Encontre uma função f e um número a tais que
y = sen(x^2)

Passo 1

Beleza, vamos lá! O enunciado pede para encontrarmos uma função e um número que satisfaçam a equação:
Aqui, estamos lidando com cálculo integral e funções. Para resolver isso, vamos precisar usar o conceito de derivadas e integrais, especialmente a Regra Fundamental do Cálculo, que conecta derivadas e integrais.

Passo 2

Primeiro, vamos entender o que essa equação está nos dizendo. Temos uma função dentro de uma integral que, ao somar com 6, dá . Para encontrar e , vamos derivar ambos os lados da equação em relação a . Isso vai simplificar a integral.

Passo 3

A derivada de ambos os lados em relação a :
Pelo Teorema Fundamental do Cálculo, a derivada da integral com limite superior é simplesmente a função dentro da integral avaliada em , multiplicada pela derivada do limite superior (que é 1). Então:

Passo 4

Vamos calcular a derivada do lado direito:
Então, temos:

Passo 5

Agora, vamos resolver para :
Então, encontramos que:

Passo 6

Agora que temos , precisamos encontrar o valor de . Voltando à equação original:
Simplificando a integral:
Então, a equação fica:

Passo 7

Vamos calcular a integral:
Aplicando os limites de integração:
Simplificando:

Passo 8

Agora, vamos isolar os termos constantes:
Resolvendo para :

Resposta

A função é e o número é .

Exercícios relacionados

Suponha que a série diverge e que . Classifique cada afirmação abaixo em falsa ou verdadeira. Se for verdadeira, justifique adequadamente; se for falsa, justifique através de um exemplo.
b. é sempre convergente.
<html><head></head><body><div>Uma empresa criou uma linha de montagem para fabricar um novo modelo de telefone celular. A taxa de produção dos telefones é</div>
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<div>Quantos telefones serão produzidos durante o terceiro mês?</div>
</body></html>
<html><head></head><body><div>Sabe-se que, daqui a <span class="math-inline " data-overflow=“visible”>
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.025ex;” xmlns=“
http://www.w3.org/2000/svg” width=“0.817ex” height=“1.441ex” role=“img” focusable=“false” viewBox=“0 -626 361 637”><title>t</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D461” d=“M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z”></path></g></g></g></svg></mjx-container></span> anos, o valor de um terreno estará aumentando à taxa de <span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.566ex;” xmlns=“http://www.w3.org/2000/svg” width=“5.079ex” height=“2.283ex” role=“img” focusable=“false” viewBox=“0 -759 2244.8 1009”><title>V’(t)</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“msup”><g data-mml-node=“mi”><path data-c=“1D449” d=“M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z”></path></g><g data-mml-node=“TeXAtom” transform=“translate(861.3,363) scale(0.707)” data-mjx-texclass=“ORD”><g data-mml-node=“mi”><path data-c=“2032” d=“M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z”></path></g></g></g><g data-mml-node=“mo” transform=“translate(1105.8,0)”><path data-c=“28” d=“M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z”></path></g><g data-mml-node=“mi” transform=“translate(1494.8,0)”><path data-c=“1D461” d=“M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z”></path></g><g data-mml-node=“mo” transform=“translate(1855.8,0)”><path data-c=“29” d=“M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z”></path></g></g></g></svg></mjx-container></span> reais por ano. Escreva uma expressão para o aumento total do valor do terreno durante os próximos cinco anos.</div>
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Encontre a derivada da função.