<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/4010a717-9d5a-4246-bb9a-5ef73ba40b70/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls4a ws0">D E R IV<span class="_0 blank"></span>A Ç ÃO<span class="_1 blank"> </span>P<span class="_0 blank"></span>A RC I A L<span class="_2 blank"> </span>&<span class="_2 blank"> </span>D I F ER E N C I A B IL ID A D E</div><div class="t m0 x2 h2 y2 ff1 fs0 fc1 sc0 ls4a ws0">C A L C U L A N D O<span class="_2 blank"> </span>D E R IV<span class="_0 blank"></span>A DA S</div><div class="t m0 x3 h3 y3 ff2 fs1 fc0 sc0 ls4a ws1">1.<span class="_3 blank"> </span>Em cada caso, calcule as deriv<span class="_0 blank"></span>adas <span class="ff3 ls0">z<span class="ff4 fs0 ls1 v1">x</span><span class="ls4a ws2">; z<span class="ff4 fs0 ls2 v1">y</span>; z<span class="ff4 fs0 ws3 v1">xx </span>; z<span class="ff4 fs0 ws4 v1">yy<span class="_4 blank"> </span></span></span></span><span class="ls3">e<span class="ff3 ls0">z</span></span><span class="ff4 fs0 ws4 v1">y x </span><span class="ff3">:</span></div><div class="t m0 x4 h4 y4 ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls4">z</span><span class="ff5 ws6">= 3<span class="ff3 ws7">x<span class="ff6 fs0 ls5 v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 ls8 v2">3</span></span></span></span><span class="ws8">(b) <span class="ff3 ls4">z</span><span class="ff5 ws9">=<span class="_5 blank"> </span>arctg (<span class="ff3 wsa">y =x</span><span class="ls9">)</span></span><span class="wsb">(c) <span class="ff3 lsa">z<span class="ff5 lsb">=</span><span class="ls4a wsc">xy <span class="ff5 wsd">exp <span class="ff7 lsc v3">\ue000</span></span><span class="ws7">x<span class="ff6 fs0 lsd v2">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lse v2">2</span></span><span class="ff7 v3">\ue001</span></span></span></span></span></span></div><div class="t m0 x4 h5 y5 ff1 fs1 fc0 sc0 ls4a ws8">(d) <span class="ff3 ls4">z</span><span class="ff5 wse">=<span class="_5 blank"> </span>sen (<span class="ff3 wsf">xy </span><span class="ws10">) + log<span class="_6 blank"> </span><span class="ff7 lsc v3">\ue000</span><span class="ff3 ws7">x<span class="ff6 fs0 lse v2">2</span><span class="ls7">y<span class="ff7 lsf v3">\ue001</span></span></span></span></span><span class="ws11">(e) <span class="ff3 lsa">z<span class="ff5 lsb">=</span></span><span class="ff7 ws12 v4">p</span><span class="ff3 ws7 v0">x<span class="ff6 fs0 ls5 v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lsd v5">2</span></span><span class="ff5 ws10">+ 1<span class="_4 blank"> </span><span class="ff1 ws13">(f )<span class="_2 blank"> </span></span></span><span class="lsa">z</span><span class="ff5 wse">=<span class="_5 blank"> </span>arccos (</span><span class="wsf">xy <span class="ff5">)</span></span></span></span></div><div class="t m0 x3 h3 y6 ff2 fs1 fc0 sc0 ls4a ws14">2.<span class="_3 blank"> </span>Em cada caso, calcule a deriv<span class="_0 blank"></span>ada indicada da função <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls10">f</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span>)</span>.</div><div class="t m0 x4 h6 y7 ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls11">x</span></span><span class="ff5 ws16">arcsen (<span class="ff3 ls12">x<span class="ff8 ls6">\ue000</span><span class="ls7">y</span></span><span class="wse">) ;<span class="_7 blank"> </span><span class="ff3 ws7">f<span class="ff4 fs0 ls13 v1">x</span></span><span class="ws12">(1<span class="ff3 ls14">;</span>1<span class="ff3 ls15">=</span><span class="ws17">2) </span></span></span></span><span class="ws8">(b) <span class="ff3 ls4">z</span><span class="ff5 ws16">=<span class="_5 blank"> </span>exp (<span class="ff3 ws18">xy </span><span class="wse">) sec (<span class="ff3 ws19">x=y </span>) ;<span class="_7 blank"> </span><span class="ff3 ws7">f<span class="ff4 fs0 ls16 v1">y</span></span><span class="ws12">(0<span class="ff3 ls14">;</span>1)</span></span></span></span></div><div class="t m0 x4 h7 y8 ff1 fs1 fc0 sc0 ls4a wsb">(c) <span class="ff3 lsa">z<span class="ff5 lsb">=</span></span><span class="ff7 ws12 v4">p</span><span class="ff3 ws7">x<span class="ff6 fs0 ls5 v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls17 v5">2</span><span class="ff5 ls18">;</span></span>f<span class="ff4 fs0 ws1a v1">xy </span><span class="ff5 ws12">(1</span><span class="ls14">;</span><span class="ff5 ws1b">0) <span class="ff2 ls19">e</span></span>f<span class="ff4 fs0 ws4 v1">y x<span class="_8 blank"> </span></span><span class="ff5 ws12">(1</span><span class="ls1a">;</span><span class="ff5 ws1c">0) </span></span><span class="ws8">(d) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls4a wsc">xy <span class="ff5 ws16">ln (</span><span class="ws1d">x=y <span class="ff5 wse">) ;<span class="_7 blank"> </span></span><span class="ws7">f<span class="ff4 fs0 ls16 v1">y</span><span class="ff5 ws12">(1</span><span class="ls1a">;</span><span class="ff5">1)</span></span></span></span></span></span></div><div class="t m0 x3 h8 y9 ff2 fs1 fc0 sc0 ls4a ws14">3.<span class="_3 blank"> </span>Considere a função <span class="ff3 ls1b">'<span class="ff5 ls1c">:<span class="ff9 ls1d">R<span class="ff6 fs0 ls1e v2">2</span><span class="ff8 ls1f">!</span><span class="ls20">R</span></span></span></span><span class="ws1e">de\u2026<span class="_9 blank"></span>nida<span class="_2 blank"> </span>p or:</span></div><div class="t m0 x5 h9 ya ff3 fs1 fc0 sc0 ls21">'<span class="ff5 ls4a ws12">(</span><span class="ls4a ws15">x;<span class="_6 blank"> </span>y<span class="ff5 ws1f">) = <span class="ff7 v6">8</span></span></span></div><div class="t m0 x6 ha yb ff7 fs1 fc0 sc0 ls4a">></div><div class="t m0 x6 ha yc ff7 fs1 fc0 sc0 ls4a"><</div><div class="t m0 x6 ha yd ff7 fs1 fc0 sc0 ls4a">></div><div class="t m0 x6 ha ye ff7 fs1 fc0 sc0 ls4a">:</div><div class="t m0 x7 hb yf ff5 fs1 fc0 sc0 ls4a ws20">exp( <span class="ff8 ws12 v7">\ue000</span><span class="v7">1</span></div><div class="t m0 x8 hc y10 ff3 fs1 fc0 sc0 ls4a ws7">x<span class="ff6 fs0 ls5 v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls22 v5">2</span></span><span class="ff5 ws12 v7">)</span><span class="ls23 v7">;</span><span class="ff2 ws21 v7">se <span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ls24">)</span><span class="ff8">6</span><span class="ws6">= (0<span class="ff3 ls1a">;</span>0)</span></span></span></div><div class="t m0 x7 h3 y11 ff5 fs1 fc0 sc0 ls15">0<span class="ff3 ls23">;<span class="ff2 ls4a ws22">se </span></span><span class="ls4a ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ws23">) = (0<span class="ff3 ls14">;</span><span class="ws24">0) <span class="ff3">:</span></span></span></span></div><div class="t m0 x4 h3 y12 ff2 fs1 fc0 sc0 ls4a ws14">Calcule, caso existam, as deriv<span class="_0 blank"></span>adas parciais <span class="ff3 ls25">'<span class="ff4 fs0 ls13 v8">x</span></span><span class="ff5 ws12">(0<span class="ff3 ls14">;</span><span class="ws24">0) <span class="ff3 ws2">; '<span class="ff4 fs0 ls16 v8">y</span></span></span>(0<span class="ff3 ls1a">;</span><span class="ws25">0) <span class="ff3">:</span></span></span></div><div class="t m0 x3 hb y13 ff2 fs1 fc0 sc0 ls4a ws26">4.<span class="_3 blank"> </span>Se <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ws1f">) = <span class="ff3 ws7">x<span class="ff6 fs0 ls5 v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 lse v2">3</span></span></span></span></span><span class="ws14">, as expressõe<span class="_a blank"> </span>s<span class="_b blank"> </span><span class="ff3 ws27 v7">@ f</span></span></div><div class="t m0 x9 hd y14 ff3 fs1 fc0 sc0 ls4a ws27">@ x<span class="_5 blank"> </span><span class="ff7 lsc v9">\ue000</span><span class="ws7 v7">x</span><span class="ff6 fs0 lsd va">2</span><span class="ff5 ls6 v7">+</span><span class="ls7 v7">y</span><span class="ff6 fs0 lse va">2</span><span class="ws28 v7">;<span class="_6 blank"> </span>y </span><span class="ff7 ls27 v9">\ue001</span><span class="ff2 ls28 v7">e</span><span class="vb">@</span></div><div class="t m0 xa hd y14 ff3 fs1 fc0 sc0 ls4a ws27">@ x<span class="_5 blank"> </span><span class="ff7 ws12 v9">\ue002</span><span class="ls26 v7">f</span><span class="ff7 lsc v9">\ue000</span><span class="ws7 v7">x</span><span class="ff6 fs0 lsd va">2</span><span class="ff5 ls6 v7">+</span><span class="ls7 v7">y</span><span class="ff6 fs0 lse va">2</span><span class="ws28 v7">;<span class="_6 blank"> </span>y</span><span class="ff7 ws29 v9">\ue001\ue003 </span><span class="ff2 ws1 v7">são iguais ou não?</span></div><div class="t m0 x3 he y15 ff2 fs1 fc0 sc0 ls4a ws14">5.<span class="_3 blank"> </span>Mostre que a função <span class="ff3 ls4">z<span class="ff5 ls29">=</span><span class="ls4a wsf v7">xy </span></span><span class="ff6 fs0 va">2</span></div><div class="t m0 xb hc y16 ff3 fs1 fc0 sc0 ls4a ws7">x<span class="ff6 fs0 ls5 v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls2a v5">2</span></span><span class="ff2 ws14 v7">satisfaz à equação diferencial </span><span class="v7">xz</span><span class="ff4 fs0 ls2b vc">x</span><span class="ff5 ls6 v7">+</span><span class="wsa v7">y z</span><span class="ff4 fs0 ls2c vc">y</span><span class="ff5 lsb v7">=</span><span class="ws2a v7">z :</span></div><div class="t m0 x3 hb y17 ff2 fs1 fc0 sc0 ls4a ws2b">6.