https://one.snvedu.cn/Recent content in 首页 on 专升本备考手册Hugozh-cnSun, 08 Nov 2026 00:00:00 +0000https://one.snvedu.cn/docs/english/vocabulary-grammar/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/vocabulary-grammar/<h2 id="-模块说明">📌 模块说明<a class="anchor" href="#-%e6%a8%a1%e5%9d%97%e8%af%b4%e6%98%8e" aria-label="Link"></a></h2> <p>词汇 + 语法选择题：词汇靠<strong>滚动重复</strong>，语法靠<strong>体系梳理 + 刷题巩固</strong>。语法题的正确率大头来自「先找标志词 → 再套规则」的固定流程。</p> <h2 id="-核心考点">📖 核心考点<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e8%80%83%e7%82%b9" aria-label="Link"></a></h2> <h3 id="1-时态标志词速查做题第一眼看这里">1. 时态标志词速查（做题第一眼看这里）<a class="anchor" href="#1-%e6%97%b6%e6%80%81%e6%a0%87%e5%bf%97%e8%af%8d%e9%80%9f%e6%9f%a5%e5%81%9a%e9%a2%98%e7%ac%ac%e4%b8%80%e7%9c%bc%e7%9c%8b%e8%bf%99%e9%87%8c" aria-label="Link"></a></h3> <table> <thead> <tr> <th>标志词</th> <th>时态</th> </tr> </thead> <tbody> <tr> <td>ago, yesterday, last&hellip;, in 2020</td> <td>一般过去时</td> </tr> <tr> <td>already, yet, just, since, so far, recently</td> <td>现在完成时</td> </tr> <tr> <td>at this time yesterday, look!, now</td> <td>进行时</td> </tr> <tr> <td>by + 过去时间 / by the time + 过去从句</td> <td>过去完成时</td> </tr> <tr> <td>tomorrow, next&hellip;, in + 将来时间</td> <td>一般将来时</td> </tr> </tbody> </table> <h3 id="2-虚拟语气公式表背表直接套">2. 虚拟语气公式表（背表直接套）<a class="anchor" href="#2-%e8%99%9a%e6%8b%9f%e8%af%ad%e6%b0%94%e5%85%ac%e5%bc%8f%e8%a1%a8%e8%83%8c%e8%a1%a8%e7%9b%b4%e6%8e%a5%e5%a5%97" aria-label="Link"></a></h3> <table> <thead> <tr> <th>虚拟对象</th> <th>if 从句</th> <th>主句</th> </tr> </thead> <tbody> <tr> <td>对现在</td> <td>did / were</td> <td>would + do</td> </tr> <tr> <td>对过去</td> <td>had done</td> <td>would have done</td> </tr> <tr> <td>对将来</td> <td>should do / were to do</td> <td>would + do</td> </tr> </tbody> </table> <h2 id="-题型库历年考法全覆盖完整解法版">🧩 题型库：历年考法全覆盖（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1时态题标志词定位法">题型 1：时态题（标志词定位法）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e6%97%b6%e6%80%81%e9%a2%98%e6%a0%87%e5%bf%97%e8%af%8d%e5%ae%9a%e4%bd%8d%e6%b3%95" aria-label="Link"></a></h3> <p>例：He ____ (live) here since 2010. 解析：标志词 <strong>since</strong> → 现在完成时 → <strong>has lived</strong> ✅ 口诀：见 since / so far / in the past few years，闭眼选完成时。</p>https://one.snvedu.cn/docs/gaoshu/functions-limits-continuity/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/functions-limits-continuity/<blockquote class=''> <p>🎯 浙江考纲定位：一元微积分板块（占分约 80%），本章是<strong>填空题第一高频区</strong>。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解函数概念，会求定义域、值域，理解函数的单调性、奇偶性、周期性</li> <li>理解极限概念，掌握极限四则运算法则</li> <li>掌握两个重要极限，会用等价无穷小替换求极限</li> <li>理解函数连续性概念，会判断间断点及其类型</li> <li>了解闭区间上连续函数的性质（最值、介值定理）</li> </ul> <h2 id="-核心公式与方法库">📖 核心公式与方法库<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e5%85%ac%e5%bc%8f%e4%b8%8e%e6%96%b9%e6%b3%95%e5%ba%93" aria-label="Link"></a></h2> <h3 id="1-等价无穷小速查x0必须背到条件反射">1. 等价无穷小速查（x→0，必须背到条件反射）<a class="anchor" href="#1-%e7%ad%89%e4%bb%b7%e6%97%a0%e7%a9%b7%e5%b0%8f%e9%80%9f%e6%9f%a5x0%e5%bf%85%e9%a1%bb%e8%83%8c%e5%88%b0%e6%9d%a1%e4%bb%b6%e5%8f%8d%e5%b0%84" aria-label="Link"></a></h3> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>∼</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>∼</mo><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>∼</mo><mi>arcsin</mi><mo>⁡</mo><mi>x</mi><mo>∼</mo><mi>arctan</mi><mo>⁡</mo><mi>x</mi><mo>∼</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><msup><mi>e</mi><mi>x</mi></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x \sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x \sim \ln(1+x) \sim e^x-1</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo>∼</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo separator="true">,</mo><mspace width="2em"/><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>α</mi></msup><mo>−</mo><mn>1</mn><mo>∼</mo><mi>α</mi><mi>x</mi><mo separator="true">,</mo><mspace width="2em"/><msup><mi>a</mi><mi>x</mi></msup><mo>−</mo><mn>1</mn><mo>∼</mo><mi>x</mi><mi>ln</mi><mo>⁡</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">1-\cos x \sim \frac{x^2}{2},\qquad (1+x)^{\alpha}-1 \sim \alpha x,\qquad a^x-1 \sim x\ln a</annotation></semantics></math></span><h3 id="2-求极限方法优先级">2. 求极限方法优先级<a class="anchor" href="#2-%e6%b1%82%e6%9e%81%e9%99%90%e6%96%b9%e6%b3%95%e4%bc%98%e5%85%88%e7%ba%a7" aria-label="Link"></a></h3> <p>直接代入 → 化简变形（有理化/拆项/对数和差转乘积）→ 等价替换 → 洛必达 → 夹逼/定积分定义</p>https://one.snvedu.cn/exam/2018/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2018/<h1 id="2018-年浙江专升本真题">2018 年浙江专升本真题<a class="anchor" href="#2018-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/english-words/days-1-10/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/days-1-10/<blockquote class=''> <p>用法：遮住「中文」列自测；想不起的画 △。动词 ⭐ 优先多看。</p> </blockquote><h2 id="day-1a">Day 1（A）<a class="anchor" href="#day-1a" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>ability ⭐</td> <td>n.</td> <td>能力</td> </tr> <tr> <td>absorb</td> <td>v.</td> <td>吸收</td> </tr> <tr> <td>abstract</td> <td>adj./n.</td> <td>抽象的；摘要</td> </tr> <tr> <td>accept ⭐</td> <td>v.</td> <td>接受</td> </tr> <tr> <td>accident</td> <td>n.</td> <td>事故</td> </tr> <tr> <td>accompany</td> <td>v.</td> <td>陪伴</td> </tr> <tr> <td>accomplish</td> <td>v.</td> <td>完成</td> </tr> <tr> <td>account</td> <td>n./v.</td> <td>账户；解释</td> </tr> <tr> <td>accurate</td> <td>adj.</td> <td>精确的</td> </tr> <tr> <td>accuse</td> <td>v.</td> <td>指控</td> </tr> <tr> <td>achieve ⭐</td> <td>v.</td> <td>达到，实现</td> </tr> <tr> <td>acknowledge</td> <td>v.</td> <td>承认</td> </tr> <tr> <td>acquire</td> <td>v.</td> <td>获得</td> </tr> <tr> <td>act ⭐</td> <td>v./n.</td> <td>行动；法案</td> </tr> <tr> <td>action</td> <td>n.</td> <td>行动</td> </tr> <tr> <td>active</td> <td>adj.</td> <td>积极的</td> </tr> <tr> <td>adapt</td> <td>v.</td> <td>适应；改编</td> </tr> <tr> <td>add ⭐</td> <td>v.</td> <td>添加</td> </tr> <tr> <td>adequate</td> <td>adj.</td> <td>足够的</td> </tr> <tr> <td>adjust</td> <td>v.</td> <td>调整</td> </tr> <tr> <td>admire</td> <td>v.</td> <td>钦佩</td> </tr> <tr> <td>admit</td> <td>v.</td> <td>承认；准许进入</td> </tr> <tr> <td>adult</td> <td>n./adj.</td> <td>成年人</td> </tr> <tr> <td>advance</td> <td>v./n.</td> <td>前进；进步</td> </tr> <tr> <td>advantage</td> <td>n.</td> <td>优势</td> </tr> </tbody> </table> <h2 id="day-2a">Day 2（A）<a class="anchor" href="#day-2a" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>adventure</td> <td>n.</td> <td>冒险</td> </tr> <tr> <td>advertisement</td> <td>n.</td> <td>广告</td> </tr> <tr> <td>advice</td> <td>n.</td> <td>建议</td> </tr> <tr> <td>affair</td> <td>n.</td> <td>事务</td> </tr> <tr> <td>affect ⭐</td> <td>v.</td> <td>影响</td> </tr> <tr> <td>afford ⭐</td> <td>v.</td> <td>负担得起</td> </tr> <tr> <td>agency</td> <td>n.</td> <td>代理机构</td> </tr> <tr> <td>aggressive</td> <td>adj.</td> <td>好斗的；进取的</td> </tr> <tr> <td>agree ⭐</td> <td>v.</td> <td>同意</td> </tr> <tr> <td>agriculture</td> <td>n.</td> <td>农业</td> </tr> <tr> <td>aim</td> <td>n./v.</td> <td>目标；瞄准</td> </tr> <tr> <td>airline</td> <td>n.</td> <td>航空公司</td> </tr> <tr> <td>alarm</td> <td>n./v.</td> <td>警报</td> </tr> <tr> <td>alive</td> <td>adj.</td> <td>活着的</td> </tr> <tr> <td>allow ⭐</td> <td>v.</td> <td>允许</td> </tr> <tr> <td>alone</td> <td>adj./adv.</td> <td>单独的</td> </tr> <tr> <td>alphabet</td> <td>n.</td> <td>字母表</td> </tr> <tr> <td>alternative</td> <td>n./adj.</td> <td>可选方案</td> </tr> <tr> <td>altitude</td> <td>n.</td> <td>高度</td> </tr> <tr> <td>amaze</td> <td>v.</td> <td>使惊讶</td> </tr> <tr> <td>ambition</td> <td>n.</td> <td>雄心</td> </tr> <tr> <td>amount</td> <td>n.</td> <td>数量</td> </tr> <tr> <td>amuse</td> <td>v.</td> <td>使愉快</td> </tr> <tr> <td>analyze ⭐</td> <td>v.</td> <td>分析</td> </tr> <tr> <td>ancient</td> <td>adj.</td> <td>古代的</td> </tr> </tbody> </table> <h2 id="day-3a">Day 3（A）<a class="anchor" href="#day-3a" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>anger</td> <td>n.</td> <td>愤怒</td> </tr> <tr> <td>angle</td> <td>n.</td> <td>角度</td> </tr> <tr> <td>angry</td> <td>adj.</td> <td>生气的</td> </tr> <tr> <td>announce</td> <td>v.</td> <td>宣布</td> </tr> <tr> <td>annual</td> <td>adj.</td> <td>每年的</td> </tr> <tr> <td>anxiety</td> <td>n.</td> <td>焦虑</td> </tr> <tr> <td>anxious</td> <td>adj.</td> <td>焦虑的</td> </tr> <tr> <td>apart</td> <td>adv.</td> <td>分开</td> </tr> <tr> <td>apartment</td> <td>n.</td> <td>公寓</td> </tr> <tr> <td>apologize</td> <td>v.</td> <td>道歉</td> </tr> <tr> <td>apparent</td> <td>adj.</td> <td>明显的</td> </tr> <tr> <td>appeal</td> <td>v./n.</td> <td>呼吁；吸引</td> </tr> <tr> <td>appear ⭐</td> <td>v.</td> <td>出现；似乎</td> </tr> <tr> <td>appearance</td> <td>n.</td> <td>外貌</td> </tr> <tr> <td>appetite</td> <td>n.</td> <td>食欲</td> </tr> <tr> <td>apply ⭐</td> <td>v.</td> <td>申请；应用</td> </tr> <tr> <td>appoint</td> <td>v.</td> <td>任命</td> </tr> <tr> <td>appreciate</td> <td>v.</td> <td>感激；欣赏</td> </tr> <tr> <td>approach</td> <td>v./n.</td> <td>接近；方法</td> </tr> <tr> <td>approve</td> <td>v.</td> <td>批准</td> </tr> <tr> <td>architect</td> <td>n.</td> <td>建筑师</td> </tr> <tr> <td>argue ⭐</td> <td>v.</td> <td>争论</td> </tr> <tr> <td>arise</td> <td>v.</td> <td>出现</td> </tr> <tr> <td>arrange</td> <td>v.