<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/6bb5cda6-b074-4792-8818-7df173dc0e00/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls3a ws0">4.<span class="_0 blank"></span>-1<span class="_1 blank"> </span>F<span class="_2 blank"></span>un<span class="_0 blank"></span>ç<span class="_0 blank"></span>õe<span class="_0 blank"></span>s D<span class="_0 blank"></span>e<span class="_0 blank"></span>r<span class="_0 blank"></span>iv<span class="_3 blank"></span>áv<span class="_4 blank"></span>ei<span class="_0 blank"></span>s</div><div class="t m0 x1 h3 y2 ff1 fs1 fc0 sc0 ls3a ws1">4.1A <span class="ff2 ws2">Em cada caso, encon<span class="_0 blank"></span>tre a deriv<span class="_4 blank"></span>ada da funç<span class="_5 blank"> </span>ão <span class="ff3 ls0">y<span class="ff4 ls1">=</span><span class="ls2">f</span></span><span class="ff4 ws3">(<span class="ff3">x</span>)</span>, usando a de\u2026<span class="_2 blank"></span>nição.</span></div><div class="t m0 x2 h4 y3 ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls4 v1">2</span><span class="ff4 ws5">+ 1<span class="_6 blank"> </span></span></span></span><span class="ws6">(b) <span class="ff3 ls0">y</span><span class="ff4 ws7">= 2<span class="ff3 ws3">x<span class="ff5 fs2 ls5 v1">3</span></span></span><span class="ws8">(c) <span class="ff3 ls6">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls7 v1">2</span><span class="ff6 ls8">\ue000<span class="ff4 ls9">5</span></span></span></span><span class="ws9">(d) <span class="ff3 ls0">y</span><span class="ff4 ws7">= 2<span class="ff3 ws3">x<span class="ff5 fs2 ls7 v1">2</span><span class="ff6 ls8">\ue000</span></span><span class="lsa">3<span class="ff3 lsb">x</span></span></span></span>(e) <span class="ff3 ls6">y<span class="ff4 lsc">=<span class="ls3a v2">1</span></span></span></span></span></div><div class="t m0 x3 h3 y4 ff3 fs1 fc0 sc0 lsd">x<span class="ff4 ls3a ws5">+ 1</span></div><div class="t m0 x4 h5 y5 ff1 fs1 fc0 sc0 ls3a wsa">4.1B <span class="ff2 wsb">Seja <span class="ff3 lse">f</span><span class="ws2">a função de\u2026<span class="_3 blank"></span>nida em <span class="ff7 lsf">R</span><span class="wsc">por<span class="_7 blank"> </span><span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = <span class="ff8 v3">8</span></span></span></span></span></span></div><div class="t m0 x5 h6 y6 ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 x5 h7 y7 ff8 fs1 fc0 sc0 ls11">:<span class="ff6 ls3a ws3 v4">\ue000<span class="ff3">x<span class="ff2 ws2">, para </span><span class="ls12">x<span class="ff6 ls3">\ue014</span></span><span class="ff4">0</span></span></span></div><div class="t m0 x6 h3 y8 ff4 fs1 fc0 sc0 ls3a ws3">2<span class="ff2 ws2">, para <span class="ff3 wse">x > </span></span>0</div><div class="t m0 x7 h3 y5 ff3 fs1 fc0 sc0 ls3a">:</div><div class="t m0 x1 h8 y9 ff2 fs1 fc0 sc0 ls3a ws2">(a) Calcule <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff6">\ue000</span><span class="wsf">1) </span></span>(b) Existem as deriv<span class="_4 blank"></span>adas <span class="ff3 ls15">f</span><span class="ff9 fs2 v1">0</span></div><div class="t m0 x8 h9 ya ff5 fs2 fc0 sc0 ls16">+<span class="ff4 fs1 ls3a ws10 v5">(0) <span class="ff2 ls17">e<span class="ff3 ls13">f</span></span></span><span class="ff9 ls3a v6">0</span></div><div class="t m0 x9 ha ya ff9 fs2 fc0 sc0 ls18">\ue000<span class="ff4 fs1 ls3a ws3 v5">(0)<span class="ff2 ws11">?<span class="_8 blank"> </span>(c) <span class="ff3 ls19">f</span><span class="ws2">é deriv<span class="_4 blank"></span>ável em <span class="ff3 ls1a">x</span><span class="ff4 ws7">= 0?</span></span></span></span></div><div class="t m0 x4 hb yb ff1 fs1 fc0 sc0 ls3a ws12">4.1C <span class="ff2 ws13">Seja <span class="ff3 ls1b">f<span class="ff4 ls1c">:<span class="ff7 ls1d">R<span class="ff6 ls1e">!</span><span class="lsf">R</span></span></span></span><span class="ws2">a função dada p<span class="_5 blank"> </span>or <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff6">j<span class="ff3">x</span><span class="ls1f">j</span></span><span class="ls8">+</span><span class="ff3">x</span></span>.</span></span></div><div class="t m0 x1 h8 yc ff2 fs1 fc0 sc0 ls3a ws2">(a) Existe <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws15">(0)? </span>(b) Existe <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ls20">)</span></span><span class="ws16">para <span class="ff3 ls12">x</span><span class="ff6 ws3">6<span class="ff4 ws7">= 0</span></span></span>?<span class="_8 blank"> </span>(c)<span class="_9 blank"> </span>Como se de\u2026<span class="_2 blank"></span>ne<span class="_7 blank"> </span>a função <span class="ff3 ls15">f<span class="ff9 fs2 ls21 v1">0</span></span><span class="ff4">?</span></div><div class="t m0 x4 h3 yd ff1 fs1 fc0 sc0 ls3a ws17">4.1D <span class="ff2 ws2">In<span class="_0 blank"></span>v<span class="_0 blank"></span>estigue a deriv<span class="_4 blank"></span>abilidade da função dada no p<span class="_a blank"> </span>on<span class="_0 blank"></span>to indicado.</span></div><div class="t m0 x2 hc ye ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls12">x</span><span class="ff4 ws7">= 0;<span class="_9 blank"> </span><span class="ff3 ls10">f</span><span class="ws3">(<span class="ff3">x</span><span class="wsd">) = <span class="ff8 v3">8</span></span></span></span></div><div class="t m0 xa h6 yf ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 xa h6 y10 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 xb hd y11 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls22 v1">2</span><span class="ff2 ws2">, se </span><span class="ls12">x<span class="ff6 ls3">\ue014</span></span><span class="ff4">0</span></div><div class="t m0 xb h3 y12 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff2 ws2">, se </span><span class="wse">x > <span class="ff4">0</span></span></div><div class="t m0 xc hc ye ff2 fs1 fc0 sc0 ls3a ws6">(b) <span class="ff3 ls12">x</span><span class="ff4 ws7">= 1;<span class="_9 blank"> </span><span class="ff3 ls10">f</span><span class="ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff8 v3">8</span></span></span></span></div><div class="t m0 xd h6 yf ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 xd h6 y10 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 xe h3 y13 ff6 fs1 fc0 sc0 ls23">p<span class="ff3 ls3a ws3 v7">x<span class="ff2 ws2">, se <span class="ff4 ls24">0</span><span class="ff3 wse">< x < <span class="ff4">1</span></span></span></span></div><div class="t m0 xe h3 y12 ff4 fs1 fc0 sc0 lsa">2<span class="ff3 lsd">x<span class="ff6 ls8">\ue000</span></span>1<span class="ff2 ls3a ws2">, se </span><span class="ls24">1<span class="ff6 ls3">\ue014<span class="ff3 ls3a wse">x < <span class="ff4">2</span></span></span></span></div><div class="t m0 x2 hc y14 ff2 fs1 fc0 sc0 ls3a ws8">(c) <span class="ff3 ls12">x</span><span class="ff4 ws7">= 1;<span class="_9 blank"> </span><span class="ff3 ls10">f</span><span class="ws3">(<span class="ff3">x</span><span class="wsd">) = <span class="ff8 v3">8</span></span></span></span></div><div class="t m0 xa h6 y15 ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 xa h6 y16 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 xb h3 y17 ff6 fs1 fc0 sc0 ls23">p<span class="ff3 ls3a ws3 v7">x<span class="ff2 ws2">, se <span class="ff4 ls24">0</span><span class="ff3 wse">< x < <span class="ff4">1</span></span></span></span></div><div class="t m0 xb he y18 ff5 fs2 fc0 sc0 ls3a">1</div><div class="t m0 xb hf y19 ff5 fs2 fc0 sc0 ls25">2<span class="ff4 fs1 ls3a ws3 v1">(<span class="ff3 lsd">x</span><span class="ws18">+<span class="_b blank"> </span>1) <span class="ff2 ws2">, se </span><span class="ls26">1<span class="ff6 ls3">\ue014</span></span><span class="ff3 wse">x < </span>2</span></span></div><div class="t m0 xc h3 y14 ff2 fs1 fc0 sc0 ls3a ws6">(d) <span class="ff3 ls12">x</span><span class="ff4 ws7">= 0;<span class="_9 blank"> </span><span class="ff3 ls10">f</span><span class="ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff6">j<span class="ff3">x</span>j</span></span></span></div><div class="t m0 x4 h10 y1a ff1 fs1 fc0 sc0 ls3a ws19">4.1E <span class="ff2 ws2">Existe algum p<span class="_5 blank"> </span>onto no qual a função <span class="ff3 ls0">y<span class="ff4 ls3">=</span></span><span class="ff8 v8">\ue00c</span></span></div><div class="t m0 xf h11 y1b ff8 fs1 fc0 sc0 ls27">\ue00c<span class="ff3 ls3a ws3 v9">x</span><span class="ff5 fs2 ls7 va">2</span><span class="ff6 ls8 v9">\ue000<span class="ff4 lsa">4<span class="ff3 ls3a ws3">x<span class="ff8 v8">\ue00c</span></span></span></span></div><div class="t m0 x10 h6 y1b ff8 fs1 fc0 sc0 ls28">\ue00c<span class="ff2 ls3a ws2 v9">não é deriv<span class="_4 blank"></span>ável?<span class="_c blank"> </span>P<span class="_0 blank"></span>or quê?</span></div><div class="t m0 x4 h3 y1c ff1 fs1 fc0 sc0 ls3a ws1a">4.1F <span class="ff2 wsb">Seja <span class="ff3 lse">f</span><span class="ws2">uma função deriv<span class="_d blank"></span>áv<span class="_0 blank"></span>el em <span class="ff3 ls12">x</span><span class="ff4 ws7">= 1<span class="_7 blank"> </span></span>tal que<span class="_c blank"> </span><span class="ff4">lim</span></span></span></div><div class="t m0 x11 h12 y1d ffa fs2 fc0 sc0 ls29">h<span class="ff9 ls3a ws1b">!<span class="ff5">0</span></span></div><div class="t m0 x12 h3 y1e ff3 fs1 fc0 sc0 ls10">f<span class="ff4 ls3a ws5">(1 + </span><span class="ls2a">h<span class="ff4 ls3a">)</span></span></div><div class="t m0 x13 h13 y1f ff3 fs1 fc0 sc0 ls2b">h<span class="ff4 ls3a ws7 v2">= 5</span><span class="ls2c v2">:<span class="ff2 ls3a ws1c">Calcule <span class="ff3 ls2">f</span><span class="ff4 ws10">(1) </span><span class="ls2d">e<span class="ff3 ls15">f<span class="ff9 fs2 ls14 v1">0</span></span></span><span class="ff4 ws1d">(1) <span class="ff3">:</span></span></span></span></div><div class="t m0 x4 hb y20 ff1 fs1 fc0 sc0 ls3a ws1e">4.1G <span class="ff2 wsc">Sup onha<span class="_e blank"> </span>que<span class="_e blank"> </span><span class="ff3 ls2e">f</span><span class="ws1f">seja uma função deriv<span class="_4 blank"></span>ável em <span class="ff7 ls2f">R</span><span class="ws20">, satisfazendo <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3 ls30">a</span><span class="ls8">+<span class="ff3 ls31">b</span></span><span class="ws14">) = <span class="ff3 ls10">f</span></span>(<span class="ff3">a</span><span class="ws21">) + <span class="ff3 ls10">f</span></span>(<span class="ff3 ls32">b</span><span class="ws22">) +</span></span></span></span></span></div><div class="t m0 x1 hb y21 ff4 fs1 fc0 sc0 ls3a ws3">5<span class="ff3">ab<span class="ff2 ls33">,</span><span class="ff6">8</span><span class="ws23">a; b<span class="_f blank"> </span><span class="ff6 ls34">2<span class="ff7 ls2f">R</span></span><span class="ff2 ws24">. Se </span></span></span>lim</div><div class="t m0 x14 h12 y22 ffa fs2 fc0 sc0 ls29">h<span class="ff9 ls3a ws1b">!<span class="ff5">0</span></span></div><div class="t m0 x15 h3 y23 ff3 fs1 fc0 sc0 ls10">f<span class="ff4 ls3a ws3">(</span><span class="ls35">h<span class="ff4 ls3a">)</span></span></div><div class="t m0 x16 h13 y24 ff3 fs1 fc0 sc0 ls36">h<span class="ff4 ls3a ws7 v2">= 3<span class="ff2 ws2">, determine <span class="ff3 ls10">f</span><span class="ff4 ws10">(0) </span><span class="ls17">e<span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ls37">)</span><span class="ff3">:</span></span></span></span></div><div class="t m0 x4 hc y25 ff1 fs1 fc0 sc0 ls3a ws25">4.1H <span class="ff2 ws26">Calcule <span class="ff3 ls38">a</span><span class="ls39">e<span class="ff3 ls31">b</span></span><span class="ws27">,<span class="_10 blank"> </span>de mo<span class="_5 blank"> </span>do que a função <span class="ff3 ls2">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws28">) = <span class="ff8 v3">8</span></span></span></span></span></div><div class="t m0 x17 h6 y26 ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 x17 h6 y27 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 x18 h14 y28 ff4 fs1 fc0 sc0 ls3a ws3">3<span class="ff3">x<span class="ff5 fs2 ls22 v1">2</span><span class="ff2 ws2">, se </span><span class="ls12">x<span class="ff6 ls3">\ue014</span></span></span>1</div><div class="t m0 x18 h3 y29 ff3 fs1 fc0 sc0 ls3a ws29">ax <span class="ff4 ls8">+</span><span class="ls31">b</span><span class="ff2 ws2">, se </span><span class="wse">x > <span class="ff4">1</span></span></div><div class="t m0 x19 h3 y25 ff2 fs1 fc0 sc0 ls3a ws27">seja deriv<span class="_4 blank"></span>ável em</div><div class="t m0 x1 h3 y2a ff3 fs1 fc0 sc0 ls12">x<span class="ff4 ls3a ws7">= 1<span class="ff3">:</span></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 x1a y2b w2 h15" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg2.png"><div class="t m0 x1 h3 y2c ff2 fs1 fc0 sc0 ls3a ws2b">18 <span class="ffb fs2 ws2c">D E R IV<span class="_d blank"></span>A<span class="_a blank"> </span>DA S<span class="_11 blank"> </span>C O M P L E M<span class="_a blank"> </span>E N T O S<span class="_e blank"> </span>4</span></div><div class="t m0 x4 h3 y2d ff1 fs1 fc0 sc0 ls3a ws2d">4.1I <span class="ff2 ws2e">Em cada caso, determine as equações das retas tangente e normal ao grá\u2026<span class="_2 blank"></span>co de <span class="ff3 ls13">f</span><span class="ws2f">, no</span></span></div><div class="t m0 x1 h3 y2e ff2 fs1 fc0 sc0 ls3a ws2">p<span class="_5 blank"> </span>onto cuja abscissa é fornecida.</div><div class="t m0 x2 h16 y2f ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = </span><span class="ff3">x<span class="ff5 fs2 ls3b v1">2<span class="ffa ls3c">=<span class="ff5 ls22">3</span></span></span><span class="ws30">;<span class="_12 blank"> </span>x </span></span><span class="ws7">= 8<span class="_6 blank"> </span></span></span><span class="ws9">(b) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff3">x<span class="ff9 fs2 ls3d v1">\ue000<span class="ff5 ls3b">3<span class="ffa">=</span><span class="ls3e">4</span></span></span><span class="ws30">;<span class="_12 blank"> </span>x </span></span><span class="ws7">= 16<span class="_6 blank"> </span></span></span><span class="ws8">(c) <span class="ff3 ls2">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff6 ls23 vb">p</span><span class="ff3 ws31">x;<span class="_12 blank"> </span>x </span><span class="ws7">= 3<span class="ff3">:</span></span></span></span></span></span></div><div class="t m0 x4 h14 y30 ff1 fs1 fc0 sc0 ls3a ws32">4.1J <span class="ff2 ws33">Determine a equação da reta tangen<span class="_0 blank"></span>te à paráb<span class="_5 blank"> </span>ola <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls22 v1">2</span></span></span><span class="ws34">, com inclinação <span class="ff3 ls3f">m<span class="ff4 ls3">=</span></span><span class="ff6 ws3">\ue000<span class="ff4">8<span class="ff3 ls40">:</span><span class="ff2">F<span class="_4 blank"></span>aça</span></span></span></span></span></div><div class="t m0 x1 h3 y31 ff2 fs1 fc0 sc0 ls3a ws2">um grá\u2026<span class="_2 blank"></span>co ilustrando a situação.