<span class="_3 blank"> </span>V<span class="_c blank"></span>eri\u2026<span class="_9 blank"></span>que que a função <span class="ff3 ls2d">u</span><span class="ff5 ws12">(<span class="ff3 ws2c">x; t</span><span class="ws2d">) =<span class="_d blank"> </span><span class="v7">1</span></span></span></div><div class="t m0 xc hf y18 ff8 fs1 fc0 sc0 ls2e">p<span class="ff3 ls2f vd">t</span><span class="ff5 ls4a wsd ve">exp </span><span class="ff7 ls30 vb">\ue012</span><span class="ls31 ve">\ue000</span><span class="ff3 ls4a ws7 vf">x<span class="ff6 fs0 v2">2</span></span></div><div class="t m0 xd h10 y19 ff5 fs1 fc0 sc0 ls4a ws12">4<span class="ff3 ws2e">k t<span class="_e blank"> </span><span class="ff7 ls30 v10">\ue013</span><span class="ff2 ls32 v7">,</span><span class="ws2f v7">t > </span></span><span class="ls33 v7">0<span class="ff2 ls34">e<span class="ff3 ls35">k</span><span class="ls4a ws2b">uma constan<span class="_0 blank"></span>te não nula, satisfaz a</span></span></span></div><div class="t m0 x4 h3 y1a ffa fs1 fc0 sc0 ls4a ws30">e<span class="_0 blank"></span>quação de tr<span class="_0 blank"></span>ansmissão de c<span class="_c blank"></span>alor<span class="_b blank"> </span><span class="ff3 ls36">u<span class="ff4 fs0 ls37 v1">t</span><span class="ff8 ls6">\ue000</span><span class="ls4a ws2e">k u<span class="ff4 fs0 ws31 v1">xx </span><span class="ff5 ws6">= 0</span>:</span></span></div><div class="t m0 x3 h8 y1b ff2 fs1 fc0 sc0 ls4a ws32">7.<span class="_3 blank"> </span>No espaço <span class="ff9 ls1d">R<span class="ff6 fs0 lse v2">2</span></span><span class="ws33">, o <span class="ffa ws34">op<span class="_0 blank"></span>er<span class="_0 blank"></span>ador de L<span class="_0 blank"></span>aplac<span class="_0 blank"></span>e<span class="_3 blank"> </span><span class="ff5 ls2e">\ue001<span class="ff3 ls38">;</span></span><span class="ff2 ws1e">é<span class="_f blank"> </span>de\u2026<span class="_9 blank"></span>nido<span class="_f blank"> </span>p or:<span class="_10 blank"> </span><span class="ff5 ws35">\ue001 = <span class="ff3 ws7">@<span class="ff4 fs0 ws36 v1">xx </span></span><span class="ls39">+</span><span class="ff3 ws7">@<span class="ff4 fs0 ws4 v1">y y<span class="_11 blank"> </span></span></span></span><span class="ws37">.<span class="_12 blank"> </span>V<span class="_c blank"></span>eri\u2026<span class="_9 blank"></span>que que as funções</span></span></span></span></div><div class="t m0 x4 h11 y1c ff3 fs1 fc0 sc0 ls2d">u<span class="ff5 ls4a ws12">(</span><span class="ls4a ws15">x;<span class="_6 blank"> </span>y<span class="ff5 ws16">)<span class="_5 blank"> </span>=<span class="_5 blank"> </span>arctan (</span><span class="wsa">y =x<span class="ff5 ls3a">)<span class="ff2 ls19">e</span></span></span></span>u<span class="ff5 ls4a ws12">(</span><span class="ls4a ws15">x;<span class="_6 blank"> </span>y <span class="ff5 ws1f">) = </span><span class="ws7">e<span class="ff4 fs0 ls13 v2">x</span><span class="ff5 ws38">cos </span><span class="ls3b">y</span><span class="ff2 ws39">satisfazem a <span class="ffa ws3a">Equação de L<span class="_0 blank"></span>aplac<span class="_0 blank"></span>e<span class="_f blank"> </span><span class="ff5 ls2e">\ue001<span class="ff3 ls3c">u</span><span class="ls4a ws6">= 0<span class="ff3">:</span></span></span></span></span></span></span></div><div class="t m0 x3 h8 y1d ff2 fs1 fc0 sc0 ls4a ws23">8.<span class="_3 blank"> </span>Sob que condições a função <span class="ff3 ls2d">u</span><span class="ff5 ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y </span><span class="ws3c">) = <span class="ff3 ws7">Ax<span class="ff6 fs0 ls3d v2">2</span></span><span class="ls3e">+</span><span class="ff3 ws3d">B xy<span class="_13 blank"> </span></span><span class="ls3f">+</span><span class="ff3 ws3e">C<span class="_11 blank"> </span>y <span class="ff6 fs0 ls3d v2">2</span></span><span class="ls3f">+</span><span class="ff3 ws3f">D x<span class="_e blank"> </span></span><span class="ls3e">+</span><span class="ff3 ws40">E y<span class="_13 blank"> </span></span><span class="ls3e">+<span class="ff3 ls40">F</span></span></span></span>satisfaz à equação de Laplace?</div><div class="t m0 x3 h12 y1e ff2 fs1 fc0 sc0 ls4a ws41">9.<span class="_3 blank"> </span>Se <span class="ff3 ls2d">u</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ls41">)</span></span><span class="ls42">e<span class="ff3 ls43">v</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ls41">)</span></span><span class="ws42">são funções com deriv<span class="_0 blank"></span>adas parciais con<span class="_0 blank"></span>tínuas até a 2<span class="fs0 ls44 v2">a</span>ordem e satisfazem às</span></div><div class="t m0 x4 h3 y1f ff2 fs1 fc0 sc0 ls4a ws43">equações <span class="ff3 ls36">u<span class="ff4 fs0 ls45 v1">x</span><span class="ff5 lsb">=</span><span class="ls4a ws7">v<span class="ff4 fs0 ls46 v1">y</span></span></span><span class="ls47">e<span class="ff3 ls36">u<span class="ff4 fs0 ls2c v1">y</span><span class="ff5 lsb">=</span></span></span><span class="ff8 ws12">\ue000<span class="ff3 ws7">v<span class="ff4 fs0 ls1 v1">x</span></span></span><span class="ws14">, mostre que <span class="ff3 ls48">u</span><span class="ls19">e<span class="ff3 ls49">v</span></span>atendem à equação de Laplace.</span></div><div class="t m0 x2 h2 y20 ff1 fs0 fc1 sc0 ls4a ws0">R E G R A<span class="_2 blank"> </span>DA<span class="_1 blank"> </span>C A D E IA</div><div class="t m0 x2 h3 y21 ff2 fs1 fc0 sc0 ls4a ws14">Admita que to<span class="_a blank"> </span>das as deriv<span class="_0 blank"></span>adas en<span class="_0 blank"></span>volvidas existam e sejam con<span class="_0 blank"></span>tínuas no domínio da função.</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 y22 w2 h13" alt="" src="https://files.passeidireto.com/4010a717-9d5a-4246-bb9a-5ef73ba40b70/bg2.png"><div class="t m0 x3 h3 y23 ff2 fs1 fc0 sc0 ls4a ws26">1.<span class="_3 blank"> </span>Se <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ws16">)<span class="_5 blank"> </span>=<span class="_5 blank"> </span>sen (<span class="_a blank"> </span><span class="ff3 ws45">x=y </span><span class="ws10">) + ln<span class="_6 blank"> </span>(<span class="ff3 wsa">y=x</span></span></span>)</span><span class="ws14">, mostre que <span class="ff3 ws7">xf<span class="ff4 fs0 ls4b v1">x</span><span class="ff5 ls6">+</span><span class="wsa">y f<span class="ff4 fs0 ls2c v1">y</span><span class="ff5 ws6">= 0</span>:</span></span></span></div><div class="t m0 x3 h3 y24 ff2 fs1 fc0 sc0 ls4a ws46">2.<span class="_3 blank"> </span>Se <span class="ff3 ls4c">f<span class="ff5 ls1c">:<span class="ff9 ls4d">R<span class="ff8 ls1f">!</span><span class="ls4e">R</span></span></span></span><span class="ws47">é deriv<span class="_0 blank"></span>áv<span class="_0 blank"></span>el, mostre que as funções <span class="ff3 ls4f">'</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ws1f">) = <span class="ff3 ls26">f</span></span>(<span class="ff3 ls12">x<span class="ff8 ls50">\ue000</span><span class="ls51">y</span></span><span class="ls52">)</span></span><span class="ls53">e<span class="ff3 ls54 ws44"> </span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ws1f">) = <span class="ff3 ls26">f</span></span>(<span class="ff3 wsf">xy </span><span class="ls55">)</span></span>satisf<span class="_a blank"> </span>azem às</span></div><div class="t m0 x4 h3 y25 ff2 fs1 fc0 sc0 ls4a ws48">relações: <span class="ff3 ls56">'<span class="ff4 fs0 ls4b v8">x</span><span class="ff5 ls6">+</span><span class="ls25">'<span class="ff4 fs0 ls2c v8">y</span></span></span><span class="ff5 ws6">= 0<span class="_2 blank"> </span></span><span class="ls19">e</span><span class="ff3 ws44">x <span class="_a blank"> </span><span class="ff4 fs0 ls4b v8">x</span><span class="ff8 ls6">\ue000</span>y<span class="_11 blank"> </span> <span class="ff4 fs0 ls2c v8">y</span><span class="ff5 ws6">= 0</span>:</span></div><div class="t m0 x3 h14 y26 ff2 fs1 fc0 sc0 ls4a ws49">3.<span class="_3 blank"> </span>Calcule <span class="ff3 ls57">z<span class="ffb fs0 ls58 v2">0</span></span><span class="ff5 ws12">(<span class="ff3 ls59">t</span><span class="ls5a">)<span class="ff3 ls5b">;</span></span></span><span class="ws1">nos seguin<span class="_0 blank"></span>tes casos:</span></div><div class="t m0 x4 h15 y27 ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls4a wsa">y e<span class="ff4 fs0 ls4b v2">x</span><span class="ff5 ls6">+</span><span class="ws7">xe<span class="ff4 fs0 ls5c v2">y</span><span class="ff5 ls5d">;</span><span class="ls5e">x<span class="ff5 lsb">=</span><span class="ls5f">t<span class="ff2 ls19">e</span><span class="ls60">y</span></span></span><span class="ff5 ws4a">=<span class="_5 blank"> </span>sen </span><span class="ls61">t</span></span></span></span><span class="ws4b">(b) <span class="ff3 lsa">z</span><span class="ff5 ws24">=<span class="_5 blank"> </span>ln <span class="ff7 lsc v3">\ue000</span><span class="ws10">1 + <span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 lse v2">2</span><span class="ff7 ls62 v3">\ue001</span></span><span class="ls5d">;<span class="ff3 ls5e">x</span></span></span><span class="ws25">=<span class="_5 blank"> </span>ln <span class="ff3 ls63">t<span class="ff2 ls19">e</span><span class="ls60">y</span></span><span class="lsb">=</span><span class="ff3 ws7">e<span class="ff4 fs0 v2">t</span></span></span></span></span></span></div><div class="t m0 x4 h7 y28 ff1 fs1 fc0 sc0 ls4a wsb">(c) <span class="ff3 lsa">z<span class="ff5 lsb">=</span></span><span class="ff7 ws12 v4">p</span><span class="ff3 ws7">x<span class="ff6 fs0 ls5 v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls17 v5">2</span><span class="ff5 ls18">;</span><span class="ls64">x<span class="ff5 lsb">=</span><span class="ls59">t<span class="ff6 fs0 ls65 v2">3</span><span class="ff2 ls19">e</span><span class="ls60">y</span></span></span></span><span class="ff5 ws4c">=<span class="_5 blank"> </span>cos </span><span class="ls66">t</span></span><span class="ws4b">(d) <span class="ff3 lsa">z<span class="ff5 lsb">=</span><span class="ls36">u<span class="ff6 fs0 lse v2">2</span><span class="ls67">v<span class="ff5 ls6">+</span><span class="ls4a ws4d">v w <span class="ff6 fs0 lsd v2">2</span><span class="ff5 ls6">+</span><span class="ws4e">uv w <span class="ff6 fs0 lse v2">3</span><span class="ff5 ls5d">;</span><span class="ls68">u<span class="ff5 lsb">=</span><span class="ls69">t<span class="ff6 fs0 lse v2">2</span></span></span><span class="ws4f">;<span class="_3 blank"> </span>v <span class="ff5 lsb">=</span><span class="ls5f">t<span class="ff2 ls47">e</span><span class="ls6a">w<span class="ff5 lsb">=</span><span class="ls69">t</span></span></span><span class="ff6 fs0 v2">3</span></span></span></span></span></span></span></span></div><div class="t m0 x3 hb y29 ff2 fs1 fc0 sc0 ls4a ws50">4.