 v6">x</span><span class="ff5 ls3b v8">3</span><span class="ff6 fs0 ls6 v6">+<span class="ff3 ls36 ws3">x</span></span><span class="ff5 ls9 v8">2</span><span class="ff9 fs0 ls3c v7">\ue001</span><span class="ff6 fs0 ws33 v6">= lim</span></span></div><div class="t m0 x14 h10 y27 ff8 fs1 fc0 sc0 ls39">x<span class="ffb ls36 ws31">!\ue0061 <span class="ff3 fs0 ws3 v6">x</span><span class="ff5 ls4 v8">3</span><span class="ff6 fs0 wse v6">(1 + 1<span class="ff3 ws3">=x<span class="ff6 ws34">) = </span><span class="ff7">\ue0061</span>:</span></span></span></div><div class="t m0 x0 h2 y28 ff1 fs0 fc0 sc0 ls36 ws35">02. <span class="ff2 ws36">Uma placa circular tem o formato da região <span class="ff3 ws3">x<span class="ff5 fs1 ls3d v1">2</span><span class="ff6 ls3e">+</span><span class="lse">y<span class="ff5 fs1 ls3f v1">2</span><span class="ff7 ls40">\ue014<span class="ff6 ls41">1</span></span></span></span>e é aquecida de tal mo<span class="_3 blank"> </span>do que a</span></div><div class="t m0 x0 h2 y29 ff2 fs0 fc0 sc0 ls36 ws2">temp eratura<span class="_16 blank"> </span>no<span class="_16 blank"> </span>p on<span class="_4 blank"></span>to<span class="_16 blank"> </span><span class="ff6 ws3">(<span class="ff3 ws37">x;<span class="_2 blank"> </span>y </span><span class="ls42">)</span></span><span class="ls43">é<span class="ff3 ls44">T</span></span><span class="ff6 ws3">(<span class="ff3 ws37">x;<span class="_2 blank"> </span>y </span><span class="ls45 ws38">)=2<span class="_17 blank"></span><span class="ff3 ls36 ws3">x<span class="ff5 fs1 ls46 v1">2</span><span class="ff6 ls47">+</span><span class="lse">y<span class="ff5 fs1 ls46 v1">2</span><span class="ff7 ls47">\ue000</span>y</span><span class="ff2 ws39">.<span class="_d blank"> </span>Determine as temperaturas nos p<span class="_3 blank"> </span>ontos</span></span></span></span></div><div class="t m0 x0 h4 y2a ff2 fs0 fc0 sc0 ls36 ws1">mais quen<span class="_4 blank"></span>tes e mais frios da placa.</div><div class="t m0 x0 h3 y2b ff1 fs0 fc0 sc0 ls36 ws3">Solução:</div><div class="t m0 x1 h4 y2c ff2 fs0 fc0 sc0 ls36 ws19">No in<span class="_4 blank"></span>terior da placa, a função temp<span class="_3 blank"> </span>eratura <span class="ff3 ls48">T</span><span class="ff6 ws3">(<span class="ff3 ws37">x;<span class="_2 blank"> </span>y </span><span class="ls49">)</span></span><span class="wsa">tem um único p<span class="_3 blank"> </span>onto crítico, determinado</span></div><div class="t m0 x0 h4 y2d ff2 fs0 fc0 sc0 ls36 ws2">p elo<span class="_1 blank"> </span>sistema:<span class="_b blank"> </span><span class="ff6 ls19">4<span class="ff3 ls10">x</span><span class="ls36 wsa">= 0<span class="_9 blank"> </span></span></span><span class="ls4a">e<span class="ff6 ls19">2<span class="ff3 ls4b">y<span class="ff7 ls4c">\ue000</span></span><span class="ls36 ws5">1 = 0</span></span></span><span class="ws3a">, isto é,<span class="_1 blank"> </span><span class="ff3 ls1a">P<span class="ff5 fs1 ls1b v2">1</span></span><span class="ff6 ws3">(0<span class="ff3 ls1c">;</span>1<span class="ff3 ls19">=</span><span class="ws17">2) <span class="ff3 ls4d">:</span></span></span><span class="ws3b">No b<span class="_3 blank"> </span>ordo da placa, a análise será feita via</span></span></div><div class="t m0 x0 h2 y2e ff2 fs0 fc0 sc0 ls36 ws3c">Multiplicadores de Lagrange, onde o vínculo é dado p<span class="_3 blank"> </span>or <span class="ff3 ls4e">g</span><span class="ff6 ws3">(<span class="ff3 ws4">x;<span class="_2 blank"> </span>y </span><span class="ws3d">) = </span><span class="ff3">x<span class="ff5 fs1 ls4f v1">2</span></span><span class="ls50">+<span class="ff3 lse">y<span class="ff5 fs1 ls51 v1">2</span></span><span class="ff7">\ue000</span></span><span class="ws3d">1 = 0</span></span><span class="ws3e">.