</td> <td>安排</td> </tr> <tr> <td>arrest</td> <td>v./n.</td> <td>逮捕</td> </tr> </tbody> </table> <h2 id="day-4ab">Day 4（A~B）<a class="anchor" href="#day-4ab" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>arrival</td> <td>n.</td> <td>到达</td> </tr> <tr> <td>artist</td> <td>n.</td> <td>艺术家</td> </tr> <tr> <td>ashamed</td> <td>adj.</td> <td>羞愧的</td> </tr> <tr> <td>aspect</td> <td>n.</td> <td>方面</td> </tr> <tr> <td>assess</td> <td>v.</td> <td>评估</td> </tr> <tr> <td>assist</td> <td>v.</td> <td>协助</td> </tr> <tr> <td>associate</td> <td>v.</td> <td>联系</td> </tr> <tr> <td>assume</td> <td>v.</td> <td>假定</td> </tr> <tr> <td>astonish</td> <td>v.</td> <td>使震惊</td> </tr> <tr> <td>athlete</td> <td>n.</td> <td>运动员</td> </tr> <tr> <td>atmosphere</td> <td>n.</td> <td>大气；氛围</td> </tr> <tr> <td>attach</td> <td>v.</td> <td>附上</td> </tr> <tr> <td>attack</td> <td>v./n.</td> <td>攻击</td> </tr> <tr> <td>attempt</td> <td>n./v.</td> <td>尝试</td> </tr> <tr> <td>attend ⭐</td> <td>v.</td> <td>出席；上（学）</td> </tr> <tr> <td>attention</td> <td>n.</td> <td>注意</td> </tr> <tr> <td>attitude</td> <td>n.</td> <td>态度</td> </tr> <tr> <td>attract</td> <td>v.</td> <td>吸引</td> </tr> <tr> <td>audience</td> <td>n.</td> <td>观众</td> </tr> <tr> <td>authority</td> <td>n.</td> <td>权威；当局</td> </tr> <tr> <td>automatic</td> <td>adj.</td> <td>自动的</td> </tr> <tr> <td>autumn</td> <td>n.</td> <td>秋天</td> </tr> <tr> <td>available</td> <td>adj.</td> <td>可获得的</td> </tr> <tr> <td>average</td> <td>adj./n.</td> <td>平均的</td> </tr> <tr> <td>avoid ⭐</td> <td>v.</td> <td>避免</td> </tr> </tbody> </table> <h2 id="day-5b">Day 5（B）<a class="anchor" href="#day-5b" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>awake</td> <td>adj.</td> <td>醒着的</td> </tr> <tr> <td>award</td> <td>n./v.</td> <td>奖品；授予</td> </tr> <tr> <td>aware</td> <td>adj.</td> <td>意识到的</td> </tr> <tr> <td>awful</td> <td>adj.</td> <td>糟糕的</td> </tr> <tr> <td>awkward</td> <td>adj.</td> <td>尴尬的</td> </tr> <tr> <td>background</td> <td>n.</td> <td>背景</td> </tr> <tr> <td>backward</td> <td>adj./adv.</td> <td>向后的</td> </tr> <tr> <td>balance</td> <td>n./v.</td> <td>平衡</td> </tr> <tr> <td>ban</td> <td>v./n.</td> <td>禁止</td> </tr> <tr> <td>band</td> <td>n.</td> <td>乐队；带子</td> </tr> <tr> <td>bare</td> <td>adj.</td> <td>赤裸的</td> </tr> <tr> <td>bargain</td> <td>n./v.</td> <td>讨价还价</td> </tr> <tr> <td>barrier</td> <td>n.</td> <td>障碍</td> </tr> <tr> <td>base</td> <td>n./v.</td> <td>基础；以…为据</td> </tr> <tr> <td>basin</td> <td>n.</td> <td>盆地</td> </tr> <tr> <td>battery</td> <td>n.</td> <td>电池</td> </tr> <tr> <td>battle</td> <td>n.</td> <td>战斗</td> </tr> <tr> <td>beach</td> <td>n.</td> <td>海滩</td> </tr> <tr> <td>bear ⭐</td> <td>v./n.</td> <td>忍受；熊</td> </tr> <tr> <td>beat</td> <td>v.</td> <td>打败；跳动</td> </tr> <tr> <td>behave</td> <td>v.</td> <td>表现</td> </tr> <tr> <td>behavior</td> <td>n.</td> <td>行为</td> </tr> <tr> <td>belief</td> <td>n.</td> <td>信念</td> </tr> <tr> <td>believe ⭐</td> <td>v.</td> <td>相信</td> </tr> <tr> <td>belong ⭐</td> <td>v.</td> <td>属于</td> </tr> </tbody> </table> <h2 id="day-6b">Day 6（B）<a class="anchor" href="#day-6b" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>below</td> <td>prep./adv.</td> <td>在下面</td> </tr> <tr> <td>belt</td> <td>n.</td> <td>腰带</td> </tr> <tr> <td>bend</td> <td>v.</td> <td>弯曲</td> </tr> <tr> <td>benefit</td> <td>n./v.</td> <td>益处</td> </tr> <tr> <td>bet</td> <td>v./n.</td> <td>打赌</td> </tr> <tr> <td>beyond</td> <td>prep.</td> <td>超出</td> </tr> <tr> <td>bill</td> <td>n.</td> <td>账单</td> </tr> <tr> <td>bind</td> <td>v.</td> <td>捆绑</td> </tr> <tr> <td>birth</td> <td>n.</td> <td>出生</td> </tr> <tr> <td>bite</td> <td>v.</td> <td>咬</td> </tr> <tr> <td>bitter</td> <td>adj.</td> <td>苦的；痛苦的</td> </tr> <tr> <td>blame</td> <td>v.</td> <td>责备</td> </tr> <tr> <td>blank</td> <td>adj./n.</td> <td>空白的</td> </tr> <tr> <td>blanket</td> <td>n.</td> <td>毛毯</td> </tr> <tr> <td>bleed</td> <td>v.</td> <td>流血</td> </tr> <tr> <td>blend</td> <td>v.</td> <td>混合</td> </tr> <tr> <td>blind</td> <td>adj.</td> <td>盲的</td> </tr> <tr> <td>block</td> <td>v./n.</td> <td>阻塞；街区</td> </tr> <tr> <td>blood</td> <td>n.</td> <td>血</td> </tr> <tr> <td>blow</td> <td>v.</td> <td>吹</td> </tr> <tr> <td>board</td> <td>n.</td> <td>木板；董事会</td> </tr> <tr> <td>boil</td> <td>v.</td> <td>煮沸</td> </tr> <tr> <td>bold</td> <td>adj.</td> <td>大胆的</td> </tr> <tr> <td>bomb</td> <td>n./v.</td> <td>炸弹</td> </tr> <tr> <td>border</td> <td>n.</td> <td>边界</td> </tr> </tbody> </table> <h2 id="day-7b">Day 7（B）<a class="anchor" href="#day-7b" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>bore</td> <td>v.</td> <td>使厌倦</td> </tr> <tr> <td>borrow ⭐</td> <td>v.</td> <td>借入</td> </tr> <tr> <td>boss</td> <td>n.</td> <td>老板</td> </tr> <tr> <td>bother</td> <td>v.</td> <td>打扰</td> </tr> <tr> <td>bottom</td> <td>n.</td> <td>底部</td> </tr> <tr> <td>bow</td> <td>v./n.</td> <td>鞠躬；弓</td> </tr> <tr> <td>brain</td> <td>n.</td> <td>大脑</td> </tr> <tr> <td>brake</td> <td>n./v.</td> <td>刹车</td> </tr> <tr> <td>branch</td> <td>n.</td> <td>树枝；分支</td> </tr> <tr> <td>brand</td> <td>n.</td> <td>品牌</td> </tr> <tr> <td>brave</td> <td>adj.</td> <td>勇敢的</td> </tr> <tr> <td>breath</td> <td>n.</td> <td>呼吸</td> </tr> <tr> <td>breathe</td> <td>v.</td> <td>呼吸</td> </tr> <tr> <td>breed</td> <td>v.</td> <td>繁殖</td> </tr> <tr> <td>brick</td> <td>n.</td> <td>砖</td> </tr> <tr> <td>bride</td> <td>n.</td> <td>新娘</td> </tr> <tr> <td>brief</td> <td>adj.</td> <td>简短的</td> </tr> <tr> <td>bright</td> <td>adj.</td> <td>明亮的</td> </tr> <tr> <td>brilliant</td> <td>adj.</td> <td>杰出的</td> </tr> <tr> <td>broad</td> <td>adj.</td> <td>宽的</td> </tr> <tr> <td>broadcast</td> <td>v./n.</td> <td>广播</td> </tr> <tr> <td>brush</td> <td>n./v.</td> <td>刷子；刷</td> </tr> <tr> <td>budget</td> <td>n.</td> <td>预算</td> </tr> <tr> <td>bunch</td> <td>n.</td> <td>束</td> </tr> <tr> <td>burden</td> <td>n.</td> <td>负担</td> </tr> </tbody> </table> <h2 id="day-8bc">Day 8（B~C）<a class="anchor" href="#day-8bc" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>burst</td> <td>v./n.</td> <td>爆发</td> </tr> <tr> <td>bury</td> <td>v.</td> <td>埋葬</td> </tr> <tr> <td>business</td> <td>n.</td> <td>生意；事务</td> </tr> <tr> <td>calculate</td> <td>v.</td> <td>计算</td> </tr> <tr> <td>calendar</td> <td>n.</td> <td>日历</td> </tr> <tr> <td>calm</td> <td>adj./v.</td> <td>平静的</td> </tr> <tr> <td>camp</td> <td>n./v.</td> <td>营地</td> </tr> <tr> <td>campaign</td> <td>n.</td> <td>运动；战役</td> </tr> <tr> <td>canal</td> <td>n.</td> <td>运河</td> </tr> <tr> <td>cancer</td> <td>n.</td> <td>癌症</td> </tr> <tr> <td>candidate</td> <td>n.</td> <td>候选人</td> </tr> <tr> <td>capable</td> <td>adj.</td> <td>有能力的</td> </tr> <tr> <td>capital</td> <td>n./adj.</td> <td>首都；资金的</td> </tr> <tr> <td>captain</td> <td>n.</td> <td>队长；船长</td> </tr> <tr> <td>capture</td> <td>v.</td> <td>捕获</td> </tr> <tr> <td>carbon</td> <td>n.</td> <td>碳</td> </tr> <tr> <td>career</td> <td>n.</td> <td>事业</td> </tr> <tr> <td>cargo</td> <td>n.</td> <td>货物</td> </tr> <tr> <td>carpet</td> <td>n.</td> <td>地毯</td> </tr> <tr> <td>carve</td> <td>v.</td> <td>雕刻</td> </tr> <tr> <td>case</td> <td>n.</td> <td>情况；案件</td> </tr> <tr> <td>cash</td> <td>n.</td> <td>现金</td> </tr> <tr> <td>cast</td> <td>v.</td> <td>投掷</td> </tr> <tr> <td>castle</td> <td>n.</td> <td>城堡</td> </tr> <tr> <td>casual</td> <td>adj.</td> <td>随意的</td> </tr> </tbody> </table> <h2 id="day-9c">Day 9（C）<a class="anchor" href="#day-9c" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>catalog</td> <td>n.</td> <td>目录</td> </tr> <tr> <td>category</td> <td>n.</td> <td>类别</td> </tr> <tr> <td>cattle</td> <td>n.</td> <td>牛</td> </tr> <tr> <td>cause ⭐</td> <td>n./v.</td> <td>原因；导致</td> </tr> <tr> <td>cautious</td> <td>adj.</td> <td>谨慎的</td> </tr> <tr> <td>cease</td> <td>v.</td> <td>停止</td> </tr> <tr> <td>ceiling</td> <td>n.</td> <td>天花板</td> </tr> <tr> <td>celebrate</td> <td>v.</td> <td>庆祝</td> </tr> <tr> <td>cell</td> <td>n.</td> <td>细胞；牢房</td> </tr> <tr> <td>central</td> <td>adj.</td> <td>中心的</td> </tr> <tr> <td>century</td> <td>n.</td> <td>世纪</td> </tr> <tr> <td>ceremony</td> <td>n.</td> <td>典礼</td> </tr> <tr> <td>certain</td> <td>adj.</td> <td>确定的</td> </tr> <tr> <td>certificate</td> <td>n.</td> <td>证书</td> </tr> <tr> <td>chain</td> <td>n.</td> <td>链条</td> </tr> <tr> <td>chairman</td> <td>n.</td> <td>主席</td> </tr> <tr> <td>challenge</td> <td>n./v.</td> <td>挑战</td> </tr> <tr> <td>champion</td> <td>n.</td> <td>冠军</td> </tr> <tr> <td>channel</td> <td>n.</td> <td>频道；渠道</td> </tr> <tr> <td>character</td> <td>n.</td> <td>性格；角色</td> </tr> <tr> <td>characteristic</td> <td>n./adj.</td> <td>特征</td> </tr> <tr> <td>charge</td> <td>v./n.</td> <td>收费；指控</td> </tr> <tr> <td>charity</td> <td>n.</td> <td>慈善</td> </tr> <tr> <td>charm</td> <td>n.</td> <td>魅力</td> </tr> <tr> <td>chart</td> <td>n.</td> <td>图表</td> </tr> </tbody> </table> <h2 id="day-10c">Day 10（C）<a class="anchor" href="#day-10c" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>chase</td> <td>v.</td> <td>追赶</td> </tr> <tr> <td>cheat</td> <td>v.</td> <td>欺骗</td> </tr> <tr> <td>cheer</td> <td>v./n.</td> <td>欢呼</td> </tr> <tr> <td>chemical</td> <td>adj./n.</td> <td>化学的</td> </tr> <tr> <td>chief</td> <td>adj./n.