<span class="_3 blank"> </span>Calcule <span class="ff3 ws27 v7">@ w</span></div><div class="t m0 xe h16 y2a ff3 fs1 fc0 sc0 ls4a ws27">@ x<span class="_14 blank"> </span><span class="ff2 ls6b v7">e</span><span class="vb">@ w</span></div><div class="t m0 xf hc y2a ff3 fs1 fc0 sc0 ls4a ws27">@ y<span class="_5 blank"> </span><span class="ls5b v7">;</span><span class="ff2 ws1 v7">nos seguin<span class="_0 blank"></span>tes casos:</span></div><div class="t m0 x4 h15 y2b ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls6a">w<span class="ff5 lsb">=</span><span class="ls36">u<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ls6">+</span><span class="ls6c">v<span class="ff6 fs0 lse v2">3</span><span class="ff5 ls5d">;</span><span class="ls68">u</span></span></span></span><span class="ff5 ws6">= 3<span class="ff3 ls12">x<span class="ff8 ls6">\ue000</span><span class="ls3b">y<span class="ff2 ls19">e</span><span class="ls6d">v</span></span></span><span class="lsb">=<span class="ff3 ls12">x</span></span><span class="ws10">+ 2<span class="ff3 ls6e">y</span></span></span><span class="ws4b">(b) <span class="ff3 ls6a">w</span><span class="ff5 ws24">=<span class="_5 blank"> </span>ln <span class="ff7 lsc v3">\ue000</span><span class="ff3 ls69">t<span class="ff6 fs0 ls5 v2">2</span></span><span class="ls6">+</span><span class="ff3 ws7">s<span class="ff6 fs0 lse v2">2</span><span class="ff7 ls62 v3">\ue001</span></span><span class="ls5d">;<span class="ff3 ls6f">t</span><span class="lsb">=</span></span><span class="ff3 ws7">x<span class="ff6 fs0 ls5 v2">3</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 ls65 v2">2</span><span class="ff2 ls19">e</span><span class="ls70">s</span></span></span><span class="ws6">= 3<span class="ff3">xy</span></span></span></span></div><div class="t m0 x4 h17 y2c ff1 fs1 fc0 sc0 ls4a wsb">(c) <span class="ff3 ls6a">w</span><span class="ff5 ws6">= 3<span class="ff3 ls71">u</span><span class="ws10">+ 7<span class="ff3 ls72">v</span><span class="ls5d">;<span class="ff3 ls68">u</span><span class="lsb">=</span></span><span class="ff3 ws7">x<span class="ff6 fs0 lse v2">2</span><span class="ls73">y<span class="ff2 ls47">e</span><span class="ls6d">v</span></span></span><span class="lsb">=<span class="ff8 ls2e v11">p</span></span><span class="ff3 ws51">xy </span></span></span><span class="ws4b">(d) <span class="ff3 ls6a">w</span><span class="ff5 wse">=<span class="_5 blank"> </span>cos (<span class="ff3 ls74">\ue018</span><span class="ls6">+<span class="ff3 ls75">\ue011</span></span>) ;<span class="_7 blank"> </span><span class="ff3 ls76">\ue018</span><span class="lsb">=<span class="ff3 ls12">x</span><span class="ls6">+<span class="ff3 ls73">y<span class="ff2 ls47">e</span><span class="ls77">\ue011</span></span></span>=<span class="ff8 ls2e v11">p</span></span><span class="ff3">xy</span></span></span></div><div class="t m0 x3 h18 y2d ff2 fs1 fc0 sc0 ls4a ws52">5.<span class="_3 blank"> </span>Considere a função <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y </span><span class="ws53">) = <span class="ff7 ls78 vb">Z</span><span class="ff4 fs0 v12">y</span></span></span></div><div class="t m0 x10 h19 y2e ff4 fs0 fc0 sc0 ls4a">x</div><div class="t m0 x11 h15 y2d ff5 fs1 fc0 sc0 ls4a wsd">exp <span class="ff7 lsc v3">\ue000</span><span class="ff3 ls69">t<span class="ff6 fs0 lse v2">2</span><span class="ff7 ls62 v3">\ue001</span><span class="ls4a ws7">dt<span class="ff2 ws54">.<span class="_14 blank"> </span>Calcule as deriv<span class="_0 blank"></span>adas parciais <span class="ff3 ws7">f<span class="ff4 fs0 ls79 v1">s</span><span class="ws55">; f<span class="ff4 fs0 ls7a v1">r</span></span></span><span class="ls7b">e</span><span class="ff3 ws7">f<span class="ff4 fs0 ws56 v1">rs<span class="_11 blank"> </span></span></span><span class="ws52">, no caso em</span></span></span></span></div><div class="t m0 x4 h8 y2f ff2 fs1 fc0 sc0 ls4a ws57">que <span class="ff3 ls5e">x<span class="ff5 lsb">=</span><span class="ls4a ws3f">r s<span class="ff6 fs0 ls7c v2">4</span></span></span><span class="ls19">e<span class="ff3 ls60">y<span class="ff5 lsb">=</span><span class="ls7d">r<span class="ff6 fs0 lse v2">4</span><span class="ls4a">s:</span></span></span></span></div><div class="t m0 x3 h3 y30 ff2 fs1 fc0 sc0 ls4a ws58">6.<span class="_3 blank"> </span>Sejam <span class="ff3 ws59">~<span class="_15 blank"></span>r <span class="ff5 ls7e">=</span>x</span></div><div class="t m0 x12 h3 y31 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x13 h3 y30 ff3 fs1 fc0 sc0 ls7f">i<span class="ff5 ls80">+</span><span class="ls4a">y</span></div><div class="t m0 x14 h3 y31 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x15 h3 y30 ff3 fs1 fc0 sc0 ls81">j<span class="ff2 ls4a ws54">o v<span class="_0 blank"></span>etor p<span class="_16 blank"> </span>osição do p<span class="_a blank"> </span>onto <span class="ff3 ls82">P</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_13 blank"> </span>y </span><span class="ls83">)</span></span><span class="ls84">e<span class="ff3 ls85">r<span class="ff5 ls7e">=</span></span></span><span class="ff8 ws12">k<span class="ff3 ws5a">~<span class="_17 blank"></span>r <span class="ff8 ls86">k</span><span class="ls87">:</span><span class="ff2 ws54">Dada uma função </span><span class="ls88">f<span class="ff5 ls89">:<span class="ff9 ls4d">R<span class="ff8 ls1f">!</span><span class="ls1d">R</span></span></span><span class="ls8a">;</span></span><span class="ff2 ws12">duas</span></span></span></span></div><div class="t m0 x4 h3 y32 ff2 fs1 fc0 sc0 ls4a ws14">v<span class="_0 blank"></span>êzes deriv<span class="_0 blank"></span>ável, represen<span class="_0 blank"></span>te p<span class="_a blank"> </span>or <span class="ff3 ls8b">g</span><span class="ws5b">a função <span class="ff3 ls8c">g</span><span class="ff5 ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y </span><span class="ws3c">) = <span class="ff3 ls10">f</span></span>(<span class="ff3 ls7d">r</span><span class="ls3a">)</span></span><span class="ws1">e mostre que</span></span></div><div class="t m0 x16 h1a y33 ff5 fs1 fc0 sc0 ls2e">\ue001<span class="ff3 ls8d">g</span><span class="lsb">=<span class="ff3 ls8e">g<span class="ff4 fs0 ls4a ws56 v1">rr<span class="_2 blank"> </span></span></span><span class="ls8f">+<span class="ff6 fs0 ls4a v13">1</span></span></span></div><div class="t m0 x17 h1b y34 ff4 fs0 fc0 sc0 ls90">r<span class="ff3 fs1 ls4a ws7 v2">g</span><span class="ls91 v14">r</span><span class="ff3 fs1 ls4a v2">:</span></div><div class="t m0 x3 h3 y35 ff2 fs1 fc0 sc0 ls4a ws26">7.<span class="_3 blank"> </span>Se <span class="ff3 ls5e">x<span class="ff5 lsb">=</span><span class="ls92">r</span></span><span class="ff5 ws38">cos <span class="ff3 ls93">\ue012</span></span><span class="ls19">e<span class="ff3 ls60">y<span class="ff5 lsb">=</span><span class="ls92">r</span></span></span><span class="ff5 ws5c">sen <span class="ff3 ls93">\ue012</span></span><span class="ls19">e<span class="ff3 ls2d">u</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ls3a">)</span></span><span class="ls19">e<span class="ff3 ls94">v</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ls3a">)</span></span><span class="ws1">têm deriv<span class="_0 blank"></span>adas parciais con<span class="_0 blank"></span>tínuas, mostre que:</span></div><div class="t m0 x18 h3 y36 ff3 fs1 fc0 sc0 ls4a ws27">@ u</div><div class="t m0 x18 h16 y37 ff3 fs1 fc0 sc0 ls4a ws27">@ r<span class="_b blank"> </span><span class="ff5 ls95 v7">=</span><span class="ff5 vb">1</span></div><div class="t m0 x19 h3 y37 ff3 fs1 fc0 sc0 ls4a">r</div><div class="t m0 x1a h3 y36 ff3 fs1 fc0 sc0 ls4a ws5d">@ v</div><div class="t m0 x1a h16 y37 ff3 fs1 fc0 sc0 ls4a ws27">@ \ue012<span class="_7 blank"> </span><span class="ff2 ls96 v7">e</span><span class="vb">@ v</span></div><div class="t m0 x1b h16 y37 ff3 fs1 fc0 sc0 ls4a ws27">@ r<span class="_b blank"> </span><span class="ff5 ls97 v7">=<span class="ff8 ls98">\ue000</span></span><span class="ff5 vb">1</span></div><div class="t m0 x1c h3 y37 ff3 fs1 fc0 sc0 ls4a">r</div><div class="t m0 x1d h3 y36 ff3 fs1 fc0 sc0 ls4a ws5d">@ u</div><div class="t m0 x1d hc y37 ff3 fs1 fc0 sc0 ls4a ws27">@ \ue012<span class="_6 blank"> </span><span class="v7">:</span></div><div class="t m0 x3 h3 y38 ff2 fs1 fc0 sc0 ls4a ws26">8.<span class="_3 blank"> </span>Se <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls10">f</span></span><span class="ff5 ws12">(<span class="ff3 ls12">x<span class="ff8 ls6">\ue000</span><span class="ls4a wsa">y ;<span class="_6 blank"> </span>y<span class="_18 blank"> </span><span class="ff8 ls6">\ue000</span><span class="ws7">x</span></span></span>)</span><span class="ws14">, use a Regra da Cadeia e mostre que <span class="ff3 ls99">z<span class="ff4 fs0 ls4b v1">x</span><span class="ff5 ls6">+</span><span class="ls0">z<span class="ff4 fs0 ls2c v1">y</span></span></span><span class="ff5 ws6">= 0<span class="ff3">:</span></span></span></div><div class="t m0 x3 h1c y39 ff2 fs1 fc0 sc0 ls4a ws1e">9.<span class="_3 blank"> </span>Sup onha<span class="_2 blank"> </span>que<span class="_2 blank"> </span><span class="ff3 ls21">'</span><span class="ff5 ws12">(<span class="ff3 ls69">t</span><span class="ls3a">)</span></span><span class="ws14">seja deriv<span class="_0 blank"></span>áv<span class="_0 blank"></span>el, com <span class="ff3 ls25">'<span class="ffb fs0 ls9a v2">0</span></span><span class="ff5 ws23">(1) = 4</span><span class="ws5e">.