<span class="_5 blank"> </span>A equação</span></div><div class="t m0 x0 h11 y2f ff2 fs0 fc0 sc0 ls36 ws3f">v<span class="_4 blank"></span>etorial <span class="ff7 ls52">r<span class="ff3 ls53">T<span class="ff6 ls6">+</span><span class="ls54">\ue015</span></span>r<span class="ff3 ls55">g<span class="ff6 ls56">=</span><span class="ls36 v9">~</span></span></span></div><div class="t m0 x15 h4 y2f ff6 fs0 fc0 sc0 ls57">0<span class="ff2 ls36 ws1">dá origem ao sistema algébrico:</span></div><div class="t m0 x13 h5 y30 ff9 fs0 fc0 sc0 ls36">8</div><div class="t m0 x13 h5 y31 ff9 fs0 fc0 sc0 ls36">></div><div class="t m0 x13 h5 y32 ff9 fs0 fc0 sc0 ls36">></div><div class="t m0 x13 h5 y33 ff9 fs0 fc0 sc0 ls36">></div><div class="t m0 x13 h5 y34 ff9 fs0 fc0 sc0 ls36"><</div><div class="t m0 x13 h5 y35 ff9 fs0 fc0 sc0 ls36">></div><div class="t m0 x13 h5 y36 ff9 fs0 fc0 sc0 ls36">></div><div class="t m0 x13 h5 y37 ff9 fs0 fc0 sc0 ls36">></div><div class="t m0 x13 h5 y38 ff9 fs0 fc0 sc0 ls36">:</div><div class="t m0 x16 h4 y39 ff6 fs0 fc0 sc0 ls36 ws3">4<span class="ff3 ls12">x</span><span class="wse">+ 2<span class="ff3 ws40">\ue015x </span><span class="wsa">= 0<span class="_18 blank"> </span><span class="ff2">(I)</span></span></span></div><div class="t m0 x16 h4 y3a ff6 fs0 fc0 sc0 ls36 ws3">2<span class="ff3 ls5">y<span class="ff7 ls6">\ue000<span class="ff6 ws41">1+2<span class="_19 blank"></span><span class="ff3 ls36 ws42">\ue015y <span class="ff6 wsa">= 0<span class="_8 blank"> </span><span class="ff2 ws2">(I I)</span></span></span></span></span></span></div><div class="t m0 x16 h2 y3b ff3 fs0 fc0 sc0 ls36 ws3">x<span class="ff5 fs1 ls3b v1">2</span><span class="ff6 ls6">+</span><span class="lse">y<span class="ff5 fs1 ls58 v1">2</span></span><span class="ff6 wsa">= 1<span class="_1a blank"> </span><span class="ff2 ws43">(I I I)</span></span></div><div class="t m0 x0 h4 y3c ff2 fs0 fc0 sc0 ls36 ws44">e de (I) deduzimos que <span class="ff3 ls59">x</span><span class="ff6 ws45">= 0 </span><span class="ws46">ou <span class="ff3 ls5a">\ue015<span class="ff6 ls5b">=</span></span><span class="ff7 ws3">\ue000<span class="ff6">2<span class="ff3 ls5c">:</span></span></span><span class="ws47">Se <span class="ff3 ls59">x</span><span class="ff6 ws45">= 0</span></span></span>,<span class="_b blank"> </span>obtemos de (I<span class="_3 blank"> </span>I<span class="_3 blank"> </span>I) <span class="ff3 ls5d">y<span class="ff6 ls5e">=</span></span><span class="ff7 ws3">\ue006<span class="ff6 ls5f">1</span></span>e selecionamos</div><div class="t m0 x0 h4 y3d ff2 fs0 fc0 sc0 ls36 ws2">os<span class="_b blank"> </span>pontos<span class="_1b blank"> </span><span class="ff3 ls1a">P<span class="ff5 fs1 ls1d v2">2</span></span><span class="ff6 ws3">(0<span class="ff3 ls1e">;</span><span class="ws48">1) </span></span><span class="ls60">e<span class="ff3 ls1a">P<span class="ff5 fs1 ls1b v2">3</span></span></span><span class="ff6 ws3">(0<span class="ff3 ls1c">;</span><span class="ff7">\ue000</span><span class="ws21">1) <span class="ff3 ls61">:</span></span></span><span class="ws49">Se <span class="ff3 ls62">\ue015<span class="ff6 ls63">=</span></span><span class="ff7 ws3">\ue000<span class="ff6">2<span class="ff3 ls61">;</span></span></span><span class="ws4a">lev<span class="_1c blank"></span>amos esse v<span class="_1c blank"></span>alor em (I<span class="_3 blank"> </span>I) e obtemos <span class="ff3 ls64">y<span class="ff6 ls63">=</span></span><span class="ff7 ws3">\ue000<span class="ff6 ls19">1</span><span class="ff3">=<span class="ff6 ls19">2</span></span></span>.</span></span></div><div class="t m0 x0 h12 y3e ff2 fs0 fc0 sc0 ls36 ws4b">Com esse v<span class="_1c blank"></span>alor de <span class="ff3 ls65">y</span><span class="ws2">obtemos<span class="_1 blank"> </span>de<span class="_16 blank"> </span>(I I I)<span class="_1 blank"> </span><span class="ff3 ls66">x<span class="ff6 ls67">=</span></span><span class="ff7 ws3">\ue006<span class="ls52 va">p</span><span class="ff6 ls19">3</span><span class="ff3">=<span class="ff6 ls68">2</span></span></span></span>e, assim,<span class="_16 blank"> </span>selecionamos os p<span class="_3 blank"> </span>ontos <span class="ff3 ls1a">P<span class="ff5 fs1 ls9 v2">4</span><span class="ff6 ls69">(<span class="ffb fs1 ls6a v8">p</span></span></span><span class="ff5 fs1 v3">3</span></div><div class="t m0 x17 h13 y3f ff5 fs1 fc0 sc0 ls6b">2<span class="ff3 fs0 ls1c v1">;<span class="ff7 ls2e">\ue000</span></span><span class="ls36 v5">1</span></div><div