</div><div class="t m0 x4 hd y32 ff1 fs1 fc0 sc0 ls3a ws35">4.1K <span class="ff2 ws2">Determine a equação da reta normal à curv<span class="_4 blank"></span>a <span class="ff3 ls0">y<span class="ff4 ls3">=</span></span><span class="ff6 ws3">\ue000<span class="ff3">x<span class="ff5 fs2 ls22 v1">3</span><span class="lsa">=</span><span class="ff4">6</span></span></span>, com inclinação <span class="ff3 ls3f">m</span><span class="ff4 ws7">= 8<span class="ff3 ws3">=</span><span class="lsa">9</span><span class="ff3">:</span></span></span></div><div class="t m0 x4 hc y33 ff1 fs1 fc0 sc0 ls3a ws36">4.1L <span class="ff2 ws37">Se <span class="ff3 ls41">y<span class="ff4 ls42">=</span><span class="ls10">f</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ls43">)</span></span><span class="ws38">é a função de\u2026<span class="_3 blank"></span>nida por <span class="ff3 ls44">y<span class="ff4 ls42">=</span></span><span class="ff8 v3">8</span></span></span></div><div class="t m0 x1b h6 y34 ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 x1b h6 y35 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 x1c h3 y36 ff6 fs1 fc0 sc0 ls45">p<span class="ff3 lsd vc">x</span><span class="ls46 vc">\ue000<span class="ff4 ls3a ws3">2<span class="ff2 ws2">, se <span class="ff3 ls12">x<span class="ff6 ls3">\ue015</span></span></span>2</span></span></div><div class="t m0 x1c h17 y37 ff6 fs1 fc0 sc0 ls3a ws3">\ue000<span class="ls23 vd">p</span><span class="ff4 ls47">2</span><span class="ls8">\ue000</span><span class="ff3 ws39">x; <span class="ff2 ws3a">se </span><span class="ls12">x</span></span><span class="ls3">\ue014</span><span class="ff4">2</span></div><div class="t m0 x3 h3 y33 ff2 fs1 fc0 sc0 ls3a ws38">, encon<span class="_0 blank"></span>tre as equações</div><div class="t m0 x1 h3 y38 ff2 fs1 fc0 sc0 ls3a ws2">das retas tangen<span class="_0 blank"></span>te e normal ao grá\u2026<span class="_2 blank"></span>co de <span class="ff3 ls13">f</span>, no p<span class="_a blank"> </span>on<span class="_0 blank"></span>to de abscissa <span class="ff3 ls12">x</span><span class="ff4 ws7">= 2<span class="ff3">:</span></span></div><div class="t m0 x4 h14 y39 ff1 fs1 fc0 sc0 ls3a ws3b">4.1M <span class="ff2 ws3c">Determine a equação da reta que tangencia o grá\u2026<span class="_2 blank"></span>co da função <span class="ff3 ls48">y<span class="ff4 ls49">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls4a v1">2</span></span></span>e é paralela à</span></div><div class="t m0 x1 h3 y3a ff2 fs1 fc0 sc0 ls3a ws3d">reta <span class="ff3 ls0">y</span><span class="ff4 ws7">= 4<span class="ff3 lsd">x</span><span class="ws5">+ 2<span class="ff3">:</span></span></span></div><div class="t m0 x4 h3 y3b ff1 fs1 fc0 sc0 ls3a ws3e">4.1N <span class="ff2 ws3f">V<span class="_4 blank"></span>eri\u2026<span class="_2 blank"></span>que que a reta tangente ao grá\u2026<span class="_2 blank"></span>co da função <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws40">) = 1</span><span class="ff3">=x</span></span>,<span class="_c blank"> </span>no ponto de abscissa</span></div><div class="t m0 x1 h3 y3c ff3 fs1 fc0 sc0 ls12">x<span class="ff4 ls3">=</span><span class="ls3a ws3">a<span class="ff2 ws2">, in<span class="_0 blank"></span>tercepta o eixo <span class="ff3 ls4b">x</span><span class="wsc">no<span class="_e blank"> </span>p onto<span class="_e blank"> </span><span class="ff3 ls4c">A</span><span class="ff4 ws3">(2<span class="ff3 ws41">a; </span><span class="ws18">0) <span class="ff3">:</span></span></span></span></span></span></div><div class="t m0 x4 h18 y3d ff1 fs1 fc0 sc0 ls3a ws42">4.1O <span class="ff2 ws43">Determine as retas horizon<span class="_0 blank"></span>tais que são tangen<span class="_0 blank"></span>tes ao grá\u2026<span class="_2 blank"></span>co da função <span class="ff3 ls4d">g</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_13 blank"> </span></span><span class="ff3 v2">x</span><span class="ff5 fs2 ve">3</span></span></span></div><div class="t m0 x1d h19 y3e ff4 fs1 fc0 sc0 ls4e">3<span class="ls4f v2">+</span><span class="ff3 ls3a ws3 vf">x<span class="ff5 fs2 v1">2</span></span></div><div class="t m0 x1e h1a y3e ff4 fs1 fc0 sc0 ls4e">2<span class="ff6 ls3a v2">\ue000</span></div><div class="t m0 x1 h3 y3f ff4 fs1 fc0 sc0 lsa">2<span class="ff3 lsd">x<span class="ff6 ls8">\ue000</span></span><span class="ls3a ws3">1<span class="ff3">:</span></span></div><div class="t m0 x4 hc y40 ff1 fs1 fc0 sc0 ls3a ws44">4.1P <span class="ff2 ws2">Considere a função <span class="ff3 ls19">f</span><span class="wsc">de\u2026<span class="_3 blank"></span>nida<span class="_e blank"> </span>p or<span class="_7 blank"> </span><span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff8 v3">8</span></span></span></span></span></div><div class="t m0 x1f h6 y41 ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 x1f h6 y42 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 x20 hd y43 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls3e v1">2</span><span class="ff2 ws2">, se </span><span class="ls1a">x<span class="ff6 ls3">\ue014</span></span><span class="ff4">1</span></div><div class="t m0 x20 h3 y44 ff4 fs1 fc0 sc0 ls3a ws3">2<span class="ff2 ws2">, se <span class="ff3 wse">x > </span></span>1</div><div class="t m0 xd h3 y40 ff3 fs1 fc0 sc0 ls3a">:</div><div class="t m0 x1 h3 y45 ff2 fs1 fc0 sc0 ls3a ws2">(a) Esb<span class="_5 blank"> </span>o<span class="_5 blank"> </span>ce o grá\u2026<span class="_3 blank"></span>co de <span class="ff3 ls50">f</span><span class="ws6">(b) <span class="ff3 lse">f</span><span class="ws45">é con<span class="_d blank"></span>tínua em <span class="ff3 ls12">x</span><span class="ff4 ws7">= 1?<span class="_8 blank"> </span></span><span class="ws8">(c) <span class="ff3 ls19">f</span><span class="ws2">é deriv<span class="_4 blank"></span>ável em <span class="ff3 ls1a">x</span><span class="ff4 ws7">= 1?</span></span></span></span></span></div><div class="t m0 x4 hc y46 ff1 fs1 fc0 sc0 ls3a ws46">4.1Q <span class="ff2 ws2">Repita o exercício preceden<span class="_0 blank"></span>te, considerando agora <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff8 v3">8</span></span></span></span></div><div class="t m0 xd h6 y47 ff8 fs1 fc0 sc0 ls3a"><</div><div class="t m0 xd h6 y48 ff8 fs1 fc0 sc0 ls3a">:</div><div class="t m0 xe hd y49 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls3e v1">2</span><span class="ff2 ws2">, se </span><span class="ls1a">x<span class="ff6 ls3">\ue014</span></span><span class="ff4">1</span></div><div class="t m0 xe h3 y4a ff4 fs1 fc0 sc0 ls3a ws3">1<span class="ff2 ws2">, se <span class="ff3 wse">x > </span></span>1</div><div class="t m0 x21 h3 y46 ff3 fs1 fc0 sc0 ls3a">:</div><div class="t m0 x4 hb y4b ff1 fs1 fc0 sc0 ls3a ws47">4.1R <span class="ff2 wsb">Seja <span class="ff3 lse">f</span><span class="ws2">a função de\u2026<span class="_3 blank"></span>nida em <span class="ff7 ls51">R</span><span class="wsc">por<span class="_7 blank"> </span><span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff3 ls52">x</span></span><span class="ff6">j<span class="ff3">x</span>j</span></span>.</span></span></span></div><div class="t m0 x1 h8 y4c ff2 fs1 fc0 sc0 ls3a ws2">(a) Determine <span class="ff3 ls15">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span>)</span>, para <span class="ff3 ls12">x</span><span class="ff6 ws3">6<span class="ff4 ws7">= 0<span class="ff3 ls53">:</span></span></span>(b) Existe <span class="ff3 ls13">f<span class="ff9 fs2 ls54 v1">0</span></span><span class="ff4 ws15">(0)? </span>(c) Esb<span class="_5 blank"> </span>o<span class="_5 blank"> </span>ce o grá\u2026<span class="_3 blank"></span>co de <span class="ff3 lse">f</span>e o de <span class="ff3 ls13">f<span class="ff9 fs2 ls21 v1">0</span><span class="ls3a">:</span></span></div><div class="t m0 x1 h2 y4d ff1 fs0 fc0 sc0 ls3a ws48">4.<span class="_0 blank"></span>0<span class="_1 blank"> </span>Re<span class="_d blank"></span>gra<span class="_d blank"></span>s Bás<span class="_d blank"></span>icas d<span class="_d blank"></span>e De<span class="_0 blank"></span>r<span class="_0 blank"></span>iv<span class="_3 blank"></span>aç<span class="_0 blank"></span>ão</div><div class="t m0 x22 h1b y4e ff1 fs1 fc0 sc0 ls3a ws1">4.2A <span class="ff2 ws49">Se <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls4 v1">2</span><span class="ff6 ls8">\ue000<span class="ls23 v8">p</span></span><span class="ff4 ws5">1 + </span><span class="ls55">u<span class="ff5 fs2 ls56 v10">2</span></span></span></span><span class="ls17">e<span class="ff3 ls57">u<span class="ff4 ls58">=</span><span class="lsd v2">x</span></span></span><span class="ff4 ws5 v2">+ 1</span></span></div><div class="t m0 x23 h1c y4f ff3 fs1 fc0 sc0 lsd">x<span class="ff6 ls8">\ue000<span class="ff4 ls59">1<span class="ff2 ls3a ws2 v2">, calcule<span class="_10 blank"> </span></span></span></span><span class="ls3a vf">dy</span></div><div class="t m0 xf h1a y4f ff3 fs1 fc0 sc0 ls3a ws4a">dx <span class="v2">:</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="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x24 y50 w3 h1d" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg3.png"><div class="t m0 x24 h3 y2c ffb fs2 fc0 sc0 ls3a ws4b">C Á L C U LO<span class="_e blank"> </span>D E<span class="_f blank"> </span>U M A<span class="_e blank"> </span>V<span class="_d blank"></span>A R IÁ<span class="_d blank"></span>V E L<span class="_14 blank"> </span>M A R IV<span class="_d blank"></span>A L D O<span class="_e blank"> </span>P<span class="_f blank"> </span>M ATO S<span class="_15 blank"> </span><span class="ff2 fs1">19</span></div><div class="t m0 x4 h4 y51 ff1 fs1 fc0 sc0 ls3a wsa">4.2B <span class="ff2 ws49">Se <span class="ff3 ls0">y<span class="ff4 ls58">=</span><span class="lsd v2">x</span></span><span class="ff4 ws5 v2">+ 1</span></span></div><div class="t m0 x14 h19 y52 ff3 fs1 fc0 sc0 lsd">x<span class="ff6 ls8">\ue000<span class="ff4 ls59">1<span class="ff2 ls3a ws2 v2">, v<span class="_0 blank"></span>eri\u2026<span class="_2 blank"></span>que que <span class="ff4 ws4c">(1 <span class="ff6 ls8">\ue000</span><span class="ff3 ws3">x</span><span class="ls5a">)<span class="ff3 ls5b v2">d</span><span class="ff5 fs2 ls22 ve">2</span></span><span class="ff3 v2">y</span></span></span></span></span></div><div class="t m0 x25 h1e y52 ff3 fs1 fc0 sc0 ls3a ws3">dx<span class="ff5 fs2 ls5c v10">2</span><span class="ff4 ws4d v2">=<span class="_f blank"> </span>2 </span><span class="vf">dy</span></div><div class="t m0 x26 h1f y52 ff3 fs1 fc0 sc0 ls3a ws4a">dx <span class="v2">:</span></div><div class="t m0 x4 h4 y53 ff1 fs1 fc0 sc0 ls3a ws4e">4.2C <span class="ff2 wsc">Sup onha<span class="_7 blank"> </span>que<span class="_13 blank"> </span><span class="ff3 ls5d">x<span class="ff4 ls5e">=</span><span class="ls52">x</span></span><span class="ff4 ws3">(<span class="ff3 ls5f">t</span><span class="ls60">)</span></span><span class="ws4f">seja uma função deriv<span class="_4 blank"></span>áv<span class="_0 blank"></span>el em <span class="ff7 ls2f">R</span><span class="ws50">.<span class="_16 blank"> </span>Se <span class="ff3 ls61">y<span class="ff4 ls62">=<span class="ls3a v2">1</span></span></span></span></span></span></div><div class="t m0 x27 h1a y54 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls4 v10">2</span><span class="ff4 ws51">+<span class="_b blank"> </span>1 <span class="ff2 ws4f v2">, veri\u2026<span class="_2 blank"></span>que que,</span></span></div><div class="t m0 x1 h4 y55 ff6 fs1 fc0 sc0 ls3a ws3">8<span class="ff3 ls63">t</span><span class="ls34">2<span class="ff7 ls2f">R</span></span><span class="ff2 ws2">, tem-se<span class="_c blank"> </span><span class="ff3 v2">dy</span></span></div><div class="t m0 x28 h1c y56 ff3 fs1 fc0 sc0 ls3a ws52">dt <span class="ff4 ls3 v2">=</span><span class="ff6 ws3 v2">\ue000<span class="ff4">2<span class="ff3 ws53">xy <span class="ff5 fs2 ls64 v1">2</span><span class="v2">dx</span></span></span></span></div><div class="t m0 x29 h1a y56 ff3 fs1 fc0 sc0 ls3a ws54">dt <span class="v2">:</span></div><div class="t m0 x4 h14 y57 ff1 fs1 fc0 sc0 ls3a ws55">4.2D <span class="ff2 wsc">Sup onha<span class="_c blank"> </span>que<span class="_13 blank"> </span><span class="ff3 ls65">x<span class="ff4 ls66">=</span><span class="ls52">x</span></span><span class="ff4 ws3">(<span class="ff3 ls67">t</span><span class="ls68">)</span></span><span class="ws56">seja uma função deriv<span class="_d blank"></span>ável até a segunda ordem.