<span class="_b blank"> </span>Se <span class="ff3 ls8c">g</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ws1f">) = <span class="ff3 ls4f">'</span></span>(<span class="ff3 ws19">x=y </span>)</span></span>, calcule <span class="ff3 ws7">g<span class="ff4 fs0 ls9b v1">x</span><span class="ff5 ws12">(1</span><span class="ls1a">;</span><span class="ff5 ws1b">1) </span></span><span class="ls19">e</span><span class="ff3 ws7">g<span class="ff4 fs0 ls16 v1">y</span><span class="ff5 ws12">(1</span><span class="ls14">;</span><span class="ff5 ws24">1) </span>:</span></span></div><div class="t m0 x2 h2 y3a ff1 fs0 fc1 sc0 ls4a ws5f">F U<span class="_16 blank"> </span>N<span class="_16 blank"> </span>Ç<span class="_a blank"> </span>Õ<span class="_16 blank"> </span>E<span class="_16 blank"> </span>S<span class="_2 blank"> </span>D I F<span class="_16 blank"> </span>E<span class="_a blank"> </span>R<span class="_16 blank"> </span>E<span class="_a blank"> </span>N<span class="_16 blank"> </span>C<span class="_16 blank"> </span>I Á<span class="_0 blank"></span>V E<span class="_16 blank"> </span>IS</div><div class="t m0 x3 h3 y3b ff2 fs1 fc0 sc0 ls4a ws14">1.<span class="_3 blank"> </span>Use o Lema F<span class="_c blank"></span>undamen<span class="_0 blank"></span>tal e mostre que a f<span class="_a blank"> </span>unção <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls10">f</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ls3a">)</span></span>é dife<span class="_a blank"> </span>renciáv<span class="_0 blank"></span>el no domínio indicado.</div><div class="t m0 x4 h4 y3c ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls4a ws7">x<span class="ff6 fs0 ls17 v2">2</span><span class="ls51">y<span class="ff6 fs0 ls17 v2">4</span><span class="ff5 ls5d">;</span><span class="ls9c">D<span class="ff5 lsb">=<span class="ff9 ls1d">R<span class="ff6 fs0 ls9d v2">2</span></span></span></span></span></span></span><span class="ws8">(b) <span class="ff3 ls4">z</span><span class="ff5 ws24">=<span class="_5 blank"> </span>ln <span class="ff7 lsc v3">\ue000</span><span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 lse v2">2</span><span class="ff7 ls62 v3">\ue001</span></span><span class="ls5d">;<span class="ff3 ls9c">D</span><span class="lsb">=<span class="ff9 ls1d">R<span class="ff6 fs0 lse v2">2</span><span class="ff8 ls9e">n</span></span></span></span></span><span class="ws12">(0<span class="ff3 ls1a">;</span><span class="ws25">0) <span class="ff3">:</span></span></span></span></span></div><div class="t m0 x3 h1d y3d ff2 fs1 fc0 sc0 ls4a ws60">2.<span class="_3 blank"> </span>Seja <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls26">f</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ls3a">)</span></span><span class="ws14">a f<span class="_a blank"> </span>unção de\u2026<span class="_9 blank"></span>nida em <span class="ff9 ls1d">R<span class="ff6 fs0 ls7c v2">2</span></span><span class="ws1e">p or:</span></span></div><div class="t m0 x1e h9 y3e ff3 fs1 fc0 sc0 ls26">f<span class="ff5 ls4a ws12">(</span><span class="ls4a ws15">x;<span class="_6 blank"> </span>y<span class="ff5 ws1f">) = <span class="ff7 v6">8</span></span></span></div><div class="t m0 x1f ha y3f ff7 fs1 fc0 sc0 ls4a">></div><div class="t m0 x1f ha y40 ff7 fs1 fc0 sc0 ls4a"><</div><div class="t m0 x1f ha y41 ff7 fs1 fc0 sc0 ls4a">></div><div class="t m0 x1f ha y42 ff7 fs1 fc0 sc0 ls4a">:</div><div class="t m0 x20 h8 y43 ff5 fs1 fc0 sc0 ls4a ws12">3<span class="ff3 ws7">x<span class="ff6 fs0 ls17 v2">2</span>y</span></div><div class="t m0 x21 hc y44 ff3 fs1 fc0 sc0 ls4a ws7">x<span class="ff6 fs0 lsd v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls22 v5">2</span></span><span class="ff2 ws61 v7">, se<span class="_7 blank"> </span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ls24">)</span><span class="ff8">6</span><span class="ws6">= (0<span class="ff3 ls1a">;</span>0)</span></span></span></div><div class="t m0 x22 h3 y45 ff5 fs1 fc0 sc0 ls4a ws12">0<span class="ff2 ws62">, se<span class="_7 blank"> </span></span>(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ws23">) = (0<span class="ff3 ls1a">;</span>0)</span></div><div class="t m0 x23 h3 y3e ff3 fs1 fc0 sc0 ls4a">:</div><div class="t m0 x4 h3 y21 ff2 fs1 fc0 sc0 ls4a ws1">Pro<span class="_0 blank"></span>ve que:</div><div class="t m0 x24 h3 y46 ff2 fs1 fc0 sc0 ls4a">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 y47 w3 h1e" alt="" src="https://files.passeidireto.com/4010a717-9d5a-4246-bb9a-5ef73ba40b70/bg3.png"><div class="t m0 x4 h6 y23 ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff2 ws64">A função <span class="ff3 ls9f">f</span><span class="ws14">é con<span class="_0 blank"></span>tínua na origem.</span></span></div><div class="t m0 x4 h6 y48 ff1 fs1 fc0 sc0 ls4a ws65">(b) <span class="ff2 ws14">As deriv<span class="_0 blank"></span>adas parciais <span class="ff3 ws7">f<span class="ff4 fs0 lsa0 v1">x</span></span><span class="ls19">e</span><span class="ff3 ws7">f<span class="ff4 fs0 lsa1 v1">y</span></span><span class="ws1e">existem<span class="_2 blank"> </span>em<span class="_2 blank"> </span>to do<span class="_2 blank"> </span><span class="ff9 lsa2">R<span class="ff6 fs0 ls17 v2">2</span></span></span>, mas não são con<span class="_0 blank"></span>tínuas em <span class="ff5 ws12">(0<span class="ff3 ls1a">;</span><span class="ws24">0) <span class="ff3">:</span></span></span></span></div><div class="t m0 x4 h6 y49 ff1 fs1 fc0 sc0 ls4a ws66">(c) <span class="ff2 ws64">A função <span class="ff3 lsa3">f</span><span class="ws14">não é diferenciáv<span class="_0 blank"></span>el n<span class="_a blank"> </span>a origem.<span class="_b blank"> </span>P<span class="_0 blank"></span>or que isso não contradiz o Lema F<span class="_c blank"></span>undamen<span class="_0 blank"></span>tal?</span></span></div><div class="t m0 x3 h3 y4a ff2 fs1 fc0 sc0 ls4a ws14">3.<span class="_3 blank"> </span>Estude a diferenciabilidade da função <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls26">f</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ls5a">)<span class="ff3 lsa4">;</span></span></span><span class="ws1e">no<span class="_2 blank"> </span>p on<span class="_0 blank"></span>to<span class="_2 blank"> </span><span class="ff3 lsa5">P<span class="ff6 fs0 ls65 v1">0</span></span><span class="ws12">indicado:</span></span></div><div class="t m0 x4 h1f y4b ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls11">x</span></span><span class="ff5 ws16">exp (<span class="ff8 ws12">\ue000<span class="ff3 ls7">y</span></span><span class="wse">) ;<span class="_7 blank"> </span><span class="ff3 lsa5">P<span class="ff6 fs0 lsa6 v1">0</span></span><span class="ws6">= (1<span class="ff3 ls14">;</span><span class="ws1c">0) </span></span></span></span><span class="ws8">(b) <span class="ff3 ls4">z<span class="ff5 lsb">=</span></span><span class="ff7 v15">\ue00c</span></span></div><div class="t m0 x25 h20 y4c ff7 fs1 fc0 sc0 lsa7">\ue00c<span class="ff3 ls4a wsf v8">xy </span><span class="ff6 fs0 ls17 v16">2</span><span class="ls4a v17">\ue00c</span></div><div class="t m0 x26 ha y4c ff7 fs1 fc0 sc0 lsa8">\ue00c<span class="ff5 ls5d v8">;<span class="ff3 lsa5">P<span class="ff6 fs0 lsa6 v1">0</span><span class="ff5 ls4a ws6">= (0</span><span class="ls14">;<span class="ff5 ls4a">1)</span></span></span></span></div><div class="t m0 x4 h21 y4d ff1 fs1 fc0 sc0 ls4a wsb">(c) <span class="ff3 lsa">z<span class="ff5 lsb">=</span></span><span class="ff7 ws12 v15">p</span><span class="ff8 ws12">j<span class="ff3 ls7">y</span><span class="ls14">j</span><span class="ff5 ws38">cos <span class="ff3 ws7">x</span><span class="ls5d">;</span><span class="ff3 ws7">P<span class="ff6 fs0 ls1e v1">0</span></span><span class="ws6">= (0<span class="ff3 ls14">;</span><span class="ws67">0) </span></span></span></span><span class="ws8">(d) <span class="ff3 ls4">z<span class="ff5 lsb">=</span></span><span class="ff7 ws12 v15">p</span><span class="ff8 ws12">j<span class="ff3 wsf">xy </span>j<span class="ff5 ls18">;<span class="ff3 lsa5">P<span class="ff6 fs0 ls1e v1">0</span></span><span class="ls4a ws6">= (0<span class="ff3 ls1a">;</span>0)</span></span></span></span></div><div class="t m0 x3 h3 y4e ff2 fs1 fc0 sc0 ls4a ws14">4.<span class="_3 blank"> </span>Calcule a diferencial das funções seguintes:</div><div class="t m0 x4 h6 y4f ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="ws23">) = 5<span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">3</span></span><span class="ws10">+ 4<span class="ff3 ws7">x<span class="ff6 fs0 lse v2">2</span><span class="lsa9">y<span class="ff8 ls6">\ue000</span></span></span></span></span>2<span class="ff3 ls7">y<span class="ff6 fs0 lsaa v2">3</span></span></span><span class="ws8">(b) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y ;<span class="_6 blank"> </span>z </span><span class="ws1f">) = <span class="ff3 ws7">e<span class="ff4 fs0 lsab v2">x</span><span class="wsa">y z<span class="_19 blank"> </span><span class="ff2 ws68">(c) </span><span class="ls26">f</span></span></span></span>(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y</span><span class="ws16">)<span class="_5 blank"> </span>=<span class="_5 blank"> </span>arctan (<span class="ff3 wsa">y =x</span>)</span></span></span></div><div class="t m0 x3 h3 y50 ff2 fs1 fc0 sc0 ls4a ws69">5.<span class="_3 blank"> </span>Certa função diferenciáv<span class="_0 blank"></span>el <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls26">f</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="lsac">)</span></span>satisfaz às condições:<span class="_b blank"> </span><span class="ff3 ls10">f</span><span class="ff5 ws12">(1<span class="ff3 ls1a">;</span><span class="ws23">2) = 3<span class="ff3 ws6a">; f<span class="ff4 fs0 ls9b v1">x</span></span></span>(1<span class="ff3 ls1a">;</span><span class="ws23">2) = 5<span class="_2 blank"> </span></span></span><span class="lsad">e</span><span class="ff3 ws7">f<span class="ff4 fs0 ls16 v1">y</span><span class="ff5 ws12">(1</span><span class="ls14">;</span><span class="ff5 ws23">2) = 8</span></span>.