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math.html
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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="ko" xml:lang="ko">
<head>
<meta charset="utf-8" />
<meta name="generator" content="quarto-1.3.361" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<meta name="author" content="이광춘" />
<title>chatGPT</title>
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
}
pre.numberSource { margin-left: 3em; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
</style>
<!-- htmldependencies:E3FAD763 -->
<script id="quarto-search-options" type="application/json">{
"location": "navbar",
"copy-button": false,
"collapse-after": 3,
"panel-placement": "end",
"type": "overlay",
"limit": 20,
"language": {
"search-no-results-text": "일치 없음",
"search-matching-documents-text": "일치된 문서",
"search-copy-link-title": "검색 링크 복사",
"search-hide-matches-text": "추가 검색 결과 숨기기",
"search-more-match-text": "추가 검색결과",
"search-more-matches-text": "추가 검색결과",
"search-clear-button-title": "제거",
"search-detached-cancel-button-title": "취소",
"search-submit-button-title": "검색",
"search-label": "검색"
}
}</script>
<script type="text/javascript">
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-229551680-1', 'auto');
ga('send', {
hitType: 'pageview',
'anonymizeIp': true,
});
</script>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
<link rel="stylesheet" href="css/quarto.css" />
</head>
<body>
<div id="quarto-search-results"></div>
<header id="quarto-header" class="headroom fixed-top">
<nav class="navbar navbar-expand-lg navbar-dark ">
<div class="navbar-container container-fluid">
<div class="navbar-brand-container">
<a class="navbar-brand" href="/index.html">
<span class="navbar-title">chatGPT</span>
</a>
</div>
<div id="quarto-search" class="" title="검색"></div>
<button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#navbarCollapse"
aria-controls="navbarCollapse" aria-expanded="false" aria-label="탐색 전환"
onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">
<span class="navbar-toggler-icon"></span>
</button>
<div class="collapse navbar-collapse" id="navbarCollapse">
<ul class="navbar-nav navbar-nav-scroll me-auto">
<li class="nav-item dropdown ">
<a class="nav-link dropdown-toggle" href="#" id="nav-menu-ai" role="button" data-bs-toggle="dropdown" aria-expanded="false" rel="" target="">
<span class="menu-text">생성 AI</span>
</a>
<ul class="dropdown-menu" aria-labelledby="nav-menu-ai">
<li>
<a class="dropdown-item" href="/architecture.html" rel="" target="">
<span class="dropdown-text">chatGPT 이해</span></a>
</li>
<li>
<a class="dropdown-item" href="/trends.html" rel="" target="">
<span class="dropdown-text">추세 트렌드</span></a>
</li>
<li>
<a class="dropdown-item" href="/why_llm.html" rel="" target="">
<span class="dropdown-text">왜 거대언어모형인가?</span></a>
</li>
<li>
<a class="dropdown-item" href="/nlp_LLM.html" rel="" target="">
<span class="dropdown-text">NLP와 LLM</span></a>
</li>
<li>
<a class="dropdown-item" href="/gpt4_performance.html" rel="" target="">
<span class="dropdown-text">GPT-4 성능</span></a>
</li>
<li>
<a class="dropdown-item" href="/gpt-security.html" rel="" target="">
<span class="dropdown-text">챗GPT 보안사고</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/interface.html" rel="" target="">
<span class="dropdown-text">인터페이스</span></a>
</li>
<li>
<a class="dropdown-item" href="/open_source.html" rel="" target="">
<span class="dropdown-text">오픈소스 S/W LLM</span></a>
</li>
</ul>
</li>
<li class="nav-item dropdown ">
<a class="nav-link dropdown-toggle" href="#" id="nav-menu-chatgpt-응용사례" role="button" data-bs-toggle="dropdown" aria-expanded="false" rel="" target="">
<span class="menu-text">chatGPT 응용사례</span>
</a>
<ul class="dropdown-menu" aria-labelledby="nav-menu-chatgpt-응용사례">
<li>
<a class="dropdown-item" href="/use-cases.html" rel="" target="">
<span class="dropdown-text">활용사례</span></a>
</li>
<li>
<a class="dropdown-item" href="/intro_avi.html" rel="" target="">
<span class="dropdown-text">법인 R 소개영상</span></a>
</li>
<li>
<a class="dropdown-item" href="/macbook.html" rel="" target="">
<span class="dropdown-text">맥북 투척사건</span></a>
</li>
<li>
<a class="dropdown-item" href="/office.html" rel="" target="">
<span class="dropdown-text">오피스 전쟁</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/news_article.html" rel="" target="">
<span class="dropdown-text">보도자료 작성</span></a>
</li>
<li>
<a class="dropdown-item" href="/intro_book.html" rel="" target="">
<span class="dropdown-text">데이터 과학 책</span></a>
</li>
<li>
<a class="dropdown-item" href="/intro_paper.html" rel="" target="">
<span class="dropdown-text">논문 초록</span></a>
</li>
<li>
<a class="dropdown-item" href="/rnd.html" rel="" target="">
<span class="dropdown-text">학술연구(R&D)</span></a>
</li>
<li>
<a class="dropdown-item" href="/yuji.html" rel="" target="">
<span class="dropdown-text">멤버 Yuji</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/stable_diffusion.html" rel="" target="">
<span class="dropdown-text">Stable Diffusion</span></a>
</li>
<li>
<a class="dropdown-item" href="/text_to_images.html" rel="" target="">
<span class="dropdown-text">Text-to-Images</span></a>
</li>
<li>
<a class="dropdown-item" href="/text2image.html" rel="" target="">