class="t m0 x18 h9 y3f ff5 fs1 fc0 sc0 ls2f">2<span class="ff6 fs0 ls36 v1">)</span></div><div class="t m0 x0 h14 y40 ff2 fs0 fc0 sc0 ls6c">e<span class="ff3 ls1a">P<span class="ff5 fs1 ls4 v2">5</span><span class="ff6 ls36 ws3">(<span class="ff7 ls30">\ue000<span class="ffb fs1 ls6a v8">p</span></span><span class="ff5 fs1 v3">3</span></span></span></div><div class="t m0 x19 h13 y41 ff5 fs1 fc0 sc0 ls6b">2<span class="ff3 fs0 ls1c v1">;<span class="ff7 ls2e">\ue000</span></span><span class="ls36 v5">1</span></div><div class="t m0 x1a h9 y41 ff5 fs1 fc0 sc0 ls2f">2<span class="ff6 fs0 ls36 ws3 v1">)<span class="ff2 ws1">.<span class="_b blank"> </span>Um c<span class="_3 blank"> </span>álculo direto nos lev<span class="_1c blank"></span>a a:</span></span></div><div class="t m0 x1b h4 y42 ff3 fs0 fc0 sc0 ls48">T<span class="ff6 ls36 ws3">(</span><span class="ls1a">P<span class="ff5 fs1 ls4 v2">1</span><span class="ff6 ls29">)</span></span>T<span class="ff6 ls36 ws3">(</span><span class="ls1a">P<span class="ff5 fs1 ls4 v2">2</span><span class="ff6 ls6d">)</span><span class="ls44">T<span class="ff6 ls36 ws3">(</span></span>P<span class="ff5 fs1 ls4 v2">3</span><span class="ff6 ls29">)</span></span>T<span class="ff6 ls36 ws3">(<span class="ff3">P<span class="ff5 fs1 ls9 v2">4</span></span><span class="ls6d">)</span></span><span class="ls44">T<span class="ff6 ls36 ws3">(</span><span class="ls1a">P<span class="ff5 fs1 ls4 v2">5</span><span class="ff6 ls36">)</span></span></span></div><div class="t m0 x1c h8 y43 ff7 fs0 fc0 sc0 ls30">\ue000<span class="ff5 fs1 ls36 v3">1</span></div><div class="t m0 x1d h13 y44 ff5 fs1 fc0 sc0 ls6e">4<span class="ff6 fs0 ls36 ws4c v1">0 2<span class="_1d blank"> </span></span><span class="ls36 v5">9</span></div><div class="t m0 x1e hc y44 ff5 fs1 fc0 sc0 ls36">4</div><div class="t m0 x1f hc y45 ff5 fs1 fc0 sc0 ls36">9</div><div class="t m0 x1f hc y44 ff5 fs1 fc0 sc0 ls36">4</div><div class="t m0 x0 h4 y46 ff2 fs0 fc0 sc0 ls36 wsd">Observ<span class="_1c blank"></span>ando a tab<span class="_3 blank"> </span>ela deduzimos que a temp<span class="_3 blank"> </span>eratura mais quente é 9/4 e ocorre nos p<span class="_3 blank"> </span>ontos <span class="ff3 ls1a">P<span class="ff5 fs1 ls6f v2">4</span></span>e</div><div class="t m0 x0 h4 y47 ff3 fs0 fc0 sc0 ls1a">P<span class="ff5 fs1 ls4 v2">5</span><span class="ff2 ls36 ws1">, enquan<span class="_4 blank"></span>to a temp<span class="_3 blank"> </span>eratura mais fria é <span class="ff7 ws3">\ue000<span class="ff6 ls19">1</span><span class="ff3">=<span class="ff6 ls57">4</span></span></span>e o<span class="_3 blank"> </span>corre no p<span class="_3 blank"> </span>onto in<span class="_1c blank"></span>terior <span class="ff3 ws3">P<span class="ff5 fs1 ls9 v2">1</span>:</span></span></div><div class="t m0 x0 h4 y48 ff1 fs0 fc0 sc0 ls36 ws0">03. <span class="ff2 ws2">Sup onha<span class="_1 blank"> </span>que<span class="_1 blank"> </span><span class="ff3 ls70">u</span><span class="ls6c">e<span class="ff3 ls71">v</span></span><span class="ws1">sejam funções de <span class="ff3 ls72">x</span><span class="ls6c">e<span class="ff3 ls73">y</span></span>de\u2026<span class="_0 blank"></span>nidas implicitamen<span class="_4 blank"></span>te p<span class="_3 blank"> </span>elo sistema:</span></span></div><div class="t m0 x20 h5 y49 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x20 h5 y4a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x20 h5 y4b ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x20 h5 y4c ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x20 h5 y4d ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x20 h5 y4e ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x21 h2 y4f ff3 fs0 fc0 sc0 ls74">u<span class="ff6 ls6">+</span><span class="ls75">v<span class="ff6 ls23">=</span><span class="ls36 ws3">x<span class="ff5 fs1 ls3b v1">2</span><span class="ff6 ls6">+</span><span class="lse">y</span><span class="ff5 fs1 v1">2</span></span></span></div><div class="t m0 x21 h4 y50 ff3 fs0 fc0 sc0 ls36 ws4d">uv <span class="ff6 ls23">=</span><span class="ff7 ws3">\ue000<span class="ff6 ls19">3</span></span>xy</div><div class="t m0 x0 h4 y51 ff2 fs0 fc0 sc0 ls36 ws4e">e que <span class="ff3 ls76">u</span><span class="ff6 ws3">(1<span class="ff3 ls1e">;</span><span class="ws5">1) = 3<span class="_1 blank"> </span></span></span><span class="ls77">e<span class="ff3 ls78">v</span></span><span class="ff6 ws3">(1<span class="ff3 ls1e">;</span><span class="ws4f">1) = </span><span class="ff7">\ue000</span>1</span><span class="ws1">.