</div><div class="t m0 x4 h3 y51 ff2 fs1 fc0 sc0 ls4a ws14">Calcule os v<span class="_0 blank"></span>alores apro<span class="_0 blank"></span>ximados de <span class="ff3 ls26">f</span><span class="ff5 ws12">(1<span class="ff3 ws7">:</span>1<span class="ff3 ls14">;</span>1<span class="ff3 ws7">:</span><span class="ws6b">8) </span></span><span class="ls19">e<span class="ff3 ls26">f</span></span><span class="ff5 ws12">(1<span class="ff3 ws7">:</span><span class="ls15">3<span class="ff3 ls1a">;</span>1</span><span class="ff3 ws7">:</span><span class="ws24">8) <span class="ff3">:</span></span></span></div><div class="t m0 x3 h3 y52 ff2 fs1 fc0 sc0 ls4a ws47">6.<span class="_3 blank"> </span>As dimensões de uma caixa retangular são <span class="ff5 ls15">5</span><span class="ff3 ws6c">m; <span class="ff5 ws12">6</span><span class="lsae">m</span></span><span class="lsaf">e</span><span class="ff5 ws12">8<span class="ff3 ws7">m</span></span>, com p<span class="_a blank"> </span>ossível de <span class="ff5 ws12">0<span class="ff3 ws7">:</span>01<span class="ff3 lsb0">m</span></span>em cada dimensão.</div><div class="t m0 x4 h3 y53 ff2 fs1 fc0 sc0 ls4a ws14">Calcule o v<span class="_0 blank"></span>alor apro<span class="_0 blank"></span>ximado do volume da caixa e o possível erro.</div><div class="t m0 x3 h3 y54 ff2 fs1 fc0 sc0 ls4a ws14">7.<span class="_3 blank"> </span>Use a diferencial e calcule, com duas casas decimais, o v<span class="_0 blank"></span>alor de:<span class="_b blank"> </span><span class="ff5 ws16">sen [1<span class="ff3 ws7">:</span><span class="wse">99 ln (1<span class="ff3 ws7">:</span><span class="ws6d">03)] <span class="ff3">:</span></span></span></span></div><div class="t m0 x2 h2 y55 ff1 fs0 fc1 sc0 ls4a ws0">D E R IV<span class="_0 blank"></span>A DA<span class="_2 blank"> </span>D<span class="_16 blank"> </span>I R E C IO N A L ,<span class="_2 blank"> </span>G R A D IE N T E<span class="_2 blank"> </span>&<span class="_2 blank"> </span>P L A N O<span class="_2 blank"> </span>TAN G<span class="_16 blank"> </span>E N T E</div><div class="t m0 x3 h3 y56 ff2 fs1 fc0 sc0 ls4a ws14">1.<span class="_3 blank"> </span>Calcule a deriv<span class="_0 blank"></span>ada direcional da função <span class="ff3 lsa">z<span class="ff5 lsb">=</span><span class="ls26">f</span></span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y</span><span class="lsb1">)<span class="ff3 ls5b">;</span></span></span><span class="ws1e">no<span class="_2 blank"> </span>p onto<span class="_2 blank"> </span><span class="ff3 lsa5">P<span class="ff6 fs0 lse v1">0</span></span></span>, na direção indicada.</div><div class="t m0 x4 h6 y57 ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls4a ws7">x<span class="ff6 fs0 ls5 v2">3</span><span class="ff5 ws10">+ 5</span>x<span class="ff6 fs0 lse v2">2</span><span class="wsa">y ;<span class="_3 blank"> </span>P<span class="ff6 fs0 lsb2 v1">0</span><span class="ff5 ws12">(2</span><span class="ls14">;</span><span class="ff5 ws1b">1) <span class="ff2 ws1">na direção da reta </span></span><span class="ls60">y<span class="ff5 lsb">=</span></span>x:</span></span></span></div><div class="t m0 x4 h6 y58 ff1 fs1 fc0 sc0 ls4a ws65">(b) <span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="lsb3">y</span></span><span class="ff5 ws16">exp (<span class="ff3 wsf">xy </span><span class="ls5a">)</span><span class="ff3 ws2">; P<span class="ff6 fs0 lsb4 v1">0</span></span><span class="ws12">(0<span class="ff3 ls1a">;</span><span class="ws6b">0) <span class="ff2 ws14">na direção da reta<span class="_5 blank"> </span><span class="ff3 ws6e">~<span class="_1a blank"></span>v <span class="ff5 ws6">= 4</span></span></span></span></span></span></div><div class="t m0 x27 h3 y59 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 xa h3 y58 ff3 fs1 fc0 sc0 lsb5">i<span class="ff5 ls4a ws10">+ 3</span></div><div class="t m0 x28 h3 y59 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x29 h3 y58 ff3 fs1 fc0 sc0 ls4a ws6f">j :</div><div class="t m0 x4 h22 y5a ff1 fs1 fc0 sc0 ls4a ws66">(c) <span class="ff3 ls4">z<span class="ff5 ls97">=</span><span class="ls4a ws7">x<span class="ff6 fs0 lsd v2">2</span><span class="ff8 ls6">\ue000</span><span class="ls7">y<span class="ff6 fs0 lse v2">2</span></span><span class="ws2">; P<span class="ff6 fs0 lsb2 v1">0</span><span class="ff5 ws12">(2</span><span class="ls14">;</span><span class="ff5 ws1b">3) <span class="ff2 ws14">na direção tangen<span class="_0 blank"></span>te à curv<span class="_0 blank"></span>a <span class="ff5 ws12">2<span class="ff3 ls12">x</span><span class="ws10">+ 5<span class="ff3 ls7">y<span class="ff6 fs0 ls1e v2">2</span></span><span class="ws6">= 15</span></span></span>, no p<span class="_a blank"> </span>onto <span class="ff5 ws12">(0<span class="ff3 ls1a">;<span class="ff8 ls2e v3">p</span></span>3)<span class="ff3">:</span></span></span></span></span></span></span></div><div class="t m0 x3 hb y5b ff2 fs1 fc0 sc0 ls4a ws14">2.<span class="_3 blank"> </span>Calcule a deriv<span class="_0 blank"></span>ada direcional<span class="_b blank"> </span><span class="ff3 ws27 v7">@ f</span></div><div class="t m0 x2a hc y5c ff3 fs1 fc0 sc0 ls4a ws70">@ ~<span class="_17 blank"></span>u<span class="_b blank"> </span><span class="ff2 ws1 v7">nos seguintes casos:</span></div><div class="t m0 x4 h6 y5d ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y;<span class="_6 blank"> </span>z </span><span class="ws1f">) = <span class="ff3 ws7">e<span class="ffb fs0 lsb6 v2">\ue000<span class="ff4 ls16">y</span></span></span><span class="ws5c">sen <span class="ff3 ls12">x</span><span class="ls8f">+</span><span class="ff6 fs0 v13">1</span></span></span></span></div><div class="t m0 x2b h23 y5e ff6 fs0 fc0 sc0 lsb7">3<span class="ff3 fs1 ls4a ws7 v2">e</span><span class="ffb lsb8 v18">\ue000</span><span class="lsb9 v18">3<span class="ff4 ls16">y<span class="ff5 fs1 ls4a ws16 v19">sen 3<span class="ff3 ls12">x<span class="ff5 ls6">+</span><span class="ls57">z</span></span></span><span class="ff6 lse">2<span class="ff5 fs1 ls5d v19">;<span class="ff3 lsa5">P</span></span><span class="lsb2 v1a">0</span><span class="ff5 fs1 ls4a ws12 v19">(<span class="ff3 wsa">\ue019 =<span class="ff5 ls15">3</span><span class="ls1a">;<span class="ff5 ls15">0</span>;</span><span class="ff5 ws71">1) <span class="ff2 lsba">e</span></span><span class="ws72">~<span class="_17 blank"></span>u <span class="ff5 lsb">=<span class="ff8 ls98">\ue000</span></span><span class="ff6 fs0 v13">1</span></span></span></span></span></span></span></div><div class="t m0 x2c h24 y5e ff6 fs0 fc0 sc0 ls4a">2</div><div class="t m0 x2d h3 y5f ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x2e h25 y5d ff3 fs1 fc0 sc0 lsbb">i<span class="ff5 ls8f">+<span class="ffb fs0 lsbc v1b">p<span class="ff6 ls4a v1c">2</span></span></span></div><div class="t m0 x2f h26 y5e ff6 fs0 fc0 sc0 lsbd">2<span class="ff3 fs1 ls4a v1d">~</span></div><div class="t m0 x30 h27 y5d ff3 fs1 fc0 sc0 lsbe">j<span class="ff5 lsbf">+<span class="ff6 fs0 ls4a v13">1</span></span></div><div class="t m0 x31 h28 y5e ff6 fs0 fc0 sc0 lsc0">2<span class="ff3 fs1 ls4a v17">~</span></div><div class="t m0 x32 h3 y5d ff3 fs1 fc0 sc0 ls4a ws2e">k :</div><div class="t m0 x4 h6 y60 ff1 fs1 fc0 sc0 ls4a ws65">(b) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y;<span class="_6 blank"> </span>z </span><span class="ws1f">) = <span class="ff3 ws7">x<span class="ff6 fs0 lse v2">2</span><span class="lsc1">y</span></span><span class="ws10">+ 3<span class="ff3 wsa">y z<span class="_16 blank"> </span><span class="ff6 fs0 lse v2">2</span></span><span class="ls5d">;<span class="ff3 lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span></span></span></span>(1<span class="ff3 ls14">;</span><span class="ff8">\ue000</span>1<span class="ff3 ls14">;</span><span class="ws73">1) <span class="ff2 lsba">e</span><span class="ff3 ws72">~<span class="_17 blank"></span>u <span class="ff5 ls95">=</span><span class="ff6 fs0 v13">1</span></span></span></span></div><div class="t m0 x33 h24 y61 ff6 fs0 fc0 sc0 ls4a">3</div><div class="t m0 x34 h3 y62 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x17 h27 y60 ff3 fs1 fc0 sc0 lsb5">i<span class="ff8 lsbf">\ue000<span class="ff6 fs0 ls4a v13">2</span></span></div><div class="t m0 x35 h26 y61 ff6 fs0 fc0 sc0 lsc2">3<span class="ff3 fs1 ls4a v1d">~</span></div><div class="t m0 x36 h27 y60 ff3 fs1 fc0 sc0 lsbe">j<span class="ff5 ls8f">+<span class="ff6 fs0 ls4a v13">2</span></span></div><div class="t m0 x37 h28 y61 ff6 fs0 fc0 sc0 lsc0">3<span class="ff3 fs1 ls4a v17">~</span></div><div class="t m0 x38 h3 y60 ff3 fs1 fc0 sc0 ls4a ws2e">k :</div><div class="t m0 x4 h4 y63 ff1 fs1 fc0 sc0 ls4a ws66">(c) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y;<span class="_6 blank"> </span>z </span><span class="ws74">)<span class="_5 blank"> </span>=<span class="_5 blank"> </span>ln <span class="ff7 lsc v3">\ue000</span><span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 ls5 v2">2</span></span>+<span class="ff3 ls57">z<span class="ff6 fs0 ls17 v2">2</span><span class="ff7 ls62 v3">\ue001</span></span><span class="ls5d">;<span class="ff3 lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span></span></span></span>(1<span class="ff3 ls1a">;</span><span class="ls15">1<span class="ff3 ls1a">;</span></span><span class="ws71">1) <span class="ff2 lsba">e</span><span class="ff3 ws72">~<span class="_17 blank"></span>u <span class="ff5 lsb">=<span class="ff8 ls98">\ue000</span></span><span class="ff6 fs0 v13">2</span></span></span></span></div><div class="t m0 x36 h24 y64 ff6 fs0 fc0 sc0 ls4a">3</div><div class="t m0 x1c h3 y65 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x39 h27 y63 ff3 fs1 fc0 sc0 lsb5">i<span class="ff5 lsbf">+<span class="ff6 fs0 ls4a v13">1</span></span></div><div class="t m0 x3a h26 y64 ff6 fs0 fc0 sc0 lsc2">3<span class="ff3 fs1 ls4a v1d">~</span></div><div class="t m0 x3b h27 y63 ff3 fs1 fc0 sc0 lsbe">j<span class="ff5 ls8f">+<span class="ff6 fs0 ls4a v13">2</span></span></div><div class="t m0 x3c h28 y64 ff6 fs0 fc0 sc0 lsc3">3<span class="ff3 fs1 ls4a v17">~</span></div><div class="t m0 x3d h3 y63 ff3 fs1 fc0 sc0 ls4a ws2e">k :</div><div class="t m0 x3 h3 y66 ff2 fs1 fc0 sc0 ls4a ws14">3.