<span class="dropdown-text">`bitGPT` Text-to-Image</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/samsung.html" rel="" target="">
<span class="dropdown-text">삼전 주가예측</span></a>
</li>
<li>
<a class="dropdown-item" href="/calculator.html" rel="" target="">
<span class="dropdown-text">계산기 프로그램</span></a>
</li>
<li>
<a class="dropdown-item" href="/rcoding.html" rel="" target="">
<span class="dropdown-text">GPT R 코딩개발</span></a>
</li>
<li>
<a class="dropdown-item" href="/rcoding-copilot.html" rel="" target="">
<span class="dropdown-text">부조종사 R 코딩개발</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/math.html" rel="" target="">
<span class="dropdown-text">수학: MIT / 수능</span></a>
</li>
<li>
<a class="dropdown-item" href="/middle_school.html" rel="" target="">
<span class="dropdown-text">Shiny 앱</span></a>
</li>
<li>
<a class="dropdown-item" href="/inaugural.html" rel="" target="">
<span class="dropdown-text">대통령 취임사</span></a>
</li>
<li>
<a class="dropdown-item" href="/translation.html" rel="" target="">
<span class="dropdown-text">번역(영한, 한영)</span></a>
</li>
<li>
<a class="dropdown-item" href="/gg_grouping.html" rel="" target="">
<span class="dropdown-text">챗GPT 팀구성</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/biz_number_API.html" rel="" target="">
<span class="dropdown-text">사업자등록증 API (국세청)</span></a>
</li>
</ul>
</li>
<li class="nav-item dropdown ">
<a class="nav-link dropdown-toggle" href="#" id="nav-menu-prompt-eng." role="button" data-bs-toggle="dropdown" aria-expanded="false" rel="" target="">
<span class="menu-text">Prompt Eng.</span>
</a>
<ul class="dropdown-menu" aria-labelledby="nav-menu-prompt-eng.">
<li>
<a class="dropdown-item" href="/prompt.html" rel="" target="">
<span class="dropdown-text">Prompt Engineering</span></a>
</li>
<li>
<a class="dropdown-item" href="/prompt_adv.html" rel="" target="">
<span class="dropdown-text">Advanced Prompt Engineering</span></a>
</li>
<li>
<a class="dropdown-item" href="/prompt_in_practice.html" rel="" target="">
<span class="dropdown-text">프롬프트 실무</span></a>
</li>
<li>
<a class="dropdown-item" href="/prompt_for_develoopers.html" rel="" target="">
<span class="dropdown-text">(개발자) 프롬프트 공학</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/prompt_data_science.html" rel="" target="">
<span class="dropdown-text">데이터 과학 프롬프트</span></a>
</li>
<li>
<a class="dropdown-item" href="/interview.html" rel="" target="">
<span class="dropdown-text">데이터 과학 인터뷰</span></a>
</li>
</ul>
</li>
<li class="nav-item dropdown ">
<a class="nav-link dropdown-toggle" href="#" id="nav-menu-" role="button" data-bs-toggle="dropdown" aria-expanded="false" rel="" target="">
<span class="menu-text">자동화/개발</span>
</a>
<ul class="dropdown-menu" aria-labelledby="nav-menu-">
<li>
<a class="dropdown-item" href="/ide.html" rel="" target="">
<span class="dropdown-text">통합개발환경(IDE)</span></a>
</li>
<li>
<a class="dropdown-item" href="/reticulate.html" rel="" target="">
<span class="dropdown-text">파이썬 환경구축</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/openAI_GPT.html" rel="" target="">
<span class="dropdown-text">OpenAI GPT</span></a>
</li>
<li>
<a class="dropdown-item" href="/langchain_arch.html" rel="" target="">
<span class="dropdown-text">LangCahin 아키텍쳐</span></a>
</li>
<li>
<a class="dropdown-item" href="/whisper.html" rel="" target="">
<span class="dropdown-text">오정보 음성인식(Whisper)</span></a>
</li>
<li>
<a class="dropdown-item" href="/news_release.html" rel="" target="">
<span class="dropdown-text">오정보 요약</span></a>
</li>
<li>
<a class="dropdown-item" href="/misinfo_embedding.html" rel="" target="">
<span class="dropdown-text">오정보 임베딩</span></a>
</li>
<li><hr class="dropdown-divider"></li>
<li>
<a class="dropdown-item" href="/palmer_penguins.html" rel="" target="">
<span class="dropdown-text">펭귄 데이터셋</span></a>
</li>
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<h1 class="title">chatGPT</h1>
<p class="subtitle lead">수학문제</p>
</div>
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<div class="quarto-title-meta-author">
<div class="quarto-title-meta-heading">저자</div>
<div class="quarto-title-meta-heading">소속</div>
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<p class="author"><a href="https://www.linkedin.com/in/kwangchunlee/">이광춘</a> </p>
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<a href="https://github.com/bit2r">
한국 R 사용자회
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<nav id="TOC" role="doc-toc">
<h2 id="toc-title">목차</h2>
<ul>
<li><a href="#페이스북" id="toc-페이스북"><span class="header-section-number">1</span> 페이스북</a></li>
<li><a href="#수식-표현" id="toc-수식-표현"><span class="header-section-number">2</span> 수식 표현</a>
<ul>
<li><a href="#latex-설치" id="toc-latex-설치"><span class="header-section-number">2.1</span> <code>LaTeX</code> 설치</a></li>
<li><a href="#헬로월드" id="toc-헬로월드"><span class="header-section-number">2.2</span> 헬로월드</a></li>
<li><a href="#mathpix-패키지" id="toc-mathpix-패키지"><span class="header-section-number">2.3</span> <code>mathpix</code> 패키지</a></li>
<li><a href="#pdf-.png" id="toc-pdf-.png"><span class="header-section-number">2.4</span> <code>pdf</code> → <code>.png</code></a></li>
</ul></li>
<li><a href="#mit-입학시험" id="toc-mit-입학시험"><span class="header-section-number">3</span> MIT 입학시험</a>
<ul>