</td> <td>首要的；首领</td> </tr> <tr> <td>circle</td> <td>n./v.</td> <td>圆圈</td> </tr> <tr> <td>circuit</td> <td>n.</td> <td>电路</td> </tr> <tr> <td>circumstance</td> <td>n.</td> <td>环境</td> </tr> <tr> <td>citizen</td> <td>n.</td> <td>公民</td> </tr> <tr> <td>civil</td> <td>adj.</td> <td>公民的</td> </tr> <tr> <td>civilization</td> <td>n.</td> <td>文明</td> </tr> <tr> <td>claim</td> <td>v./n.</td> <td>声称</td> </tr> <tr> <td>clarify</td> <td>v.</td> <td>澄清</td> </tr> <tr> <td>classic</td> <td>adj./n.</td> <td>经典的</td> </tr> <tr> <td>classify</td> <td>v.</td> <td>分类</td> </tr> <tr> <td>clerk</td> <td>n.</td> <td>职员</td> </tr> <tr> <td>client</td> <td>n.</td> <td>客户</td> </tr> <tr> <td>climate</td> <td>n.</td> <td>气候</td> </tr> <tr> <td>climb</td> <td>v.</td> <td>攀登</td> </tr> <tr> <td>clinic</td> <td>n.</td> <td>诊所</td> </tr> <tr> <td>cloth</td> <td>n.</td> <td>布</td> </tr> <tr> <td>clothing</td> <td>n.</td> <td>服装</td> </tr> <tr> <td>clue</td> <td>n.</td> <td>线索</td> </tr> <tr> <td>coach</td> <td>n./v.</td> <td>教练；辅导</td> </tr> <tr> <td>coal</td> <td>n.</td> <td>煤</td> </tr> </tbody> </table>https://one.snvedu.cn/docs/english-basics/01-sentence/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/01-sentence/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>英语句子的三件套：<strong>谁 + 干什么 + （对谁/什么）</strong></li> <li>七个「人」的说法：I, you, he, she, it, we, they</li> <li>最重要的一组动词：<strong>am / is / are</strong>（＝中文的「是」）</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-句子三件套">1. 句子三件套<a class="anchor" href="#1-%e5%8f%a5%e5%ad%90%e4%b8%89%e4%bb%b6%e5%a5%97" aria-label="Link"></a></h3> <p>中文说「<strong>我 吃 苹果</strong>」，英文一模一样：</p>https://one.snvedu.cn/daily/math/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/daily/math/<blockquote class=''> <p>每天一题：先独立做（限 8 分钟）→ 打开解析 → 对号题型库/禁术编号。题目按章节由浅入深排列。</p> </blockquote><h2 id="day-1">Day 1<a class="anchor" href="#day-1" aria-label="Link"></a></h2> <p>求 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>x</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\dfrac{1-\cos 2x}{x^2}</annotation></semantics></math></span></p> <details> <summary>解析</summary> <p><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>x</mi><mo>∼</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1-\cos 2x \sim \frac{(2x)^2}{2}=2x^2</annotation></semantics></math></span> → 极限 = <strong>2</strong>。考点：等价无穷小（禁术 6）。</p> </details>https://one.snvedu.cn/bank/choice-1/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/choice-1/<blockquote class=''> <p>限时：每题 90 秒。先选，再看解析。全卷考点覆盖 ch1~ch9。</p> </blockquote><h2 id="1-hahahugoshortcode13s0hbhb--">1. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>sin</mi><mo>⁡</mo><mn>3</mn><mi>x</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\dfrac{\sin 2x}{\sin 3x}</annotation></semantics></math></span><link rel="stylesheet" href="https://one.snvedu.cn/katex/katex.min.css" /> = ?<a class="anchor" href="#1-hahahugoshortcode13s0hbhb--" aria-label="Link"></a></h2> <p>A. 1　B. 2/3　C. 3/2　D. 0</p> <details><summary>答案：B</summary> 等价替换：2x/3x = 2/3（禁术 6） </details> <h2 id="2-hahahugoshortcode13s1hbhb--">2. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></msub><msup><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>x</mi></mfrac></mstyle><mo fence="true">)</mo></mrow><mrow><mn>2</mn><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\lim_{x \to \infty}\left(1+\dfrac1x\right)^{2x}</annotation></semantics></math></span> = ?<a class="anchor" href="#2-hahahugoshortcode13s1hbhb--" aria-label="Link"></a></h2> <p>A. e　B. e²　C. 2e　D. 1</p>https://one.snvedu.cn/docs/gaoshu/derivatives-differentials/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/derivatives-differentials/<blockquote class=''> <p>🎯 浙江考纲定位：一元微积分板块。计算题必出求导，填空常考切线与微分。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解导数定义（会用定义求导、判断可导性与连续性的关系）</li> <li>掌握基本初等函数导数公式、四则运算求导、复合函数求导（链式法则）</li> <li>会求隐函数、参数方程确定的函数的导数</li> <li>理解高阶导数，会求简单函数的二阶导数</li> <li>理解微分概念，会求微分</li> </ul> <h2 id="-核心公式库">📖 核心公式库<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e5%85%ac%e5%bc%8f%e5%ba%93" aria-label="Link"></a></h2> <h3 id="1-基本导数表背到条件反射">1. 基本导数表（背到条件反射）<a class="anchor" href="#1-%e5%9f%ba%e6%9c%ac%e5%af%bc%e6%95%b0%e8%a1%a8%e8%83%8c%e5%88%b0%e6%9d%a1%e4%bb%b6%e5%8f%8d%e5%b0%84" aria-label="Link"></a></h3> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>a</mi></msup><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>a</mi><msup><mi>x</mi><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msup><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><msup><mi>a</mi><mi>x</mi></msup><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mi>ln</mi><mo>⁡</mo><mi>a</mi><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">(x^a)&#x27;=ax^{a-1},\quad (a^x)&#x27;=a^x\ln a,\quad (e^x)&#x27;=e^x</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>ln</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>a</mi></msub><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mi>ln</mi><mo>⁡</mo><mi>a</mi></mrow></mfrac><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><msqrt><mi>x</mi></msqrt><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mi>x</mi></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\ln x)&#x27;=\frac{1}{x},\quad (\log_a x)&#x27;=\frac{1}{x\ln a},\quad (\sqrt{x})&#x27;=\frac{1}{2\sqrt{x}}</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mi>tan</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msup><mrow><mi>sec</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>x</mi><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mi>cot</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mo>−</mo><msup><mrow><mi>csc</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>x</mi></mrow><annotation encoding="application/x-tex">(\sin x)&#x27;=\cos x,\quad (\cos x)&#x27;=-\sin x,\quad (\tan x)&#x27;=\sec^2 x,\quad (\cot x)&#x27;=-\csc^2 x</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>sec</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>sec</mi><mo>⁡</mo><mi>x</mi><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mi>arctan</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo separator="true">,</mo><mspace width="1em"/><mo stretchy="false">(</mo><mi>arcsin</mi><mo>⁡</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>1</mn><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">(\sec x)&#x27;=\sec x\tan x,\quad (\arctan x)&#x27;=\frac{1}{1+x^2},\quad (\arcsin x)&#x27;=\frac{1}{\sqrt{1-x^2}}</annotation></semantics></math></span><h3 id="2-四则与结构公式">2. 四则与结构公式<a class="anchor" href="#2-%e5%9b%9b%e5%88%99%e4%b8%8e%e7%bb%93%e6%9e%84%e5%85%ac%e5%bc%8f" aria-label="Link"></a></h3> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>v</mi><msup><mo stretchy="false">)</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msup><mi>u</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>v</mi><mo>+</mo><mi>u</mi><msup><mi>v</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mspace width="2em"/><msup><mrow><mo fence="true">(</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo fence="true">)</mo></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mrow><msup><mi>u</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>v</mi><mo>−</mo><mi>u</mi><msup><mi>v</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><msup><mi>v</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">(uv)&#x27;=u&#x27;v+uv&#x27;,\qquad \left(\frac{u}{v}\right)&#x27;=\frac{u&#x27;v-uv&#x27;}{v^2}</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mtext> （链式，洋葱从外往里剥）</mtext></mrow><annotation encoding="application/x-tex">\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\ \text{（链式，洋葱从外往里剥）}</annotation></semantics></math></span><h2 id="-题型库历年考法全覆盖完整解法版">🧩 题型库：历年考法全覆盖（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1复合函数求导计算题常客">题型 1：复合函数求导（计算题常客）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e5%a4%8d%e5%90%88%e5%87%bd%e6%95%b0%e6%b1%82%e5%af%bc%e8%ae%a1%e7%ae%97%e9%a2%98%e5%b8%b8%e5%ae%a2" aria-label="Link"></a></h3> <p><strong>方法</strong>：逐层剥洋葱，每层用「外层导 × 内层导」，层数多时宁写两步不要心算。</p>https://one.snvedu.cn/docs/english/reading-comprehension/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/reading-comprehension/<h2 id="-模块说明">📌 模块说明<a class="anchor" href="#-%e6%a8%a1%e5%9d%97%e8%af%b4%e6%98%8e" aria-label="Link"></a></h2> <p>阅读是英语的得分大头（约 40% 分数），核心策略：<strong>先看题干，再回原文定位</strong>。正确答案 90% 都能在原文找到「同义改写」。</p> <h2 id="-题型库四大题型完整例题">🧩 题型库：四大题型完整例题<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%9b%9b%e5%a4%a7%e9%a2%98%e5%9e%8b%e5%ae%8c%e6%95%b4%e4%be%8b%e9%a2%98" aria-label="Link"></a></h2> <h3 id="题型-1细节题">题型 1：细节题<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e7%bb%86%e8%8a%82%e9%a2%98" aria-label="Link"></a></h3> <p>例：</p>https://one.snvedu.cn/exam/2019/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2019/<h1 id="2019-年浙江专升本真题">2019 年浙江专升本真题<a class="anchor" href="#2019-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/english-words/days-11-20/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/days-11-20/<blockquote class=''> <p>用法同前：遮中文自测，动词 ⭐ 优先，想不起画 △。