<span class="_17 blank"> </span>Se <span class="ff3 ls69">y<span class="ff4 ls66">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls22 v1">3</span></span></span>,</span></span></div><div class="t m0 x1 h18 y58 ff2 fs1 fc0 sc0 ls3a ws57">v<span class="_0 blank"></span>eri\u2026<span class="_2 blank"></span>que que<span class="_10 blank"> </span><span class="ff3 ls5b v2">d</span><span class="ff5 fs2 ls3e ve">2</span><span class="ff3 v2">y</span></div><div class="t m0 x2a h20 y59 ff3 fs1 fc0 sc0 ls3a ws3">dt<span class="ff5 fs2 ls6a v10">2</span><span class="ff4 ws7 v2">= 6</span><span class="ls52 v2">x</span><span class="ff8 ls6b v11">\ue012</span><span class="vf">dx</span></div><div class="t m0 x2b h20 y59 ff3 fs1 fc0 sc0 ls3a ws54">dt <span class="ff8 ls6c v11">\ue013</span><span class="ff5 fs2 v12">2</span></div><div class="t m0 x2c h18 y58 ff4 fs1 fc0 sc0 ls3a ws5">+ 3<span class="ff3 ws3">x<span class="ff5 fs2 ls6d v1">2</span><span class="ls5b v2">d</span><span class="ff5 fs2 ls3e ve">2</span><span class="v2">x</span></span></div><div class="t m0 x2d h1a y59 ff3 fs1 fc0 sc0 ls3a ws3">dt<span class="ff5 fs2 ls6e v10">2</span><span class="v2">:</span></div><div class="t m0 x4 h8 y5a ff1 fs1 fc0 sc0 ls3a ws58">4.2E <span class="ff2 wsc">Sab endo-se<span class="_c blank"> </span>que<span class="_c blank"> </span><span class="ff3 ls4d">g</span><span class="ff4 ws3">(<span class="ff6">\ue000</span><span class="ws59">1) = 2<span class="ff3 ws5a">;<span class="_18 blank"> </span>f </span>(2) = </span><span class="ff6">\ue000</span><span class="lsa">3</span><span class="ff3 ws5b">;<span class="_18 blank"> </span>g <span class="ff9 fs2 ls14 v1">0</span></span>(<span class="ff6">\ue000</span><span class="ws59">1) = </span><span class="ff6">\ue000</span><span class="lsa">1</span><span class="ff3">=</span><span class="ls6f">3</span></span><span class="ls70">e<span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span></span><span class="ff4 ws59">(2) = 6</span><span class="ws5c">,<span class="_10 blank"> </span>determine as</span></span></div><div class="t m0 x1 h3 y5b ff2 fs1 fc0 sc0 ls3a ws2">equações das retas tangen<span class="_0 blank"></span>te e normal à curv<span class="_4 blank"></span>a <span class="ff3 ls71">h</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff3 ls10">f</span></span>(<span class="ff3 ls72">g</span>(<span class="ff3">x</span>))</span>, em <span class="ff3 ls12">x<span class="ff4 ls3">=</span></span><span class="ff6 ws3">\ue000<span class="ff4 lsa">1</span><span class="ff3">:</span></span></div><div class="t m0 x4 h21 y5c ff1 fs1 fc0 sc0 ls3a ws5d">4.2F <span class="ff2 ws5e">Se <span class="ff3 ls73">h</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws5f">) = [<span class="ff3 ls2">f</span></span>(<span class="ff3">x</span>)]<span class="ff5 fs2 ls74 v13">3</span><span class="ls75">+<span class="ff3 ls10">f<span class="ff8 ls76 v14">\ue000</span><span class="ls3a">x<span class="ff5 fs2 ls22 v1">3</span><span class="ff8 ls76 v14">\ue001</span></span></span></span></span><span class="ws60">,<span class="_10 blank"> </span>calcule <span class="ff3 ls35">h<span class="ff9 fs2 ls54 v1">0</span></span><span class="ff4 ws3">(2)</span><span class="wsc">,<span class="_12 blank"> </span>sabendo<span class="_10 blank"> </span>que<span class="_c blank"> </span><span class="ff3 ls2">f</span><span class="ff4 ws5f">(2) = 1<span class="ff3 ws61">;<span class="_19 blank"> </span>f <span class="ff9 fs2 ls14 v1">0</span></span><span class="ws62">(2) =<span class="_12 blank"> </span>7 </span></span><span class="ws63">e que</span></span></span></span></div><div class="t m0 x1 h8 y5d ff3 fs1 fc0 sc0 ls13">f<span class="ff9 fs2 ls14 v1">0</span><span class="ff4 ls3a ws64">(8) = <span class="ff6 ws3">\ue000</span><span class="lsa">3</span><span class="ff3">:</span></span></div><div class="t m0 x4 h3 y5e ff1 fs1 fc0 sc0 ls3a ws65">4.2G <span class="ff2 ws66">Use a Regra da Cadeia para mostrar que a deriv<span class="_d blank"></span>ada de uma função par é uma função</span></div><div class="t m0 x1 h3 y5f ff2 fs1 fc0 sc0 ls3a ws2">ímpar e que a deriv<span class="_4 blank"></span>ada de um<span class="_5 blank"> </span>a função ímpar é uma função par.</div><div class="t m0 x4 h3 y60 ff1 fs1 fc0 sc0 ls3a ws67">4.2H <span class="ff2 ws2">Calcule a deriv<span class="_4 blank"></span>ada de primeira ordem de<span class="_7 blank"> </span>cada uma das funções abaixo.</span></div><div class="t m0 x2 h4 y61 ff2 fs1 fc0 sc0 ls3a ws68">(a) <span class="ff3 ls0">y<span class="ff4 ls58">=</span><span class="ls3a v2">\ue019</span></span></div><div class="t m0 x2e h1c y62 ff3 fs1 fc0 sc0 ls77">x<span class="ff4 ls3a ws23 v2">+<span class="_b blank"> </span>ln 2<span class="_1a blank"> </span><span class="ff2 ws9">(b) <span class="ff3 ls6">y<span class="ff4 ls78">=</span><span class="ls3a v2">\ue019</span></span></span></span></div><div class="t m0 x2f h1c y62 ff3 fs1 fc0 sc0 ls77">x<span class="ff4 ls3a ws23 v2">+<span class="_b blank"> </span>ln 2<span class="_1b blank"> </span><span class="ff2 ws69">(c) <span class="ff3 ls0">y<span class="ff4 ls78">=<span class="ls3a v2">1</span></span></span></span></span></div><div class="t m0 xe h22 y62 ff4 fs1 fc0 sc0 ls79">4<span class="ff6 ls7a v2">\ue000</span><span class="ff5 fs2 ls3a v15">1</span></div><div class="t m0 x30 h23 y63 ff5 fs2 fc0 sc0 ls7b">3<span class="ff3 fs1 lsd v1">x<span class="ff4 ls8">+<span class="ff3 ls3a ws3">x</span></span></span><span class="ls7 vb">2</span><span class="ff6 fs1 ls8 v1">\ue000<span class="ff4 lsa">0<span class="ff3 ls3a ws3">:</span>5<span class="ff3 ls3a ws3">x</span></span></span><span class="ls3a vb">4</span></div><div class="t m0 x2 h24 y64 ff2 fs1 fc0 sc0 ls3a ws9">(d) <span class="ff3 ls0">y<span class="ff4 ls58">=<span class="ls3a ws5 v2">1 + </span><span class="ff6 ls45 v16">p</span></span><span class="ls3a v2">x</span></span></div><div class="t m0 x2e h25 y65 ff4 fs1 fc0 sc0 ls7c">1<span class="ff6 ls46">\ue000<span class="ls45 vb">p</span><span class="ff3 ls7d">x<span class="ff2 ls3a ws69 v2">(e) </span><span class="ls0 v2">y</span></span></span><span class="ls3 v2">=<span class="ff3 ls52">x<span class="ff4 ls3a ws6a">arcsen </span><span class="ls7e">x<span class="ff2 ls3a ws6b">(f )<span class="_e blank"> </span></span><span class="ls0">y<span class="ff4 ls58">=<span class="ff8 ls76 v17">\ue000</span></span><span class="ls3a ws3 vb">x</span><span class="ff5 fs2 ls7 v18">2</span></span></span></span></span><span class="ls3a ws5 v19">+ 1</span><span class="ff8 ls7f v1a">\ue001</span><span class="ls3a ws6c v19">arctg <span class="ff3 ls80">x<span class="ff6 ls8">\ue000</span><span class="ls3a">x</span></span></span></div><div class="t m0 x31 h3 y66 ff4 fs1 fc0 sc0 ls3a">2</div><div class="t m0 x2 h4 y67 ff2 fs1 fc0 sc0 ls3a ws4">(g) <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">e<span class="ffa fs2 ls81 v1">x</span><span class="ff4 ws6d">cos </span><span class="ls82">x</span></span></span><span class="ws9">(h) <span class="ff3 ls6">y<span class="ff4 ls83">=<span class="ls3a v2">1</span></span></span></span></div><div class="t m0 x2f h1c y68 ff3 fs1 fc0 sc0 ls84">x<span class="ff4 ls3a ws6e v2">+<span class="_b blank"> </span>2 ln </span><span class="lsd v2">x<span class="ff6 ls7a">\ue000<span class="ff4 ls3a ws23 v2">ln <span class="ff3">x</span></span></span></span></div><div class="t m0 x32 h26 y68 ff3 fs1 fc0 sc0 ls85">x<span class="ff2 ls3a ws6f v2">(i) </span><span class="ls0 v2">y<span class="ff4 ls3a ws70">=<span class="_f blank"> </span>(3 <span class="ff6 ls8">\ue000</span><span class="ws71">2 sen <span class="ff3 ws3">x<span class="ff4">)<span class="ff5 fs2 v13">5</span></span></span></span></span></span></div><div class="t m0 x2 h27 y69 ff2 fs1 fc0 sc0 ls3a ws72">(j) <span class="ff3 ls0">y</span><span class="ff4 ws7">= 2<span class="ff3 lsd">x</span><span class="ws73">+<span class="_b blank"> </span>5 cos<span class="ff5 fs2 ls86 v1">3</span><span class="ff3 ls87">x</span></span></span><span class="ws74">(k) <span class="ff3 ls6">y<span class="ff4 ls3">=<span class="ff8 ls88 v1b">r</span><span class="ls3a ws71 v2">3 sen </span></span><span class="lsd v2">x<span class="ff6 ls8">\ue000<span class="ff4 ls3a ws73">2 cos <span class="ff3">x</span></span></span></span></span></span></div><div class="t m0 x33 h28 y6a ff4 fs1 fc0 sc0 ls89">5<span class="ff2 ls3a ws6f v2">(l) <span class="ff3 ls0">y<span class="ff4 ls3">=<span class="ff6 ls23 vd">p</span></span><span class="ls3a ws3">xe<span class="ffa fs2 ls8a v10">x</span><span class="ff4 ls8">+</span>x</span></span></span></div><div class="t m0 x2 h4 y6b ff2 fs1 fc0 sc0 ls3a ws75">(m) <span class="ff3 ls6">y</span><span class="ff4 ws73">=<span class="_f blank"> </span>arccos (<span class="ff3 ws3">e<span class="ffa fs2 ls8b v1">x</span></span><span class="ls8c">)</span></span><span class="ws9">(n) <span class="ff3 ls6">y</span><span class="ff4 ws73">=<span class="_f blank"> </span>sen (3<span class="ff3 ws3">x</span><span class="ws5">) + cos(<span class="_1c blank"> </span><span class="ff3 v2">x</span></span></span></span></div><div class="t m0 x34 h29 y6c ff4 fs1 fc0 sc0 ls8d">5<span class="ls3a ws5 v2">) + tg<span class="_1d blank"> </span>(</span><span class="ff6 ls23 v16">p</span><span class="ff3 ls3a ws3 v2">x</span><span class="ls8e v2">)<span class="ff2 ls3a ws68">(o) <span class="ff3 ls0">y<span class="ff4 ls78">=</span></span></span></span><span class="ls3a ws5 vf">1 + cos<span class="_1d blank"> </span>(2<span class="ff3 ws3">x<span class="ff4">)</span></span></span></div><div class="t m0 xe h3 y6c ff4 fs1 fc0 sc0 ls47">1<span class="ff6 ls8">\ue000</span><span class="ls3a ws76">cos (2<span class="ff3 ws3">x</span>)</span></div><div class="t m0 x2 h2a y6d ff2 fs1 fc0 sc0 ls3a ws9">(p) <span class="ff3 ls0">y</span><span class="ff4 ws77">=<span class="_f blank"> </span>arctg <span class="ff8 ls8f v16">\ue012</span><span class="ws5 v2">1 + <span class="ff3">x</span></span></span></div><div class="t m0 x35 h20 y6e ff4 fs1 fc0 sc0 ls47">1<span class="ff6 ls8">\ue000<span class="ff3 ls90">x<span class="ff8 ls91 v11">\ue013</span><span class="ff2 ls3a ws74 v2">(q) </span><span class="ls6 v2">y</span></span></span><span class="ls3a ws78 v2">=<span class="_f blank"> </span>ln (sen <span class="ff3 ws3">x<span class="ff4 ls92">)</span><span class="ff2 ws79">(r) </span><span class="ls0">y</span><span class="ff4 ws7">= ln<span class="ff5 fs2 ls93 v1c">2</span></span><span class="lsd">x</span><span class="ff4 ws7a">+<span class="_b blank"> </span>ln (ln </span>x<span class="ff4">)</span></span></span></div><div class="t m0 x4 h2b y6f ff1 fs1 fc0 sc0 ls3a ws7b">4.2I <span class="ff2 ws2">V<span class="_4 blank"></span>eri\u2026<span class="_2 blank"></span>que que a função <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">xe<span class="ff9 fs2 ls3d v1">\ue000<span class="ffa ls94">x</span></span></span></span>é solução da equação <span class="ff3 ws7c">xy <span class="ff9 fs2 ls95 v1">0</span><span class="ff4 ws70">=<span class="_f blank"> </span>(1 <span class="ff6 ls8">\ue000</span></span><span class="ws3">x<span class="ff4 ls37">)</span><span class="ws7d">y :</span></span></span></span></div><div class="t m0 x4 h4 y70 ff1 fs1 fc0 sc0 ls3a ws7e">4.2J <span class="ff2 ws2">V<span class="_4 blank"></span>eri\u2026<span class="_2 blank"></span>que que a função <span class="ff3 ls6">y<span class="ff4 ls96">=<span class="ls3a v2">1</span></span></span></span></div><div class="t m0 x36 h13 y71 ff4 fs1 fc0 sc0 ls3a ws5">1 + <span class="ff3 lsd">x</span><span class="ws23">+<span class="_b blank"> </span>ln <span class="ff3 ls97">x</span><span class="ff2 ws2 v2">é solução da equação <span class="ff3 ws7f">xy <span class="ff9 fs2 ls95 v1">0</span><span class="ff4 ws7">= (</span><span class="ls98">y</span></span></span><span class="v2">ln <span class="ff3 lsd">x<span class="ff6 ls8">\ue000</span></span><span class="ws80">1) <span class="ff3 ws7d">y :</span></span></span></span></div><div class="t m0 x4 h2c y72 ff1 fs1 fc0 sc0 ls3a ws81">4.2K <span class="ff2 ws82">Se <span class="ff3 ls99">a</span><span class="ls9a">e<span class="ff3 ls9b">b</span></span><span class="ws83">são constan<span class="_0 blank"></span>tes quaisquer, v<span class="_0 blank"></span>eri\u2026<span class="_2 blank"></span>que que a função <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">ae<span class="ff9 fs2 ls3d v1">\ue000<span class="ffa ls9c">x</span></span><span class="ff4 ls9d">+</span>be<span class="ff9 fs2 ls3d v1">\ue000<span class="ff5 ls3b">2<span class="ffa ls9e">x</span></span></span></span></span>é solução da</span></span></div><div class="t m0 x1 h8 y2a ff2 fs1 fc0 sc0 ls3a ws84">equação <span class="ff3 ls9f">y</span><span class="ff9 fs2 ws85 v1">00 </span><span class="ff4 ws5">+ 3<span class="ff3 ls9f">y<span class="ff9 fs2 lsa0 v1">0</span></span>+ 2<span class="ff3 ls6">y</span><span class="ws7">= 0<span class="ff3">:</span></span></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="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x1a y73 w2 h2d" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg4.png"><div class="t m0 x1 h3 y2c ff2 fs1 fc0 sc0 ls3a ws2b">20 <span class="ffb fs2 ws2c">D E R IV<span class="_d blank"></span>A<span class="_a blank"> </span>DA S<span class="_11 blank"> </span>C O M P L E M<span class="_a blank"> </span>E N T O S<span class="_e blank"> </span>4</span></div><div class="t m0 x4 h3 y2d ff1 fs1 fc0 sc0 ls3a ws86">4.2L <span class="ff2 ws87">Os grá\u2026<span class="_2 blank"></span>cos da coluna da esqu<span class="_5 blank"> </span>erda são das deriv<span class="_4 blank"></span>adas das f<span class="_5 blank"> </span>unções cujos grá\u2026<span class="_3 blank"></span>cos estão na</span></div><div class="t m0 x1 h3 y2e ff2 fs1 fc0 sc0 ls3a ws2">coluna da direita.<span class="_c blank"> </span>F<span class="_4 blank"></span>aça a corresp<span class="_5 blank"> </span>ondência, numerando, con<span class="_d blank"></span>venien<span class="_d blank"></span>temen<span class="_0 blank"></span>te, a coluna da direita.</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 x1 y74 w4 h2e" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg5.png"><div class="t m0 x24 h3 y2c ffb fs2 fc0 sc0 ls3a ws4b">C Á L C U LO<span class="_e blank"> </span>D E<span class="_f blank"> </span>U M A<span class="_e blank"> </span>V<span class="_d blank"></span>A R IÁ<span class="_d blank"></span>V E L<span class="_14 blank"> </span>M A R IV<span class="_d blank"></span>A L D O<span class="_e blank"> </span>P<span class="_f blank"> </span>M ATO S<span class="_15 blank"> </span><span class="ff2 fs1">21</span></div><div class="t m0 x1 h2 y75 ff1 fs0 fc0 sc0 ls3a ws88">4.<span class="_0 blank"></span>1<span class="_1 blank"> </span>Re<span class="_d blank"></span>gra d<span class="_d blank"></span>a Ca<span class="_0 blank"></span>de<span class="_d blank"></span>ia e De<span class="_d blank"></span>riv<span class="_3 blank"></span>aç<span class="_d blank"></span>ão Im<span class="_d blank"></span>pl<span class="_0 blank"></span>íc<span class="_0 blank"></span>ita</div><div class="t m0 x22 h3 y76 ff1 fs1 fc0 sc0 ls3a ws89">4.3A <span class="ff2 ws8a">Cada uma das equações abaixo de\u2026<span class="_3 blank"></span>ne, implicitamen<span class="_0 blank"></span>te, <span class="ff3 lsa1">y</span>como função de <span class="ff3 ws3">x</span><span class="ws8b">. Encontre</span></span></div><div class="t m0 x37 h3 y77 ff3 fs1 fc0 sc0 ls3a">dy</div><div class="t m0 x37 h1a y78 ff3 fs1 fc0 sc0 ls3a ws8c">dx <span class="ff2 v2">.