<li><a href="#산술-arithmetic" id="toc-산술-arithmetic"><span class="header-section-number">3.1</span> 산술 (Arithmetic)</a></li>
<li><a href="#ai-추론" id="toc-ai-추론"><span class="header-section-number">3.2</span> AI 추론</a>
<ul>
<li><a href="#번-문제" id="toc-번-문제">1번 문제</a></li>
<li><a href="#번-문제-1" id="toc-번-문제-1">2번 문제</a></li>
<li><a href="#번-문제-2" id="toc-번-문제-2">3번 문제</a></li>
<li><a href="#번-문제-3" id="toc-번-문제-3">4번 문제</a></li>
<li><a href="#번-문제-4" id="toc-번-문제-4">5번 문제</a></li>
<li><a href="#번-문제-5" id="toc-번-문제-5">6번 문제</a></li>
<li><a href="#번-문제-6" id="toc-번-문제-6">7번 문제</a></li>
</ul></li>
</ul></li>
<li><a href="#수능-짝수형-문제" id="toc-수능-짝수형-문제"><span class="header-section-number">4</span> 수능 짝수형 문제</a>
<ul>
<li><a href="#번-문제-7" id="toc-번-문제-7"><span class="header-section-number">4.1</span> 2번 문제</a></li>
<li><a href="#번-문제-8" id="toc-번-문제-8"><span class="header-section-number">4.2</span> 3번 문제</a></li>
</ul></li>
</ul>
</nav>
<p>GPT-4의 구체적인 기능이나 개선 사항은 아직 공개되지 않았지만, 일반적으로 후속 모델은 이전 모델보다 성능이 개선되는 것이 대부분이다. GPT-4의 추론능력이 GPT-3.5와 비교하여 어떤 부분이 개선되었는지 다음과 같이 예측해볼 수 있다:</p>
<ul>
<li><p>학습 데이터셋 크기: GPT-4는 더 큰 학습 데이터셋을 사용해 훈련될 것으로 예상되며, 이를 통해 다양한 주제에 대한 이해도와 표현력이 향상될 것으로 기대.</p></li>
<li><p>모델 크기: GPT-4는 아마도 더 많은 파라미터를 가지고 있을 것으로 예상되며, 이를 통해 더욱 복잡한 패턴과 문맥을 인식하고 처리할 수 있을 것으로 기대.</p></li>
<li><p>최적화 알고리즘: GPT-4는 새로운 최적화 알고리즘을 사용하여 더욱 효율적으로 학습함으로써 모델이 더 빠르게 수렴하고, 더 나은 성능을 발휘할 수 있을 것으로 예상.</p></li>
<li><p>일반화 및 추론: GPT-4는 이전 버전보다 더 나은 일반화와 추론 능력을 가질 것으로 예상되며, 더욱 정확한 답변과 논리적인 응답을 생성할 수 있을 것으로 기대.</p></li>
<li><p>적응력: GPT-4는 사용자와의 상호 작용을 통해 더욱 빠르게 적응하고, 더 나은 결과를 제공할 수 있을 것으로 기대.</p></li>
</ul>
<p>특히, GPT-4는 추론역량(Reasoning)이 이전 모델과 비교하여 향상된 것이 확인된다.</p>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/reasoning_facebook.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-1"><img src="images/reasoning_facebook.jpg" class="img-fluid" /></a></p>
</figure>
</div>
<section id="페이스북" class="level1" data-number="1">
<h1 data-number="1"><span class="header-section-number">1</span> 페이스북</h1>
<p>페이스북 페친이 올린 기계가 풀기 어려웠던 고난도 수학문제를 풀어보자.</p>
<div class="panel-tabset">
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<div id="tabset-1-1" class="tab-pane active" role="tabpanel" aria-labelledby="tabset-1-1-tab">
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<p><a href="images/reasoning_fb_question.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-5"><img src="images/reasoning_fb_question.jpg" class="img-fluid" /></a></p>
</figure>
</div>
</div>
<div id="tabset-1-2" class="tab-pane" role="tabpanel" aria-labelledby="tabset-1-2-tab">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/reasoning_fb_gpt35.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-6"><img src="images/reasoning_fb_gpt35.jpg" class="img-fluid" /></a></p>
</figure>
</div>
</div>
<div id="tabset-1-3" class="tab-pane" role="tabpanel" aria-labelledby="tabset-1-3-tab">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/reasoning_fb_gpt4.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-7"><img src="images/reasoning_fb_gpt4.jpg" class="img-fluid" /></a></p>
</figure>
</div>
</div>
</div>
</div>
</section>
<section id="수식-표현" class="level1" data-number="2">
<h1 data-number="2"><span class="header-section-number">2</span> 수식 표현</h1>
<section id="latex-설치" class="level2" data-number="2.1">
<h2 data-number="2.1"><span class="header-section-number">2.1</span> <code>LaTeX</code> 설치</h2>
<p><code>tinytex::install_tinytex()</code> 명령어로 <code>$\LaTeX$</code>을 설치한다.</p>
<div class="cell">
<details>
<summary>코드</summary>
<div class="sourceCode" id="cb1"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>tinytex<span class="sc">::</span><span class="fu">install_tinytex</span>()</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="co"># 설치위치 확인</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>tinytex<span class="sc">::</span><span class="fu">tinytex_root</span>()</span></code></pre></div>
</details>
</div>
</section>
<section id="헬로월드" class="level2" data-number="2.2">
<h2 data-number="2.2"><span class="header-section-number">2.2</span> 헬로월드</h2>
<p>코드를 작성하여 문서(PDF)를 작성하는 사례를 구현해보자. 한글을 PDF 문서로 작성하는 것은 다른 문제로 한글을 PDF로 표현할 경우 글꼴을 비롯한 한글문서 고유한 특성을 반영할 것이 필요하기 때문에 자세한 사항은 <a href="https://r2bit.com/quarto/">데이터 과학 글쓰기</a>를 참고한다.</p>
<div class="cell">
<details>
<summary>코드</summary>
<div class="sourceCode" id="cb2"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">writeLines</span>(<span class="fu">c</span>(</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="st">'</span><span class="sc">\\</span><span class="st">documentclass{article}'</span>,</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> <span class="st">'</span><span class="sc">\\</span><span class="st">begin{document}'</span>, <span class="st">'Hello world!'</span>, <span class="st">'</span><span class="sc">\\</span><span class="st">end{document}'</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>), <span class="st">'pdf/test_eng.tex'</span>)</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a>tinytex<span class="sc">::</span><span class="fu">pdflatex</span>(<span class="st">'pdf/test_eng.tex'</span>)</span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] "pdf/test_eng.pdf"</span></span></code></pre></div>
</details>
</div>