<span class="_a blank"> </span>Calcule as deriv<span class="_6 blank"></span>adas <span class="ff3 ls79">u<span class="ff8 fs1 ls7a v2">x</span></span><span class="ls6c">e</span><span class="ff3 ws3">v<span class="ff8 fs1 ls7a v2">x</span></span><span class="ws2">no<span class="_1 blank"> </span>p onto<span class="_1 blank"> </span><span class="ff3 ws3">P<span class="ff5 fs1 ls1d v2">0</span><span class="ff6">(1</span><span class="ls1e">;</span><span class="ff6 ws17">1) </span>:</span></span></span></div><div class="t m0 x22 h4 y52 ff2 fs0 fc0 sc0 ls36">2</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y53 w2 h15" alt="" src="https://files.passeidireto.com/fdd3c887-ff5c-440e-ac61-9d4e18aef435/bg3.png"><div class="t m0 x0 h3 y24 ff1 fs0 fc0 sc0 ls36 ws3">Solução:</div><div class="t m0 x1 h2 y25 ff2 fs0 fc0 sc0 ls36 ws50">Sejam <span class="ff3 ls7b">F</span><span class="ff6 ws3">(<span class="ff3 ws51">x; y<span class="_3 blank"> </span>; u; v<span class="_3 blank"> </span></span><span class="ws24">) = <span class="ff3 ls7c">u</span><span class="ls7d">+<span class="ff3 ls7e">v</span><span class="ff7">\ue000</span></span></span><span class="ff3">x<span class="ff5 fs1 ls7f v1">2</span><span class="ff7 ls7d">\ue000</span><span class="lse">y<span class="ff5 fs1 ls80 v1">2</span></span></span></span><span class="ls81">e<span class="ff3 ls82">G</span></span><span class="ff6 ws3">(<span class="ff3 ws51">x; y<span class="_10 blank"> </span>; u; v<span class="_3 blank"> </span></span><span class="ws34">) = <span class="ff3 ws52">uv </span><span class="ws53">+ 3<span class="ff3 ws7">xy </span></span></span></span><span class="ws54">,<span class="_9 blank"> </span>en<span class="_4 blank"></span>tão no p<span class="_3 blank"> </span>on<span class="_4 blank"></span>to <span class="ff3 ws3">Q<span class="ff5 fs1 ls4 v2">0</span><span class="ff6">(1</span><span class="ls1c">;</span><span class="ff6">1</span><span class="ls1c">;</span><span class="ff6">3</span><span class="ls1c">;</span><span class="ff7">\ue000<span class="ff6">1)</span></span></span>,</span></div><div class="t m0 x0 h4 y54 ff2 fs0 fc0 sc0 ls36 ws3">temos:</div><div class="t m0 x23 h16 y55 ff3 fs0 fc0 sc0 ls83">J<span class="ff6 ls84">=</span><span class="ls85 vb">@<span class="ff6 ls36 ws3">(<span class="ff3 ws2">F ;<span class="_2 blank"> </span>G</span>)</span></span></div><div class="t m0 x24 h17 y56 ff3 fs0 fc0 sc0 ls86">@<span class="ff6 ls36 ws3">(</span><span class="ls36 ws55">u;<span class="_2 blank"> </span>v <span class="ff6 ls87">)<span class="ls23 vb">=</span></span><span class="ff9 vc">\ue00c</span></span></div><div class="t m0 x25 h5 y57 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x25 h5 y58 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x25 h5 y59 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x25 h5 y5a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x25 h5 y5b ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x26 h4 y5c ff3 fs0 fc0 sc0 ls88">F<span class="ff8 fs1 ls89 v2">u</span>F<span class="ff8 fs1 ls36 v2">v</span></div><div class="t m0 x26 h4 y5d ff3 fs0 fc0 sc0 ls8a">G<span class="ff8 fs1 ls8b v2">u</span>G<span class="ff8 fs1 ls36 v2">v</span></div><div class="t m0 x27 h5 y5e ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x27 h5 y57 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x27 h5 y58 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x27 h5 y59 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x27 h5 y5a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x27 h5 y5b ff9 fs0 fc0 sc0 ls8c">\ue00c<span class="ff5 fs1 ls8d vd">(<span class="ff8 ls36 ws56">Q<span class="ffc fs2 ls8e ve">0</span><span class="ff5">)</span></span></span></div><div class="t m0 x28 