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6 | 6 | <description>Recent content in Log on shyの考研日志</description> |
7 | 7 | <generator>Hugo -- gohugo.io</generator> |
8 | 8 | <language>zh-cn</language> |
9 | | - <lastBuildDate>Sat, 16 Aug 2025 23:48:21 +0800</lastBuildDate><atom:link href="https://auroraemiya.github.io/npee_learning_log.github.io/categories/log/index.xml" rel="self" type="application/rss+xml" /><item> |
| 9 | + <lastBuildDate>Sun, 17 Aug 2025 23:52:05 +0800</lastBuildDate><atom:link href="https://auroraemiya.github.io/npee_learning_log.github.io/categories/log/index.xml" rel="self" type="application/rss+xml" /><item> |
| 10 | + <title>0817学习日志</title> |
| 11 | + <link>https://auroraemiya.github.io/npee_learning_log.github.io/posts/0817/</link> |
| 12 | + <pubDate>Sun, 17 Aug 2025 23:52:05 +0800</pubDate> |
| 13 | + |
| 14 | + <guid>https://auroraemiya.github.io/npee_learning_log.github.io/posts/0817/</guid> |
| 15 | + <description><h2 id="数学-朝花夕拾">数学-朝花夕拾 |
| 16 | +</h2><p>关于一元函数的不定积分和积分</p> |
| 17 | +<ol> |
| 18 | +<li> |
| 19 | +<p>关于原函数的存在条件</p> |
| 20 | +<ol> |
| 21 | +<li>连续函数一定有原函数</li> |
| 22 | +<li>含有第一类间断点和无穷间断点的函数在包含改间断点的区间内不存在原函数</li> |
| 23 | +<li>震荡间断点可能有可能没有</li> |
| 24 | +</ol> |
| 25 | +</li> |
| 26 | +<li> |
| 27 | +<p>定积分的极限形式 |
| 28 | +</p> |
| 29 | +$$ |
| 30 | + \begin{align} |
| 31 | + \int_b^af(x)dx&=\lim\limits_{\lambda\to0}\sum\limits_{k=1}^n{f(\xi_k)\Delta x_k},\xi\in[x_{k-1},x_k],\Delta x_i = x_i - x_{i-1}\\ |
| 32 | + \int_b^af(x)dx&=\lim\limits_{n\to \infty}\sum\limits_{i=1}^nf(a+\frac{b-a}{n}i)\frac{b-a}{n}\\ |
| 33 | + \int_0^1f(x)dx&=\lim\limits_{n\to \infty}\sum\limits_{i=1}^nf(\frac{i}{n})\frac{1}{n} |
| 34 | + \end{align} |
| 35 | + $$</li> |
| 36 | +<li> |
| 37 | +<p>定积分存在定理</p> |
| 38 | +<ol> |
| 39 | +<li>充分条件 |
| 40 | +<ol> |
| 41 | +<li>闭区间连续</li> |
| 42 | +<li>闭区间单调</li> |
| 43 | +<li>闭区间有界，有限间断点（不包含无穷间断点）</li> |
| 44 | +<li>闭区间有有限个第一类间断点</li> |
| 45 | +</ol> |
| 46 | +</li> |
| 47 | +<li>必要条件 |
| 48 | +<ol> |
| 49 | +<li>闭区间有界</li> |
| 50 | +</ol> |
| 51 | +</li> |
| 52 | +</ol> |
| 53 | +</li> |
| 54 | +<li> |
| 55 | +<p>几个重要的积分不等式 |
| 56 | +</p> |
| 57 | +$$ |
| 58 | + \begin{align} |
| 59 | + \left|\int_b^af(x)dx\right|&\leq\int_b^a\left|f(x)\right|\\ |
| 60 | + mL\leq\int_b^af(x)dx&\leq ML,(m\leq f(x)\leq M,L=|b-a|) |
| 61 | + \end{align} |
| 62 | + $$</li> |
| 63 | +<li> |
| 64 | +<p>变限积分性质</p> |
| 65 | +<ol> |
| 66 | +<li>f(x)可积$\Rightarrow F(x)=\int_a^xf(t)dt$ 连续</li> |
| 67 | +<li>f(x)连续$\Rightarrow F(x)=\int_a^xf(t)dt$ 可导且$F&rsquo;(x)=f(x)$</li> |
| 68 | +<li>跳跃间断点，则$F(x)$在该处不可导</li> |
| 69 | +<li>可取间断点，则$F(x)$在该处可导，但$F&rsquo;(x)\neq f(x)$</li> |
| 70 | +</ol> |
| 71 | +</li> |
| 72 | +<li> |
| 73 | +<p>反常积分</p> |
| 74 | +<ol> |
| 75 | +<li> |
| 76 | +$$ |
| 77 | + \int_0^1\frac{1}{x^p}dx\begin{cases} |
| 78 | + \text{收敛},0<p<1,\\ |
| 79 | + \\ |
| 80 | + \text{发散},p\geq1.\\ |
| 81 | + \end{cases} |
| 82 | + $$</li> |
| 83 | +<li> |
| 84 | +$$ |
| 85 | + \int_{1}^{+\infty}\frac{1}{x^p}dx\begin{cases} |
| 86 | + \text{发散},0<p<1,\\ |
| 87 | + \\ |
| 88 | + \text{收敛},p\geq1.\\ |
| 89 | + \end{cases} |
| 90 | + $$</li> |
| 91 | +<li> |
| 92 | +$$ |
| 93 | + \int_0^1\frac{\ln x}{x^p}dx\begin{cases} |
| 94 | + \text{收敛},0<p<1,\\ |
| 95 | + \text{发散},p\geq1.\\ |
| 96 | + \end{cases} |
| 97 | + $$</li> |
| 98 | +</ol> |
| 99 | +</li> |
| 100 | +</ol> |
| 101 | +</description> |
| 102 | + </item> |
| 103 | + <item> |
10 | 104 | <title>0816学习日志</title> |
11 | 105 | <link>https://auroraemiya.github.io/npee_learning_log.github.io/posts/0816/</link> |
12 | 106 | <pubDate>Sat, 16 Aug 2025 23:48:21 +0800</pubDate> |
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