</section>
<section id="mathpix-패키지" class="level2" data-number="2.3">
<h2 data-number="2.3"><span class="header-section-number">2.3</span> <code>mathpix</code> 패키지</h2>
<p>OCR 작업을 위해서 과거 <a href="https://github.com/tesseract-ocr/tesseract">tesseract</a>이 많이 사용되었으나 최근에는 AI 기술을 적용한 다양한 OCR 시스템이 소개되고 있다. 수식을 기계가 인식할 수 있는 형태로 인식하는 것은 다른 문제로 다양한 도구가 소개되고 있으나 <a href="https://mathpix.com/ocr">mathpix</a>도 성능이 좋은 도구중 하나로 과학기술분야 널리 사용되고 있다.</p>
<div class="columns">
<div class="column">
<div class="cell">
<details>
<summary>코드</summary>
<div class="sourceCode" id="cb3"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(mathpix)</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mathpix</span>(<span class="st">"images//integral.jpg"</span>)</span></code></pre></div>
</details>
</div>
</div><div class="column">
<p><span class="math display">\[
\int \frac { 4 x } { \sqrt { x ^ { 2 } + 1 } } d x
\]</span></p>
</div>
</div>
</section>
<section id="pdf-.png" class="level2" data-number="2.4">
<h2 data-number="2.4"><span class="header-section-number">2.4</span> <code>pdf</code> → <code>.png</code></h2>
<p>2023년 수능 PDF 수학문제 파일을 이미지(<code>.png</code>) 파일로 변환시키자.</p>
<div class="columns">
<div class="column">
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="pdf/math_01.png" class="lightbox" data-gallery="quarto-lightbox-gallery-2"><img src="pdf/math_01.png" class="img-fluid" /></a></p>
</figure>
</div>
</div><div class="column">
<div class="cell">
<details>
<summary>코드</summary>
<div class="sourceCode" id="cb4"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(pdftools)</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="fu">pdf_convert</span>(<span class="st">"data/2교시_수학영역_문제지.pdf"</span>, <span class="at">format =</span> <span class="st">"png"</span>, <span class="at">pages=</span><span class="dv">1</span>, <span class="at">filenames =</span> <span class="st">"pdf/math_01.png"</span>)</span></code></pre></div>
</details>
</div>
<div class="cell">
<details>
<summary>코드</summary>
<div class="sourceCode" id="cb5"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>math_01_png <span class="ot"><-</span> mathpix<span class="sc">::</span><span class="fu">mathpix</span>(<span class="st">"pdf/math_01.png"</span>, <span class="at">insert =</span> <span class="cn">FALSE</span>)</span></code></pre></div>
</details>
</div>
<p><span class="math display">\[
g ( x ) = x ^ { 2 } f ( x )
\]</span></p>
<div class="cell">
<details>
<summary>코드</summary>
<div class="sourceCode" id="cb6"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>math_01_png</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a><span class="co">#> [1] "$$\n g ( x ) = x ^ { 2 } f ( x ) \n$$"</span></span></code></pre></div>
</details>
</div>
</div>
</div>
</section>
</section>
<section id="mit-입학시험" class="level1" data-number="3">
<h1 data-number="3"><span class="header-section-number">3</span> MIT 입학시험</h1>
<p><span class="column-margin"><a href="https://web.archive.org/web/20120403061216/http://libraries.mit.edu/archives/exhibits/exam/index.html">Massachusetts Institute of Technology - Entrance Examination, 1869-70</a></span></p>
<p>MIT 입학시험문제가 공개되어 있지만, 이미지에 문자가 담긴 형태라 이를 OCR 기계가 읽을 수 있는 형태로 변환하는 작업을 수행한 후에 GPT-4 를 통해 추론역량을 확인해보자.</p>
<section id="산술-arithmetic" class="level2" data-number="3.1">
<h2 data-number="3.1"><span class="header-section-number">3.1</span> 산술 (Arithmetic)</h2>
<p>MIT 입학시험 산술문제를 기계판독하고 오류가 있는 부분을 수정하여 마이크로소프트 빙 AI, OpenAI GPT-4를 통해 추론능력을 살펴보자.</p>
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<ol type="1">
<li>Find the sum, then the difference, and then the product of <span class="math inline">\(3 \frac{5}{9}\)</span> and <span class="math inline">\(1 \frac{7}{24}\)</span>. Divide <span class="math inline">\(3 \frac{5}{9}\)</span> by <span class="math inline">\(1 \frac{7}{24}\)</span>.</li>
<li>Multiply 73 thousandths by 19 hundredths.</li>
<li>Divide 2880 by .0036 .</li>
<li>Find the value in decimals of <span class="math inline">\(\frac{1}{5}+\frac{8}{4}\)</span>.</li>
<li>What part of the month of August is <span class="math inline">\(\frac{7}{18}\)</span> minutes?</li>
<li>How many degrees in .01 of a circumference?</li>
<li>By selling a house and lot for <span class="math inline">\(\$ 5,790\)</span>, the owner lost <span class="math inline">\(3 \frac{1}{2}\)</span> per cent. What was their cost?</li>
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<li>Sum: <span class="math inline">\(3 \frac{5}{9}+1 \frac{7}{24}=\frac{32}{9}+\frac{31}{24}=\frac{256}{72}+\frac{93}{72}=\frac{349}{72}\)</span> or <span class="math inline">\(4 \frac{61}{72}\)</span> <br> Diff: <span class="math inline">\(3 \frac{5}{9}-1 \frac{7}{24}=\frac{163}{72}\)</span> or <span class="math inline">\(2 \frac{19}{72}\)</span> <br> Prod: <span class="math inline">\(3 \frac{5}{9} \cdot 1 \frac{7}{24}=\frac{124}{27}\)</span> or <span class="math inline">\(4 \frac{16}{27}\)</span> <br> Div: <span class="math inline">\(3 \frac{5}{9} \div 1 \frac{7}{24}=\frac{32}{9} \div \frac{31}{24}\)</span> <span class="math inline">\(= \frac{32}{9} \cdot \frac{24}{31} = \frac{256}{93}\)</span> or <span class="math inline">\(2 \frac{70}{99}\)</span></li>