h7 y55 ff6 fs0 fc0 sc0 ls23">=<span class="ff9 ls36 v4">\ue00c</span></div><div class="t m0 xf h5 y57 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 xf h5 y58 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 xf h5 y59 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 xf h5 y5a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 xf h5 y5b ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x29 h4 y5c ff6 fs0 fc0 sc0 ls36 ws57">1 1</div><div class="t m0 x29 h4 y5d ff3 fs0 fc0 sc0 ls36 ws58">v u</div><div class="t m0 x1e h5 y5e ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x1e h5 y57 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x1e h5 y58 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x1e h5 y59 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x1e h5 y5a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x1e h5 y5b ff9 fs0 fc0 sc0 ls8c">\ue00c<span class="ff5 fs1 ls8d vd">(<span class="ff8 ls36 ws56">Q<span class="ffc fs2 ls8f ve">0</span><span class="ff5">)</span></span></span></div><div class="t m0 x2a h4 y55 ff6 fs0 fc0 sc0 ls23">=<span class="ff3 ls74">u<span class="ff7 ls90">\ue000</span><span class="ls91">v</span></span><span class="ls36 wse">=<span class="_9 blank"> </span>3 + 1<span class="_9 blank"> </span>=<span class="_f blank"> </span>4</span></div><div class="t m0 x0 h4 y5f ff2 fs0 fc0 sc0 ls36 ws39">e, de fato, o sistema de\u2026<span class="_0 blank"></span>ne <span class="ff3 ls92">u</span><span class="ls93">e<span class="ff3 ls94">v</span></span>como funções de <span class="ff3 ls95">x</span><span class="ls96">e<span class="ff3 lse">y</span></span>, em uma vizinhança de <span class="ff3 ls1a">P<span class="ff5 fs1 ls4 v2">0</span></span>.<span class="_d blank"> </span>O cálculo</div><div class="t m0 x0 h4 y60 ff2 fs0 fc0 sc0 ls36 ws59">das deriv<span class="_1c blank"></span>adas <span class="ff3 ls79">u<span class="ff8 fs1 ls97 v2">x</span></span><span class="ls98">e</span><span class="ff3 ws3">v<span class="ff8 fs1 ls99 v2">x</span></span><span class="ws5a">p<span class="_3 blank"> </span>o<span class="_3 blank"> </span>de ser feito utilizando regras de deriv<span class="_1c blank"></span>ação ou diretamente por deriv<span class="_1c blank"></span>ação</span></div><div class="t m0 x0 h4 y61 ff2 fs0 fc0 sc0 ls36 ws8">implícita.<span class="_b blank"> </span>Aqui usaremos as regras de deriv<span class="_1c blank"></span>ação.<span class="_a blank"> </span>T<span class="_6 blank"></span>emos:</div><div class="t m0 x2b h16 y62 ff3 fs0 fc0 sc0 ls79">u<span class="ff8 fs1 lsb v2">x</span><span class="ff6 ls18">=<span class="ff7 ls9a">\ue000</span><span class="ls36 vb">1</span></span></div><div class="t m0 x2c h4 y63 ff3 fs0 fc0 sc0 ls36">J</div><div class="t m0 x2d h4 y64 ff3 fs0 fc0 sc0 ls86">@<span class="ff6 ls36 ws3">(</span><span class="ls36 ws2">F ;<span class="_2 blank"> </span>G<span class="ff6">)</span></span></div><div class="t m0 x2e h18 y63 ff3 fs0 fc0 sc0 ls85">@<span class="ff6 ls36 ws3">(</span><span class="ls36 ws5b">x;<span class="_2 blank"> </span>v <span class="ff6 ls9b">)<span class="ls36 ws3 vb">(<span class="ff3">Q<span class="ff5 fs1 ls4 v2">0</span><span class="ff6 ws24">) = <span class="ff7 ls2e">\ue000</span><span class="vb">1</span></span></span></span></span></span></div><div class="t m0 x2f h4 y63 ff6 fs0 fc0 sc0 ls36">4</div><div class="t m0 x30 h5 y65 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x30 h5 y66 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x30 h5 y67 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x30 h5 y68 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x30 h5 y69 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x30 h5 y6a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x31 h4 y6b ff3 fs0 fc0 sc0 ls88">F<span class="ff8 fs1 ls9c v2">x</span><span class="ff6 ls36">1</span></div><div class="t m0 x31 h4 y6c ff3 fs0 fc0 sc0 ls8a">G<span class="ff8 fs1 ls9d v2">x</span><span class="ls36">u</span></div><div class="t m0 x32 h5 y65 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x32 h5 y66 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x32 h5 y67 