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cyrxdzj committed Oct 26, 2023
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2 changes: 1 addition & 1 deletion 2021-07-02-LuoguMonthlyCompetition202105Summary/index.html
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2 changes: 1 addition & 1 deletion 2021-07-07-HowToUseLuoguCard/index.html
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2 changes: 1 addition & 1 deletion 2021-07-09-Grade7Term2FinalExamSummary/index.html
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2 changes: 1 addition & 1 deletion 2021-07-09-Grade7Term2MidtermExamSummary/index.html
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2 changes: 1 addition & 1 deletion 2021-07-22-LuoguMonthlyCompetition202107Summary/index.html
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2 changes: 1 addition & 1 deletion 2023-05-21-TalkingAboutComputerSecurityFromTheIntelligentClassromSystem/index.html
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2 changes: 1 addition & 1 deletion 2023-09-22-CyrxNote-Math-001/index.html
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8 changes: 4 additions & 4 deletions 2023-09-22-CyrxNoteIndex/index.html
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Expand Up @@ -490,7 +490,7 @@ <h1 class="description center-align post-title">cyrxdzj的文化课学习笔记
<td>数学</td>
<td><a href="/2023-09-22-CyrxNote-Math-001/">cyrxdzj的文化课学习笔记 数学001 基本不等式</a></td>
<td>2023-09-22 21:58:37</td>
<td>2023-10-16 11:48:44</td>
<td>2023-10-16 17:41:27</td>
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<td>数学</td>
Expand All @@ -502,13 +502,13 @@ <h1 class="description center-align post-title">cyrxdzj的文化课学习笔记
<td>数学</td>
<td><a href="/2023-10-04-CyrxNote-Math-003/">cyrxdzj的文化课学习笔记 数学003 一元二次不等式</a></td>
<td>2023-10-04 20:57:51</td>
<td>2023-10-08 18:45:00</td>
<td>2023-10-16 17:56:40</td>
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<tr>
<td>数学</td>
<td><a href="/2023-10-09-CyrxNote-Math-004/">cyrxdzj的文化课学习笔记 数学004 函数</a></td>
<td>2023-10-09 07:01:12</td>
<td>2023-10-11 18:25:36</td>
<td>2023-10-26 07:25:57</td>
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<td>物理</td>
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2 changes: 1 addition & 1 deletion 2023-09-23-CyrxNote-Math-002/index.html
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2 changes: 1 addition & 1 deletion 2023-09-28-CyrxNote-Chemistry-001/index.html
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2 changes: 1 addition & 1 deletion 2023-10-04-CyrxNote-Math-003/index.html
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25 changes: 22 additions & 3 deletions 2023-10-09-CyrxNote-Math-004/index.html
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<div class="info-break-policy">
<i class="far fa-file-word fa-fw"></i>文章字数:&nbsp;&nbsp;
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Expand Down Expand Up @@ -526,7 +526,7 @@ <h3 id="4、仔细分析更复杂的函数"><a href="#4、仔细分析更复杂
<p><img src="../post_image/CyrxNote-Math-004/CyrxNote-Math-004-02.PNG" alt=""></p>
<p>同时，我们可以得到 $f(\frac{1}{x})=f(-\frac{1}{x})$，这一条待会有用。</p>
<p>接下来，细心推出以下式子：</p>
<p>$f(\frac{1}{x})\\=\frac{1+\frac{1}{x}^2}{1-\frac{1}{x}^2}\\=\frac{x^2+1}{x^2}\div\frac{x^2-1}{x^2}\\=\frac{1+x^2}{x^2-1}$</p>
<p>$f(\frac{1}{x})\\=\frac{1+(\frac{1}{x})^2}{1-(\frac{1}{x})^2}\\=\frac{x^2+1}{x^2}\div\frac{x^2-1}{x^2}\\=\frac{1+x^2}{x^2-1}$</p>
<p>至此，函数 $f(\frac{1}{x})$ 计算完毕。</p>
<p>而函数 $f(x)=\frac{1+x^2}{1-x^2}$，可以很容易地发现 $f(\frac{1}{x})=-f(x)$，B 选项正确，C 选项错误。图象如下：</p>
<p><img src="../post_image/CyrxNote-Math-004/CyrxNote-Math-004-03.PNG" alt=""></p>
Expand Down Expand Up @@ -621,6 +621,25 @@ <h3 id="5、找规律"><a href="#5、找规律" class="headerlink" title="5、
<p><img src="../post_image/CyrxNote-Math-004/CyrxNote-Math-004-06.PNG" alt=""></p>
<p>当然考试时没有计算机的。</p>
<p>对于这类问题，可以稍微爆算几轮，发现规律，同时根据函数解析式验证规律。</p>
<h3 id="6、将函数分成若干个部分，以分析图象"><a href="#6、将函数分成若干个部分，以分析图象" class="headerlink" title="6、将函数分成若干个部分，以分析图象"></a>6、将函数分成若干个部分，以分析图象</h3><p>例如，指出函数 $f(x)=\vert x+2\vert$ 和 $f(x)=\vert x^2-3x+2\vert$ 的单调区间。</p>
<p>首先，我们先看看 $f_1(x)=x+2$ 的图象。很显然，这就是一个一次函数的图象。同时，我们可以发现 $f(x)=\vert f_1(x)\vert$（废话）。</p>
<p>然后，我们发现，绝对值符号，就是将函数图象在 $x$ 轴以下的部分翻转过来。因此，$f(x)$ 的图象如图红色实线：</p>
<p><img src="../post_image/CyrxNote-Math-004/CyrxNote-Math-004-07.PNG" alt=""></p>
<p>因此，函数在 $(-\infty,-2)$ 单调递减，$[-2,+\infty)$ 单调递增。</p>
<p>我们再看看 $f_2(x)=x^2-3x+2$，这是一个二次函数，对称轴为 $x=1.5$，当 $x=1$ 和 $x=2$ 时函数值为 $0$。而绝对值符号就是将图象在 $x$ 轴以下的部分翻转过来，因此函数图象如下：</p>
<p><img src="../post_image/CyrxNote-Math-004/CyrxNote-Math-004-08.PNG" alt=""></p>
<p>这个图象分析稍微复杂一点。</p>
<ol>
<li><p>对于 $(-\infty,1)$，一开始（代表 $f_2(x)$ 的蓝线，下同）就是单调递减，也没被翻转，因此现在还是单调递减；</p>
</li>
<li><p>对于 $[1,1.5]$，一开始是单调递减，但由于被翻转了，所以现在是单调递增；</p>
</li>
<li><p>对于 $(1.5,2)$，一开始是单调递增，但由于被翻转了，所以现在是单调递减；</p>
</li>
<li><p>对于 $[2,+\infty)$，一开始就是单调递增，也没被翻转，因此现在还是单调递增。</p>
</li>
</ol>
<p>分析完毕。</p>


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2 changes: 1 addition & 1 deletion 2023-10-12-CyrxNote-Physics-001/index.html
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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