<li><span class="math inline">\(\frac{73}{1000} \cdot \frac{19}{100}=\frac{1387}{100000}\)</span> or .01387 or 1387 hundred thousandths</li>
<li><span class="math inline">\(\frac{2880}{.0036}=\frac{28800000}{36}=800,000\)</span> or 8 hundred thousandths</li>
<li><span class="math inline">\(\quad \frac{1}{5}+\frac{3}{9}=\frac{4}{20}+\frac{15}{20}=\frac{19}{20} \cdot \frac{5}{5}=\frac{95}{100}=.95\)</span> <br> A August has 31 days <span class="math inline">\(=744\)</span> hours <span class="math inline">\(=44640\)</span> min <br> one minute <span class="math inline">\(=\frac{1}{44640}\)</span> part of Angust <br> So <span class="math inline">\(\frac{7}{18} \min =\frac{7}{18}\left(\frac{1}{44640}\right)=\frac{7}{803520}\)</span> part of August</li>
<li>1 circumference <span class="math inline">\(=360^{\circ} ;\)</span> 0.01 of of cercumferene <span class="math inline">\(=3.6^{\circ}\)</span></li>
<li>lose of <span class="math inline">\(3 \frac{1}{2} \%\)</span> makes selling price <span class="math inline">\(=100 \%-3 \frac{1}{2} \%=96.5 \%\)</span> of cost <span class="math inline">\(8 / 5790=96.5 \%\)</span> of cost; <span class="math inline">\(5790=\frac{96.5}{100}\)</span> of cost; <br> cost <span class="math inline">\(=5790 \cdot \frac{100}{96,5}=\$ 6000\)</span></li>
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</section>
<section id="ai-추론" class="level2" data-number="3.2">
<h2 data-number="3.2"><span class="header-section-number">3.2</span> AI 추론</h2>
<section id="번-문제" class="level3 unnumbered">
<h3 class="unnumbered">1번 문제</h3>
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<section id="bing-ai" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<ol type="1">
<li>Sure! Let’s first convert the mixed numbers to improper fractions. <span class="math inline">\(3 \frac{5}{9}\)</span> is equal to <span class="math inline">\(\frac{32}{9}\)</span> and <span class="math inline">\(1 \frac{7}{24}\)</span> is equal to <span class="math inline">\(\frac{31}{24}\)</span>. <br> Now we can find the sum: <span class="math inline">\(\frac{32}{9} + \frac{31}{24} = \frac{(32)(8) + (31)(3)}{(9)(8)} = \frac{256 + 93}{72} = \frac{349}{72}\)</span>. <br> The difference is: <span class="math inline">\(\frac{32}{9} - \frac{31}{24} = \frac{(32)(8) - (31)(3)}{(9)(8)} = \frac{256 - 93}{72} = \frac{163}{72}\)</span>. <br> The product is: <span class="math inline">\(\left(\dfrac{32}{9}\right)\left(\dfrac{31}{24}\right) = \dfrac{(32)(31)}{(9)(24)} = \dfrac{992}{216}\)</span>. <br> To divide <span class="math inline">\(3 \frac{5}{9}\)</span> by <span class="math inline">\(1 \frac{7}{24}\)</span>, we can multiply <span class="math inline">\(3 \frac{5}{9}\)</span> by the reciprocal of <span class="math inline">\(1 \frac{7}{24}\)</span>. The reciprocal of <span class="math inline">\(\dfrac{31}{24}\)</span> is <span class="math inline">\(\dfrac{24}{31}\)</span>. <br> So the quotient is: <span class="math inline">\(\left(\dfrac{32}{9}\right)\left(\dfrac{24}{31}\right) = \dfrac{(32)(24)}{(9)(31)} = \dfrac{768}{279}\)</span>.</li>
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<section id="openai" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>Let’s first convert the mixed numbers into improper fractions: <span class="math inline">\(3 \frac{5}{9} = \frac{3 \times 9 + 5}{9} = \frac{27 + 5}{9} = \frac{32}{9}\)</span></p>
<p><span class="math inline">\(1 \frac{7}{24} = \frac{1 \times 24 + 7}{24} = \frac{24 + 7}{24} = \frac{31}{24}\)</span></p>
<p>Find the sum, difference, product, and division of the two improper fractions: Sum:</p>
<p><span class="math inline">\(\frac{32}{9} + \frac{31}{24} = \frac{32 \times 24}{9 \times 24} + \frac{31 \times 9}{24 \times 9} = \frac{768}{216} + \frac{279}{216} = \frac{1047}{216}\)</span></p>
<p>Difference:</p>
<p><span class="math inline">\(\frac{32}{9} - \frac{31}{24} = \frac{32 \times 24}{9 \times 24} - \frac{31 \times 9}{24 \times 9} = \frac{768}{216} - \frac{279}{216} = \frac{489}{216}\)</span></p>
<p>Product:</p>
<p><span class="math inline">\(\frac{32}{9} \times \frac{31}{24} = \frac{32 \times 31}{9 \times 24} = \frac{992}{216}\)</span></p>
<p>Division:</p>
<p><span class="math inline">\(\frac{32}{9} \div \frac{31}{24} = \frac{32}{9} \times \frac{24}{31} = \frac{32 \times 24}{9 \times 31} = \frac{768}{279}\)</span></p>
<p>The corrected results are:</p>
<p>Sum: <span class="math inline">\(\frac{1047}{216}\)</span> Difference: <span class="math inline">\(\frac{489}{216}\)</span> Product: <span class="math inline">\(\frac{992}{216}\)</span> Division: <span class="math inline">\(\frac{768}{279}\)</span></p>
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</section>
<section id="번-문제-1" class="level3 unnumbered">
<h3 class="unnumbered">2번 문제</h3>
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<section id="bing-ai-1" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<p>Sure! To multiply 73 thousandths by 19 hundredths, you can convert them to decimals first. 73 thousandths is equivalent to 0.073 and 19 hundredths is equivalent to 0.19. Multiplying these two numbers gives us a result of 0.01387.</p>
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<section id="openai-1" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>To multiply decimals, first multiply the numbers as if they were whole numbers, and then count the total number of decimal places in both numbers.</p>