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x32 h5 y68 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x32 h5 y69 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x32 h5 y6a ff9 fs0 fc0 sc0 ls8c">\ue00c<span class="ff5 fs1 ls8d vd">(<span class="ff8 ls36 ws56">Q<span class="ffc fs2 ls8f ve">0</span><span class="ff5">)</span></span></span></div><div class="t m0 x33 h19 y62 ff6 fs0 fc0 sc0 ls23">=<span class="ff7 ls30">\ue000</span><span class="ls36 vb">1</span></div><div class="t m0 x34 h4 y63 ff6 fs0 fc0 sc0 ls36">4</div><div class="t m0 x35 h5 y65 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x35 h5 y66 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x35 h5 y67 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x35 h5 y68 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x35 h5 y69 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x35 h5 y6a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x36 h4 y6b ff7 fs0 fc0 sc0 ls36 ws3">\ue000<span class="ff6 ws5c">2 1</span></div><div class="t m0 x37 h4 y6c ff6 fs0 fc0 sc0 ls36 ws5d">3 3</div><div class="t m0 x38 h5 y65 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y66 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y67 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y68 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y69 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y6a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x39 h16 y62 ff6 fs0 fc0 sc0 ls84">=<span class="ls36 vb">9</span></div><div class="t m0 x3a h4 y63 ff6 fs0 fc0 sc0 ls36">4</div><div class="t m0 x0 h4 y6d ff2 fs0 fc0 sc0 ls36">e</div><div class="t m0 x4 h16 y6e ff3 fs0 fc0 sc0 ls36 ws3">v<span class="ff8 fs1 lsb v2">x</span><span class="ff6 ls23">=<span class="ff7 ls9e">\ue000</span><span class="ls36 vb">1</span></span></div><div class="t m0 x3b h4 y6f ff3 fs0 fc0 sc0 ls36">J</div><div class="t m0 x2c h4 y70 ff3 fs0 fc0 sc0 ls85">@<span class="ff6 ls36 ws3">(</span><span class="ls36 ws2">F ;<span class="_2 blank"> </span>G<span class="ff6">)</span></span></div><div class="t m0 x3c h18 y6f ff3 fs0 fc0 sc0 ls85">@<span class="ff6 ls36 ws3">(</span><span class="ls36 ws51">u; x<span class="ff6 ls9f">)<span class="ls36 ws3 vb">(<span class="ff3">Q<span class="ff5 fs1 ls4 v2">0</span><span class="ff6 ws24">) = <span class="ff7 ls30">\ue000</span><span class="vb">1</span></span></span></span></span></span></div><div class="t m0 x3d h4 y6f ff6 fs0 fc0 sc0 ls36">4</div><div class="t m0 x2f h5 y71 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x2f h5 y72 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x2f h5 y73 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x2f h5 y74 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x2f h5 y75 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x2f h5 y76 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x30 h4 y77 ff6 fs0 fc0 sc0 lsa0">1<span class="ff3 lsa1">F<span class="ff8 fs1 ls36 v2">x</span></span></div><div class="t m0 x30 h4 y78 ff3 fs0 fc0 sc0 ls36 ws58">v G<span class="ff8 fs1 v2">x</span></div><div class="t m0 x3e h5 y71 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x3e h5 y72 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x3e h5 y73 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x3e h5 y74 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x3e h5 y75 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x3e h5 y76 ff9 fs0 fc0 sc0 lsa2">\ue00c<span class="ff5 fs1 lsa3 vd">(<span class="ff8 ls36 ws56">Q<span class="ffc fs2 ls8e ve">0</span><span class="ff5">)</span></span></span></div><div class="t m0 x3f h16 y6e ff6 fs0 fc0 sc0 ls23">=<span class="ff7 ls2e">\ue000</span><span class="ls36 vb">1</span></div><div class="t m0 x8 h4 y6f ff6 fs0 fc0 sc0 ls36">4</div><div class="t m0 x34 h5 y71 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x34 h5 y72 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x34 h5 y73 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x34 h5 y74 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x34 h5 y75 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x34 h5 y76 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x40 h4 y77 ff6 