</p> </blockquote><h2 id="day-11c">Day 11（C）<a class="anchor" href="#day-11c" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>coast</td> <td>n.</td> <td>海岸</td> </tr> <tr> <td>code</td> <td>n.</td> <td>代码；密码</td> </tr> <tr> <td>collapse</td> <td>v./n.</td> <td>倒塌</td> </tr> <tr> <td>collar</td> <td>n.</td> <td>衣领</td> </tr> <tr> <td>colleague</td> <td>n.</td> <td>同事</td> </tr> <tr> <td>collect ⭐</td> <td>v.</td> <td>收集</td> </tr> <tr> <td>collision</td> <td>n.</td> <td>碰撞</td> </tr> <tr> <td>column</td> <td>n.</td> <td>柱；专栏</td> </tr> <tr> <td>comb</td> <td>n./v.</td> <td>梳子；梳理</td> </tr> <tr> <td>combine ⭐</td> <td>v.</td> <td>结合</td> </tr> <tr> <td>comedy</td> <td>n.</td> <td>喜剧</td> </tr> <tr> <td>command</td> <td>v./n.</td> <td>命令</td> </tr> <tr> <td>comment</td> <td>n./v.</td> <td>评论</td> </tr> <tr> <td>commerce</td> <td>n.</td> <td>商业</td> </tr> <tr> <td>commit</td> <td>v.</td> <td>犯（错）；承诺</td> </tr> <tr> <td>communicate ⭐</td> <td>v.</td> <td>交流</td> </tr> <tr> <td>community</td> <td>n.</td> <td>社区</td> </tr> <tr> <td>companion</td> <td>n.</td> <td>同伴</td> </tr> <tr> <td>company</td> <td>n.</td> <td>公司；陪伴</td> </tr> <tr> <td>compare ⭐</td> <td>v.</td> <td>比较</td> </tr> <tr> <td>compel</td> <td>v.</td> <td>强迫</td> </tr> <tr> <td>compete</td> <td>v.</td> <td>竞争</td> </tr> <tr> <td>complain</td> <td>v.</td> <td>抱怨</td> </tr> <tr> <td>complex</td> <td>adj.</td> <td>复杂的</td> </tr> <tr> <td>compose</td> <td>v.</td> <td>组成；创作</td> </tr> </tbody> </table> <h2 id="day-12c">Day 12（C）<a class="anchor" href="#day-12c" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>comprehension</td> <td>n.</td> <td>理解</td> </tr> <tr> <td>concentrate</td> <td>v.</td> <td>集中</td> </tr> <tr> <td>concept</td> <td>n.</td> <td>概念</td> </tr> <tr> <td>concern</td> <td>v./n.</td> <td>关心；担忧</td> </tr> <tr> <td>conclude</td> <td>v.</td> <td>得出结论</td> </tr> <tr> <td>concrete</td> <td>adj.</td> <td>具体的</td> </tr> <tr> <td>conduct</td> <td>v./n.</td> <td>实施；行为</td> </tr> <tr> <td>conference</td> <td>n.</td> <td>会议</td> </tr> <tr> <td>confident</td> <td>adj.</td> <td>自信的</td> </tr> <tr> <td>confirm</td> <td>v.</td> <td>确认</td> </tr> <tr> <td>conflict</td> <td>n.</td> <td>冲突</td> </tr> <tr> <td>confuse</td> <td>v.</td> <td>使困惑</td> </tr> <tr> <td>congratulate</td> <td>v.</td> <td>祝贺</td> </tr> <tr> <td>connect ⭐</td> <td>v.</td> <td>连接</td> </tr> <tr> <td>conquer</td> <td>v.</td> <td>征服</td> </tr> <tr> <td>conscience</td> <td>n.</td> <td>良心</td> </tr> <tr> <td>conscious</td> <td>adj.</td> <td>有意识的</td> </tr> <tr> <td>consent</td> <td>n./v.</td> <td>同意</td> </tr> <tr> <td>consequence</td> <td>n.</td> <td>后果</td> </tr> <tr> <td>consider ⭐</td> <td>v.</td> <td>考虑</td> </tr> <tr> <td>consist</td> <td>v.</td> <td>由…组成</td> </tr> <tr> <td>constant</td> <td>adj.</td> <td>持续的</td> </tr> <tr> <td>construct</td> <td>v.</td> <td>建造</td> </tr> <tr> <td>consult</td> <td>v.</td> <td>咨询</td> </tr> <tr> <td>consume</td> <td>v.</td> <td>消耗</td> </tr> </tbody> </table> <h2 id="day-13c">Day 13（C）<a class="anchor" href="#day-13c" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>contact</td> <td>v./n.</td> <td>联系</td> </tr> <tr> <td>contain ⭐</td> <td>v.</td> <td>包含</td> </tr> <tr> <td>contemporary</td> <td>adj.</td> <td>当代的</td> </tr> <tr> <td>content</td> <td>n./adj.</td> <td>内容；满足的</td> </tr> <tr> <td>contest</td> <td>n.</td> <td>竞赛</td> </tr> <tr> <td>context</td> <td>n.</td> <td>上下文</td> </tr> <tr> <td>continent</td> <td>n.</td> <td>大陆</td> </tr> <tr> <td>continue ⭐</td> <td>v.</td> <td>继续</td> </tr> <tr> <td>contract</td> <td>n.</td> <td>合同</td> </tr> <tr> <td>contrary</td> <td>adj./n.</td> <td>相反的</td> </tr> <tr> <td>contrast</td> <td>n./v.</td> <td>对比</td> </tr> <tr> <td>contribute</td> <td>v.</td> <td>贡献</td> </tr> <tr> <td>control ⭐</td> <td>v./n.</td> <td>控制</td> </tr> <tr> <td>convince</td> <td>v.</td> <td>说服</td> </tr> <tr> <td>cooperate</td> <td>v.</td> <td>合作</td> </tr> <tr> <td>cope</td> <td>v.</td> <td>应对</td> </tr> <tr> <td>copper</td> <td>n.</td> <td>铜</td> </tr> <tr> <td>copyright</td> <td>n.</td> <td>版权</td> </tr> <tr> <td>cord</td> <td>n.</td> <td>绳索</td> </tr> <tr> <td>corporation</td> <td>n.</td> <td>公司</td> </tr> <tr> <td>correspond</td> <td>v.</td> <td>对应；通信</td> </tr> <tr> <td>cottage</td> <td>n.</td> <td>小屋</td> </tr> <tr> <td>cotton</td> <td>n.</td> <td>棉花</td> </tr> <tr> <td>cough</td> <td>v./n.</td> <td>咳嗽</td> </tr> <tr> <td>council</td> <td>n.</td> <td>委员会</td> </tr> </tbody> </table> <h2 id="day-14c">Day 14（C）<a class="anchor" href="#day-14c" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>count ⭐</td> <td>v.</td> <td>数；重要</td> </tr> <tr> <td>counter</td> <td>n.</td> <td>柜台；反驳</td> </tr> <tr> <td>couple</td> <td>n.</td> <td>一对</td> </tr> <tr> <td>courage</td> <td>n.</td> <td>勇气</td> </tr> <tr> <td>court</td> <td>n.</td> <td>法院；球场</td> </tr> <tr> <td>cover ⭐</td> <td>v./n.</td> <td>覆盖</td> </tr> <tr> <td>crack</td> <td>n./v.</td> <td>裂缝</td> </tr> <tr> <td>crash</td> <td>v./n.</td> <td>坠毁</td> </tr> <tr> <td>create ⭐</td> <td>v.</td> <td>创造</td> </tr> <tr> <td>creature</td> <td>n.</td> <td>生物</td> </tr> <tr> <td>credit</td> <td>n.</td> <td>信用；学分</td> </tr> <tr> <td>crime</td> <td>n.</td> <td>犯罪</td> </tr> <tr> <td>crisis</td> <td>n.</td> <td>危机</td> </tr> <tr> <td>critic</td> <td>n.</td> <td>批评家</td> </tr> <tr> <td>critical</td> <td>adj.</td> <td>批评的；关键的</td> </tr> <tr> <td>criticize</td> <td>v.</td> <td>批评</td> </tr> <tr> <td>crop</td> <td>n.</td> <td>庄稼</td> </tr> <tr> <td>crowd</td> <td>n.</td> <td>人群</td> </tr> <tr> <td>crown</td> <td>n.</td> <td>王冠</td> </tr> <tr> <td>cruel</td> <td>adj.</td> <td>残酷的</td> </tr> <tr> <td>crush</td> <td>v.</td> <td>压碎</td> </tr> <tr> <td>cultivate</td> <td>v.</td> <td>培养</td> </tr> <tr> <td>culture</td> <td>n.</td> <td>文化</td> </tr> <tr> <td>curious</td> <td>adj.</td> <td>好奇的</td> </tr> <tr> <td>currency</td> <td>n.</td> <td>货币</td> </tr> </tbody> </table> <h2 id="day-15cd">Day 15（C~D）<a class="anchor" href="#day-15cd" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>current</td> <td>adj./n.</td> <td>当前的；水流</td> </tr> <tr> <td>curtain</td> <td>n.</td> <td>窗帘</td> </tr> <tr> <td>curve</td> <td>n./v.</td> <td>曲线</td> </tr> <tr> <td>custom</td> <td>n.</td> <td>习俗</td> </tr> <tr> <td>customer</td> <td>n.</td> <td>顾客</td> </tr> <tr> <td>cycle</td> <td>n./v.</td> <td>循环；骑行</td> </tr> <tr> <td>daily</td> <td>adj./adv.</td> <td>每天的</td> </tr> <tr> <td>damage</td> <td>v./n.</td> <td>损坏</td> </tr> <tr> <td>damp</td> <td>adj.</td> <td>潮湿的</td> </tr> <tr> <td>dare</td> <td>v.</td> <td>敢</td> </tr> <tr> <td>data</td> <td>n.</td> <td>数据</td> </tr> <tr> <td>dawn</td> <td>n.</td> <td>黎明</td> </tr> <tr> <td>deadline</td> <td>n.</td> <td>截止日期</td> </tr> <tr> <td>deaf</td> <td>adj.</td> <td>聋的</td> </tr> <tr> <td>deal ⭐</td> <td>v./n.</td> <td>处理；交易</td> </tr> <tr> <td>debate</td> <td>n./v.</td> <td>辩论</td> </tr> <tr> <td>debt</td> <td>n.</td> <td>债务</td> </tr> <tr> <td>decade</td> <td>n.</td> <td>十年</td> </tr> <tr> <td>deceive</td> <td>v.</td> <td>欺骗</td> </tr> <tr> <td>declare</td> <td>v.</td> <td>宣布</td> </tr> <tr> <td>decline</td> <td>v./n.</td> <td>下降；拒绝</td> </tr> <tr> <td>decorate</td> <td>v.</td> <td>装饰</td> </tr> <tr> <td>decrease</td> <td>v./n.</td> <td>减少</td> </tr> <tr> <td>defeat</td> <td>v./n.</td> <td>击败</td> </tr> <tr> <td>defend</td> <td>v.</td> <td>保卫</td> </tr> </tbody> </table> <h2 id="day-16d">Day 16（D）<a class="anchor" href="#day-16d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>define</td> <td>v.</td> <td>定义</td> </tr> <tr> <td>definite</td> <td>adj.</td> <td>明确的</td> </tr> <tr> <td>degree</td> <td>n.</td> <td>程度；学位</td> </tr> <tr> <td>delay</td> <td>v./n.</td> <td>推迟</td> </tr> <tr> <td>deliberate</td> <td>adj./v.</td> <td>故意的；深思</td> </tr> <tr> <td>delicate</td> <td>adj.</td> <td>精致的；脆弱的</td> </tr> <tr> <td>deliver ⭐</td> <td>v.</td> <td>递送；发表</td> </tr> <tr> <td>demand ⭐</td> <td>v./n.</td> <td>要求；需求</td> </tr> <tr> <td>democracy</td> <td>n.</td> <td>民主</td> </tr> <tr> <td>demonstrate</td> <td>v.</td> <td>证明；演示</td> </tr> <tr> <td>dense</td> <td>adj.</td> <td>密集的</td> </tr> <tr> <td>deny</td> <td>v.</td> <td>否认</td> </tr> <tr> <td>depart</td> <td>v.</td> <td>出发</td> </tr> <tr> <td>deposit</td> <td>n./v.</td> <td>存款；押金</td> </tr> <tr> <td>depress</td> <td>v.</td> <td>使沮丧</td> </tr> <tr> <td>depth</td> <td>n.</td> <td>深度</td> </tr> <tr> <td>derive</td> <td>v.</td> <td>源于</td> </tr> <tr> <td>describe ⭐</td> <td>v.</td> <td>描述</td> </tr> <tr> <td>desert</td> <td>n./v.</td> <td>沙漠；抛弃</td> </tr> <tr> <td>deserve</td> <td>v.</td> <td>值得</td> </tr> <tr> <td>design ⭐</td> <td>v./n.</td> <td>设计</td> </tr> <tr> <td>desire</td> <td>v./n.</td> <td>渴望</td> </tr> <tr> <td>despair</td> <td>n./v.</td> <td>绝望</td> </tr> <tr> <td>desperate</td> <td>adj.</td> <td>绝望的</td> </tr> <tr> <td>despite</td> <td>prep.</td> <td>尽管</td> </tr> </tbody> </table> <h2 id="day-17d">Day 17（D）<a class="anchor" href="#day-17d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>destroy</td> <td>v.</td> <td>破坏</td> </tr> <tr> <td>detail</td> <td>n.