<p>73 thousandths can be written as 0.073, and 19 hundredths can be written as 0.19.</p>
<p>Now, multiply the numbers as if they were whole numbers:</p>
<p>73 * 19 = 1387</p>
<p>There are a total of 5 decimal places in both numbers combined (3 decimal places in 0.073 and 2 decimal places in 0.19). So, we need to place the decimal point 5 places from the right in the product:</p>
<p>0.001387</p>
<p>Therefore, the product of 73 thousandths and 19 hundredths is 0.001387.</p>
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<section id="번-문제-2" class="level3 unnumbered">
<h3 class="unnumbered">3번 문제</h3>
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<section id="bing-ai-2" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<p>Sure! Here’s a step-by-step guide on how to divide 2880 by 0.0036:</p>
<ol type="1">
<li>Write the problem as a long division problem: 2880 ÷ 0.0036</li>
<li>Move the decimal point in the divisor (0.0036) to the right until it becomes a whole number (36). Move the decimal point in the dividend (2880) the same number of places to the right: 28800000 ÷ 36</li>
<li>Perform long division to get a result of 800000.</li>
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<section id="openai-2" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>To divide 2880 by 0.0036, you can first divide both numbers by the same power of 10 to make the divisor a whole number. In this case, since the divisor has four decimal places, you can multiply both numbers by 10^4:</p>
<p>2880 * 10^4 = 28800000 0.0036 * 10^4 = 36</p>
<p>Now, divide 28800000 by 36:</p>
<p>28800000 ÷ 36 = 800000</p>
<p>So, 2880 divided by 0.0036 is 800000.</p>
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<section id="번-문제-3" class="level3 unnumbered">
<h3 class="unnumbered">4번 문제</h3>
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<section id="bing-ai-3" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<p>Sure! To find the value in decimals of <span class="math inline">\(\frac{1}{5}+\frac{8}{4}\)</span>, you can first convert the fractions to decimals. <span class="math inline">\(\frac{1}{5}\)</span> is equivalent to 0.2 and <span class="math inline">\(\frac{8}{4}\)</span> is equivalent to 2. Adding these two numbers gives us a result of 2.2.</p>
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<section id="openai-3" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>First, let’s simplify each fraction and convert them to decimals:</p>
<p><span class="math inline">\(\frac{1}{5}\)</span> as a decimal is 0.2.</p>
<p><span class="math inline">\(\frac{8}{4}\)</span> can be simplified to <span class="math inline">\(\frac{2}{1}\)</span>, which as a decimal is 2.</p>
<p>Now, add the two decimal values:</p>
<p><span class="math inline">\(0.2 + 2 = 2.2\)</span></p>
<p>So, the value of <span class="math inline">\(\frac{1}{5}+\frac{8}{4}\)</span> in decimals is 2.2.</p>
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<section id="번-문제-4" class="level3 unnumbered">
<h3 class="unnumbered">5번 문제</h3>
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<section id="bing-ai-4" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<p>I’m sorry but your question is a bit unclear. Could you please clarify what you mean by “what part of the month of August is <span class="math inline">\(\frac{7}{18}\)</span> minutes”? Are you asking for a specific date and time in August that corresponds to <span class="math inline">\(\frac{7}{18}\)</span> minutes?</p>
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<section id="openai-4" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>To determine what part of the month of August is <span class="math inline">\(\frac{7}{18}\)</span> minutes, we first need to know the total number of minutes in the month of August.</p>
<p>August has 31 days, and each day has 24 hours with 60 minutes per hour. Therefore, there are:</p>
<p><span class="math inline">\(31 \times 24 \times 60 = 44640\)</span> minutes in August.</p>
<p>Now, we can find what part of the month of August is <span class="math inline">\(\frac{7}{18}\)</span> minutes by dividing <span class="math inline">\(\frac{7}{18}\)</span> by the total number of minutes:</p>
<p><span class="math inline">\(\frac{7}{18} \div 44640 = \frac{7}{18 \times 44640}\)</span></p>
<p>Now, simplify the fraction:</p>
<p><span class="math inline">\(\frac{7}{802080}\)</span></p>
<p>So, <span class="math inline">\(\frac{7}{18}\)</span> minutes is <span class="math inline">\(\frac{7}{802080}\)</span> of the month of August.</p>
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<section id="번-문제-5" class="level3 unnumbered">
<h3 class="unnumbered">6번 문제</h3>
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<section id="bing-ai-5" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<p>Sure! A full circumference is equivalent to 360 degrees. To find the number of degrees in 0.01 of a circumference, you can multiply 360 by 0.01 to get a result of 3.6 degrees.</p>
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<section id="openai-5" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>To find out how many degrees are in 0.01 of a circumference, you need to consider that a full circumference corresponds to 360 degrees.</p>