fs0 fc0 sc0 lsa4">1<span class="ff7 ls36 ws3">\ue000<span class="ff6">2</span></span></div><div class="t m0 x35 h4 y78 ff7 fs0 fc0 sc0 ls36 ws3">\ue000<span class="ff6 ws5d">1 3</span></div><div class="t m0 x38 h5 y71 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y72 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y73 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y74 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y75 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x38 h5 y76 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x39 h16 y6e ff6 fs0 fc0 sc0 ls23">=<span class="ff7 ls2e">\ue000</span><span class="ls36 vb">1</span></div><div class="t m0 x41 h4 y6f ff6 fs0 fc0 sc0 ls36">4</div><div class="t m0 x0 h4 y79 ff1 fs0 fc0 sc0 ls36 ws0">04. <span class="ff2 ws1">Considere a transformação <span class="ff3 ls48">T</span><span class="ff6 ws3">(<span class="ff3 ws37">x;<span class="_2 blank"> </span>y </span><span class="ws5">) = (<span class="ff3 ls12">x<span class="ff7 ls6">\ue000</span><span class="ls36 ws5e">y ;<span class="_2 blank"> </span>x<span class="_e blank"> </span></span></span><span class="ls6">+<span class="ff3 lse">y</span></span></span>)</span>.</span></div><div class="t m0 x1 h4 y7a ff2 fs0 fc0 sc0 ls36 ws1">(a) Calcule o jacobiano <span class="ff3 lsa5">J</span>da transformação e de sua inv<span class="_1c blank"></span>ersa;</div><div class="t m0 x1 h4 y7b ff2 fs0 fc0 sc0 ls36 ws5f">(b) Seja a região <span class="ff3 lsa6">R</span><span class="ff8 fs1 ws60 v2">xy </span><span class="ff6 lsa7">:</span><span class="ff7 ws3">j<span class="ff3">x</span><span class="lsa8">j<span class="ff6 lsa9">+</span></span>j<span class="ff3 lse">y</span><span class="ws61">j \ue014 </span><span class="ff6">2</span></span>.<span class="_a blank"> </span>Determine a imagem <span class="ff3 lsaa">R</span><span class="ff8 fs1 ws62 v2">uv </span><span class="ws63">da região <span class="ff3 lsa6">R</span><span class="ff8 fs1 ws64 v2">xy </span><span class="ws2">p ela<span class="_16 blank"> </span>transfor-</span></span></div><div class="t m0 x0 h4 y7c ff2 fs0 fc0 sc0 ls36 ws65">mação <span class="ff3 lsab">T</span><span class="ws1">e calcule a razão en<span class="_4 blank"></span>tre as áreas de <span class="ff3 lsa6">R</span><span class="ff8 fs1 ws66 v2">xy </span><span class="ls77">e<span class="ff3 lsaa">R</span></span><span class="ff8 fs1 ws67 v2">uv </span><span class="ff3">:</span></span></div><div class="t m0 x0 h3 y7d ff1 fs0 fc0 sc0 ls36 ws3">Solução:</div><div class="t m0 x1 h1a y7e ff2 fs0 fc0 sc0 ls36 ws1">(a) Consideremos <span class="ff6 ws3">(<span class="ff3 ws68">u;<span class="_2 blank"> </span>v </span><span class="ws24">) = <span class="ff3 ls48">T</span></span>(<span class="ff3 ws37">x;<span class="_2 blank"> </span>y </span><span class="lsac">)<span class="ff3 lsd">;</span></span></span><span class="ws2">de<span class="_1 blank"> </span>mo do<span class="_1 blank"> </span>que<span class="_1 blank"> </span><span class="ff9 v4">8</span></span></div><div class="t m0 x42 h5 y7f ff9 fs0 fc0 sc0 ls36"><</div><div class="t m0 x42 h5 y80 ff9 fs0 fc0 sc0 ls36">:</div><div class="t m0 x1e h4 y81 ff3 fs0 fc0 sc0 lsad">u<span class="ff6 ls18">=</span><span class="ls12">x<span class="ff7 ls6">\ue000</span><span class="ls36">y</span></span></div><div class="t m0 x1e h4 y82 ff3 fs0 fc0 sc0 ls75">v<span class="ff6 ls23">=</span><span class="ls12">x<span class="ff6 ls6">+</span><span class="ls36 ws5e">y :</span></span></div><div class="t m0 x0 h4 y83 ff2 fs0 fc0 sc0 ls36 ws3">T<span class="_6 blank"></span>emos:</div><div class="t m0 x2e h16 y84 ff3 fs0 fc0 sc0 ls83">J<span class="ff6 lsae">=</span><span class="ls85 vb">@<span class="ff6 ls36 ws3">(<span class="ff3 ws68">u;<span class="_2 blank"> </span>v </span>)</span></span></div><div class="t m0 x25 h17 y85 ff3 fs0 fc0 sc0 ls85">@<span class="ff6 ls36 ws3">(</span><span class="ls36 ws4">x;<span class="_2 blank"> </span>y <span class="ff6 lsaf">)<span class="ls23 vb">=</span></span><span class="ff9 vc">\ue00c</span></span></div><div class="t m0 x43 h5 y86 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x43 h5 y87 