<span class="_3 blank"> </span>Calcule o v<span class="_0 blank"></span>alor máximo da deriv<span class="_c blank"></span>ada direcional da função <span class="ff3 ls6a">w<span class="ff5 lsb">=</span><span class="ls10">f</span></span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y ;<span class="_6 blank"> </span>z </span><span class="ls3a">)</span></span><span class="ws1e">no<span class="_2 blank"> </span>p on<span class="_0 blank"></span>to<span class="_1 blank"> </span><span class="ff3 lsa5">P<span class="ff6 fs0 lsa6 v1">0</span></span><span class="ff5">:</span></span></div><div class="t m0 x4 h4 y21 ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff3 ls6a">w<span class="ff5 lsb">=<span class="ff7 lsc v3">\ue000</span></span><span class="ls4a ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lsd v2">2</span><span class="ff5 ls6">+</span><span class="ls57">z<span class="ff6 fs0 lse v2">2</span><span class="ff7 lsc v3">\ue001</span><span class="ffb fs0 lsb8 v1d">\ue000<span class="ff6 lsb2">1</span></span><span class="ff5 ls5d">;</span><span class="lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span></span></span><span class="ff5 ws12">(1</span><span class="ls14">;</span><span class="ff5 ws12">2</span><span class="ls14">;</span><span class="ff8 ws12">\ue000<span class="ff5 ws71">3) </span></span></span></span><span class="ws75">(b) <span class="ff3 ls6a">w<span class="ff5 lsb">=</span><span class="ls4a ws7">e<span class="ff4 fs0 ls13 v2">x</span><span class="ff5 ws76">cos (</span><span class="wsa">y z <span class="ff5 wse">) ;<span class="_7 blank"> </span></span></span>P<span class="ff6 fs0 lsb4 v1">0</span><span class="ff5 ws12">(1</span><span class="ls1a">;<span class="ff5 ls15">0</span></span><span class="ws77">;<span class="_6 blank"> </span>\ue019 <span class="ff5 ls5a">)</span>:</span></span></span></span></div><div class="t m0 x24 h3 y46 ff2 fs1 fc0 sc0 ls4a">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"><img fetchpriority="low" loading="lazy" class="bi x3e y67 w4 h29" alt="" src="https://files.passeidireto.com/4010a717-9d5a-4246-bb9a-5ef73ba40b70/bg4.png"><div class="t m0 x3 h8 y23 ff2 fs1 fc0 sc0 ls4a ws78">4.<span class="_3 blank"> </span>Seja <span class="ff3 lsc4">z<span class="ff5 lsc5">=</span><span class="ls26">f</span></span><span class="ff5 ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y</span><span class="lsc6">)</span></span><span class="ws79">um<span class="_a blank"> </span>a função diferenciáv<span class="_0 blank"></span>el em cada p<span class="_16 blank"> </span>on<span class="_0 blank"></span>to do círculo <span class="ff3 ws7">x<span class="ff6 fs0 lsc7 v2">2</span><span class="ff5 lsc8">+</span><span class="ls7">y<span class="ff6 fs0 lsc9 v2">2</span></span><span class="ff5 ws7a">= 1</span></span>.<span class="_1b blank"> </span>Mostre que a</span></div><div class="t m0 x4 h3 y68 ff2 fs1 fc0 sc0 ls4a ws14">deriv<span class="_0 blank"></span>ada direcional de <span class="ff3 ls9f">f</span><span class="ws1e">no<span class="_2 blank"> </span>p on<span class="_0 blank"></span>to<span class="_2 blank"> </span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ls3a">)</span></span><span class="ws14">na direção da tangen<span class="_0 blank"></span>te ao círculo é<span class="_1 blank"> </span><span class="ff8 ws12">\ue000<span class="ff3 wsa">y f<span class="ff4 fs0 ls4b v1">x</span><span class="ff5 ls6">+</span><span class="ws7">xf<span class="ff4 fs0 ls2 v1">y</span>:</span></span></span></span></span></div><div class="t m0 x3 h3 y69 ff2 fs1 fc0 sc0 ls4a ws14">5.<span class="_3 blank"> </span>Encon<span class="_0 blank"></span>tre o plano tangente e a reta normal à sup<span class="_a blank"> </span>erfície dada, no p<span class="_a blank"> </span>onto indicado:</div><div class="t m0 x4 h6 y6a ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff3 lsca">S<span class="ff5 lscb">:</span><span class="ls4">z<span class="ff5 lsb">=</span><span class="ls4a ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff8 ls50">\ue000</span><span class="ls51">y<span class="ff6 fs0 ls17 v2">2</span><span class="ff5 ls5d">;</span></span>P<span class="ff6 fs0 lsb4 v1">0</span><span class="ff5 ws12">(1</span><span class="ls1a">;<span class="ff5 ls15">1</span>;</span><span class="ff5 ws24">0) </span>:</span></span></span></div><div class="t m0 x4 h6 y6b ff1 fs1 fc0 sc0 ls4a ws65">(b) <span class="ff3 lsca">S<span class="ff5 lscb">:</span><span class="ls4a ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ws10">+ 2</span><span class="ls7">y<span class="ff6 fs0 ls5 v2">2</span></span><span class="ff5 ws10">+ 3</span><span class="ls57">z<span class="ff6 fs0 lsa6 v2">2</span></span><span class="ff5 ws6">= 6;<span class="_7 blank"> </span></span><span class="lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span><span class="ff5 ws12">(1</span><span class="ls14">;</span><span class="ff5 ws12">1</span><span class="ls14">;</span><span class="ff5 ws24">1) </span>:</span></span></div><div class="t m0 x4 h5 y6c ff1 fs1 fc0 sc0 ls4a ws66">(c) <span class="ff3 lscc">S<span class="ff5 ls1c">:</span><span class="ls4">z<span class="ff5 lsb">=</span><span class="ls4a ws7">x<span class="ff7 ws12 v4">p</span>x<span class="ff6 fs0 lsd v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lse v5">2</span><span class="ff5 ls5d">;</span><span class="lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span></span><span class="ff5 ws12">(3</span><span class="ls14">;</span><span class="ff8 ws12">\ue000<span class="ff5">4</span></span><span class="ls14">;</span><span class="ff5 ws7b">15) </span>:</span></span></span></div><div class="t m0 x4 h5 y6d ff1 fs1 fc0 sc0 ls4a ws65">(d) <span class="ff3 lsca">S<span class="ff5 lscb">:</span><span class="ls4">z<span class="ff5 lsb">=</span></span></span><span class="ff7 ws12 v4">p</span><span class="ff5 lscd">9<span class="ff8 ls6">\ue000</span></span><span class="ff3 ws7">x<span class="ff6 fs0 ls5 v5">2</span><span class="ff8 ls6">\ue000</span><span class="ls7">y<span class="ff6 fs0 ls17 v5">2</span><span class="ff5 ls5d">;</span></span>P<span class="ff6 fs0 lsb4 v1">0</span><span class="ff5 ws12">(<span class="ff8">\ue000</span><span class="ls15">1</span></span><span class="ls1a">;<span class="ff5 ls15">2</span>;</span><span class="ff5 ws24">2) </span>:</span></div><div class="t m0 x3 h8 y6e ff2 fs1 fc0 sc0 ls4a ws7c">6.<span class="_3 blank"> </span>Seja <span class="ff3 lsce">\ue00d</span><span class="ws7d">a curv<span class="_0 blank"></span>a em <span class="ff9 ls1d">R<span class="ff6 fs0 lscf v2">3</span></span><span class="ws1e">descrita<span class="_1c blank"> </span>p or:<span class="_b blank"> </span><span class="ff3 ls5e">x</span><span class="ff5 ws7e">=<span class="_5 blank"> </span>sen <span class="ff3 ws7f">t;<span class="_1d blank"> </span>y </span>=<span class="_5 blank"> </span>sen <span class="ff3 lsd0">t</span></span><span class="lsd1">e<span class="ff3 ls4">z</span></span><span class="ff5 wse">=<span class="_5 blank"> </span>cos 2<span class="ff3 ws80">t; </span><span class="lsd2">0<span class="ff8 lsb">\ue014<span class="ff3 ls6f">t</span>\ue014</span><span class="ls15">2</span></span><span class="ff3 wsa">\ue019 :<span class="_1c blank"> </span></span></span></span>Mostre que a curv<span class="_0 blank"></span>a</span></div><div class="t m0 x4 h8 y6f ff3 fs1 fc0 sc0 lsd3">\ue00d<span class="ff2 ls4a ws81">está con<span class="_0 blank"></span>tida no parab<span class="_16 blank"> </span>olóide <span class="ff3 ws7">x<span class="ff6 fs0 lsd4 v2">2</span><span class="ff5 lsd5">+</span><span class="ls7">y<span class="ff6 fs0 lsd4 v2">2</span><span class="ff5 lsd5">+</span><span class="ls4">z</span></span><span class="ff5 ws6">= 1<span class="_2 blank"> </span></span></span>e determine a reta tangente e o plano normal à curv<span class="_c blank"></span>a</span></div><div class="t m0 x4 h3 y70 ff2 fs1 fc0 sc0 ls4a ws1e">no<span class="_2 blank"> </span>p on<span class="_0 blank"></span>to<span class="_1 blank"> </span>corresp onden<span class="_0 blank"></span>te<span class="_1 blank"> </span>a<span class="_2 blank"> </span><span class="ff3 ls6f">t<span class="ff5 lsb">=</span><span class="ls4a wsa">\ue019 =<span class="ff5 ls15">4</span>:</span></span></div><div class="t m0 x3 h3 y71 ff2 fs1 fc0 sc0 ls4a ws49">7.<span class="_3 blank"> </span>Calcule <span class="ff8 ls2e">r<span class="ff3 lsa3">f</span></span><span class="ws14">e v<span class="_0 blank"></span>eri\u2026<span class="_9 blank"></span>que em cada caso que este vetor é normal as curv<span class="_c blank"></span>as ou sup<span class="_16 blank"> </span>erfícies de nív<span class="_0 blank"></span>el.