</span></div><div class="t m0 x2 h4 y79 ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls9f">y<span class="ff5 fs2 lsa2 v1">3</span><span class="ff4 lsa3">=</span><span class="lsd">x<span class="ff4 ls8">+</span><span class="lsa4">y</span></span></span><span class="ws8d">(b) <span class="ff3 v2">y</span></span></div><div class="t m0 x38 h1c y7a ff3 fs1 fc0 sc0 lsd">x<span class="ff6 ls8">\ue000</span><span class="lsa5">y<span class="ff4 lsa6 v2">+</span><span class="ls3a vf">x</span></span></div><div class="t m0 x36 h2f y7a ff3 fs1 fc0 sc0 lsa7">y<span class="ff4 ls3 v2">=</span><span class="ff6 ls23 v16">p</span><span class="lsa8 v2">x<span class="ff2 ls3a ws8">(c) <span class="ff6 ls23 v2">p</span><span class="ff3 lsd">x<span class="ff4 ls8">+</span><span class="ls0">y<span class="ff4 lsa3">=<span class="ff6 ls23 vb">p</span></span><span class="lsa9">y</span></span></span><span class="ff4 ws5">+ 1</span></span></span></div><div class="t m0 x2 h30 y7b ff2 fs1 fc0 sc0 ls3a ws9">(d) <span class="ff4 ws73">4 cos <span class="ff3 ls52">x</span><span class="ws8e">sen <span class="ff3 ls0">y</span><span class="ws7">= 1<span class="_1e blank"> </span></span></span></span><span class="ws8">(e) <span class="ff3 ws8f">xy <span class="ff4 ws90">=<span class="_f blank"> </span>cotg (</span><span class="ws53">xy <span class="ff4 lsaa">)</span></span></span><span class="ws6b">(f )<span class="_e blank"> </span><span class="ff6 ls23 v6">p</span><span class="ff3 ws8f">xy <span class="ff4 ws91">=<span class="_f blank"> </span>1 + </span><span class="ws3">x<span class="ff5 fs2 ls22 v1">2</span>y</span></span></span></span></div><div class="t m0 x4 h31 y7c ff1 fs1 fc0 sc0 ls3a wsa">4.3B <span class="ff2 ws49">Se <span class="ff3 lsab">n</span><span class="ws2">é um n<span class="_0 blank"></span>úmero natural, qual é a deriv<span class="_4 blank"></span>ada de ordem <span class="ff3 lsab">n</span><span class="ws92">da função <span class="ff3 ls6">y</span><span class="ff4 ws7">= (<span class="ff3 ws93">ax </span><span class="ls8">+<span class="ff3 ls31">b</span></span><span class="ws3">)<span class="ffa fs2 lsac v13">n</span>?</span></span></span></span></span></div><div class="t m0 x4 h14 y7d ff1 fs1 fc0 sc0 ls3a ws94">4.3C <span class="ff2 ws95">Determine as retas tangen<span class="_0 blank"></span>te e normal à circunferência <span class="ff3 ws3">x<span class="ff5 fs2 lsad v1">2</span><span class="ff4 lsae">+</span><span class="ls9f">y<span class="ff5 fs2 lsa2 v1">2</span></span><span class="ff4 ws7">= 25</span></span><span class="wsc">,<span class="_f blank"> </span>no<span class="_b blank"> </span>p onto<span class="_b blank"> </span><span class="ff3 lsaf">P<span class="ff5 fs2 lsa2 v1d">0</span></span><span class="ff4 ws7">= (3<span class="ff3 lsb0">;</span><span class="ws18">4) <span class="ff3">:</span></span></span></span></span></div><div class="t m0 x4 h18 y7e ff1 fs1 fc0 sc0 ls3a ws96">4.3D <span class="ff2 ws97">Mesma questão preceden<span class="_0 blank"></span>te,<span class="_1f blank"> </span>considerando agora a hip<span class="_5 blank"> </span>érb<span class="_5 blank"> </span>ole<span class="_19 blank"> </span><span class="ff3 ws3 v2">x</span><span class="ff5 fs2 ve">2</span></span></div><div class="t m0 x39 h19 y7f ff4 fs1 fc0 sc0 ls3a ws98">16 <span class="ff6 lsb1 v2">\ue000</span><span class="ff3 ls9f vf">y</span><span class="ff5 fs2 v1e">2</span></div><div class="t m0 x31 h1a y7f ff4 fs1 fc0 sc0 lsb2">9<span class="ls3a ws99 v2">=<span class="_1f blank"> </span>1 <span class="ff2 lsb3">e<span class="ff3 lsaf">P<span class="ff5 fs2 lsb4 v1d">0</span></span></span>=</span></div><div class="t m0 x1 h3 y80 ff4 fs1 fc0 sc0 ls3a ws3">(<span class="ff6">\ue000</span><span class="lsa">5<span class="ff3 lsb5">;</span>9</span><span class="ff3">=</span><span class="ws80">4) <span class="ff3">:</span></span></div><div class="t m0 x4 h3 y81 ff1 fs1 fc0 sc0 ls3a ws9a">4.3E <span class="ff2 wsc">Sup onha<span class="_7 blank"> </span>que<span class="_e blank"> </span><span class="ff3 lsb6">f</span><span class="ws9b">seja uma função deriv<span class="_d blank"></span>ável em seu domínio <span class="ff3 lsb7">D</span>e que, para todo <span class="ff3 lsb8">x</span><span class="ws9c">em <span class="ff3 lsb9">D</span>,</span></span></span></div><div class="t m0 x1 h4 y82 ff2 fs1 fc0 sc0 ls3a ws9d">satisfaça <span class="ff3 ws9e">xf <span class="ff4 ws3">(<span class="ff3">x</span><span class="ws5">) + sen<span class="_1d blank"> </span>[</span></span><span class="ls2">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">)] = 4</span></span></span><span class="ws9f">.<span class="_c blank"> </span>Se <span class="ff3 lsd">x</span><span class="ff4 ws76">+<span class="_b blank"> </span>cos [<span class="ff3 ls10">f</span><span class="ws3">(<span class="ff3">x</span><span class="wsa0">)] </span><span class="ff6">6</span><span class="ws7">= 0</span></span></span><span class="ws2">, mostre que <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_20 blank"> </span></span><span class="ff6 v2">\ue000<span class="ff3 ls10">f</span></span><span class="v2">(<span class="ff3">x<span class="ff4">)</span></span></span></span></span></span></div><div class="t m0 x3a h1a y83 ff3 fs1 fc0 sc0 lsd">x<span class="ff4 ls3a ws76">+<span class="_b blank"> </span>cos [</span><span class="ls10">f<span class="ff4 ls3a ws3">(<span class="ff3">x</span><span class="wsa1">)] <span class="ff3 v2">:</span></span></span></span></div><div class="t m0 x4 h3 y84 ff1 fs1 fc0 sc0 ls3a wsa2">4.3F <span class="ff2 wsa3">P<span class="_0 blank"></span>ara cada uma das funções <span class="ff3 lsba">f</span>de\u2026<span class="_2 blank"></span>nidas abaixo,<span class="_16 blank"> </span>compro<span class="_d blank"></span>ve a existência da in<span class="_d blank"></span>v<span class="_0 blank"></span>ersa <span class="ff3 lsbb">g</span>,</span></div><div class="t m0 x1 h3 y85 ff2 fs1 fc0 sc0 ls3a wsa4">determine o domínio desta última e uma expressão que a de\u2026<span class="_2 blank"></span>n<span class="_5 blank"> </span>a explicitamen<span class="_0 blank"></span>te.<span class="_c blank"> </span>Esb<span class="_5 blank"> </span>o<span class="_5 blank"> </span>ce os grá\u2026<span class="_3 blank"></span>cos</div><div class="t m0 x1 h3 y86 ff2 fs1 fc0 sc0 ls3a ws49">de <span class="ff3 lse">f</span><span class="ls17">e</span><span class="ff3 ws7d">g :</span></div><div class="t m0 x2 h17 y87 ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = </span><span class="ff3">x<span class="ff5 fs2 ls4 v1">2</span><span class="ff6 ls8">\ue000</span></span>4<span class="ff3 ws30">;<span class="_12 blank"> </span>x <span class="ff6 ls3">\ue015</span></span><span class="lsbc">0</span></span><span class="ws9">(b) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = </span><span class="ff3">x<span class="ff5 fs2 ls4 v1">2</span><span class="ff6 ls8">\ue000</span></span>4<span class="ff3 ws30">;<span class="_12 blank"> </span>x <span class="ff6 lsa3">\ue014</span></span><span class="lsbd">0</span></span><span class="wsa5">c) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff6">\ue000<span class="ls23 vd">p</span></span><span class="ls47">1<span class="ff6 ls8">\ue000</span></span><span class="ff3 ws31">x;<span class="_12 blank"> </span>x <span class="ff6 ls3">\ue014</span></span>1</span></span></span></div><div class="t m0 x2 h4 y88 ff2 fs1 fc0 sc0 ls3a ws9">(d) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_21 blank"> </span><span class="ff3 v2">x</span></span></span></div><div class="t m0 x3b h19 y89 ff3 fs1 fc0 sc0 lsd">x<span class="ff4 ls3a ws51">+<span class="_b blank"> </span>1 </span><span class="ls3a wse v2">;<span class="_12 blank"> </span>x > <span class="ff6 ws3">\ue000<span class="ff4 ls9">1</span><span class="ff2 ws8">(e) <span class="ff3 ls2">f</span></span><span class="ff4">(<span class="ff3">x</span><span class="ws7">) =<span class="_21 blank"> </span></span><span class="ff3 v2">x</span><span class="ff5 fs2 ve">2</span></span></span></span></div><div class="t m0 x3c h19 y89 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls4 v10">2</span><span class="ff4 ws51">+<span class="_b blank"> </span>1 </span><span class="ws30 v2">;<span class="_12 blank"> </span>x <span class="ff6 ls3">\ue015<span class="ff4 ls9">0<span class="ff2 ls3a ws6b">(f )<span class="_e blank"> </span><span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_21 blank"> </span></span></span></span></span></span></span><span class="vf">x</span><span class="ff5 fs2 v1e">2</span></div><div class="t m0 x3d h1a y89 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls4 v10">2</span><span class="ff4 ws51">+<span class="_b blank"> </span>1 </span><span class="ws30 v2">;<span class="_12 blank"> </span>x <span class="ff6 ls3">\ue014<span class="ff4 ls3a">0</span></span></span></div><div class="t m0 x4 h3 y8a ff1 fs1 fc0 sc0 ls3a wsa6">4.3G <span class="ff2 wsa7">P<span class="_0 blank"></span>or meio de restrições adequadas, faça com que cada uma das funções dadas abaixo gere</span></div><div class="t m0 x1 h3 y72 ff2 fs1 fc0 sc0 ls3a wsa8">duas funções in<span class="_0 blank"></span>v<span class="_0 blank"></span>ertív<span class="_d blank"></span>eis <span class="ff3 ws3">f<span class="ff5 fs2 lsbe v1d">1</span></span><span class="lsbf">e</span><span class="ff3 ws3">f<span class="ff5 fs2 ls3e v1d">2</span></span>,<span class="_7 blank"> </span>determinando, em seguida, as resp<span class="_5 blank"> </span>ectiv<span class="_4 blank"></span>as inv<span class="_d blank"></span>ersas <span class="ff3 ws3">g<span class="ff5 fs2 lsbe v1d">1</span></span><span class="lsbf">e</span><span class="ff3 ws3">g<span class="ff5 fs2 ls3e v1d">2</span></span><span class="ws27">. Calcule</span></div><div class="t m0 x1 h3 y2a ff2 fs1 fc0 sc0 ls3a ws2">as deriv<span class="_4 blank"></span>adas dessas inv<span class="_d blank"></span>ersas e esb<span class="_a blank"> </span>o<span class="_5 blank"> </span>ce os grá\u2026<span class="_3 blank"></span>cos das funções <span class="ff3 ws3">f<span class="ff5 fs2 ls3e v1d">1</span><span class="wsa9">; f<span class="ff5 fs2 ls3e v1d">2</span>; g<span class="ff5 fs2 ls56 v1d">1</span></span></span><span class="ls17">e</span><span class="ff3 ws3">g<span class="ff5 fs2 ls22 v1d">2</span></span>, em cada caso.</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x1a y73 w2 h2d" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg6.png"><div class="t m0 x1 h3 y2c ff2 fs1 fc0 sc0 ls3a ws2b">22 <span class="ffb fs2 ws2c">D E R IV<span class="_d blank"></span>A<span class="_a blank"> </span>DA S<span class="_11 blank"> </span>C O M P L E M<span class="_a blank"> </span>E N T O S<span class="_e blank"> </span>4</span></div><div class="t m0 x2 h1b y8b ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls4 v1">2</span><span class="ff6 ls8">\ue000</span><span class="ff4">2</span><span class="lsd">x<span class="ff6 ls8">\ue000<span class="ff4 ls9">3</span></span></span></span></span><span class="ws9">(b) <span class="ff3 ls6">y<span class="ff4 ls3">=</span></span><span class="ff6 ws3">\ue000<span class="ff3">x<span class="ff5 fs2 ls4 v1">2</span><span class="ff4 ls8">+</span><span class="lsd">x</span><span class="ff4 ws5">+ 2<span class="_6 blank"> </span></span></span></span><span class="ws8">(c) <span class="ff3 ls0">y<span class="ff4 lsa3">=<span class="ff6 ls23 v8">p</span><span class="ls7c">1<span class="ff6 ls8">\ue000</span></span></span><span class="ls3a ws3">x<span class="ff5 fs2 ls5 v10">2</span></span></span></span>(d) <span class="ff3 ls0">y<span class="ff4 ls3">=</span></span><span class="ff6 ws3">\ue000<span class="ls23 v8">p</span><span class="ff4 ls47">4</span><span class="ls8">\ue000</span><span class="ff3">x<span class="ff5 fs2 v10">2</span></span></span></span></div><div class="t m0 x4 h4 y8c ff1 fs1 fc0 sc0 ls3a wsaa">4.3H <span class="ff2 wsab">V<span class="_4 blank"></span>eri\u2026<span class="_2 blank"></span>que que a função <span class="ff3 ls0">y<span class="ff4 lsa3">=</span><span class="ls10">f</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_22 blank"> </span><span class="ff3 v2">x</span></span></span></span></div><div class="t m0 x25 h32 y8d ff6 fs1 fc0 sc0 ls45">p<span class="ff4 ls3a ws5 v1f">1 + <span class="ff3 ws3">x<span class="ff5 fs2 ls64 v10">2</span><span class="ff2 wsab v14">, de\u2026<span class="_3 blank"></span>nida em <span class="ff7 lsc0">R</span>, tem como in<span class="_d blank"></span>versa a função</span></span></span></div><div class="t m0 x1 h4 y8e ff3 fs1 fc0 sc0 ls12">x<span class="ff4 ls3">=</span><span class="ls4d">g<span class="ff4 ls3a ws3">(</span><span class="ls9f">y<span class="ff4 ls3a ws7">) =<span class="_23 blank"> </span><span class="ff3 v2">y</span></span></span></span></div><div class="t m0 x3e h33 y8f ff8 fs1 fc0 sc0 ls3a ws3">p<span class="ff4 ls47 v20">1<span class="ff6 ls8">\ue000<span class="ff3 ls9f">y<span class="ff5 fs2 ls6d v10">2</span></span></span></span><span class="ff2 ws2 v21">, de\u2026<span class="_2 blank"></span>nida para <span class="ff6 ws3">j<span class="ff3 ls9f">y</span><span class="lsc1">j<span class="ff3 ls3"><</span></span><span class="ff4">1<span class="ff3">:</span></span></span></span></div><div class="t m0 x4 h4 y90 ff1 fs1 fc0 sc0 ls3a wsac">4.3I <span class="ff2 wsad">Qual a in<span class="_0 blank"></span>v<span class="_d blank"></span>ersa da fun<span class="_5 blank"> </span>ção <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_13 blank"> </span><span class="v2">1</span></span></span></span></div><div class="t m0 x3f h1c y91 ff3 fs1 fc0 sc0 lsc2">x<span class="ff4 lsc3 v2">?<span class="ff2 ls3a wsad">E da função <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_21 blank"> </span><span class="ff3 v2">x</span></span></span></span></span></div><div class="t m0 x40 h1a y91 ff3 fs1 fc0 sc0 lsd">x<span class="ff4 ls3a ws51">+<span class="_b blank"> </span>1 <span class="lsc3 v2">?</span><span class="ff2 wsc v2">Esp eci\u2026<span class="_2 blank"></span>que<span class="_e blank"> </span>os<span class="_f blank"> </span>domínios</span></span></div><div class="t m0 x1 h3 y92 ff2 fs1 fc0 sc0 ls3a ws2">e as imagens, esb<span class="_5 blank"> </span>o<span class="_5 blank"> </span>çando, também, os grá\u2026<span class="_3 blank"></span>cos.