<p>To find the number of degrees in 0.01 of a circumference, multiply 360 degrees by 0.01:</p>
<p>360° × 0.01 = 3.6°</p>
<p>So, there are 3.6 degrees in 0.01 of a circumference.</p>
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<section id="번-문제-6" class="level3 unnumbered">
<h3 class="unnumbered">7번 문제</h3>
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<section id="bing-ai-6" class="level4 unnumbered">
<h4 class="unnumbered">Bing AI</h4>
<p>Sure! Let’s say the owner’s cost was <span class="math inline">\(x\)</span>. If they lost <span class="math inline">\(3 \frac{1}{2}\)</span> per cent on the sale, then their selling price was <span class="math inline">\(100% - 3 \frac{1}{2}% = 96.5%\)</span> of their cost. So we can write the equation: <span class="math inline">\(0.965x = 5790\)</span>. Solving for <span class="math inline">\(x\)</span>, we find that the owner’s cost was $ <span class="math inline">\(5994.81\)</span>.</p>
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<section id="openai-6" class="level4 unnumbered">
<h4 class="unnumbered">OpenAI</h4>
<p>To find the original cost of the house and lot, we set up and solve the equation as follows:</p>
<p>Let the cost of the house and lot be denoted by C. The owner lost <span class="math inline">\(3\frac{1}{2}%\)</span> on the sale, which means they sold it for <span class="math inline">\(100% - 3\frac{1}{2}%\)</span> of its original value. First, convert <span class="math inline">\(3\frac{1}{2}%\)</span> to a decimal:</p>
<p><span class="math inline">\(3\frac{1}{2}% = 3.5%\)</span></p>
<p>Now, subtract this percentage from <span class="math inline">\(100%\)</span> to find the percentage of the cost at which the house was sold:</p>
<p><span class="math inline">\(100% - 3.5% = 96.5%\)</span></p>
<p>Now, we know that <span class="math inline">\(96.5%\)</span> of the cost is equal to the selling price, which is $<span class="math inline">\(5,790\)</span>. We can write this as an equation:</p>
<p>$0.965C = <span class="math inline">\(5,790\)</span></p>
<p>Now, we can find the cost (C) by dividing both sides of the equation by 0.965:</p>
<p>$C = $</p>
<p>$C <span class="math inline">\(5,995.85\)</span></p>
<p>So, the original cost of the house and lot was approximately $<span class="math inline">\(5,995.85\)</span>. In LaTeX format, the solution is as follows:</p>
<p>Let the cost be denoted by <span class="math inline">\(C\)</span>. The equation representing the loss is: [0.965C = 5790]</p>
<p>To find the cost, divide both sides by 0.965: [C = ]</p>
<p>Therefore, the original cost of the house and lot was approximately $5,995.85.</p>
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<section id="수능-짝수형-문제" class="level1" data-number="4">
<h1 data-number="4"><span class="header-section-number">4</span> 수능 짝수형 문제</h1>
<p>2022학년도 대학수학능력시험 문제지 수학영역 짝수형에 나와 있는 수능 문제를 추론능력을 활용하여 풀어보자.</p>
<p><span class="column-margin"><a href="https://chat.openai.com/share/9296c614-0b6d-4fae-a900-5f8970aa22a0">챗GPT 수능문제풀이</a></span></p>
<section id="번-문제-7" class="level2" data-number="4.1">
<h2 data-number="4.1"><span class="header-section-number">4.1</span> 2번 문제</h2>
<blockquote>
<p>함수 <span class="math inline">\(f(x)=x^3+3 x^2+x-1\)</span> 에 대하여 <span class="math inline">\(f^{\prime}(1)\)</span> 의 값은?</p>
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<details>
<summary>코드</summary>
<div class="sourceCode" id="cb7"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(openai)</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>solve_math_02 <span class="ot"><-</span> <span class="fu">create_completion</span>(</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> <span class="at">model=</span><span class="st">"text-davinci-003"</span>,</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> <span class="at">prompt =</span> <span class="st">'</span><span class="sc">\\</span><span class="st">text { 2. 함수 } f(x)=x^3+3 x^2+x-1 </span><span class="sc">\\</span><span class="st">text { 에 대하여 } f^{</span><span class="sc">\\</span><span class="st">prime}(1) </span><span class="sc">\\</span><span class="st">text { 의 값은? } and explain the answer'</span>,</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> <span class="at">max_tokens=</span><span class="dv">1024</span>,</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a> <span class="at">openai_api_key =</span> <span class="fu">Sys.getenv</span>(<span class="st">"OPENAI_API_KEY"</span>)</span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(solve_math_02<span class="sc">$</span>choices<span class="sc">$</span>text)</span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a><span class="co">#> f'(1)은 함수 f(x)의 미분값이므로 </span></span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a><span class="co">#> f'(1) = 3x^2 + 6x + 1 = 8</span></span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a><span class="co">#> 따라서 f'(1)의 값은 8입니다. </span></span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a><span class="co">#> </span></span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a><span class="co">#> 그 이유는 함수 f'(x)는 x값에 따라 변하는 값이기 때문입니다. 그래서 x값이 1일 때 f'(1)의 값은 3·12 + 6·1 + 1 = 8 입니다.</span></span></code></pre></div>