</span></div><div class="t m0 x4 h6 y72 ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y</span><span class="ws1f">) = <span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 lse v2">2</span><span class="ls4a">:</span></span></span></span></span></div><div class="t m0 x4 h4 y73 ff1 fs1 fc0 sc0 ls4a ws65">(b) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y</span><span class="ws82">)<span class="_5 blank"> </span>=<span class="_5 blank"> </span>exp <span class="ff7 lsc v3">\ue000</span><span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">2</span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 lse v2">2</span><span class="ff7 ls62 v3">\ue001</span><span class="ls4a">:</span></span></span></span></span></div><div class="t m0 x4 h6 y74 ff1 fs1 fc0 sc0 ls4a ws66">(c) <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y;<span class="_6 blank"> </span>z </span><span class="ws23">) = 2<span class="ff3 ws7">x<span class="ff6 fs0 ls5 v2">2</span></span><span class="ws10">+ 2<span class="ff3 ls7">y<span class="ff6 fs0 ls5 v2">2</span><span class="ff8 ls6">\ue000</span><span class="ls4a ws2a">xz :</span></span></span></span></span></div><div class="t m0 x3 h1d y75 ff2 fs1 fc0 sc0 ls4a ws14">8.<span class="_3 blank"> </span>Mostre que a reta normal à esfera <span class="ff3 ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lsd v2">2</span><span class="ff5 ls6">+</span><span class="ls57">z<span class="ff6 fs0 ls1e v2">2</span><span class="ff5 lsb">=</span><span class="lsd6">r<span class="ff6 fs0 ls17 v2">2</span></span></span></span></span><span class="ws1e">,<span class="_2 blank"> </span>no<span class="_2 blank"> </span>p onto<span class="_1c blank"> </span><span class="ff3 lsa5">P<span class="ff6 fs0 ls65 v1">0</span></span></span>passa p<span class="_a blank"> </span>ela centro da esfera.</div><div class="t m0 x3 he y76 ff2 fs1 fc0 sc0 ls4a ws83">9.<span class="_3 blank"> </span>Considere a função <span class="ff3 ls10">f</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ws84">) =<span class="_1e blank"> </span><span class="ff3 ws7 v7">x</span><span class="ff6 fs0 lse va">2</span><span class="ff3 v7">y</span></span></span></div><div class="t m0 x3f hc y77 ff3 fs1 fc0 sc0 ls4a ws7">x<span class="ff6 fs0 lsd v5">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls22 v5">2</span></span><span class="ff2 ws85 v7">, se <span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="lsd7">)</span><span class="ff8">6</span><span class="ws84">= (0<span class="ff3 ls1a">;</span><span class="ws86">0) <span class="ff2 lsd8">e<span class="ff3 ls26">f</span></span></span></span>(0<span class="ff3 ls14">;</span><span class="ws84">0) = 0<span class="ff2 ws83">.<span class="_b blank"> </span>V<span class="_c blank"></span>eri\u2026<span class="_9 blank"></span>que que <span class="ff3 lsd9">f</span>tem deriv<span class="_0 blank"></span>ada</span></span></span></span></div><div class="t m0 x4 h3 y78 ff2 fs1 fc0 sc0 ls4a ws14">direcional na origem em qualquer direção, mas não é aí diferenciáv<span class="_0 blank"></span>el.</div><div class="t m0 x40 h3 y79 ff2 fs1 fc0 sc0 ls4a ws87">10.<span class="_3 blank"> </span>Seja <span class="ff3 ws88">~<span class="_15 blank"></span>r <span class="ff5 lsda">=</span>x</span></div><div class="t m0 x41 h3 y7a ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x42 h3 y79 ff3 fs1 fc0 sc0 lsdb">i<span class="ff5 lsdc">+</span><span class="ls4a">y</span></div><div class="t m0 x43 h3 y7a ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x44 h2a y79 ff3 fs1 fc0 sc0 lsdd">j<span class="ff5 lsdc">+</span><span class="ls4a ws7">z<span class="v1e">~</span></span></div><div class="t m0 x45 h8 y79 ff3 fs1 fc0 sc0 lsde">k<span class="ff2 ls4a ws89">o v<span class="_0 blank"></span>etor p<span class="_a blank"> </span>osição de um p<span class="_16 blank"> </span>on<span class="_0 blank"></span>to <span class="ff3 ls82">P</span><span class="ff5 ws12">(<span class="ff3 wsa">x;<span class="_6 blank"> </span>y ;<span class="_6 blank"> </span>z </span><span class="lsdf">)</span></span><span class="ws8a">do <span class="ff9 ls1d">R<span class="ff6 fs0 lse0 v2">3</span></span></span>e represen<span class="_0 blank"></span>te p<span class="_16 blank"> </span>or <span class="ff3 lse1">r</span><span class="ws8b">sua norma.</span></span></div><div class="t m0 x4 h3 y7b ff2 fs1 fc0 sc0 ls4a ws26">Se <span class="ff3 ls26">f</span><span class="ff5 ws12">(<span class="ff3 ls59">t</span><span class="ls3a">)</span></span><span class="ws14">é uma função real deriv<span class="_0 blank"></span>áv<span class="_0 blank"></span>el, mostre que:</span></div><div class="t m0 x21 hb y7c ff8 fs1 fc0 sc0 ls2e">r<span class="ff3 ls26">f<span class="ff5 ls4a ws12">(</span><span class="lsd6">r<span class="ff5 ls4a ws1f">) = </span><span class="lse2">f<span class="ffb fs0 ls58 v13">0</span><span class="ff5 ls4a ws12">(</span><span class="ls7d">r<span class="ff5 lse3">)</span><span class="ls4a ws7 v7">~<span class="_15 blank"></span>r</span></span></span></span></span></div><div class="t m0 x46 hc y7d ff3 fs1 fc0 sc0 lse4">r<span class="ls4a v7">:</span></div><div class="t m0 x4 h3 y7e ff2 fs1 fc0 sc0 ls4a ws14">Usando essa fórm<span class="_0 blank"></span>ula, calcule <span class="ff8 lse5">r</span><span class="ff5 ws12">(<span class="ff3 ls7d">r</span><span class="lsb1">)<span class="ff3 ls23">;<span class="ff8 lse5">r</span></span></span>(1<span class="ff3 ws8c">=r </span><span class="lse6">)</span></span><span class="ls19">e<span class="ff8 lse5">r</span></span><span class="ff5 ws8d">(ln <span class="ff3 ls7d">r</span>)</span></div><div class="t m0 x40 h3 y7f ff2 fs1 fc0 sc0 ls4a ws1">11.<span class="_3 blank"> </span>Determine a reta tangen<span class="_0 blank"></span>te à curv<span class="_0 blank"></span>a <span class="ff3 lse7">\ue00d</span><span class="ws14">, no p<span class="_a blank"> </span>onto <span class="ff3 lsa5">P<span class="ff6 fs0 ls65 v1">0</span></span><span class="ws12">indicado.</span></span></div><div class="t m0 x4 h2b y80 ff1 fs1 fc0 sc0 ls4a ws5">(a) <span class="ff3 lse8">\ue00d<span class="ff5 lscb">:</span></span><span class="ff7 v1f">\ue00c</span></div><div class="t m0 x47 ha y81 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x47 ha y82 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x47 ha y83 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x47 ha y84 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x47 ha y85 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x48 h8 y86 ff5 fs1 fc0 sc0 ls4a ws12">3<span class="ff3 ws7">x<span class="ff6 fs0 ls5 v2">2</span></span><span class="ls50">+<span class="ff3 ls51">y<span class="ff6 fs0 lsd v2">2</span></span><span class="ls6">+<span class="ff3 ls4">z</span></span></span><span class="ws6">= 4</span></div><div class="t m0 x48 h8 y87 ff8 fs1 fc0 sc0 ls4a ws12">\ue000<span class="ff3 ws7">x<span class="ff6 fs0 lsd v2">2</span><span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ls6">+</span><span class="ls57">z<span class="ff6 fs0 ls1e v2">2</span></span></span><span class="ff5 ws6">= 12</span></span></div><div class="t m0 x49 h2c y88 ff5 fs1 fc0 sc0 ls5d">;<span class="ff3 lsa5">P<span class="ff6 fs0 lsb4 v1">0</span></span><span class="ls4a ws12">(1<span class="ff3 ls1a">;</span><span class="ls15">2<span class="ff3 ls1a">;</span></span><span class="ff8">\ue000</span><span class="ws8e">3) <span class="ff1 ws4b">(b) <span class="ff3 lse8">\ue00d</span></span><span class="lscb">:</span><span class="ff7 v1f">\ue00c</span></span></span></div><div class="t m0 x4a ha y81 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x4a ha y82 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x4a ha y83 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x4a ha y84 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x4a ha y85 ff7 fs1 fc0 sc0 ls4a">\ue00c</div><div class="t m0 x46 h3 y86 ff5 fs1 fc0 sc0 ls4a ws12">3<span class="ff3 ws8f">xy </span><span class="ws10">+ 2<span class="ff3 wsa">y z<span class="_18 blank"> </span></span>+ 6<span class="_1c blank"> </span>=<span class="_5 blank"> </span>0</span></div><div class="t m0 x46 h8 y87 ff3 fs1 fc0 sc0 ls4a ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff8 ls6">\ue000<span class="ff5 ls15">2</span></span><span class="ws90">xz <span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lse v2">2</span><span class="ls4">z</span></span><span class="ff5 ws6">= 1</span></span></div><div class="t m0 x4b h3 y88 ff5 fs1 fc0 sc0 ls5d">;<span class="ff3 lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span><span class="ls4a ws12">(1<span class="ff3 ls14">;</span><span class="ff8">\ue000</span>2<span class="ff3 ls14">;</span>0)</span></div><div class="t m0 x40 h8 y20 ff2 fs1 fc0 sc0 ls4a ws91">12.<span class="_3 blank"> </span>Calcule a deriv<span class="_0 blank"></span>ada direcional no ponto <span class="ff3 lsa5">P<span class="ff6 fs0 lsb4 v1">0</span></span><span class="ff5 ws12">(1<span class="ff3 ls1a">;</span><span class="ls15">2<span class="ff3 ls1a">;</span></span><span class="ws92">3) </span></span>da função <span class="ff3 lse9">w</span><span class="ff5 ws93">= 2<span class="ff3 ws7">x<span class="ff6 fs0 lsea v2">2</span><span class="ff8 lseb">\ue000</span><span class="ls7">y<span class="ff6 fs0 lsea v2">2</span></span></span><span class="lseb">+<span class="ff3 ls57">z<span class="ff6 fs0 ls17 v2">2</span><span class="lsec">;</span></span></span></span>na direção da reta</div><div class="t m0 x4 h3 y21 ff2 fs1 fc0 sc0 ls4a ws1">que passa nos p<span class="_a blank"> </span>on<span class="_0 blank"></span>tos <span class="ff3 lsed">A</span><span class="ff5 ws12">(1<span class="ff3 ls1a">;</span><span class="ls15">2<span class="ff3 ls1a">;</span></span><span class="ws6b">1) </span></span><span class="ls19">e<span class="ff3 lsee">B</span></span><span class="ff5 ws12">(3<span class="ff3 ls1a">;</span><span class="ls15">5<span class="ff3 ls1a">;</span></span><span class="ws24">0) <span class="ff3">:</span></span></span></div><div class="t m0 x24 h3 y46 ff2 fs1 fc0 sc0 ls4a">4</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y89 w1 h2d" alt="" src="https://files.passeidireto.com/4010a717-9d5a-4246-bb9a-5ef73ba40b70/bg5.png"><div class="t m0 x40 h8 y23 ff2 fs1 fc0 sc0 ls4a ws14">13.