</div><div class="t m0 x4 h14 y93 ff1 fs1 fc0 sc0 ls3a wsae">4.3J <span class="ff2 wsaf">Considere a função <span class="ff3 lsc4">y<span class="ff4 lsc5">=</span><span class="ls10">f</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsb0">) = </span><span class="ff3">x<span class="ff5 fs2 lsc6 v1">2</span><span class="ff6 lsc7">\ue000</span><span class="lsc8">x<span class="ff6 lsc7">\ue000</span></span></span><span class="lsa">2</span></span>, de\u2026<span class="_3 blank"></span>nida para <span class="ff3 lsc9">x<span class="ff6 lsc5">\ue015</span></span><span class="ff4 ws3">1<span class="ff3 lsa">=</span>2</span><span class="ws38">, e seja <span class="ff3 lsca">x<span class="ff4 lscb">=</span><span class="ls4d">g</span></span><span class="ff4 ws3">(<span class="ff3 ls9f">y</span><span class="lscc">)</span><span class="ff2">sua</span></span></span></span></div><div class="t m0 x1 h3 y94 ff2 fs1 fc0 sc0 ls3a ws3">in<span class="_0 blank"></span>v<span class="_d blank"></span>ersa.</div><div class="t m0 x1 h8 y95 ff2 fs1 fc0 sc0 ls3a ws2">(a) Qual o domínio e qual a imagem de <span class="ff3 lsbb">g<span class="ff4 lscd">?</span></span><span class="wsc">(b)<span class="_e blank"> </span>S ab endo-se<span class="_e blank"> </span>que<span class="_7 blank"> </span><span class="ff3 ls4d">g</span><span class="ff4 ws3">(<span class="ff6">\ue000</span><span class="wse">2) = 1</span></span></span>, calcule <span class="ff3 lsbb">g<span class="ff9 fs2 ls54 v1">0</span></span><span class="ff4 ws3">(<span class="ff6">\ue000</span><span class="ws18">2) <span class="ff3">:</span></span></span></div><div class="t m0 x1 h2 y96 ff1 fs0 fc0 sc0 ls3a ws0">4.<span class="_0 blank"></span>2<span class="_1 blank"> </span>Ma<span class="_d blank"></span>is F<span class="_2 blank"></span>un<span class="_d blank"></span>çõe<span class="_0 blank"></span>s E<span class="_0 blank"></span>le<span class="_0 blank"></span>m<span class="_d blank"></span>en<span class="_4 blank"></span>tar<span class="_d blank"></span>es</div><div class="t m0 x22 h3 y97 ff1 fs1 fc0 sc0 ls3a wsb1">4.4A <span class="ff2 wsb2">Considere as funções <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsb3">)<span class="_f blank"> </span>=<span class="_f blank"> </span>arctg <span class="ff3 lsce">x</span><span class="wsb4">+ arctg<span class="_1d blank"> </span>(1</span></span><span class="ff3">=x</span><span class="lscf">)</span></span><span class="lsd0">e<span class="ff3 ls72">g</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsb5">)<span class="_f blank"> </span>=<span class="_f blank"> </span>arcsen <span class="ff3 lsce">x</span><span class="wsb4">+ arccos<span class="_1d blank"> </span></span></span><span class="ff3">x</span></span><span class="wsb6">, de\u2026<span class="_2 blank"></span>nidas,</span></span></div><div class="t m0 x1 h3 y98 ff2 fs1 fc0 sc0 ls3a wsc">resp ectiv<span class="_4 blank"></span>amente,<span class="_e blank"> </span>para<span class="_7 blank"> </span><span class="ff3 wse">x > <span class="ff4 lsd1">0</span></span><span class="wsb7">e para <span class="ff3 ls12">x<span class="ff6 ls34">2</span></span><span class="ff4 ws3">[<span class="ff6">\ue000</span><span class="lsa">1<span class="ff3 lsb5">;</span></span>1]<span class="ff3">:</span></span></span></div><div class="t m0 x1 h8 y99 ff2 fs1 fc0 sc0 ls3a ws2">(a) Mostre que <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = 0<span class="ff3 lsd2">;</span></span><span class="ff6">8<span class="ff3 wse">x > </span></span>0<span class="ff3 ls2c">;</span></span><span class="wsb8">e que <span class="ff3 lsbb">g<span class="ff9 fs2 ls54 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = 0<span class="ff3 lsd2">;</span></span><span class="ff6">8<span class="ff3 ls12">x</span><span class="ls34">2</span></span>(<span class="ff6">\ue000</span><span class="lsa">1<span class="ff3 lsb5">;</span></span><span class="ws18">1) <span class="ff3">:</span></span></span></span></div><div class="t m0 x1 h3 y9a ff2 fs1 fc0 sc0 ls3a wsb9">(b) Lem<span class="_0 blank"></span>brando que as funções constan<span class="_0 blank"></span>tes são as que p<span class="_5 blank"> </span>ossuem deriv<span class="_d blank"></span>ada n<span class="_0 blank"></span>ula, deduza que <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = <span class="ff3 ws7d">\ue019 =</span></span>2</span>,</div><div class="t m0 x1 h3 y9b ff6 fs1 fc0 sc0 ls3a ws3">8<span class="ff3 wse">x > </span><span class="ff4">0<span class="ff3 ls2c">;</span><span class="ff2 wsb8">e que <span class="ff3 ls4d">g</span></span>(<span class="ff3">x</span><span class="ws14">) = <span class="ff3 ws7d">\ue019 =</span><span class="lsa">2<span class="ff3 lsd2">;</span></span></span></span>8<span class="ff3 ls12">x</span><span class="ls34">2</span><span class="ff4">[</span>\ue000<span class="ff4 lsa">1<span class="ff3 lsb5">;</span><span class="ls3a">1]<span class="ff3">:</span></span></span></div><div class="t m0 x4 h34 y9c ff1 fs1 fc0 sc0 ls3a wsba">4.4B <span class="ff2 wsbb">Se <span class="ff3 lsd3">f</span><span class="wsbc">é uma função deriv<span class="_4 blank"></span>ável, tal que <span class="ff3 ls2">f</span><span class="ff4 wsbd">(2) = 1<span class="_7 blank"> </span></span><span class="lsd4">e<span class="ff3 ls13">f<span class="ff9 fs2 ls54 v1">0</span></span></span><span class="ff4 wsbd">(2) = 1<span class="ff3 ws3">=</span><span class="lsa">2</span></span>,<span class="_13 blank"> </span>determine a equação da</span></span></div><div class="t m0 x1 h3 y9d ff2 fs1 fc0 sc0 ls3a ws2">reta tangen<span class="_0 blank"></span>te à curv<span class="_4 blank"></span>a <span class="ff3 ls0">y</span><span class="ff4 ws90">=<span class="_f blank"> </span>arctg [<span class="ff3 ls10">f</span><span class="ws3">(<span class="ff3">x</span>)]</span></span>, no p<span class="_a blank"> </span>on<span class="_d blank"></span>to d<span class="_5 blank"> </span>e abscissa <span class="ff3 ls12">x</span><span class="ff4 ws7">= 2<span class="ff3">:</span></span></div><div class="t m0 x4 h21 y9e ff1 fs1 fc0 sc0 ls3a wsbe">4.4C <span class="ff2 wsc">Sab endo-se<span class="_c blank"> </span>que<span class="_13 blank"> </span>no<span class="_c blank"> </span>p onto<span class="_13 blank"> </span><span class="ff3 ls4c">A</span><span class="ff4 ws3">(0<span class="ff3 lsb5">;</span><span class="wsbf">1) </span></span><span class="ws56">o grá\u2026<span class="_3 blank"></span>co da função <span class="ff3 ls2">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsc0">) = exp<span class="_24 blank"> </span><span class="ff8 ls76 v14">\ue000</span></span><span class="ff3">x<span class="ff5 fs2 ls7 v1">2</span></span><span class="ws5">+ 2</span><span class="ff3">x<span class="ff8 lsd5 v14">\ue001</span></span></span><span class="wsc">p ossui<span class="_13 blank"> </span>a</span></span></span></div><div class="t m0 x1 h8 y9f ff2 fs1 fc0 sc0 ls3a ws2">mesma reta tangen<span class="_0 blank"></span>te que o de uma certa função <span class="ff3 lsbb">g</span>, determine <span class="ff3 lsd6">g<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 wsc1">(0) <span class="ff3">:</span></span></div><div class="t m0 x4 h34 ya0 ff1 fs1 fc0 sc0 ls3a wsc2">4.4D <span class="ff2 wsc3">Se <span class="ff3 lsd7">f</span><span class="wsc4">é uma função deriv<span class="_d blank"></span>áv<span class="_0 blank"></span>el,<span class="_c blank"> </span>tal que <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsc5">) = 2<span class="_5 blank"> </span><span class="ff3 ws9e">xf </span></span>(<span class="ff3">x</span>)</span><span class="ws5c">,<span class="_c blank"> </span>mostre que a fun<span class="_5 blank"> </span>ção <span class="ff3 ls4d">g</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws59">) =</span></span></span></span></span></div><div class="t m0 x1 h35 ya1 ff3 fs1 fc0 sc0 ls10">f<span class="ff4 ls3a ws3">(<span class="ff3">x</span><span class="ls37">)</span><span class="ff3">e<span class="ff9 fs2 ls3d v1">\ue000<span class="ffa lsd8">x<span class="ffc fs3 lsd9 v5">2</span></span></span><span class="ff2 ws57">é constan<span class="_0 blank"></span>te.</span></span></span></div><div class="t m0 x4 h3 ya2 ff1 fs1 fc0 sc0 ls3a wsc6">4.4E <span class="ff2 wsc7">P<span class="_0 blank"></span>ara cada uma das funções de\u2026<span class="_3 blank"></span>nidas abaixo, determine o domínio e calcule a deriv<span class="_4 blank"></span>ada</span></div><div class="t m0 x1 h3 ya3 ff2 fs1 fc0 sc0 ls3a ws57">de primeira ordem.</div><div class="t m0 x2 h1b ya4 ff2 fs1 fc0 sc0 ls3a ws68">(a) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = ln(<span class="ff6 lsda v8">p</span><span class="ls7c">5<span class="ff6 ls46">\ue000</span></span></span><span class="ff3">x<span class="ff5 fs2 ls3e v10">2</span></span><span class="ls8e">)</span></span><span class="ws6">(b) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsc8">)<span class="_f blank"> </span>=<span class="_f blank"> </span>ln(sen </span><span class="ff3">x</span><span class="lsdb">)</span></span><span class="ws69">(c) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff3 ls52">x</span><span class="ws23">ln <span class="ff3 lsd">x<span class="ff6 ls8">\ue000</span><span class="ls3a">x</span></span></span></span></span></span></span></div><div class="t m0 x2 h3 ya5 ff2 fs1 fc0 sc0 ls3a ws9">(d) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsc9">)<span class="_f blank"> </span>=<span class="_f blank"> </span>ln </span><span class="ff6">j<span class="ff3">x</span><span class="lsdc">j</span></span></span><span class="ws69">(e) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = 1<span class="ff3 lsdd">=</span><span class="ws23">ln <span class="ff3 lsde">x</span></span></span></span><span class="ws6b">(f )<span class="_e blank"> </span><span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsca">)<span class="_f blank"> </span>=<span class="_f blank"> </span>ln(ln </span><span class="ff3">x</span>)</span></span></span></div><div class="t m0 x2 h36 ya6 ff2 fs1 fc0 sc0 ls3a ws68">(g) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = ln(<span class="ff8 ls88 v4">r</span><span class="ls47 v2">2<span class="ff6 ls8">\ue000<span class="ff3 ls3a">x</span></span></span></span></span></div><div class="t m0 x14 h1a ya7 ff4 fs1 fc0 sc0 ls47">3<span class="ff6 ls8">\ue000<span class="ff3 ls90">x</span></span><span class="lsdf v2">)<span class="ff2 ls3a ws6">(h) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws73">)<span class="_f blank"> </span>=<span class="_f blank"> </span>ln(cos (3<span class="ff3 lsd">x</span><span class="ws5">+ 5))<span class="_6 blank"> </span></span></span></span><span class="ws6f">(i) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = sen(ln(2<span class="ff3 lsd">x</span><span class="ws5">+ 3))</span></span></span></span></span></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="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x37 ya8 w5 h37" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg7.png"><div class="t m0 x24 h3 y2c ffb fs2 fc0 sc0 ls3a ws4b">C Á L C U LO<span class="_e blank"> </span>D E<span class="_f blank"> </span>U M A<span class="_e blank"> </span>V<span class="_d blank"></span>A R IÁ<span class="_d blank"></span>V E L<span class="_14 blank"> </span>M A R IV<span class="_d blank"></span>A L D O<span class="_e blank"> </span>P<span class="_f blank"> </span>M ATO S<span class="_15 blank"> </span><span class="ff2 fs1">23</span></div><div class="t m0 x4 h21 y2d ff1 fs1 fc0 sc0 ls3a ws1a">4.4F <span class="ff2 ws2">Considere a função <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsc9">)<span class="_f blank"> </span>=<span class="_f blank"> </span>ln <span class="ff8 ls76 v14">\ue000</span></span><span class="ff3">x<span class="ff5 fs2 ls7 v1">2</span></span><span class="ws5">+ 1<span class="ff8 ls7f v14">\ue001</span><span class="ff3">:</span></span></span></span></div><div class="t m0 x1 h3 ya9 ff2 fs1 fc0 sc0 ls3a ws2">(a) Qual o domínio de <span class="ff3 ls13">f</span><span class="ff4">?</span></div><div class="t m0 x1 h3 yaa ff2 fs1 fc0 sc0 ls3a wscb">(b) Qual é a equação da reta tangen<span class="_0 blank"></span>te ao grá\u2026<span class="_2 blank"></span>co de <span class="ff3 ls15">f</span>, no p<span class="_a blank"> </span>on<span class="_0 blank"></span>to de abscissa <span class="ff3 lse0">x<span class="ff4 lse1">=</span></span><span class="ff6 ws3">\ue000<span class="ff4 wscc">1? </span></span><span class="wscd">E<span class="_7 blank"> </span>no<span class="_7 blank"> </span>p onto</span></div><div class="t m0 x1 h3 yab ff2 fs1 fc0 sc0 ls3a wsce">de abscissa <span class="ff3 ls12">x</span><span class="ff4 ws7">= 0?</span></div><div class="t m0 x4 h3 yac ff1 fs1 fc0 sc0 ls3a wscf">4.4G <span class="ff2 lse2">O</span><span class="ffd wsd0">lo<span class="_d blank"></span>garitmo <span class="ff2 wsd1">de um n<span class="_0 blank"></span>úmero <span class="ff3 wsd2">N<span class="_c blank"> </span>> <span class="ff4 ws3">0</span></span><span class="wsd3">, em uma base <span class="ff3 wsd4">b; <span class="ff4 lse3">0</span><span class="wsd5">< b <span class="ff6 ws3">6<span class="ff4 wsd6">= 1</span></span></span></span>, é de\u2026<span class="_3 blank"></span>nido p<span class="_5 blank"> </span>or meio da</span></span></span></div><div class="t m0 x1 h3 yad ff2 fs1 fc0 sc0 ls3a ws3">eqüiv<span class="_4 blank"></span>alência</div><div class="t m0 x2d h38 yae ff4 fs1 fc0 sc0 ls3a wsd7">log<span class="ffa fs2 lse4 v9">b</span><span class="ff3 lse5">N</span><span class="ls3">=<span class="ff3 lse6">a</span></span><span class="ff6 wsd8">(<span class="_2 blank"></span>) <span class="ff3 ls31">b<span class="ffa fs2 lse7 v1c">a</span><span class="ff4 lsa3">=</span><span class="ls3a wsd9">N :</span></span></span></div><div class="t m0 x1 h4 yaf ff2 fs1 fc0 sc0 ls3a ws2">(a) Pro<span class="_0 blank"></span>v<span class="_d blank"></span>e a propriedade<span class="_7 blank"> </span>de Mudança de Base:<span class="_c blank"> </span><span class="ff4 wsd7">log <span class="ffa fs2 lse8 v9">b</span><span class="ff3 lse5">N</span><span class="ls58">=</span><span class="ws23 v2">ln <span class="ff3">N</span></span></span></div><div class="t m0 x6 h3 yb0 ff4 fs1 fc0 sc0 ls3a ws23">ln <span class="ff3">b</span></div><div class="t m0 x1 h4 yb1 ff2 fs1 fc0 sc0 ls3a ws2">(b) Se <span class="ff3 ls19">f</span><span class="wscd">é<span class="_e blank"> </span>de\u2026<span class="_3 blank"></span>nida<span class="_7 blank"> </span>por<span class="_7 blank"> </span><span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsda">)<span class="_f blank"> </span>=<span class="_f blank"> </span>log<span class="ffa fs2 lse8 v9">b</span><span class="ff3 wsdb">x; </span></span></span><span class="wsdc">para <span class="ff3 wse">x > <span class="ff4 ws3">0</span></span><span class="ws2">, mostre que <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_21 blank"> </span><span class="v2">1</span></span></span></span></span></span></div><div class="t m0 x41 h3 yb2 ff3 fs1 fc0 sc0 ls52">x<span class="ff4 ls3a ws23">ln <span class="ff3">b</span></span></div><div class="t m0 x4 h3 yb3 ff1 fs1 fc0 sc0 ls3a ws67">4.4H <span class="ff2 ws2">Calcule a deriv<span class="_4 blank"></span>ada de primeira ordem de<span class="_7 blank"> </span>cada uma das funções abaixo.