</td> <td>细节</td> </tr> <tr> <td>detect</td> <td>v.</td> <td>察觉</td> </tr> <tr> <td>determine ⭐</td> <td>v.</td> <td>决定</td> </tr> <tr> <td>develop ⭐</td> <td>v.</td> <td>发展</td> </tr> <tr> <td>device</td> <td>n.</td> <td>设备</td> </tr> <tr> <td>devote</td> <td>v.</td> <td>奉献</td> </tr> <tr> <td>diagram</td> <td>n.</td> <td>图表</td> </tr> <tr> <td>dial</td> <td>v./n.</td> <td>拨号</td> </tr> <tr> <td>dialog</td> <td>n.</td> <td>对话</td> </tr> <tr> <td>diet</td> <td>n.</td> <td>饮食</td> </tr> <tr> <td>differ</td> <td>v.</td> <td>不同</td> </tr> <tr> <td>digital</td> <td>adj.</td> <td>数字的</td> </tr> <tr> <td>dimension</td> <td>n.</td> <td>维度</td> </tr> <tr> <td>dip</td> <td>v.</td> <td>蘸</td> </tr> <tr> <td>diploma</td> <td>n.</td> <td>文凭</td> </tr> <tr> <td>direct</td> <td>adj./v.</td> <td>直接的；指导</td> </tr> <tr> <td>disappear</td> <td>v.</td> <td>消失</td> </tr> <tr> <td>disappoint</td> <td>v.</td> <td>使失望</td> </tr> <tr> <td>disaster</td> <td>n.</td> <td>灾难</td> </tr> <tr> <td>discipline</td> <td>n.</td> <td>纪律</td> </tr> <tr> <td>discount</td> <td>n.</td> <td>折扣</td> </tr> <tr> <td>discourage</td> <td>v.</td> <td>使气馁</td> </tr> <tr> <td>discover</td> <td>v.</td> <td>发现</td> </tr> <tr> <td>discuss ⭐</td> <td>v.</td> <td>讨论</td> </tr> </tbody> </table> <h2 id="day-18d">Day 18（D）<a class="anchor" href="#day-18d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>disease</td> <td>n.</td> <td>疾病</td> </tr> <tr> <td>dismiss</td> <td>v.</td> <td>解雇；解散</td> </tr> <tr> <td>disorder</td> <td>n.</td> <td>混乱</td> </tr> <tr> <td>display</td> <td>v./n.</td> <td>展示</td> </tr> <tr> <td>dispute</td> <td>v./n.</td> <td>争论</td> </tr> <tr> <td>distance</td> <td>n.</td> <td>距离</td> </tr> <tr> <td>distant</td> <td>adj.</td> <td>遥远的</td> </tr> <tr> <td>distinct</td> <td>adj.</td> <td>清晰的；不同的</td> </tr> <tr> <td>distinguish</td> <td>v.</td> <td>区分</td> </tr> <tr> <td>distribute</td> <td>v.</td> <td>分发</td> </tr> <tr> <td>district</td> <td>n.</td> <td>地区</td> </tr> <tr> <td>disturb</td> <td>v.</td> <td>打扰</td> </tr> <tr> <td>dive</td> <td>v.</td> <td>跳水</td> </tr> <tr> <td>diverse</td> <td>adj.</td> <td>多样的</td> </tr> <tr> <td>divide ⭐</td> <td>v.</td> <td>分开</td> </tr> <tr> <td>document</td> <td>n./v.</td> <td>文件</td> </tr> <tr> <td>domestic</td> <td>adj.</td> <td>国内的</td> </tr> <tr> <td>donate</td> <td>v.</td> <td>捐赠</td> </tr> <tr> <td>double</td> <td>adj./v.</td> <td>双倍的</td> </tr> <tr> <td>doubt</td> <td>v./n.</td> <td>怀疑</td> </tr> <tr> <td>dozen</td> <td>n.</td> <td>一打</td> </tr> <tr> <td>draft</td> <td>n./v.</td> <td>草稿</td> </tr> <tr> <td>drag</td> <td>v.</td> <td>拖</td> </tr> <tr> <td>drama</td> <td>n.</td> <td>戏剧</td> </tr> <tr> <td>dramatic</td> <td>adj.</td> <td>戏剧性的</td> </tr> </tbody> </table> <h2 id="day-19d">Day 19（D）<a class="anchor" href="#day-19d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>drift</td> <td>v.</td> <td>漂流</td> </tr> <tr> <td>drill</td> <td>n./v.</td> <td>钻头；训练</td> </tr> <tr> <td>drown</td> <td>v.</td> <td>溺水</td> </tr> <tr> <td>drug</td> <td>n.</td> <td>药物</td> </tr> <tr> <td>drum</td> <td>n.</td> <td>鼓</td> </tr> <tr> <td>drunk</td> <td>adj.</td> <td>醉的</td> </tr> <tr> <td>due</td> <td>adj.</td> <td>到期的；应得的</td> </tr> <tr> <td>dull</td> <td>adj.</td> <td>枯燥的</td> </tr> <tr> <td>durable</td> <td>adj.</td> <td>耐用的</td> </tr> <tr> <td>dust</td> <td>n.</td> <td>灰尘</td> </tr> <tr> <td>duty</td> <td>n.</td> <td>责任</td> </tr> <tr> <td>eager</td> <td>adj.</td> <td>渴望的</td> </tr> <tr> <td>earn ⭐</td> <td>v.</td> <td>赚得</td> </tr> <tr> <td>earthquake</td> <td>n.</td> <td>地震</td> </tr> <tr> <td>ease</td> <td>n./v.</td> <td>容易；缓解</td> </tr> <tr> <td>eastern</td> <td>adj.</td> <td>东方的</td> </tr> <tr> <td>economic</td> <td>adj.</td> <td>经济的</td> </tr> <tr> <td>economy</td> <td>n.</td> <td>经济</td> </tr> <tr> <td>edge</td> <td>n.</td> <td>边缘</td> </tr> <tr> <td>edit</td> <td>v.</td> <td>编辑</td> </tr> <tr> <td>educate</td> <td>v.</td> <td>教育</td> </tr> <tr> <td>effect ⭐</td> <td>n.</td> <td>效果</td> </tr> <tr> <td>efficient</td> <td>adj.</td> <td>高效的</td> </tr> <tr> <td>effort</td> <td>n.</td> <td>努力</td> </tr> <tr> <td>elect</td> <td>v.</td> <td>选举</td> </tr> </tbody> </table> <h2 id="day-20e">Day 20（E）<a class="anchor" href="#day-20e" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>electric</td> <td>adj.</td> <td>电的</td> </tr> <tr> <td>elegant</td> <td>adj.</td> <td>优雅的</td> </tr> <tr> <td>element</td> <td>n.</td> <td>元素</td> </tr> <tr> <td>elementary</td> <td>adj.</td> <td>基础的</td> </tr> <tr> <td>embarrass</td> <td>v.</td> <td>使尴尬</td> </tr> <tr> <td>emerge</td> <td>v.</td> <td>出现</td> </tr> <tr> <td>emergency</td> <td>n.</td> <td>紧急情况</td> </tr> <tr> <td>emotion</td> <td>n.</td> <td>情感</td> </tr> <tr> <td>emphasis</td> <td>n.</td> <td>强调</td> </tr> <tr> <td>emphasize</td> <td>v.</td> <td>强调</td> </tr> <tr> <td>employ</td> <td>v.</td> <td>雇用</td> </tr> <tr> <td>empty</td> <td>adj.</td> <td>空的</td> </tr> <tr> <td>enable</td> <td>v.</td> <td>使能够</td> </tr> <tr> <td>encounter</td> <td>v.</td> <td>遭遇</td> </tr> <tr> <td>encourage ⭐</td> <td>v.</td> <td>鼓励</td> </tr> <tr> <td>endless</td> <td>adj.</td> <td>无尽的</td> </tr> <tr> <td>endure</td> <td>v.</td> <td>忍受</td> </tr> <tr> <td>energy</td> <td>n.</td> <td>能量</td> </tr> <tr> <td>enforce</td> <td>v.</td> <td>执行</td> </tr> <tr> <td>engage</td> <td>v.</td> <td>从事；订婚</td> </tr> <tr> <td>engine</td> <td>n.</td> <td>发动机</td> </tr> <tr> <td>engineer</td> <td>n.</td> <td>工程师</td> </tr> <tr> <td>enormous</td> <td>adj.</td> <td>巨大的</td> </tr> <tr> <td>ensure</td> <td>v.</td> <td>确保</td> </tr> <tr> <td>entertain</td> <td>v.</td> <td>娱乐</td> </tr> </tbody> </table>https://one.snvedu.cn/docs/english-basics/02-nouns/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/02-nouns/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>名词分两种：可数（能数）与不可数（不能数）</li> <li>可数名词怎么变复数（5 条规则 + 常见不规则）</li> <li>a / an / the 的选择口诀</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-可数与不可数">1. 可数与不可数<a class="anchor" href="#1-%e5%8f%af%e6%95%b0%e4%b8%8e%e4%b8%8d%e5%8f%af%e6%95%b0" aria-label="Link"></a></h3> <ul> <li><strong>可数</strong>：book 书、apple 苹果、student 学生 → 能说「一个、两个、三个」</li> <li><strong>不可数</strong>：water 水、milk 牛奶、time 时间、information 信息 → 不能说「一个水」，要靠「一杯 a glass of water」这种说法</li> </ul> <h3 id="2-复数规则多数情况加-s但有变体">2. 复数规则（多数情况加 s，但有变体）<a class="anchor" href="#2-%e5%a4%8d%e6%95%b0%e8%a7%84%e5%88%99%e5%a4%9a%e6%95%b0%e6%83%85%e5%86%b5%e5%8a%a0-s%e4%bd%86%e6%9c%89%e5%8f%98%e4%bd%93" aria-label="Link"></a></h3> <table> <thead> <tr> <th>规则</th> <th>例子</th> </tr> </thead> <tbody> <tr> <td>一般 + <strong>s</strong></td> <td>book → books，cat → cats</td> </tr> <tr> <td>s / x / ch / sh 结尾 + <strong>es</strong></td> <td>bus → buses，box → boxes，watch → watches</td> </tr> <tr> <td>辅音字母 + y 结尾：<strong>y → ies</strong></td> <td>city → cities，family → families（但 boy → boys，因为 o 前是元音）</td> </tr> <tr> <td>f / fe 结尾：<strong>f → ves</strong></td> <td>leaf → leaves，life → lives</td> </tr> <tr> <td>o 结尾（部分）+ <strong>es</strong></td> <td>tomato → tomatoes，potato → potatoes（但 photo → photos）</td> </tr> </tbody> </table> <p><strong>必背不规则复数（考试超高频）</strong>：</p>https://one.snvedu.cn/bank/choice-2/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/choice-2/<blockquote class=''> <p>同上半卷：90 秒一题，先选后看。</p> </blockquote><h2 id="21-hahahugoshortcode20s0hbhb--">21. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>e</mi><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mi>x</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\dfrac{e^{3x}-1}{x}</annotation></semantics></math></span><link rel="stylesheet" href="https://one.snvedu.cn/katex/katex.min.css" /> = ?<a class="anchor" href="#21-hahahugoshortcode20s0hbhb--" aria-label="Link"></a></h2> <p>A. 1　B. 3　C. 1/3　D. 0</p> <details><summary>答案：B</summary> e^{3x}−1 ~ 3x（禁术 6） </details> <h2 id="22-曲线-hahahugoshortcode20s1hbhb-上过点-04-的切线">22. 曲线 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">y=x^2</annotation></semantics></math></span> 上过点 (0,−4) 的切线？<a class="anchor" href="#22-%e6%9b%b2%e7%ba%bf-hahahugoshortcode20s1hbhb-%e4%b8%8a%e8%bf%87%e7%82%b9-04-%e7%9a%84%e5%88%87%e7%ba%bf" aria-label="Link"></a></h2> <p>A. 恰一条　B. 恰两条　C. 三条　D. 不存在</p>https://one.snvedu.cn/daily/english/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/daily/english/<blockquote class=''> <p>每天 1 句：先自己拆（找连词 → 找谓语 → 抓主干）→ 打开解析对照 → 口头译成中文。难度对标专升本阅读真题。配套：<a href="https://one.snvedu.cn/docs/english-basics/30-sentence-analysis/">第 30 课五步法</a>。</p> </blockquote><h2 id="day-1">Day 1<a class="anchor" href="#day-1" aria-label="Link"></a></h2> <p>The book on the desk belongs to my brother.</p> <details> <summary>拆解与译文</summary> <p><strong>主干</strong>：The book belongs to my brother.｜<strong>修饰</strong>：on the desk（介词短语作定语）→「桌上的」 <strong>译文</strong>：桌上的那本书是我哥哥的。</p> </details>https://one.snvedu.cn/exam/2020/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2020/<h1 id="2020-年浙江专升本真题">2020 年浙江专升本真题<a class="anchor" href="#2020-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/english/banked-cloze/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/banked-cloze/<blockquote class=''> <p>🎯 浙江题型定位：阅读理解板块的一部分，10 题 10 分。