</details>
</div>
</div>
<div id="tabset-3-2" class="tab-pane" role="tabpanel" aria-labelledby="tabset-3-2-tab">
<p>Sure! The derivative of a function <span class="math inline">\(f(x)\)</span> is denoted by <span class="math inline">\(f’(x)\)</span> and represents the rate of change of the function at a given point. To find the value of <span class="math inline">\(f’(1)\)</span> for the function <span class="math inline">\(f(x)=x^3+3 x^2+x-1\)</span>, we first need to find its derivative.</p>
<p>The derivative of <span class="math inline">\(x^3\)</span> is <span class="math inline">\(3x^2\)</span>, the derivative of <span class="math inline">\(3x^2\)</span> is <span class="math inline">\(6x\)</span>, and the derivative of <span class="math inline">\(x\)</span> is 1. The derivative of a constant term like -1 is 0. So, the derivative of the function <span class="math inline">\(f(x)=x^3+3 x^2+x-1\)</span> is:</p>
<p><span class="math display">\[f’(x) = 3x^2 + 6x + 1\]</span></p>
<p>To find the value of <span class="math inline">\(f’(1)\)</span>, we can substitute x=1 into this equation:</p>
<p><span class="math display">\[f’(1) = 3(1)^2 + 6(1) + 1 = 10\]</span></p>
<p>So, the value of <span class="math inline">\(f’(1)\)</span> for the function <span class="math inline">\(f(x)=x^3+3 x^2+x-1\)</span> is 10.</p>
</div>
<div id="tabset-3-3" class="tab-pane" role="tabpanel" aria-labelledby="tabset-3-3-tab">
<p>함수 <span class="math inline">\(f(x)=x^3+3x^2+x-1\)</span>의 도함수를 구하고, <span class="math inline">\(x=1\)</span>일 때의 값을 구하겠습니다.</p>
<p><span class="math inline">\(f(x)\)</span>의 도함수는 <span class="math inline">\(f'(x)\)</span>로 표시되며, 각 항에 대한 미분을 적용합니다:</p>
<p><span class="math inline">\(f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(3x^2) + \frac{d}{dx}(x) - \frac{d}{dx}(1)\)</span></p>
<p>각 항에 대한 미분 결과는 다음과 같습니다:</p>
<p><span class="math inline">\(\frac{d}{dx}(x^3) = 3x^2\)</span></p>
<p><span class="math inline">\(\frac{d}{dx}(3x^2) = 6x\)</span></p>
<p><span class="math inline">\(\frac{d}{dx}(x) = 1\)</span></p>
<p><span class="math inline">\(\frac{d}{dx}(1) = 0\)</span></p>
<p>이제 도함수를 다음과 같이 작성할 수 있습니다:</p>
<p><span class="math inline">\(f'(x) = 3x^2 + 6x + 1\)</span></p>
<p>이제 <span class="math inline">\(f'(1)\)</span>을 구합니다:</p>
<p><span class="math inline">\(f'(1) = 3(1)^2 + 6(1) + 1 = 3 + 6 + 1 = 10\)</span></p>
<p>따라서, <span class="math inline">\(f^{\prime}(1)\)</span>의 값은 10입니다.</p>
</div>
<div id="tabset-3-4" class="tab-pane" role="tabpanel" aria-labelledby="tabset-3-4-tab">
<p>환영환각을 없앤 수학 엔진 울프램(Woolfram)을 사용해서 문제를 풀어보자</p>
<p>함수 <span class="math inline">\(f(x)=x^3+3 x^2+x-1\)</span> 의 도함수 <span class="math inline">\(f^{\prime}(x)\)</span> 를 구하면, <span class="math inline">\(f^{\prime}(x)=3 x^2+6 x+1\)</span> 이 됩니 다. 이제 <span class="math inline">\(f^{\prime}(1)\)</span> 의 값을 구하면 됩니다. Wolfram Alpha를 사용하여 이 계산을 수행하겠습니다. Used Wolfram 함수 <span class="math inline">\(f^{\prime}(x)=3 x^2+6 x+1\)</span> 에 <span class="math inline">\(x=1\)</span> 을 대입하면, <span class="math inline">\(f^{\prime}(1)=10\)</span> 이 됩니다. 따라서, <span class="math inline">\(f^{\prime}(1)=10\)</span> 입니다. 이는 함수 <span class="math inline">\(f(x)=x^3+3 x^2+x-1\)</span> 의 <span class="math inline">\(x=1\)</span> 에서의 기울기를 나타냅니다.</p>
<span class="math display">\[\begin{align*}
f(x) &= x^3 + 3x^2 + x - 1 \\
f'(x) &= \frac{d}{dx}(x^3 + 3x^2 + x - 1) \\
&= 3x^2 + 6x + 1 \\
f'(1) &= 3(1)^2 + 6(1) + 1 \\
&= 10
\end{align*}\]</span>
</div>
</div>
</div>
</section>
<section id="번-문제-8" class="level2" data-number="4.2">
<h2 data-number="4.2"><span class="header-section-number">4.2</span> 3번 문제</h2>
<blockquote>
<p>등차수열 <span class="math inline">\(\left\{a_n\right\}\)</span> 에 대하여 <span class="math display">\[
a_2=6, \quad a_4+a_6=36
\]</span> 일 때, <span class="math inline">\(a_{10}\)</span> 의 값은?</p>
</blockquote>
<div class="quarto-figure quarto-figure-center">
<figure>
<p><a href="images/K-SAT-math-03.jpg" class="lightbox" data-gallery="quarto-lightbox-gallery-4"><img src="images/K-SAT-math-03.jpg" class="img-fluid" /></a></p>
</figure>
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<summary>코드</summary>
<div class="sourceCode" id="cb8"><pre class="sourceCode r cell-code"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>solve_math_03 <span class="ot"><-</span> <span class="fu">create_completion</span>(</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a> <span class="at">model=</span><span class="st">"text-davinci-003"</span>,</span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a> <span class="at">prompt =</span> <span class="st">'등차수열 $\left\{a_n</span><span class="sc">\r</span><span class="st">ight\}$ 에 대하여 $$ a_2=6, \quad a_4+a_6=36 $$ 일 때, $a_{10}$ 의 값은?'</span>,</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a> <span class="at">max_tokens=</span><span class="dv">1024</span>,</span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a> <span class="at">openai_api_key =</span> <span class="fu">Sys.getenv</span>(<span class="st">"OPEN_AI_KEY"</span>)</span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(solve_math_03<span class="sc">$</span>choices<span class="sc">$</span>text)</span></code></pre></div>