</span></div><div class="t m0 x2 h39 yb4 ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = </span><span class="ff3">e<span class="ff5 fs2 wsdd v1">sen <span class="ffa lse9">x</span></span></span></span><span class="ws6">(b) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff3">e<span class="ffa fs2 lsd8 v1">x</span><span class="ffc fs3 lsea v6">2</span></span></span><span class="ws69">(c) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = (</span><span class="ff3">e<span class="ffa fs2 ls8b v1">x</span></span>)<span class="ff5 fs2 v1">2</span></span></span></span></div><div class="t m0 x2 h39 yb5 ff2 fs1 fc0 sc0 ls3a ws9">(d) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = 3<span class="ff9 fs2 lseb v1">\ue000<span class="ffa lsec">x</span></span></span></span><span class="ws69">(e) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff3">x<span class="ffa fs2 lsed v1">x</span></span></span><span class="ws6b">(f )<span class="_e blank"> </span><span class="ff3 ls2">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff3">x<span class="ff5 fs2 lsee v1">(<span class="ffa lsd8">x<span class="ffe fs3 lsef v5">x</span><span class="ff5 ls3a">)</span></span></span></span></span></span></span></div><div class="t m0 x2 h35 yb6 ff2 fs1 fc0 sc0 ls3a ws4">(g) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = (</span><span class="ff3">x<span class="ffa fs2 ls8b v1">x</span></span>)<span class="ffa fs2 lsf0 v1">x</span></span><span class="ws6">(h) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = </span><span class="ff3">x<span class="ff5 fs2 ls22 v1">2</span></span><span class="lsa">3<span class="ffa fs2 lsf1 v1">x<span class="ff5 ls3a wsdd">sen <span class="ffa lsf0">x</span></span></span></span></span><span class="ws6f">(i) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse">) = 2<span class="ffa fs2 lsd8 v1">x</span><span class="ffe fs3 v6">x</span></span></span></span></span></div><div class="t m0 x4 h3 yb7 ff1 fs1 fc0 sc0 ls3a wsde">4.4I <span class="ff2 wsdf">As funções trigonométricas hip<span class="_5 blank"> </span>erb<span class="_5 blank"> </span>ólicas - <span class="ffd ws66">seno hip<span class="_d blank"></span>erb<span class="_d blank"></span>ólic<span class="_d blank"></span>o, c<span class="_d blank"></span>osseno hip<span class="_d blank"></span>erb<span class="_d blank"></span>ólic<span class="_d blank"></span>o, tangente</span></span></div><div class="t m0 x1 h3 yb8 ffd fs1 fc0 sc0 ls3a wse0">hip<span class="_d blank"></span>erb<span class="_d blank"></span>ólic<span class="_d blank"></span>a <span class="ff2 lsf2">e</span><span class="wsdf">c<span class="_d blank"></span>otangente hip<span class="_d blank"></span>erb<span class="_d blank"></span>ólic<span class="_d blank"></span>a<span class="_13 blank"> </span><span class="ff2 wse1">- denotadas, resp<span class="_5 blank"> </span>ectiv<span class="_d blank"></span>amente, por <span class="ff4 ws3">senh<span class="ff3 lsf3">;</span>cosh<span class="ff3 lsf3">;</span><span class="wse2">tgh </span></span><span class="lsf2">e</span><span class="ff4 ws3">cotgh</span>, são</span></span></div><div class="t m0 x1 h3 yb9 ff2 fs1 fc0 sc0 ls3a wsc">de\u2026<span class="_2 blank"></span>nidas<span class="_7 blank"> </span>p elas<span class="_7 blank"> </span>expressões:</div><div class="t m0 x1a h3a yba ff4 fs1 fc0 sc0 ls3a wse3">senh <span class="ff3 ls12">x</span><span class="ls58">=</span><span class="ff3 ws3 v2">e</span><span class="ffa fs2 lsf4 ve">x</span><span class="ff6 ls46 v2">\ue000</span><span class="ff3 ws3 v2">e</span><span class="ff9 fs2 ls3d ve">\ue000<span class="ffa ls3a">x</span></span></div><div class="t m0 x42 h3b ybb ff4 fs1 fc0 sc0 lsf5">2<span class="ls3a wse4 v2">cosh <span class="ff3 ls12">x<span class="ff4 ls78">=</span><span class="ls3a ws3 v2">e</span><span class="ffa fs2 lsf4 ve">x</span></span></span><span class="ls8 vf">+<span class="ff3 ls3a ws3">e<span class="ff9 fs2 lseb v1">\ue000<span class="ffa ls3a">x</span></span></span></span></div><div class="t m0 x43 h3b ybb ff4 fs1 fc0 sc0 lsf6">2<span class="ls3a wse5 v2">tgh <span class="ff3 ls12">x<span class="ff4 ls58">=</span><span class="ls3a ws3 v2">e</span><span class="ffa fs2 ls8a ve">x</span><span class="ff6 ls8 v2">\ue000</span><span class="ls3a ws3 v2">e</span><span class="ff9 fs2 ls3d ve">\ue000<span class="ffa ls3a">x</span></span></span></span></div><div class="t m0 x44 h3b ybb ff3 fs1 fc0 sc0 ls3a ws3">e<span class="ffa fs2 lsf4 v10">x</span><span class="ff4 ls8">+</span>e<span class="ffa fs2 lsf7 v10">x</span><span class="ff4 wse6 v2">cotgh </span><span class="ls12 v2">x<span class="ff4 ls78">=</span></span><span class="vf">e</span><span class="ffa fs2 lsf4 v1e">x</span><span class="ff4 ls8 vf">+</span><span class="vf">e</span><span class="ff9 fs2 lseb v1e">\ue000<span class="ffa ls3a">x</span></span></div><div class="t m0 x45 h3c ybb ff3 fs1 fc0 sc0 ls3a ws3">e<span class="ffa fs2 lsf4 v10">x</span><span class="ff6 ls46">\ue000</span>e<span class="ffa fs2 v10">x</span></div><div class="t m0 x1 h3 ybc ff2 fs1 fc0 sc0 ls3a ws2">Com base nessas de\u2026<span class="_3 blank"></span>nições, mostre que:</div><div class="t m0 x2 h3d ybd ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff4 ws3">cosh<span class="ff5 fs2 ls86 v1c">2</span><span class="ff3 lsd">x<span class="ff6 ls8">\ue000</span></span>senh<span class="ff5 fs2 ls86 v1c">2</span><span class="ff3 ls1a">x</span><span class="ws7">= 1<span class="_6 blank"> </span></span></span><span class="wse7">(b) <span class="ff4">lim</span></span></div><div class="t m0 x46 h12 ybe ffa fs2 fc0 sc0 lsd8">x<span class="ff9 ls3a ws1b">!<span class="ff5">0</span></span></div><div class="t m0 x47 h3 ybf ff4 fs1 fc0 sc0 ls3a wse3">senh <span class="ff3">x</span></div><div class="t m0 x48 h1c yc0 ff3 fs1 fc0 sc0 lsf8">x<span class="ff4 ls3a ws7 v2">= 1<span class="_25 blank"> </span><span class="ff2 wse8">(c) <span class="ff3 v2">d</span></span></span></div><div class="t m0 x12 h1a yc0 ff3 fs1 fc0 sc0 ls3a wse9">dx <span class="ff4 wsea v2">(senh </span><span class="ws3 v2">x<span class="ff4 wseb">)<span class="_f blank"> </span>=<span class="_f blank"> </span>cosh <span class="ff3">x</span></span></span></div><div class="t m0 x2 h4 yc1 ff2 fs1 fc0 sc0 ls3a wsec">(d) <span class="ff3 v2">d</span></div><div class="t m0 x49 h1c yc2 ff3 fs1 fc0 sc0 ls3a wse9">dx <span class="ff4 wsed v2">(cosh </span><span class="ws3 v2">x<span class="ff4 wsee">)<span class="_f blank"> </span>=<span class="_f blank"> </span>senh <span class="ff3 lsf9">x</span><span class="ff2 wse8">(e) </span></span></span><span class="vf">d</span></div><div class="t m0 x46 h1c yc2 ff3 fs1 fc0 sc0 ls3a wse9">dx <span class="ff4 ws23 v2">(tgh </span><span class="ws3 v2">x<span class="ff4 wsef">)<span class="_f blank"> </span>=<span class="_f blank"> </span>(cosh </span>x<span class="ff4">)<span class="ff9 fs2 ls3d v13">\ue000<span class="ff5 lsfa">2</span></span><span class="ff2 ws6b">(f )<span class="_17 blank"> </span></span></span><span class="v2">d</span></span></div><div class="t m0 x12 h26 yc2 ff3 fs1 fc0 sc0 ls3a wsf0">dx <span class="ff4 wsf1 v2">(cotgh </span><span class="ws3 v2">x<span class="ff4 ws14">) = <span class="ff6 lsfb">\ue000</span><span class="wsc8">(senh </span></span>x<span class="ff4">)<span class="ff9 fs2 lseb v13">\ue000<span class="ff5 ls3a">2</span></span></span></span></div><div class="t m0 x1 h3 yc3 ff2 fs1 fc0 sc0 ls3a ws66">(A iden<span class="_0 blank"></span>tidade (a) e as deriv<span class="_4 blank"></span>adas são comprov<span class="_4 blank"></span>adas usando as de\u2026<span class="_3 blank"></span>nições das funções hip<span class="_5 blank"> </span>erb<span class="_5 blank"> </span>ólicas e</div><div class="t m0 x1 h3 yc4 ff2 fs1 fc0 sc0 ls3a wsf2">as regras de deriv<span class="_4 blank"></span>ação.<span class="_9 blank"> </span>Quanto ao ítem (b), use o seguin<span class="_0 blank"></span>te fato:<span class="_26 blank"> </span><span class="ff4">lim</span></div><div class="t m0 x40 h12 yc5 ffa fs2 fc0 sc0 ls29">h<span class="ff9 ls3a ws1b">!<span class="ff5">0</span></span></div><div class="t m0 x4a h2b yc6 ff3 fs1 fc0 sc0 ls3a ws3">e<span class="ffa fs2 lsfc v1">h</span><span class="ff6 ls8">\ue000</span><span class="ff4">1</span></div><div class="t m0 x7 h1a yc7 ff3 fs1 fc0 sc0 lsfd">h<span class="ff4 ls3a wsf3 v2">= lim</span></div><div class="t m0 x31 h12 yc5 ffa fs2 fc0 sc0 ls29">h<span class="ff9 ls3a ws1b">!<span class="ff5">0</span></span></div><div class="t m0 x4b h2b yc6 ff3 fs1 fc0 sc0 ls3a ws3">e<span class="ff5 fs2 ws1b v1">0+<span class="ffa lsfc">h</span></span><span class="ff6 ls8">\ue000</span>e<span class="ff5 fs2 v1">0</span></div><div class="t m0 x4c h1a yc7 ff3 fs1 fc0 sc0 lsfe">h<span class="ff4 ls3a v2">=</span></div><div class="t m0 x4d h12 yc8 ffa fs2 fc0 sc0 ls3a">d</div><div class="t m0 x37 h3e yc9 ffa fs2 fc0 sc0 ls3a wsf4">dx <span class="ff4 fs1 ws3 v1">[<span class="ff3">e</span></span><span class="ls8b vb">x</span><span class="ff4 fs1 ws3 v1">]<span class="ff8 v8">\ue00c</span></span></div><div class="t m0 x4e h6 yca ff8 fs1 fc0 sc0 lsff">\ue00c<span class="ffa fs2 lsd8 v22">x<span class="ff5 ls3a wsf5">=0 </span></span><span class="ff4 ls3a ws7 v9">= 1<span class="ff3 ws3">:<span class="ff4">)</span></span></span></div><div class="t m0 x4 h3 y2a ff1 fs1 fc0 sc0 ls3a ws7e">4.4J <span class="ff2 ws2">P<span class="_0 blank"></span>ara cada uma das funções dadas abaixo, calcule o limite quando <span class="ff3 ls12">x<span class="ff6 ls100">!</span></span><span class="ff4 ws3">0</span>.</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="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x1a ycb w2 h3f" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg8.png"><div class="t m0 x1 h3 y2c ff2 fs1 fc0 sc0 ls3a ws2b">24 <span class="ffb fs2 ws2c">D E R IV<span class="_d blank"></span>A<span class="_a blank"> </span>DA S<span class="_11 blank"> </span>C O M P L E M<span class="_a blank"> </span>E N T O S<span class="_e blank"> </span>4</span></div><div class="t m0 x2 h4 ycc ff2 fs1 fc0 sc0 ls3a ws4">(a) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_13 blank"> </span><span class="ws23 v2">sen 2<span class="ff3">x</span></span></span></span></div><div class="t m0 x28 h1c ycd ff3 fs1 fc0 sc0 ls101">x<span class="ff2 ls3a ws6 v2">(b) </span><span class="ls10 v2">f<span class="ff4 ls3a ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_13 blank"> </span><span class="ws8e v2">sen <span class="ff3">x</span></span></span></span></span></div><div class="t m0 x36 h1c ycd ff4 fs1 fc0 sc0 ls3a ws3">3<span class="ff3 ls102">x</span><span class="ff2 ws69 v2">(c) <span class="ff3 ls10">f</span></span><span class="v2">(<span class="ff3">x<span class="ff4 ws7">) =<span class="_19 blank"> </span><span class="wsf6 v2">tg </span></span><span class="v2">x</span></span></span></div><div class="t m0 x4f h3 ycd ff4 fs1 fc0 sc0 ls3a ws8e">sen <span class="ff3">x</span></div><div class="t m0 x2 h4 yce ff2 fs1 fc0 sc0 ls3a ws9">(d) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_27 blank"> </span><span class="ws76 v2">cos 2<span class="ff3">x</span></span></span></span></div><div class="t m0 x3b h40 ycf ff4 fs1 fc0 sc0 ls3a ws5">1 + sen<span class="_24 blank"> </span><span class="ff3 ls103">x</span><span class="ff2 ws69 v2">(e) <span class="ff3 ls10">f</span></span><span class="ws3 v2">(<span class="ff3">x<span class="ff4 ws7">) =<span class="_13 blank"> </span><span class="ws8e vb">sen </span><span class="ff8 ls76 v17">\ue000</span></span><span class="vb">x</span><span class="ff5 fs2 ls22 v18">2</span><span class="ff8 v17">\ue001</span></span></span></div><div class="t m0 x50 h40 ycf ff3 fs1 fc0 sc0 ls104">x<span class="ff2 ls3a ws6b v2">(f )<span class="_e blank"> </span></span><span class="ls10 v2">f<span class="ff4 ls3a ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_13 blank"> </span><span class="ws8e vb">sen </span><span class="ff8 ls76 v17">\ue000</span><span class="lsa vb">2</span></span><span class="ff3 vb">x</span><span class="ff5 fs2 ls22 v18">2</span><span class="ff8 v17">\ue001</span></span></span></div><div class="t m0 x51 h3 ycf ff4 fs1 fc0 sc0 lsa">3<span class="ff3 ls3a">x</span></div><div class="t m0 x2 h18 yd0 ff2 fs1 fc0 sc0 ls3a ws4">(g) <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws7">) =<span class="_13 blank"> </span></span><span class="v2">sen(<span class="ff3">x<span class="ff5 fs2 ls3e v1">3</span><span class="ff4">)</span></span></span></span></div><div class="t m0 x28 h1c yd1 ff3 fs1 fc0 sc0 ls3a ws3">x<span class="ff5 fs2 ls105 v10">3</span><span class="ff2 ws6 v2">(h) </span><span class="ls10 v2">f</span><span class="ff4 v2">(</span><span class="v2">x<span class="ff4 ws7">) =<span class="_28 blank"> </span></span></span><span class="ls52 vf">x</span><span class="ff4 ws8e vf">sen </span><span class="vf">x</span></div><div class="t m0 x47 h1c yd1 ff4 fs1 fc0 sc0 ls3a ws3">sen(2<span class="ff3">x<span class="ff5 fs2 ls22 v10">2</span></span><span class="ls106">)</span><span class="ff2 ws6f v2">(i) <span class="ff3 ls10">f</span></span><span class="v2">(<span class="ff3">x<span class="ff4 ws7">) =<span class="_13 blank"> </span><span class="ws8e v2">sen </span></span><span class="ls52 v2">x</span></span></span><span class="ws23 vf">sen 2<span class="ff3">x</span></span></div><div class="t m0 x10 h3 yd1 ff3 fs1 fc0 sc0 ls52">x<span class="ff4 ls3a ws23">sen 3<span class="ff3">x</span></span></div><div class="t m0 x4 hb yd2 ff1 fs1 fc0 sc0 ls3a wsf7">4.4K <span class="ff2 wsf8">Seja <span class="ff3 ls107">f<span class="ff4 ls108">:<span class="ff7 ls1d">R<span class="ff6 ls1e">!