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x43 h5 y88 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x43 h5 y89 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x43 h5 y8a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x44 h4 y8b ff3 fs0 fc0 sc0 ls79">u<span class="ff8 fs1 lsb0 v2">x</span>u<span class="ff8 fs1 ls36 v2">y</span></div><div class="t m0 x44 h4 y8c ff3 fs0 fc0 sc0 ls36 ws3">v<span class="ff8 fs1 lsb1 v2">x</span>v<span class="ff8 fs1 v2">y</span></div><div class="t m0 x45 h5 y8d ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x45 h5 y86 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x45 h5 y87 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x45 h5 y88 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x45 h5 y89 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x45 h5 y8a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x3e h7 y84 ff6 fs0 fc0 sc0 ls23">=<span class="ff9 ls36 v4">\ue00c</span></div><div class="t m0 x46 h5 y8e ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x46 h5 y87 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x46 h5 y8f ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x46 h5 y89 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x46 h5 y8a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x47 h4 y8b ff6 fs0 fc0 sc0 lsb2">1<span class="ff7 ls36 ws3">\ue000<span class="ff6">1</span></span></div><div class="t m0 x47 h4 y8c ff6 fs0 fc0 sc0 ls36 ws69">1 1</div><div class="t m0 x48 h5 y8d ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x48 h5 y8e ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x48 h5 y87 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x48 h5 y8f ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x48 h5 y89 ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x48 h5 y8a ff9 fs0 fc0 sc0 ls36">\ue00c</div><div class="t m0 x49 h4 y84 ff6 fs0 fc0 sc0 ls36 wsa">= 2<span class="ff3">:</span></div><div class="t m0 x0 h1b y90 ff2 fs0 fc0 sc0 ls36 ws15">Usando a relação <span class="ff3 lsb3">J<span class="ff9 ls3a va">\ue000</span><span class="lsb4">T<span class="ffb fs1 lsb5 v1">\ue000<span class="ff5 ls4">1</span></span><span class="ff9 ls3c va">\ue001</span></span></span><span class="ff6 wsa">= 1<span class="ff3 ws6a">=J; </span></span>encon<span class="_1c blank"></span>tramos o jacobiano da transformação inv<span class="_1c blank"></span>ersa <span class="ff3 lsb6">J<span class="ff9 ls3a va">\ue000</span><span class="lsb4">T<span class="ffb fs1 lsb5 v1">\ue000<span class="ff5 ls4">1</span></span><span class="ff9 ls3c va">\ue001</span></span></span><span class="ff6">=</span></div><div class="t m0 x0 h4 y91 ff6 fs0 fc0 sc0 ls19">1<span class="ff3 ls36 ws3">=</span>2<span class="ff3 ls36">:</span></div><div class="t m0 x1 h1c y92 ff2 fs0 fc0 sc0 ls36 ws6b">(b) A teoria nos ensina que uma aplicação linear <span class="ff3 lsb7">T<span class="ff6 lsb8">:<span class="ff4 lsa">R<span class="ff5 fs1 lsb9 v1">2</span><span class="ff7 lsba">!</span>R<span class="ff5 fs1 lsbb v1">2</span></span></span></span>transforma retas em retas e</div><div class="t m0 x0 h4 y93 ff2 fs0 fc0 sc0 ls36 ws1">que o jacobiano é a <span class="ffa ws6c">r<span class="_1c blank"></span>azão <span class="ff2 ws1">entre as áreas.<span class="_b blank"> </span>Assim,</span></span></div><div class="t m0 x4a h4 y94 ff3 fs0 fc0 sc0 lsbc">A<span class="ff6 ls36 ws3">(</span><span class="lsa6">R<span class="ff8 fs1 ls36 ws67 v2">uv </span><span class="ff6 ls36 ws24">) = </span><span class="lsbd">J<span class="ff7 ls6">\ue002</span></span></span>A<span class="ff6 ls36 ws3">(</span><span class="lsa6">R<span class="ff8 fs1 ls36 ws6d v2">xy </span><span class="ff6 ls36 ws5">) = 2</span></span>A<span class="ff6 ls36 ws3">(</span><span class="lsa6">R<span class="ff8 fs1 ls36 ws6d v2">xy </span><span class="ff6 lsbe">)</span><span class="ls36">:</span></span></div><div class="t m0 x22 h4 y52 ff2 fs0 fc0 sc0 ls36">3</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><div class="t m0 x22 h4 y52 ff2 fs0 fc0 sc0 ls36">4</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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