</p> </blockquote><h2 id="-模块说明">📌 模块说明<a class="anchor" href="#-%e6%a8%a1%e5%9d%97%e8%af%b4%e6%98%8e" aria-label="Link"></a></h2> <p>15 个备选词填 10 个空——<strong>备选多于空格 + 干扰项</strong>。本质考三件事：<strong>词性判断 + 语法功能 + 语义匹配</strong>，其中词性分类是决定成败的第一步。</p>https://one.snvedu.cn/docs/gaoshu/mean-value-applications/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/mean-value-applications/<blockquote class=''> <p>🎯 浙江考纲定位：一元微积分板块。计算题考极值/凹凸/渐近线，<strong>综合题常考最值应用与中值定理证明</strong>（证明专项见第 10 章）。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解罗尔定理、拉格朗日中值定理的条件与结论</li> <li>掌握用洛必达法则求极限</li> <li>掌握用导数判断单调性、求极值与最值</li> <li>会用导数判断曲线的凹凸性、求拐点</li> <li>会求函数的渐近线</li> </ul> <h2 id="-核心考点">📖 核心考点<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e8%80%83%e7%82%b9" aria-label="Link"></a></h2> <h3 id="1-两大中值定理">1. 两大中值定理<a class="anchor" href="#1-%e4%b8%a4%e5%a4%a7%e4%b8%ad%e5%80%bc%e5%ae%9a%e7%90%86" aria-label="Link"></a></h3> <table> <thead> <tr> <th>定理</th> <th>条件</th> <th>结论</th> </tr> </thead> <tbody> <tr> <td>罗尔</td> <td>闭区间连续 + 开区间可导 + <strong>端点值相等</strong></td> <td>存在 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math></span>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;(\xi)=0</annotation></semantics></math></span></td> </tr> <tr> <td>拉格朗日</td> <td>闭区间连续 + 开区间可导</td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(b)-f(a)=f&#x27;(\xi)(b-a)</annotation></semantics></math></span></td> </tr> </tbody> </table> <h3 id="-快速解法秒杀技巧">⚡ 快速解法（秒杀技巧）<a class="anchor" href="#-%e5%bf%ab%e9%80%9f%e8%a7%a3%e6%b3%95%e7%a7%92%e6%9d%80%e6%8a%80%e5%b7%a7" aria-label="Link"></a></h3> <ul> <li><strong>渐近线顺序</strong>：垂直（分母零点）→ 水平 → 斜（<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>=</mo><mi>lim</mi><mo>⁡</mo><mfrac><mi>f</mi><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">k=\lim\frac f x</annotation></semantics></math></span>）</li> <li><strong>图像反推</strong>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">f&#x27;</annotation></semantics></math></span> 变号零点 = 极值点；<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">f&#x27;</annotation></semantics></math></span> 的极值点 = 拐点</li> <li><strong>第二充分条件</strong>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;(x_0)=0</annotation></semantics></math></span> 且 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;&#x27;(x_0)&lt;0</annotation></semantics></math></span> 极大、<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&gt;0</annotation></semantics></math></span> 极小（选择题快）</li> </ul> <h2 id="-题型库历年考法全覆盖完整解法版">🧩 题型库：历年考法全覆盖（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1洛必达综合使用跨章高频">题型 1：洛必达综合使用（跨章高频）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e6%b4%9b%e5%bf%85%e8%be%be%e7%bb%bc%e5%90%88%e4%bd%bf%e7%94%a8%e8%b7%a8%e7%ab%a0%e9%ab%98%e9%a2%91" aria-label="Link"></a></h3> <p>例：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\frac{x-\sin x}{x^3}</annotation></semantics></math></span>：</p>https://one.snvedu.cn/docs/english-words/days-21-30/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/days-21-30/<blockquote class=''> <p>用法同前：遮中文自测，动词 ⭐ 优先，想不起画 △。</p> </blockquote><h2 id="day-21e">Day 21（E）<a class="anchor" href="#day-21e" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>enthusiasm</td> <td>n.</td> <td>热情</td> </tr> <tr> <td>entire</td> <td>adj.</td> <td>全部的</td> </tr> <tr> <td>entrance</td> <td>n.</td> <td>入口</td> </tr> <tr> <td>envelope</td> <td>n.</td> <td>信封</td> </tr> <tr> <td>environment</td> <td>n.</td> <td>环境</td> </tr> <tr> <td>equal</td> <td>adj./v.</td> <td>相等的</td> </tr> <tr> <td>equip</td> <td>v.</td> <td>装备</td> </tr> <tr> <td>error</td> <td>n.</td> <td>错误</td> </tr> <tr> <td>escape</td> <td>v./n.</td> <td>逃脱</td> </tr> <tr> <td>especially</td> <td>adv.</td> <td>尤其</td> </tr> <tr> <td>essay</td> <td>n.</td> <td>文章</td> </tr> <tr> <td>essential</td> <td>adj.</td> <td>必要的</td> </tr> <tr> <td>establish</td> <td>v.</td> <td>建立</td> </tr> <tr> <td>estimate</td> <td>v./n.</td> <td>估计</td> </tr> <tr> <td>event</td> <td>n.</td> <td>事件</td> </tr> <tr> <td>eventually</td> <td>adv.</td> <td>最终</td> </tr> <tr> <td>evidence</td> <td>n.</td> <td>证据</td> </tr> <tr> <td>evident</td> <td>adj.</td> <td>明显的</td> </tr> <tr> <td>exact</td> <td>adj.</td> <td>确切的</td> </tr> <tr> <td>examine</td> <td>v.</td> <td>检查</td> </tr> <tr> <td>exceed</td> <td>v.</td> <td>超过</td> </tr> <tr> <td>excellent</td> <td>adj.</td> <td>优秀的</td> </tr> <tr> <td>except</td> <td>prep.</td> <td>除…之外</td> </tr> <tr> <td>exchange</td> <td>v./n.</td> <td>交换</td> </tr> <tr> <td>excite</td> <td>v.</td> <td>使兴奋</td> </tr> </tbody> </table> <h2 id="day-22ef">Day 22（E~F）<a class="anchor" href="#day-22ef" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>excuse</td> <td>n./v.</td> <td>借口</td> </tr> <tr> <td>exercise</td> <td>n./v.</td> <td>锻炼</td> </tr> <tr> <td>exhaust</td> <td>v.</td> <td>耗尽；排气</td> </tr> <tr> <td>exhibit</td> <td>v./n.</td> <td>展览</td> </tr> <tr> <td>exist</td> <td>v.</td> <td>存在</td> </tr> <tr> <td>exit</td> <td>n.</td> <td>出口</td> </tr> <tr> <td>expand</td> <td>v.</td> <td>扩张</td> </tr> <tr> <td>expect ⭐</td> <td>v.</td> <td>期待</td> </tr> <tr> <td>expense</td> <td>n.</td> <td>费用</td> </tr> <tr> <td>experience</td> <td>n./v.</td> <td>经验</td> </tr> <tr> <td>experiment</td> <td>n.</td> <td>实验</td> </tr> <tr> <td>expert</td> <td>n.</td> <td>专家</td> </tr> <tr> <td>explode</td> <td>v.</td> <td>爆炸</td> </tr> <tr> <td>explore</td> <td>v.</td> <td>探索</td> </tr> <tr> <td>expose</td> <td>v.</td> <td>暴露</td> </tr> <tr> <td>express ⭐</td> <td>v.</td> <td>表达</td> </tr> <tr> <td>extend</td> <td>v.</td> <td>延伸</td> </tr> <tr> <td>extent</td> <td>n.</td> <td>程度</td> </tr> <tr> <td>extra</td> <td>adj.</td> <td>额外的</td> </tr> <tr> <td>extraordinary</td> <td>adj.</td> <td>非凡的</td> </tr> <tr> <td>extreme</td> <td>adj.</td> <td>极端的</td> </tr> <tr> <td>fabric</td> <td>n.</td> <td>织物</td> </tr> <tr> <td>facility</td> <td>n.</td> <td>设施</td> </tr> <tr> <td>factor</td> <td>n.</td> <td>因素</td> </tr> <tr> <td>fade</td> <td>v.</td> <td>褪色</td> </tr> </tbody> </table> <h2 id="day-23f">Day 23（F）<a class="anchor" href="#day-23f" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>faint</td> <td>adj./v.</td> <td>微弱的；晕倒</td> </tr> <tr> <td>fairly</td> <td>adv.</td> <td>相当地；公平地</td> </tr> <tr> <td>faith</td> <td>n.</td> <td>信念</td> </tr> <tr> <td>false</td> <td>adj.</td> <td>假的</td> </tr> <tr> <td>fame</td> <td>n.</td> <td>名声</td> </tr> <tr> <td>familiar</td> <td>adj.</td> <td>熟悉的</td> </tr> <tr> <td>fancy</td> <td>adj./n.</td> <td>花哨的；想象</td> </tr> <tr> <td>fare</td> <td>n.</td> <td>车费</td> </tr> <tr> <td>farewell</td> <td>n.</td> <td>告别</td> </tr> <tr> <td>fashion</td> <td>n.</td> <td>时尚</td> </tr> <tr> <td>fasten</td> <td>v.</td> <td>系紧</td> </tr> <tr> <td>fate</td> <td>n.</td> <td>命运</td> </tr> <tr> <td>fault</td> <td>n.</td> <td>过错</td> </tr> <tr> <td>favor</td> <td>n./v.</td> <td>支持；恩惠</td> </tr> <tr> <td>feature</td> <td>n.</td> <td>特征</td> </tr> <tr> <td>fee</td> <td>n.</td> <td>费用</td> </tr> <tr> <td>feed ⭐</td> <td>v.</td> <td>喂养</td> </tr> <tr> <td>female</td> <td>adj./n.</td> <td>女性的</td> </tr> <tr> <td>fence</td> <td>n.</td> <td>栅栏</td> </tr> <tr> <td>festival</td> <td>n.</td> <td>节日</td> </tr> <tr> <td>fetch</td> <td>v.</td> <td>去取来</td> </tr> <tr> <td>fever</td> <td>n.</td> <td>发烧</td> </tr> <tr> <td>fiction</td> <td>n.</td> <td>小说</td> </tr> <tr> <td>fierce</td> <td>adj.</td> <td>激烈的</td> </tr> <tr> <td>figure</td> <td>n./v.</td> <td>数字；认为</td> </tr> </tbody> </table> <h2 id="day-24f">Day 24（F）<a class="anchor" href="#day-24f" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>file</td> <td>n./v.</td> <td>文件；归档</td> </tr> <tr> <td>finance</td> <td>n.</td> <td>财政</td> </tr> <tr> <td>finding</td> <td>n.</td> <td>发现（研究结果）</td> </tr> <tr> <td>firm</td> <td>n./adj.</td> <td>公司；坚定的</td> </tr> <tr> <td>flexible</td> <td>adj.</td> <td>灵活的</td> </tr> <tr> <td>float</td> <td>v.</td> <td>漂浮</td> </tr> <tr> <td>fold</td> <td>v.</td> <td>折叠</td> </tr> <tr> <td>folk</td> <td>n./adj.</td> <td>民间的</td> </tr> <tr> <td>fond</td> <td>adj.</td> <td>喜爱的</td> </tr> <tr> <td>forbid</td> <td>v.</td> <td>禁止</td> </tr> <tr> <td>force ⭐</td> <td>v./n.</td> <td>强迫；力量</td> </tr> <tr> <td>forecast</td> <td>v./n.</td> <td>预报</td> </tr> <tr> <td>foreign</td> <td>adj.</td> <td>外国的</td> </tr> <tr> <td>forever</td> <td>adv.</td> <td>永远</td> </tr> <tr> <td>forgive</td> <td>v.</td> <td>原谅</td> </tr> <tr> <td>formal</td> <td>adj.</td> <td>正式的</td> </tr> <tr> <td>former</td> <td>adj.</td> <td>以前的</td> </tr> <tr> <td>formula</td> <td>n.</td> <td>公式</td> </tr> <tr> <td>fortunate</td> <td>adj.</td> <td>幸运的</td> </tr> <tr> <td>forward</td> <td>adv./v.</td> <td>向前；转发</td> </tr> <tr> <td>foundation</td> <td>n.</td> <td>基础；基金会</td> </tr> <tr> <td>fountain</td> <td>n.</td> <td>喷泉</td> </tr> <tr> <td>fraction</td> <td>n.</td> <td>分数；小部分</td> </tr> <tr> <td>frame</td> <td>n.</td> <td>框架</td> </tr> <tr> <td>freeze</td> <td>v.</td> <td>结冰</td> </tr> </tbody> </table> <h2 id="day-25fg">Day 25（F~G）<a class="anchor" href="#day-25fg" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>frequent</td> <td>adj.