</span><span class="ls109">R</span></span></span></span><span class="wsf9">uma função deriv<span class="_4 blank"></span>ável e sup<span class="_5 blank"> </span>onha que exista uma constan<span class="_0 blank"></span>te <span class="ff3 ls10a">k</span><span class="wsfa">tal que</span></span></span></div><div class="t m0 x1 h2b yd3 ff3 fs1 fc0 sc0 ls13">f<span class="ff9 fs2 ls14 v1">0</span><span class="ff4 ls3a ws3">(<span class="ff3">x</span><span class="wsfb">) = </span></span><span class="ls3a wsfc">k f<span class="_f blank"> </span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ls10b">)</span></span><span class="ls10c">;</span><span class="ff6 ws3">8<span class="ff3">x<span class="ff2 wsfd">.<span class="_29 blank"> </span>Deriv<span class="_0 blank"></span>e o quo<span class="_5 blank"> </span>cien<span class="_0 blank"></span>te <span class="ff3 wsb4">f =e<span class="ffa fs2 wsfe v1">kx<span class="_16 blank"> </span></span></span>e deduza que existe uma constan<span class="_0 blank"></span>te <span class="ff3 ls10d">C</span><span class="wsff">tal que</span></span></span></span></span></div><div class="t m0 x1 h2b yd4 ff3 fs1 fc0 sc0 ls10">f<span class="ff4 ls3a ws3">(<span class="ff3">x</span><span class="wsd">) = </span></span><span class="ls3a ws100">C e<span class="ffa fs2 wsfe v1">kx<span class="_a blank"> </span></span><span class="ff2">.</span></span></div><div class="t m0 x4 h8 yd5 ff1 fs1 fc0 sc0 ls3a ws101">4.4L <span class="ff2 wsc7">No exercício preceden<span class="_0 blank"></span>te, sup<span class="_5 blank"> </span>onha que <span class="ff3 ls10e">f</span><span class="ws102">satisfaça <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wse1">) = </span><span class="ff6">\ue000</span><span class="lsa">2</span><span class="ff3 ws103">xf </span>(<span class="ff3">x</span>)</span></span>.<span class="_16 blank"> </span>Mostre que existe</span></div><div class="t m0 x1 h39 yd6 ff2 fs1 fc0 sc0 ls3a ws2">uma constan<span class="_0 blank"></span>te <span class="ff3 ls10f">C</span>tal que <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws14">) = <span class="ff3 ws100">C e<span class="ff9 fs2 ls3d v1">\ue000<span class="ffa lsd8">x<span class="ffc fs3 ls110 v5">2</span></span></span>:</span></span></span></div><div class="t m0 x4 h8 yd7 ff1 fs1 fc0 sc0 ls3a ws104">4.4M <span class="ff2 ws105">Se <span class="ff3 ls111">f</span><span class="ws106">satisfaz <span class="ff3 ls13">f<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws107">) = <span class="ff3 lsd6">g<span class="ff9 fs2 ls14 v1">0</span></span></span>(<span class="ff3">x</span><span class="ls37">)<span class="ff3 ls10">f</span></span>(<span class="ff3">x</span><span class="ls37">)<span class="ff3 ls112">;</span></span><span class="ff6">8<span class="ff3 ls113">x</span><span class="ls114">2<span class="ff7 ls2f">R</span></span></span></span><span class="ws108">,<span class="_16 blank"> </span>mostre que existe <span class="ff3 ls115">C</span><span class="ws109">tal que <span class="ff3 ls2">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="ws107">) =</span></span></span></span></span></span></div><div class="t m0 x1 h3 yd8 ff3 fs1 fc0 sc0 ls116">C<span class="ff4 ls3a ws3">exp[</span><span class="ls4d">g<span class="ff4 ls3a ws3">(<span class="ff3">x</span>)]<span class="ff3">:</span></span></span></div><div class="t m0 x4 h3 yd9 ff1 fs1 fc0 sc0 ls3a ws10a">4.4N <span class="ff2 wsc">Esb o ce<span class="_2a blank"> </span>o<span class="_2a blank"> </span>grá\u2026<span class="_2 blank"></span>co<span class="_2a blank"> </span>da<span class="_2a blank"> </span>função<span class="_2a blank"> </span><span class="ff3 ls117">y</span><span class="ff4 ws23">=<span class="_7 blank"> </span>ln (1<span class="_b blank"> </span>+<span class="_b blank"> </span><span class="ff3 ws3">x</span><span class="ls118">)</span></span><span class="ws10b">e determine a reta normal ao grá\u2026<span class="_3 blank"></span>co, que é</span></span></div><div class="t m0 x1 h3 yda ff2 fs1 fc0 sc0 ls3a ws2">paralela à reta <span class="ff3 lsd">x</span><span class="ff4 ws5">+ 2<span class="ff3 ls6">y</span><span class="ws7">= 5<span class="ff3">:</span></span></span></div><div class="t m0 x4 h41 ydb ff1 fs1 fc0 sc0 ls3a ws46">4.4O <span class="ff2 ws2">Considere a função <span class="ff3 ls10">f</span><span class="ff4 ws3">(<span class="ff3">x</span><span class="wsd">) = </span><span class="ff6">j<span class="ff3 lsd">x</span></span><span class="ws5">+ 2</span><span class="ff6">j<span class="ff5 fs2 ls22 v13">3</span></span></span>.</span></div><div class="t m0 x4 h3 ydc ff2 fs1 fc0 sc0 ls3a ws2">(a) V<span class="_4 blank"></span>eri\u2026<span class="_2 blank"></span>que que <span class="ff3 ls19">f</span><span class="ws57">é deriv<span class="_d blank"></span>ável em qualquer <span class="ff3 ls119">x</span><span class="ws2">e ac<span class="_d blank"></span>he uma expressão para a deriv<span class="_d blank"></span>ada</span></span></div><div class="t m0 x4 h3 ydd ff2 fs1 fc0 sc0 ls3a ws2">(b) Encon<span class="_0 blank"></span>tre o p<span class="_5 blank"> </span>on<span class="_0 blank"></span>to <span class="ff3 ws3">P<span class="ff5 fs2 ls56 v1d">0</span></span>onde a tangen<span class="_0 blank"></span>te ao grá\u2026<span class="_2 blank"></span>co de <span class="ff3 ls19">f</span><span class="ws57">é horizontal;</span></div><div class="t m0 x4 h2b yde ff2 fs1 fc0 sc0 ls3a ws2">(c) Encon<span class="_0 blank"></span>tre o p<span class="_5 blank"> </span>on<span class="_0 blank"></span>to <span class="ff3 lsaf">P<span class="ff5 fs2 ls11a v1d">0</span></span>onde o ângulo da tangen<span class="_0 blank"></span>te ao grá\u2026<span class="_2 blank"></span>co d<span class="_5 blank"> </span>e <span class="ff3 ls19">f</span>com o eixo <span class="ff3 ls4b">x</span><span class="ws57">é 60<span class="ffa fs2 ls11b v1">o</span>.</span></div><div class="t m0 x4 h14 ydf ff1 fs1 fc0 sc0 ls3a ws44">4.4P <span class="ff2 ws57">Determine as retas tangen<span class="_0 blank"></span>tes à curv<span class="_4 blank"></span>a <span class="ff3 ls6">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls11a v1">2</span></span></span>que passam no p<span class="_5 blank"> </span>onto <span class="ff4 ws3">(0<span class="ff3 lsb0">;</span><span class="ff6">\ue000</span><span class="ws80">1) <span class="ff3">:</span></span></span></span></div><div class="t m0 x1 h2 ye0 ff1 fs0 fc0 sc0 ls3a ws48">4.<span class="_0 blank"></span>3<span class="_1 blank"> </span>Pr<span class="_d blank"></span>obl<span class="_0 blank"></span>em<span class="_d blank"></span>as de T<span class="_2b blank"></span>ax<span class="_d blank"></span>a de V<span class="_2b blank"></span>ari<span class="_0 blank"></span>aç<span class="_d blank"></span>ão</div><div class="t m0 x22 h3 ye1 ff1 fs1 fc0 sc0 ls3a ws10c">4.5A <span class="ff2 ws10d">Uma partícula se mo<span class="_0 blank"></span>v<span class="_d blank"></span>e<span class="_13 blank"> </span>de mo<span class="_5 blank"> </span>do que,<span class="_13 blank"> </span>no instante <span class="ff3 ls5f">t</span>, a distância percorrida é dada p<span class="_a blank"> </span>or</span></div><div class="t m0 x1 h42 ye2 ff3 fs1 fc0 sc0 ls11c">s<span class="ff4 ls3a ws3">(</span><span class="ls67">t<span class="ff4 ls3a ws7">) =<span class="_2a blank"> </span><span class="ff5 fs2 v1c">1</span></span></span></div><div class="t m0 x52 h23 ye3 ff5 fs2 fc0 sc0 ls7b">3<span class="ff3 fs1 ls5f v1">t</span><span class="ls4 vb">3</span><span class="ff6 fs1 ls8 v1">\ue000<span class="ff3 ls5f">t</span></span><span class="ls4 vb">2</span><span class="ff6 fs1 ls8 v1">\ue000<span class="ff4 ls3a ws3">3<span class="ff3 ls67">t</span><span class="ff2">.</span></span></span></div><div class="t m0 x1 h3 ye4 ff2 fs1 fc0 sc0 ls3a ws2">(a) Encon<span class="_0 blank"></span>tre as expressões que fornecem a v<span class="_0 blank"></span>elo<span class="_5 blank"> </span>cidade e a aceleração da partícula.</div><div class="t m0 x1 h3 ye5 ff2 fs1 fc0 sc0 ls3a ws57">(b) Em que instan<span class="_0 blank"></span>te a v<span class="_d blank"></span>elo<span class="_a blank"> </span>cidade é zero?</div><div class="t m0 x1 h3 y2a ff2 fs1 fc0 sc0 ls3a ws2">(c) Em que instan<span class="_0 blank"></span>te a aceleração é zero?</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x1 ye6 w4 h43" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bg9.png"><div class="t m0 x24 h3 y2c ffb fs2 fc0 sc0 ls3a ws4b">C Á L C U LO<span class="_e blank"> </span>D E<span class="_f blank"> </span>U M A<span class="_e blank"> </span>V<span class="_d blank"></span>A R IÁ<span class="_d blank"></span>V E L<span class="_14 blank"> </span>M A R IV<span class="_d blank"></span>A L D O<span class="_e blank"> </span>P<span class="_f blank"> </span>M ATO S<span class="_15 blank"> </span><span class="ff2 fs1">25</span></div><div class="t m0 x4 h14 y2d ff1 fs1 fc0 sc0 ls3a ws10e">4.5B <span class="ff2 ws10f">Uma partícula mo<span class="_0 blank"></span>v<span class="_d blank"></span>e-se sobre a paráb<span class="_a blank"> </span>ola <span class="ff3 ls0">y<span class="ff4 ls3">=</span><span class="ls3a ws3">x<span class="ff5 fs2 ls22 v1">2</span><span class="ls11d">:</span></span></span>Sab<span class="_5 blank"> </span>endo-se que suas co<span class="_5 blank"> </span>ordenadas <span class="ff3 ls52">x</span><span class="ff4 ws3">(<span class="ff3 ls67">t</span>)</span></span></div><div class="t m0 x1 h3 y2e ff2 fs1 fc0 sc0 ls3a ws57">e y<span class="ff4 ws3">(<span class="ff3 ls67">t</span><span class="ls11e">)</span></span><span class="ws2">são funções deriv<span class="_4 blank"></span>áveis, em que p<span class="_5 blank"> </span>on<span class="_0 blank"></span>to da paráb<span class="_5 blank"> </span>ola elas deslo<span class="_5 blank"> </span>cam-se à mesma taxa?</span></div><div class="t m0 x4 h4 ye7 ff1 fs1 fc0 sc0 ls3a ws110">4.5C <span class="ff2 ws2">Um p<span class="_5 blank"> </span>onto mo<span class="_d blank"></span>v<span class="_0 blank"></span>e-se ao longa da curv<span class="_4 blank"></span>a <span class="ff3 ls11f">y<span class="ff4 ls120">=<span class="ls3a v2">1</span></span></span></span></div><div class="t m0 x1f h1a ye8 ff4 fs1 fc0 sc0 ls3a ws5">1 + <span class="ff3 ws3">x<span class="ff5 fs2 ls6d v10">2</span><span class="ff2 ws2 v2">, de tal mo<span class="_5 blank"> </span>do que sua abscissa </span><span class="ls121 v2">x</span><span class="ff2 v2">v<span class="_d blank"></span>aria</span></span></div><div class="t m0 x1 h3 ye9 ff2 fs1 fc0 sc0 ls3a ws2">a uma v<span class="_0 blank"></span>elo<span class="_5 blank"> </span>cidade constan<span class="_0 blank"></span>te de <span class="ff4 lsd1">3</span><span class="ff3 ws3">cm=s</span>.<span class="_c blank"> </span>Qual será a velocidade da ordenada <span class="ff3 ls9f">y</span><span class="ws111">, quando <span class="ff3 ls12">x</span><span class="ff4 ws7">= 2<span class="_7 blank"> </span><span class="ff3 ws3">cm</span>?</span></span></div><div class="t m0 x4 h14 yea ff1 fs1 fc0 sc0 ls3a ws112">4.5D <span class="ff2 ws113">Um p<span class="_5 blank"> </span>onto mo<span class="_d blank"></span>v<span class="_0 blank"></span>e-se sobre a paráb<span class="_5 blank"> </span>ola <span class="ff3 ls122">y</span><span class="ff4 wscb">= 3<span class="ff3 ws3">x<span class="ff5 fs2 ls123 v1">2</span><span class="ff6 ls124">\ue000</span></span><span class="lsa">2</span><span class="ff3 ws3">x</span></span><span class="ws114">.<span class="_18 blank"> </span>Supondo-se que suas co<span class="_a blank"> </span>ordenadas</span></span></div><div class="t m0 x1 h8 yeb ff3 fs1 fc0 sc0 ls52">x<span class="ff4 ls3a ws3">(</span><span class="ls67">t<span class="ff4 ls125">)<span class="ff2 ls126">e</span></span><span class="ls98">y<span class="ff4 ls3a ws3">(</span><span class="ls5f">t<span class="ff4 ls127">)<span class="ff2 ls3a ws115">são funções deriv<span class="_4 blank"></span>áveis e que <span class="ff3 ws3">x<span class="ff9 fs2 ls14 v1">0</span><span class="ff4">(</span><span class="ls5f">t<span class="ff4 ls128">)</span></span><span class="ff6">6<span class="ff4 ws116">= 0</span></span></span>, em que p<span class="_5 blank"> </span>onto da parábola a velocidade da</span></span></span></span></span></div><div class="t m0 x1 h3 yec ff2 fs1 fc0 sc0 ls3a ws117">ordenada <span class="ff3 ls129">y</span><span class="ws2">será o triplo da v<span class="_0 blank"></span>elo<span class="_5 blank"> </span>cidade da abscissa <span class="ff3 ws3">x<span class="ff4">?</span></span></span></div><div class="t m0 x4 h3 yed ff1 fs1 fc0 sc0 ls3a ws118">4.5E <span class="ff2 ws119">Um cub<span class="_5 blank"> </span>o se expande de mo<span class="_5 blank"> </span>do que sua aresta v<span class="_d blank"></span>aria à razão de <span class="ff4 ws3">12<span class="ff3 lsb0">;</span><span class="ls12a">5</span><span class="ff3">cm=s</span></span><span class="ws11a">.<span class="_16 blank"> </span>E<span class="_5 blank"> </span>ncon<span class="_0 blank"></span>tre a</span></span></div><div class="t m0 x1 h3 yee ff2 fs1 fc0 sc0 ls3a ws2">taxa de v<span class="_4 blank"></span>ariação de seu volume, no instan<span class="_d blank"></span>te em que a aresta atinge <span class="ff4 ws11b">10 <span class="ff3 ws11c">cm </span></span><span class="ws57">de comprimento.</span></div><div class="t m0 x4 h3 yef ff1 fs1 fc0 sc0 ls3a ws11d">4.5F <span class="ff2 ws11e">Uma esfera aumen<span class="_0 blank"></span>ta de mo<span class="_5 blank"> </span>do que seu raio cresce à razão de <span class="ff4 ws3">2<span class="ff3 lsb0">;</span>5<span class="ff3">cm=s</span></span>.<span class="_13 blank"> </span>Quão rapidamente</span></div><div class="t m0 x1 h3 yf0 ff2 fs1 fc0 sc0 ls3a ws11f">v<span class="_4 blank"></span>aria seu volume no instan<span class="_d blank"></span>te em que o raio mede <span class="ff4 ws3">7<span class="ff3 lsb0">;</span>5<span class="ff3">cm</span><span class="ls12b">?</span></span>(o volume da esfera de raio <span class="ff3 ls12c">r</span><span class="ls12d">é<span class="ff3 ls12e">V</span></span><span class="ff4 ws3">(<span class="ff3 ls12f">r</span><span class="ws10f">) =</span></span></div><div class="t m0 x37 he yf1 ff5 fs2 fc0 sc0 ls3a">4</div><div class="t m0 x37 h23 yf2 ff5 fs2 fc0 sc0 ls130">3<span class="ff3 fs1 ls3a ws120 v1">\ue019 r </span><span class="ls22 vb">3</span><span class="ff4 fs1 ls3a ws3 v1">)<span class="ff3">:</span></span></div><div class="t m0 x4 h3 yf3 ff1 fs1 fc0 sc0 ls3a ws121">4.5G <span class="ff2 ws122">Sejam <span class="ff3 ls131">x</span><span class="ls132">e<span class="ff3 ls133">y</span></span><span class="ws123">os catetos de um triângulo retângulo e <span class="ff3 ls134">\ue012</span><span class="ws124">o ângulo op<span class="_5 blank"> </span>osto a <span class="ff3 ls9f">y</span><span class="wsc">.