<span class="_3 blank"> </span>Considere a curv<span class="_0 blank"></span>a <span class="ff3 lsef">\ue00d</span>de equações paramétricas <span class="ff3 ls5e">x<span class="ff5 lsb">=</span><span class="ls4a ws94">t;<span class="_3 blank"> </span>y <span class="ff5 lsb">=</span><span class="ls59">t<span class="ff6 fs0 ls65 v2">2</span></span></span></span><span class="ls19">e<span class="ff3 ls4">z<span class="ff5 lsb">=</span><span class="ls69">t<span class="ff6 fs0 lse v2">3</span><span class="lsf0">;</span></span></span></span><span class="ff8 ws10">\ue000 1<span class="_5 blank"> </span><span class="ff3 ws95">< t < </span><span class="ws12">1<span class="ff3">:</span></span></span></div><div class="t m0 x4 h6 y8a ff1 fs1 fc0 sc0 ls4a ws63">(a) <span class="ff2 ws1">Determine a reta tangen<span class="_0 blank"></span>te e o plano normal no p<span class="_a blank"> </span>onto <span class="ff3 lsa5">P<span class="ff6 fs0 lsb2 v1">0</span></span><span class="ff5 ws12">(2<span class="ff3 ls14">;</span>4<span class="ff3 ls14">;</span><span class="ws24">8) <span class="ff3">:</span></span></span></span></div><div class="t m0 x4 h6 y8b ff1 fs1 fc0 sc0 ls4a ws65">(b) <span class="ff2 ws1">Determine a reta tangen<span class="_0 blank"></span>te que passa no p<span class="_a blank"> </span>onto <span class="ff3 lsa5">P<span class="ff6 fs0 lsb2 v1">1</span></span><span class="ff5 ws12">(0<span class="ff3 ls14">;</span><span class="ff8">\ue000</span><span class="ls15">1<span class="ff3 ls1a">;</span></span><span class="ws25">2) <span class="ff3">:</span></span></span></span></div><div class="t m0 x4 h6 y8c ff1 fs1 fc0 sc0 ls4a ws66">(c) <span class="ff2 ws14">V<span class="_c blank"></span>eri\u2026<span class="_9 blank"></span>que se existe reta tangen<span class="_0 blank"></span>te passando no p<span class="_16 blank"> </span>on<span class="_0 blank"></span>to <span class="ff3 ws7">Q<span class="ff6 fs0 lsb2 v1">1</span><span class="ff5 ws12">(0</span><span class="ls14">;</span><span class="ff8 ws12">\ue000<span class="ff5 ls15">1</span></span><span class="ls1a">;</span><span class="ff5 ws25">3) </span>:</span></span></div><div class="t m0 x40 h15 y8d ff2 fs1 fc0 sc0 ls4a ws96">14.<span class="_3 blank"> </span>Seja <span class="ff3 lsf1">f<span class="ff5 lsf2">:<span class="ff9 ls4d">R<span class="ff8 ls1f">!</span><span class="lsf3">R</span></span></span></span><span class="ws97">uma função deriv<span class="_0 blank"></span>áv<span class="_0 blank"></span>el,<span class="_b blank"> </span>com <span class="ff3 lse2">f<span class="ffb fs0 ls58 v2">0</span></span><span class="ff5 ws12">(<span class="ff3 ls59">t</span><span class="lsf4">)<span class="ff3 lsf5">></span><span class="ls15">0<span class="ff3 lsf6">;</span></span></span><span class="ff8">8<span class="ff3 ls69">t</span></span></span><span class="ws98">.<span class="_10 blank"> </span>Se <span class="ff3 lsf7">g</span><span class="ff5 ws12">(<span class="ff3 ws15">x;<span class="_6 blank"> </span>y </span><span class="ws99">) = <span class="ff3 ls26">f<span class="ff7 lsc v3">\ue000</span><span class="ls4a ws7">x<span class="ff6 fs0 lsd v2">2</span></span></span><span class="ls6">+<span class="ff3 ls7">y<span class="ff6 fs0 lse v2">2</span><span class="ff7 lsc v3">\ue001</span></span></span></span></span><span class="ws9a">, mostre que a</span></span></span></div><div class="t m0 x4 h3 y8e ff2 fs1 fc0 sc0 ls4a ws9b">deriv<span class="_0 blank"></span>ada direcional <span class="ff3">D</span></div><div class="t m0 x4c h2e y8f ff4 fs0 fc0 sc0 ls4a ws9c">~<span class="_1f blank"></span>v <span class="ff3 fs1 ls8c v14">g<span class="ff5 ls4a ws12">(<span class="ff3 ws3b">x;<span class="_6 blank"> </span>y</span><span class="ls3a">)</span><span class="ff2 ws14">será máxima quando <span class="ff3 ws6e">~<span class="_17 blank"></span>v <span class="ff5 lsb">=</span>x</span></span></span></span></div><div class="t m0 x46 h3 y90 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x4d h3 y8e ff3 fs1 fc0 sc0 lsbb">i<span class="ff5 ls50">+</span><span class="ls4a">y</span></div><div class="t m0 x4e h3 y90 ff3 fs1 fc0 sc0 ls4a">~</div><div class="t m0 x4f h3 y8e ff3 fs1 fc0 sc0 ls4a ws6f">j :</div><div class="t m0 x40 h3 y91 ff2 fs1 fc0 sc0 ls4a ws9d">15. Se <span class="ff3 lsf8">f<span class="ff5 lsf9">:<span class="ff9 ls4d">R<span class="ff8 ls1f">!</span><span class="lsfa">R</span></span></span></span><span class="ws9e">é uma função deriv<span class="_0 blank"></span>áv<span class="_0 blank"></span>el,<span class="_14 blank"> </span>mostre que os p<span class="_a blank"> </span>lanos tangen<span class="_0 blank"></span>tes à sup<span class="_16 blank"> </span>erfície de equação</span></div><div class="t m0 x4 h3 y92 ff3 fs1 fc0 sc0 ls4">z<span class="ff5 lsb">=</span><span class="ls4a wsa">y f<span class="_5 blank"> </span><span class="ff5 ws12">(</span><span class="ws1d">x=y <span class="ff5 ls3a">)</span><span class="ff2 ws1e">passam<span class="_2 blank"> </span>to dos<span class="_2 blank"> </span>p ela<span class="_2 blank"> </span>origem.</span></span></span></div><div class="t m0 x40 h8 y93 ff2 fs1 fc0 sc0 ls4a ws9f">16.<span class="_3 blank"> </span>Determine o plano tangen<span class="_0 blank"></span>te à sup<span class="_16 blank"> </span>erfície <span class="ff3 ls4">z</span><span class="ff5 ws6">= 2<span class="ff3 ws7">x<span class="ff6 fs0 lsfb v2">2</span></span><span class="lsfc">+<span class="ff3 ls7">y<span class="ff6 fs0 lsfb v2">2</span><span class="ff8 lsfd">\ue000</span></span></span><span class="ws12">3<span class="ff3 wsf">xy </span></span></span>, paralelo ao plano de equação <span class="ff5 ws12">10<span class="ff3 lsfe">x<span class="ff8 lsfc">\ue000</span></span><span class="ls15">7<span class="ff3 lsff">y</span></span><span class="ff8">\ue000</span></span></div><div class="t m0 x4 h3 y94 ff5 fs1 fc0 sc0 ls15">2<span class="ff3 ls100">z</span><span class="ls4a ws10">+ 5<span class="_5 blank"> </span>=<span class="_5 blank"> </span>0<span class="ff3">:</span></span></div><div class="t m0 x40 h3 y95 ff2 fs1 fc0 sc0 ls4a wsa0">17.<span class="_3 blank"> </span>Determine um plano que passa nos p<span class="_a blank"> </span>on<span class="_0 blank"></span>tos <span class="ff3 ls82">P</span><span class="ff5 ws12">(5<span class="ff3 ls1a">;</span><span class="ls15">0<span class="ff3 ls1a">;</span></span><span class="wsa1">1) </span></span><span class="ls101">e<span class="ff3 ls102">Q</span></span><span class="ff5 ws12">(1<span class="ff3 ls1a">;</span><span class="ls15">0<span class="ff3 ls1a">;</span></span><span class="wsa1">3) </span></span><span class="wsa2">e que s<span class="_a blank"> </span>eja tangen<span class="_0 blank"></span>te à sup<span class="_16 blank"> </span>erfície</span></div><div class="t m0 x4 h8 y96 ff3 fs1 fc0 sc0 ls4a ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ws10">+ 2</span><span class="ls7">y<span class="ff6 fs0 ls5 v2">2</span><span class="ff5 ls6">+</span><span class="ls103">z<span class="ff6 fs0 lsa6 v2">2</span></span></span><span class="ff5 ws6">= 7<span class="ff2">.</span></span></div><div class="t m0 x40 h8 y97 ff2 fs1 fc0 sc0 ls4a wsa3">18.<span class="_3 blank"> </span>Determine os p<span class="_a blank"> </span>ontos da superfície <span class="ff3 ls104">z</span><span class="ff5 wsa4">=<span class="_1 blank"> </span>8 <span class="ff8 ls105">\ue000</span><span class="ws12">3<span class="ff3 ws7">x<span class="ff6 fs0 ls106 v2">2</span><span class="ff8 ls105">\ue000</span></span><span class="ls15">2<span class="ff3 ls7">y<span class="ff6 fs0 ls107 v2">2</span></span></span></span></span><span class="wsa5">nos quais o plano tangent<span class="_0 blank"></span>e é p<span class="_16 blank"> </span>erp<span class="_a blank"> </span>endicular à</span></div><div class="t m0 x4 h3 y98 ff2 fs1 fc0 sc0 ls4a wsa6">reta <span class="ff3 ls5e">x</span><span class="ff5 wsa7">=<span class="_5 blank"> </span>2 <span class="ff8 ls6">\ue000</span><span class="ls15">3</span><span class="ff3 wsa8">t;<span class="_3 blank"> </span>y </span><span class="ws10">=<span class="_5 blank"> </span>7 + 8<span class="ff3 wsa9">t;<span class="_3 blank"> </span>z </span></span>=<span class="_5 blank"> </span>5 <span class="ff8 ls6">\ue000</span><span class="ff3">t:</span></span></div><div class="t m0 x40 h8 y99 ff2 fs1 fc0 sc0 ls4a ws1">19.<span class="_3 blank"> </span>Determine o p<span class="_a blank"> </span>onto<span class="_1c blank"> </span>da sup<span class="_a blank"> </span>erfície <span class="ff3 ls108">z</span><span class="ff5 wsaa">= 3<span class="ff3 ws7">x<span class="ff6 fs0 ls109 v2">2</span><span class="ff8 ls10a">\ue000</span><span class="ls7">y<span class="ff6 fs0 ls10b v2">2</span></span></span></span>onde o plano tangente é paralelo ao plano de equação</div><div class="t m0 x4 h3 y9a ff5 fs1 fc0 sc0 ls4a ws12">3<span class="ff3 ls10c">x</span><span class="ls6">+<span class="ff3 lsc1">y</span></span><span class="ws10">+ 2<span class="ff3 lsa">z</span><span class="ws6">= 1<span class="ff2">.</span></span></span></div><div class="t m0 x40 h8 y9b ff2 fs1 fc0 sc0 ls4a ws1">20.<span class="_3 blank"> </span>Determine em que p<span class="_a blank"> </span>on<span class="_0 blank"></span>tos da sup<span class="_16 blank"> </span>erfície <span class="ff3 lsa">z<span class="ff5 lsb">=</span><span class="ls4a ws7">x<span class="ff6 fs0 ls5 v2">2</span><span class="ff8 ls6">\ue000</span><span class="ws8f">xy <span class="ff5 ls6">+</span><span class="ls7">y<span class="ff6 fs0 lsd v2">2</span><span class="ff8 ls6">\ue000</span></span><span class="ff5 ws12">2</span><span class="ls12">x</span><span class="ff5 ws10">+ 4</span><span class="ls3b">y</span></span></span></span><span class="ws14">o plano tangen<span class="_0 blank"></span>te é<span class="_1 blank"> </span>horizon<span class="_0 blank"></span>tal.</span></div><div class="t m0 x24 h3 y46 ff2 fs1 fc0 sc0 ls4a">5</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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