</td> <td>频繁的</td> </tr> <tr> <td>frighten</td> <td>v.</td> <td>使惊恐</td> </tr> <tr> <td>frustrate</td> <td>v.</td> <td>使沮丧</td> </tr> <tr> <td>fuel</td> <td>n./v.</td> <td>燃料</td> </tr> <tr> <td>function</td> <td>n./v.</td> <td>功能；起作用</td> </tr> <tr> <td>fund</td> <td>n./v.</td> <td>基金；资助</td> </tr> <tr> <td>fundamental</td> <td>adj.</td> <td>根本的</td> </tr> <tr> <td>funeral</td> <td>n.</td> <td>葬礼</td> </tr> <tr> <td>furnish</td> <td>v.</td> <td>布置（家具）</td> </tr> <tr> <td>furniture</td> <td>n.</td> <td>家具</td> </tr> <tr> <td>further</td> <td>adj./adv.</td> <td>进一步的</td> </tr> <tr> <td>furthermore</td> <td>adv.</td> <td>此外</td> </tr> <tr> <td>gain</td> <td>v./n.</td> <td>获得</td> </tr> <tr> <td>gallery</td> <td>n.</td> <td>画廊</td> </tr> <tr> <td>gap</td> <td>n.</td> <td>差距；缺口</td> </tr> <tr> <td>garage</td> <td>n.</td> <td>车库</td> </tr> <tr> <td>garbage</td> <td>n.</td> <td>垃圾</td> </tr> <tr> <td>gas</td> <td>n.</td> <td>气体</td> </tr> <tr> <td>gather</td> <td>v.</td> <td>聚集</td> </tr> <tr> <td>gaze</td> <td>v./n.</td> <td>凝视</td> </tr> <tr> <td>general</td> <td>adj./n.</td> <td>总体的；将军</td> </tr> <tr> <td>generous</td> <td>adj.</td> <td>慷慨的</td> </tr> <tr> <td>genius</td> <td>n.</td> <td>天才</td> </tr> <tr> <td>gentle</td> <td>adj.</td> <td>温和的</td> </tr> <tr> <td>genuine</td> <td>adj.</td> <td>真正的</td> </tr> </tbody> </table> <h2 id="day-26g">Day 26（G）<a class="anchor" href="#day-26g" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>gesture</td> <td>n.</td> <td>手势</td> </tr> <tr> <td>glance</td> <td>v./n.</td> <td>一瞥</td> </tr> <tr> <td>globe</td> <td>n.</td> <td>地球</td> </tr> <tr> <td>glory</td> <td>n.</td> <td>荣誉</td> </tr> <tr> <td>goal</td> <td>n.</td> <td>目标</td> </tr> <tr> <td>govern</td> <td>v.</td> <td>统治</td> </tr> <tr> <td>grab</td> <td>v.</td> <td>抓住</td> </tr> <tr> <td>grade</td> <td>n./v.</td> <td>年级；评分</td> </tr> <tr> <td>gradual</td> <td>adj.</td> <td>逐渐的</td> </tr> <tr> <td>graduate</td> <td>v./n.</td> <td>毕业</td> </tr> <tr> <td>grain</td> <td>n.</td> <td>谷物</td> </tr> <tr> <td>grand</td> <td>adj.</td> <td>宏大的</td> </tr> <tr> <td>grant</td> <td>v./n.</td> <td>授予；拨款</td> </tr> <tr> <td>graph</td> <td>n.</td> <td>图表</td> </tr> <tr> <td>grasp</td> <td>v.</td> <td>抓住；领会</td> </tr> <tr> <td>grateful</td> <td>adj.</td> <td>感激的</td> </tr> <tr> <td>gravity</td> <td>n.</td> <td>重力</td> </tr> <tr> <td>greet</td> <td>v.</td> <td>问候</td> </tr> <tr> <td>guarantee</td> <td>v./n.</td> <td>保证</td> </tr> <tr> <td>guard</td> <td>v./n.</td> <td>保卫</td> </tr> <tr> <td>guide</td> <td>n./v.</td> <td>指导</td> </tr> <tr> <td>guilty</td> <td>adj.</td> <td>有罪的</td> </tr> <tr> <td>habit</td> <td>n.</td> <td>习惯</td> </tr> <tr> <td>handle</td> <td>v./n.</td> <td>处理；把手</td> </tr> <tr> <td>hang</td> <td>v.</td> <td>悬挂</td> </tr> </tbody> </table> <h2 id="day-27h">Day 27（H）<a class="anchor" href="#day-27h" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>hardship</td> <td>n.</td> <td>艰难</td> </tr> <tr> <td>hardware</td> <td>n.</td> <td>硬件</td> </tr> <tr> <td>harm</td> <td>v./n.</td> <td>伤害</td> </tr> <tr> <td>harmony</td> <td>n.</td> <td>和谐</td> </tr> <tr> <td>harvest</td> <td>n./v.</td> <td>收获</td> </tr> <tr> <td>headline</td> <td>n.</td> <td>标题</td> </tr> <tr> <td>heal</td> <td>v.</td> <td>治愈</td> </tr> <tr> <td>heaven</td> <td>n.</td> <td>天堂</td> </tr> <tr> <td>height</td> <td>n.</td> <td>高度</td> </tr> <tr> <td>helpful</td> <td>adj.</td> <td>有帮助的</td> </tr> <tr> <td>hence</td> <td>adv.</td> <td>因此</td> </tr> <tr> <td>hero</td> <td>n.</td> <td>英雄</td> </tr> <tr> <td>hesitate</td> <td>v.</td> <td>犹豫</td> </tr> <tr> <td>hint</td> <td>n./v.</td> <td>暗示</td> </tr> <tr> <td>hire</td> <td>v.</td> <td>雇用</td> </tr> <tr> <td>historic</td> <td>adj.</td> <td>有历史意义的</td> </tr> <tr> <td>hollow</td> <td>adj.</td> <td>空心的</td> </tr> <tr> <td>holy</td> <td>adj.</td> <td>神圣的</td> </tr> <tr> <td>honest</td> <td>adj.</td> <td>诚实的</td> </tr> <tr> <td>honor</td> <td>n./v.</td> <td>荣誉</td> </tr> <tr> <td>horizon</td> <td>n.</td> <td>地平线</td> </tr> <tr> <td>host</td> <td>v./n.</td> <td>主办；主人</td> </tr> <tr> <td>household</td> <td>n./adj.</td> <td>家庭</td> </tr> <tr> <td>humble</td> <td>adj.</td> <td>谦虚的</td> </tr> <tr> <td>humor</td> <td>n.</td> <td>幽默</td> </tr> </tbody> </table> <h2 id="day-28hi">Day 28（H~I）<a class="anchor" href="#day-28hi" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>hunt</td> <td>v.</td> <td>打猎</td> </tr> <tr> <td>ideal</td> <td>adj./n.</td> <td>理想的</td> </tr> <tr> <td>identical</td> <td>adj.</td> <td>完全相同的</td> </tr> <tr> <td>identify</td> <td>v.</td> <td>识别</td> </tr> <tr> <td>ignorant</td> <td>adj.</td> <td>无知的</td> </tr> <tr> <td>ignore</td> <td>v.</td> <td>忽视</td> </tr> <tr> <td>illegal</td> <td>adj.</td> <td>非法的</td> </tr> <tr> <td>image</td> <td>n.</td> <td>形象</td> </tr> <tr> <td>imagination</td> <td>n.</td> <td>想象力</td> </tr> <tr> <td>imitate</td> <td>v.</td> <td>模仿</td> </tr> <tr> <td>immediate</td> <td>adj.</td> <td>立即的</td> </tr> <tr> <td>impact</td> <td>n./v.</td> <td>影响</td> </tr> <tr> <td>imply</td> <td>v.</td> <td>暗示</td> </tr> <tr> <td>import</td> <td>v./n.</td> <td>进口</td> </tr> <tr> <td>impose</td> <td>v.</td> <td>强加</td> </tr> <tr> <td>impress</td> <td>v.</td> <td>使印象深刻</td> </tr> <tr> <td>improvement</td> <td>n.</td> <td>改进</td> </tr> <tr> <td>incident</td> <td>n.</td> <td>事件</td> </tr> <tr> <td>include ⭐</td> <td>v.</td> <td>包括</td> </tr> <tr> <td>income</td> <td>n.</td> <td>收入</td> </tr> <tr> <td>increase ⭐</td> <td>v./n.</td> <td>增加</td> </tr> <tr> <td>indeed</td> <td>adv.</td> <td>的确</td> </tr> <tr> <td>independence</td> <td>n.</td> <td>独立</td> </tr> <tr> <td>independent</td> <td>adj.</td> <td>独立的</td> </tr> <tr> <td>indicate</td> <td>v.</td> <td>表明</td> </tr> </tbody> </table> <h2 id="day-29i">Day 29（I）<a class="anchor" href="#day-29i" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>individual</td> <td>n./adj.</td> <td>个人</td> </tr> <tr> <td>industrial</td> <td>adj.</td> <td>工业的</td> </tr> <tr> <td>influence</td> <td>n./v.</td> <td>影响</td> </tr> <tr> <td>inform</td> <td>v.</td> <td>通知</td> </tr> <tr> <td>initial</td> <td>adj.</td> <td>最初的</td> </tr> <tr> <td>injure</td> <td>v.</td> <td>受伤</td> </tr> <tr> <td>inner</td> <td>adj.</td> <td>内部的</td> </tr> <tr> <td>innocent</td> <td>adj.</td> <td>无辜的</td> </tr> <tr> <td>insist</td> <td>v.</td> <td>坚持</td> </tr> <tr> <td>inspect</td> <td>v.</td> <td>检查</td> </tr> <tr> <td>inspire</td> <td>v.</td> <td>激励</td> </tr> <tr> <td>install</td> <td>v.</td> <td>安装</td> </tr> <tr> <td>instance</td> <td>n.</td> <td>例子</td> </tr> <tr> <td>instead</td> <td>adv.</td> <td>反而</td> </tr> <tr> <td>institute</td> <td>n.</td> <td>学院；研究所</td> </tr> <tr> <td>instruct</td> <td>v.</td> <td>指示</td> </tr> <tr> <td>instrument</td> <td>n.</td> <td>乐器；仪器</td> </tr> <tr> <td>insurance</td> <td>n.</td> <td>保险</td> </tr> <tr> <td>intend</td> <td>v.</td> <td>打算</td> </tr> <tr> <td>intention</td> <td>n.</td> <td>意图</td> </tr> <tr> <td>interfere</td> <td>v.</td> <td>干涉</td> </tr> <tr> <td>internal</td> <td>adj.</td> <td>内部的</td> </tr> <tr> <td>international</td> <td>adj.</td> <td>国际的</td> </tr> <tr> <td>interpret</td> <td>v.</td> <td>口译；解释</td> </tr> <tr> <td>interrupt</td> <td>v.</td> <td>打断</td> </tr> </tbody> </table> <h2 id="day-30i">Day 30（I）<a class="anchor" href="#day-30i" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>interval</td> <td>n.</td> <td>间隔</td> </tr> <tr> <td>interview</td> <td>n./v.</td> <td>面试</td> </tr> <tr> <td>introduce ⭐</td> <td>v.</td> <td>介绍</td> </tr> <tr> <td>invent</td> <td>v.</td> <td>发明</td> </tr> <tr> <td>invest</td> <td>v.</td> <td>投资</td> </tr> <tr> <td>investigate</td> <td>v.</td> <td>调查</td> </tr> <tr> <td>involve</td> <td>v.</td> <td>涉及</td> </tr> <tr> <td>jam</td> <td>n.</td> <td>堵塞；果酱</td> </tr> <tr> <td>jealous</td> <td>adj.</td> <td>嫉妒的</td> </tr> <tr> <td>joint</td> <td>adj./n.</td> <td>共同的；关节</td> </tr> <tr> <td>journal</td> <td>n.</td> <td>期刊</td> </tr> <tr> <td>judge</td> <td>v./n.</td> <td>判断；法官</td> </tr> <tr> <td>justice</td> <td>n.</td> <td>正义</td> </tr> <tr> <td>keen</td> <td>adj.</td> <td>热衷的</td> </tr> <tr> <td>kick</td> <td>v.</td> <td>踢</td> </tr> <tr> <td>kindness</td> <td>n.</td> <td>善良</td> </tr> <tr> <td>knowledge</td> <td>n.</td> <td>知识</td> </tr> <tr> <td>lab</td> <td>n.</td> <td>实验室</td> </tr> <tr> <td>label</td> <td>n./v.</td> <td>标签</td> </tr> <tr> <td>labor</td> <td>n.</td> <td>劳动</td> </tr> <tr> <td>lack</td> <td>v./n.</td> <td>缺乏</td> </tr> <tr> <td>landmark</td> <td>n.</td> <td>地标</td> </tr> <tr> <td>landscape</td> <td>n.</td> <td>风景</td> </tr> <tr> <td>launch</td> <td>v.</td> <td>发射；发起</td> </tr> <tr> <td>laundry</td> <td>n.</td> <td>洗衣店</td> </tr> </tbody> </table>https://one.snvedu.cn/docs/english-basics/03-pronouns/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/03-pronouns/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>同一个「我」，有四种形态：<strong>I / me / my / mine</strong></li> <li>什么时候用哪一种（一张表全搞定）</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-四种形态对照表以我为例">1. 