<span class="_10 blank"> </span>Sup ondo-se</span></span></span></span></div><div class="t m0 x1 h8 yf4 ff2 fs1 fc0 sc0 ls3a ws125">que <span class="ff3 ls12">x</span><span class="ff4 ws7">= 12<span class="_e blank"> </span></span><span class="wsb8">e que <span class="ff3 ls135">\ue012</span><span class="ws2">decresce à razão d<span class="_5 blank"> </span>e <span class="ff4 lsa">1<span class="ff3">=</span><span class="ls3a ws126">30<span class="_12 blank"> </span>rad <span class="ff3 ws3">=s</span></span></span>, calcule <span class="ff3 ls9f">y<span class="ff9 fs2 ls14 v1">0</span></span><span class="ff4 ws3">(<span class="ff3 ls5f">t</span>)</span><span class="ws127">, quando <span class="ff3 ls136">\ue012<span class="ff4 lsa3">=</span><span class="ls3a ws7d">\ue019 =<span class="ff4 ws128">3<span class="_12 blank"> </span>rad </span>:</span></span></span></span></span></div><div class="t m0 x4 h3 yf5 ff1 fs1 fc0 sc0 ls3a ws129">4.5H <span class="ff2 ws12a">Uma escada de <span class="ff4 ls137">8<span class="ff3 ls138">m</span></span><span class="ws12b">está encostada em uma parede v<span class="_0 blank"></span>ertical.<span class="_12 blank"> </span>Se a extremidade inferior</span></span></div><div class="t m0 x1 h3 yf6 ff2 fs1 fc0 sc0 ls3a wsbd">da escada for afastada do p<span class="_5 blank"> </span>é da parede a uma velocidade constante de <span class="ff4 ls139">2</span><span class="ff3 ws3">m=s</span>, com que v<span class="_d blank"></span>elo<span class="_5 blank"> </span>cidade</div><div class="t m0 x1 h3 yf7 ff2 fs1 fc0 sc0 ls3a ws2">a extremidade sup<span class="_5 blank"> </span>erior estará descendo no instante em que a inferior estiv<span class="_d blank"></span>er a <span class="ff4 lsd1">3<span class="ff3 ls13a">m</span></span><span class="ws57">da parede?</span></div><div class="t m0 x4 h3 yf8 ff1 fs1 fc0 sc0 ls3a ws12c">4.5I <span class="ff2 ws12d">Uma viga medindo <span class="ff4 ws12e">30 <span class="ff3 ls13b">m</span></span><span class="wse1">de comprimen<span class="_0 blank"></span>to está ap<span class="_5 blank"> </span>oiada em uma parede e o seu top<span class="_5 blank"> </span>o está</span></span></div><div class="t m0 x1 h3 yf9 ff2 fs1 fc0 sc0 ls3a ws1f">se delo<span class="_5 blank"> </span>cando a uma velocidade de <span class="ff4 lsa">0<span class="ff3 lsb5">;</span><span class="ls13c">5</span></span><span class="ff3 ws3">m=s</span>.<span class="_c blank"> </span>Qual a taxa de v<span class="_d blank"></span>ariação de medida do ângulo formado</div><div class="t m0 x1 h3 yfa ff2 fs1 fc0 sc0 ls3a ws2">p<span class="_5 blank"> </span>ela viga e p<span class="_5 blank"> </span>elo chão, quando a topo da viga estiver a uma altura de <span class="ff4 ws12f">18 <span class="ff3 ws3">m</span>?</span></div><div class="t m0 x4 h3 yfb ff1 fs1 fc0 sc0 ls3a ws130">4.5J <span class="ff2 ws3f">A Lei de Bo<span class="_0 blank"></span>yle para a dilatação dos gases é dada pela equação <span class="ff3 ws131">P V<span class="_19 blank"> </span><span class="ff4 ls13d">=</span><span class="ls13e">C</span></span><span class="ws132">, ond<span class="_5 blank"> </span>e <span class="ff3 ls13f">P</span></span>é a</span></div><div class="t m0 x1 h3 yfc ff2 fs1 fc0 sc0 ls3a wsbd">pressão, medida em Newtons p<span class="_5 blank"> </span>or unidade de área, <span class="ff3 ls140">V</span>é o v<span class="_0 blank"></span>olume e <span class="ff3 ls141">C</span>é uma constan<span class="_0 blank"></span>te.<span class="_c blank"> </span>Num certo</div><div class="t m0 x1 hd yfd ff2 fs1 fc0 sc0 ls3a ws133">instan<span class="_0 blank"></span>te, a pressão é de <span class="ff4 ws3">3<span class="ff3">:</span><span class="ws134">000 <span class="ff3 ws135">N =m<span class="ff5 fs2 ls3e v1">2</span></span></span></span>,<span class="_13 blank"> </span>o volume é de <span class="ff4 ls142">5<span class="ff3 ls143">m<span class="ff5 fs2 ls5c v1">3</span></span></span>e está crescendo à taxa de <span class="ff4 ls142">2<span class="ff3 ls143">m<span class="ff5 fs2 ls22 v1">3</span><span class="ls144">=</span></span><span class="ls3a ws3">min</span></span>.</div><div class="t m0 x1 h3 yfe ff2 fs1 fc0 sc0 ls3a ws2">Qual a taxa de v<span class="_4 blank"></span>ariação da pressão nesse<span class="_7 blank"> </span>instante?</div><div class="t m0 x4 h3 y72 ff1 fs1 fc0 sc0 ls3a ws136">4.5K <span class="ff2 ws137">Expresse a taxa de crescimen<span class="_0 blank"></span>to do v<span class="_0 blank"></span>olume <span class="ff3 ls145">V</span>de uma esfera, relativ<span class="_4 blank"></span>amente à superfície</span></div><div class="t m0 x1 h3 y2a ff3 fs1 fc0 sc0 ls146">S<span class="ff2 ls3a ws2">, em função do raio </span><span class="ls147">r<span class="ff2 ls3a ws2">da esfera.<span class="_c blank"> </span>F<span class="_4 blank"></span>aça o mesmo para o raio, relativ<span class="_4 blank"></span>amente ao v<span class="_d blank"></span>olume.</span></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="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x1a yff w2 h44" alt="" src="https://files.passeidireto.com/6bb5cda6-b074-4792-8818-7df173dc0e00/bga.png"><div class="t m0 x1 h3 y2c ff2 fs1 fc0 sc0 ls3a ws2b">26 <span class="ffb fs2 ws2c">D E R IV<span class="_d blank"></span>A<span class="_a blank"> </span>DA S<span class="_11 blank"> </span>C O M P L E M<span class="_a blank"> </span>E N T O S<span class="_e blank"> </span>4</span></div><div class="t m0 x4 h45 y2d ff1 fs1 fc0 sc0 ls3a ws138">4.5L <span class="ff2 wsdf">Num reserv<span class="_4 blank"></span>atório contendo um orifício, a v<span class="_4 blank"></span>azão p<span class="_5 blank"> </span>elo orifício é de <span class="ff4 ws3">110<span class="ff6 ls23 v8">p</span></span><span class="ff3">h cm<span class="ff5 fs2 ls22 v1">3</span><span class="ws3">=s</span></span><span class="ws139">, onde <span class="ff3">h</span></span></span></div><div class="t m0 x1 h3 y2e ff2 fs1 fc0 sc0 ls3a ws13a">é a altura, em centímetros, do nível da água no reserv<span class="_4 blank"></span>atório, acima do orifício.<span class="_9 blank"> </span>O reserv<span class="_4 blank"></span>atório é</div><div class="t m0 x1 h3 y100 ff2 fs1 fc0 sc0 ls3a ws2">alimen<span class="_0 blank"></span>tado à taxa de <span class="ff4 ws11b">88 <span class="ff3 ws13b">l=<span class="_1d blank"> </span></span><span class="ws3">min</span></span>.<span class="_c blank"> </span>Calcule a altura <span class="ff3 ls148">h</span>do nível a que o reserv<span class="_4 blank"></span>atório se estabiliza.</div><div class="t m0 x4 h3 y101 ff1 fs1 fc0 sc0 ls3a ws13c">4.5M <span class="ff2 ws1f">Um balão sob<span class="_5 blank"> </span>e v<span class="_0 blank"></span>erticalmen<span class="_0 blank"></span>te com uma v<span class="_0 blank"></span>elo<span class="_5 blank"> </span>cidade <span class="ff3 ls149">v</span>e um observ<span class="_d blank"></span>ador, a certa distância</span></div><div class="t m0 x1 h4 y102 ff3 fs1 fc0 sc0 ls5b">d<span class="ff2 ls3a ws13d">, vê o balão sob um ângulo de lev<span class="_4 blank"></span>ação <span class="ff3 ls14a">\ue012</span>.<span class="_1f blank"> </span>Ache uma expressão para a taxa<span class="_10 blank"> </span><span class="ff3 v2">d\ue012</span></span></div><div class="t m0 x53 h1a y103 ff3 fs1 fc0 sc0 ls3a ws13e">dt <span class="ff2 ws13d v2">de v<span class="_4 blank"></span>ariação de <span class="ff3">\ue012</span></span></div><div class="t m0 x1 h4 y104 ff2 fs1 fc0 sc0 ls3a wsaf">em termos de <span class="ff3 ws7d">v ;<span class="_16 blank"> </span>\ue012<span class="_7 blank"> </span></span><span class="ls14b">e<span class="ff3 ls5b">d</span></span>.<span class="_16 blank"> </span>A que v<span class="_0 blank"></span>elo<span class="_5 blank"> </span>cidade sob<span class="_5 blank"> </span>e o balão se <span class="ff3 ls14c">d</span><span class="ff4 wsab">= 500<span class="_7 blank"> </span><span class="ff3 ls14d">m</span></span><span class="ls14e">e</span><span class="ff3 v2">d\ue012</span></div><div class="t m0 x54 h1a y105 ff3 fs1 fc0 sc0 ls3a ws13f">dt <span class="ff4 wsab v2">= 0</span><span class="lsb0 v2">;</span><span class="ff4 ws140 v2">02<span class="_16 blank"> </span>rad </span><span class="ws3 v2">=s<span class="ff2 wsbc">, quando</span></span></div><div class="t m0 x1 h3 y106 ff3 fs1 fc0 sc0 ls136">\ue012<span class="ff4 lsa3">=</span><span class="ls3a ws7d">\ue019 =<span class="ff4 wsa9">4 rad<span class="ff2">.</span></span></span></div><div class="t m0 x4 h3 y107 ff1 fs1 fc0 sc0 ls3a ws141">4.5N <span class="ff2 ws142">Uma b<span class="_5 blank"> </span>ola de neve derrete a uma taxa v<span class="_d blank"></span>olumétrica <span class="ff3 ws143">dV =dt<span class="_2c blank"> </span></span>prop<span class="_5 blank"> </span>orcional à sua área.<span class="_13 blank"> </span>Mostre</span></div><div class="t m0 x1 h3 y108 ff2 fs1 fc0 sc0 ls3a ws57">que o seu raio <span class="ff3 ls14f">r</span><span class="ws2">decresce a uma taxa <span class="ff3 wsc">dr =dt<span class="_e blank"> </span></span><span class="ws3">constante.</span></span></div><div class="t m0 x4 h3 y109 ff1 fs1 fc0 sc0 ls3a ws144">4.5O <span class="ff2 ws145">Um reserv<span class="_4 blank"></span>atório cônico,<span class="_10 blank"> </span>com vértice para baixo, contém água de v<span class="_d blank"></span>olume <span class="ff3 ls150">V</span>até uma</span></div><div class="t m0 x1 h3 y10a ff2 fs1 fc0 sc0 ls3a ws146">altura <span class="ff3 ls35">h</span><span class="ws147">.<span class="_9 blank"> </span>Sup<span class="_5 blank"> </span>ondo que a ev<span class="_d blank"></span>ap<span class="_5 blank"> </span>oração da água se pro<span class="_5 blank"> </span>cessa a uma taxa <span class="ff3 ws143">dV =dt<span class="_c blank"> </span></span><span class="wsc">prop orcional<span class="_13 blank"> </span>à<span class="_13 blank"> </span>sua</span></span></div><div class="t m0 x1 h3 y10b ff2 fs1 fc0 sc0 ls3a wsc">sup erfície,<span class="_e blank"> </span>mostre<span class="_7 blank"> </span>que<span class="_7 blank"> </span><span class="ff3 ls148">h</span><span class="ws2">decresce a uma taxa <span class="ff3 ws148">dh=dt </span><span class="ws3">constan<span class="_d blank"></span>te</span></span></div><div class="t m0 x4 h41 y10c ff1 fs1 fc0 sc0 ls3a ws149">4.5P <span class="ff2 ws14a">Uma piscina está sendo esv<span class="_d blank"></span>aziada de tal forma que <span class="ff3 ls12e">V</span><span class="ff4 ws3">(<span class="ff3 ls67">t</span><span class="ws73">)<span class="_12 blank"> </span>=<span class="_10 blank"> </span>300 (20<span class="_b blank"> </span><span class="ff6 ls8">\ue000<span class="ff3 ls67">t</span></span></span>)<span class="ff5 fs2 ls151 v13">2</span><span class="ff2">representa</span></span></span></div><div class="t m0 x1 h3 y10d ff2 fs1 fc0 sc0 ls3a ws14b">o n<span class="_0 blank"></span>úmero de litros de água na piscina <span class="ff3 ls152">t</span>horas ap<span class="_5 blank"> </span>ós o início da op<span class="_5 blank"> </span>eração.<span class="_28 blank"> </span>Calcule a v<span class="_d blank"></span>elo<span class="_a blank"> </span>cidade</div><div class="t m0 x1 h3 y10e ff2 fs1 fc0 sc0 ls3a ws87">(instatânea) de escoamen<span class="_0 blank"></span>to da água ao cab<span class="_5 blank"> </span>o de 8 horas e a velocidade média desse escoamento no</div><div class="t m0 x1 h3 y10f ff2 fs1 fc0 sc0 ls3a wscd">mesmo<span class="_e blank"> </span>temp o.</div><div class="t m0 x4 h3 y110 ff1 fs1 fc0 sc0 ls3a ws14c">4.5Q <span class="ff2 ws14d">Uma estátua de altura <span class="ff3 ls153">h</span>está sendo instalada sobre um p<span class="_5 blank"> </span>edestal de altura <span class="ff3 ls154">l</span>acima do</span></div><div class="t m0 x1 h3 y111 ff2 fs1 fc0 sc0 ls3a ws27">plano horizon<span class="_0 blank"></span>tal que passa p<span class="_5 blank"> </span>elo olho de um observ<span class="_d blank"></span>ador.<span class="_2d blank"> </span>Com o observ<span class="_4 blank"></span>ador a uma distância <span class="ff3 ws3">x</span>,</div><div class="t m0 x1 h3 y112 ff2 fs1 fc0 sc0 ls3a ws4f">calcule a taxa de v<span class="_4 blank"></span>ariação,<span class="_2a blank"> </span>em relação a <span class="ff3 ws3">x</span>,<span class="_2a blank"> </span>do ângulo <span class="ff3 ls155">\ue012</span>sob o qual o observ<span class="_d blank"></span>ador vê a estátua, em</div><div class="t m0 x1 h3 y113 ff2 fs1 fc0 sc0 ls3a ws2">termos de <span class="ff3 ws14e">h;<span class="_12 blank"> </span>l </span><span class="ls17">e</span><span class="ff3 ws3">x</span>.<span class="_c blank"> </span>Qual o v<span class="_d blank"></span>alor dessa taxa se <span class="ff3 ls156">h</span><span class="ff4 ws7">= 20<span class="ff3 ws14f">;<span class="_12 blank"> </span>l </span>= 5<span class="_7 blank"> </span></span><span class="ls17">e<span class="ff3 ls12">x</span></span><span class="ff4 ws7">= 50?</span></div><div class="t m0 x55 h3 y114 ff1 fs1 fc0 sc0 ls3a ws150">4.5R <span class="ff2 ws151">A \u2026<span class="_2 blank"></span>gura ao lado mostra um reserv<span class="_d blank"></span>atório cônico de <span class="ff4 ws3">10<span class="ff3">m</span></span></span></div><div class="t m0 x2 h3 y115 ff2 fs1 fc0 sc0 ls3a wsaf">de altura e <span class="ff4 lsa">4<span class="ff3 ls157">m</span></span>de raio con<span class="_0 blank"></span>tendo água, que escoa a uma v<span class="_d blank"></span>azão</div><div class="t m0 x2 h14 y116 ff2 fs1 fc0 sc0 ls3a ws152">de <span class="ff4 lsa">5</span><span class="ff3 ws3">m<span class="ff5 fs2 ls3e v1">3</span><span class="wsc">=hor a</span></span>.</div><div class="t m0 x2 h3 y117 ff2 fs1 fc0 sc0 ls3a ws2">(a) Qual a relação en<span class="_0 blank"></span>tre as v<span class="_4 blank"></span>ariáveis <span class="ff3 ls158">R</span><span class="ls2d">e<span class="ff3 ls159">H</span></span>?</div><div class="t m0 x2 h3 y118 ff2 fs1 fc0 sc0 ls3a ws2">(b) A que taxa o nív<span class="_0 blank"></span>el da água dimin<span class="_0 blank"></span>ui, quando <span class="ff3 ls15a">H</span><span class="ff4 ws7">= 6<span class="ff3 ws3">m</span></span>?</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
Compartilhar