四种形态对照表（以「我」为例）<a class="anchor" href="#1-%e5%9b%9b%e7%a7%8d%e5%bd%a2%e6%80%81%e5%af%b9%e7%85%a7%e8%a1%a8%e4%bb%a5%e6%88%91%e4%b8%ba%e4%be%8b" aria-label="Link"></a></h3> <table> <thead> <tr> <th>形态</th> <th>名字</th> <th>干什么用</th> <th>例句</th> </tr> </thead> <tbody> <tr> <td>I</td> <td>主格</td> <td><strong>做动作的人</strong>（句子的主语）</td> <td><strong>I</strong> love music.</td> </tr> <tr> <td>me</td> <td>宾格</td> <td><strong>被动作打到的人</strong>（宾语）</td> <td>He loves <strong>me</strong>.</td> </tr> <tr> <td>my</td> <td>形容词性物主代词</td> <td><strong>我的……</strong>（后面必须跟东西）</td> <td><strong>my</strong> book</td> </tr> <tr> <td>mine</td> <td>名词性物主代词</td> <td><strong>我的（东西）</strong>（后面不跟东西）</td> <td>This book is <strong>mine</strong>.</td> </tr> </tbody> </table> <h3 id="2-全家福所有人称">2. 全家福（所有人称）<a class="anchor" href="#2-%e5%85%a8%e5%ae%b6%e7%a6%8f%e6%89%80%e6%9c%89%e4%ba%ba%e7%a7%b0" aria-label="Link"></a></h3> <table> <thead> <tr> <th>中文</th> <th>主格（做动作）</th> <th>宾格（接动作）</th> <th>「…的」</th> <th>「…的东西」</th> </tr> </thead> <tbody> <tr> <td>我</td> <td>I</td> <td>me</td> <td>my</td> <td>mine</td> </tr> <tr> <td>你</td> <td>you</td> <td>you</td> <td>your</td> <td>yours</td> </tr> <tr> <td>他</td> <td>he</td> <td>him</td> <td>his</td> <td>his</td> </tr> <tr> <td>她</td> <td>she</td> <td>her</td> <td>her</td> <td>hers</td> </tr> <tr> <td>它</td> <td>it</td> <td>it</td> <td>its</td> <td>its</td> </tr> <tr> <td>我们</td> <td>we</td> <td>us</td> <td>our</td> <td>ours</td> </tr> <tr> <td>他们</td> <td>they</td> <td>them</td> <td>their</td> <td>theirs</td> </tr> </tbody> </table> <h3 id="3-判断三步法考试就考这个">3. 判断三步法（考试就考这个）<a class="anchor" href="#3-%e5%88%a4%e6%96%ad%e4%b8%89%e6%ad%a5%e6%b3%95%e8%80%83%e8%af%95%e5%b0%b1%e8%80%83%e8%bf%99%e4%b8%aa" aria-label="Link"></a></h3> <ol> <li>这个词<strong>前面有没有名词</strong>？有 → 用「…的」（my）；没有 → 用「…的东西」（mine）</li> <li>这个词<strong>是不是动作的发出者</strong>（在动词前）？是 → 主格（I / he / they）</li> <li>是动作的<strong>承受者</strong>（在动词后）？→ 宾格（me / him / them）</li> </ol> <p>例句对照：</p>https://one.snvedu.cn/bank/blank/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/blank/<blockquote class=''> <p>填空没有选项可排除——靠「结论直写」和快准。每题 ≤2 分钟。答案全折叠。</p> </blockquote><h2 id="1-hahahugoshortcode7s0hbhb--____">1. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>x</mi><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\dfrac{1-\cos x}{x\sin x}</annotation></semantics></math></span><link rel="stylesheet" href="https://one.snvedu.cn/katex/katex.min.css" /> = ____<a class="anchor" href="#1-hahahugoshortcode7s0hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：1/2</summary> 分子 ~x²/2，分母 ~x² </details> <h2 id="2-hahahugoshortcode7s1hbhb则-hahahugoshortcode7s2hbhb--____">2. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">y=xe^x</annotation></semantics></math></span>，则 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;</annotation></semantics></math></span> = ____<a class="anchor" href="#2-hahahugoshortcode7s1hbhb%e5%88%99-hahahugoshortcode7s2hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：e^x(x+2)</summary> y'=e^x(x+1)，再求导 </details> <h2 id="3-曲线-hahahugoshortcode7s3hbhb-的极大值为-____">3. 曲线 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y=x^3-3x+1</annotation></semantics></math></span> 的极大值为 ____<a class="anchor" href="#3-%e6%9b%b2%e7%ba%bf-hahahugoshortcode7s3hbhb-%e7%9a%84%e6%9e%81%e5%a4%a7%e5%80%bc%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：3</summary> f'=3x²−3=0 → x=±1；极大 f(−1)=−1+3+1=3 </details> <h2 id="4-hahahugoshortcode7s4hbhb--____">4. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>π</mi></msubsup><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_0^{\pi}\sin^2x\,dx</annotation></semantics></math></span> = ____<a class="anchor" href="#4-hahahugoshortcode7s4hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：π/2</summary> 降幂：∫(1−cos2x)/2 = π/2 </details> <h2 id="5-hahahugoshortcode7s5hbhb--____">5. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>x</mi><mi>ln</mi><mo>⁡</mo><mi>x</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\int\dfrac{dx}{x\ln x}</annotation></semantics></math></span> = ____<a class="anchor" href="#5-hahahugoshortcode7s5hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：ln|ln x|+C</summary> dx/x = d(ln x)，凑微分（考点：复合 ln） </details> <h2 id="6-曲线过点-01-且任一点切线斜率为-hahahugoshortcode7s6hbhb则曲线方程为-____">6. 曲线过点 (0,1) 且任一点切线斜率为 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">e^x</annotation></semantics></math></span>，则曲线方程为 ____<a class="anchor" href="#6-%e6%9b%b2%e7%ba%bf%e8%bf%87%e7%82%b9-01-%e4%b8%94%e4%bb%bb%e4%b8%80%e7%82%b9%e5%88%87%e7%ba%bf%e6%96%9c%e7%8e%87%e4%b8%ba-hahahugoshortcode7s6hbhb%e5%88%99%e6%9b%b2%e7%ba%bf%e6%96%b9%e7%a8%8b%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：y=e^x</summary> y'=e^x → y=e^x+C；y(0)=1 → C=0（积分几何意义） </details> <h2 id="7-hahahugoshortcode7s7hbhb-满足-hahahugoshortcode7s8hbhb-的特解为-____">7. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;+2y&#x27;+y=0</annotation></semantics></math></span> 满足 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mtext> </mtext><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y(0)=1,\ y&#x27;(0)=0</annotation></semantics></math></span> 的特解为 ____<a class="anchor" href="#7-hahahugoshortcode7s7hbhb-%e6%bb%a1%e8%b6%b3-hahahugoshortcode7s8hbhb-%e7%9a%84%e7%89%b9%e8%a7%a3%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：y=(1+x)e^{−x}</summary> 重根 r=−1；C₁=1，y'(0)=C₂−C₁=0 → C₂=1 </details> <h2 id="8-幂级数-hahahugoshortcode7s9hbhb-的收敛半径-r--____">8. 幂级数 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\sum nx^n</annotation></semantics></math></span> 的收敛半径 R = ____<a class="anchor" href="#8-%e5%b9%82%e7%ba%a7%e6%95%b0-hahahugoshortcode7s9hbhb-%e7%9a%84%e6%94%b6%e6%95%9b%e5%8d%8a%e5%be%84-r--____" aria-label="Link"></a></h2> <details><summary>答案：1</summary> R=lim|a_n/a_{n+1}| = lim n/(n+1) = 1 </details> <h2 id="9-hahahugoshortcode7s10hbhb-在-12-处的全微分-hahahugoshortcode7s11hbhb--____">9. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">z=x^2+y^2</annotation></semantics></math></span> 在 (1,2) 处的全微分 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">dz</annotation></semantics></math></span> = ____<a class="anchor" href="#9-hahahugoshortcode7s10hbhb-%e5%9c%a8-12-%e5%a4%84%e7%9a%84%e5%85%a8%e5%be%ae%e5%88%86-hahahugoshortcode7s11hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：2dx+4dy</summary> z_x=2x=2, z_y=2y=4 </details> <h2 id="10-交换次序hahahugoshortcode7s12hbhb--____">10. 交换次序：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>d</mi><mi>y</mi><msubsup><mo>∫</mo><mi>y</mi><mn>1</mn></msubsup><mi>f</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_0^1dy\int_y^1f\,dx</annotation></semantics></math></span> = ____<a class="anchor" href="#10-%e4%ba%a4%e6%8d%a2%e6%ac%a1%e5%ba%8fhahahugoshortcode7s12hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：∫₀¹dx∫₀^x f dy</summary> 画图：区域 0≤y≤x≤1 </details> <h2 id="11-hahahugoshortcode7s13hbhb-且-hahahugoshortcode7s14hbhb则-hahahugoshortcode7s15hbhb--____">11. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">f&#x27;(x)=2x</annotation></semantics></math></span> 且 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">f(1)=3</annotation></semantics></math></span>，则 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span> = ____<a class="anchor" href="#11-hahahugoshortcode7s13hbhb-%e4%b8%94-hahahugoshortcode7s14hbhb%e5%88%99-hahahugoshortcode7s15hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：x²+2</summary> f=x²+C；f(1)=1+C=3 → C=2 </details> <h2 id="12-hahahugoshortcode7s16hbhb-____填存在值不存在">12. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><msup><mi>x</mi><mn>2</mn></msup><msup><mi>e</mi><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}x^2e^{1/x}</annotation></semantics></math></span> ____（填：存在值/不存在）<a class="anchor" href="#12-hahahugoshortcode7s16hbhb-____%e5%a1%ab%e5%ad%98%e5%9c%a8%e5%80%bc%e4%b8%8d%e5%ad%98%e5%9c%a8" aria-label="Link"></a></h2> <details><summary>答案：不存在</summary> 左极限 0，右极限 +∞（e^{1/x} 信号词，必分侧） </details> <h2 id="13-hahahugoshortcode7s17hbhb--____">13. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><msup><mi>x</mi><mn>3</mn></msup><mi>cos</mi><mo>⁡</mo><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_{-\pi}^{\pi}x^3\cos x\,dx</annotation></semantics></math></span> = ____<a class="anchor" href="#13-hahahugoshortcode7s17hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：0</summary> x³ 奇 × cos 偶 = 奇函数，对称区间 → 0（禁术 12） </details> <h2 id="14-曲线-hahahugoshortcode7s18hbhb-的拐点个数为-____">14. 曲线 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=x(x-1)(x-2)</annotation></semantics></math></span> 的拐点个数为 ____<a class="anchor" href="#14-%e6%9b%b2%e7%ba%bf-hahahugoshortcode7s18hbhb-%e7%9a%84%e6%8b%90%e7%82%b9%e4%b8%aa%e6%95%b0%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：1 个</summary> y''=6x−6 在 x=1 为零且变号（三次曲线恒有一个拐点） </details> <h2 id="15-hahahugoshortcode7s19hbhb-的通解为-____">15. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo>−</mo><mn>5</mn><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mn>6</mn><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;-5y&#x27;+6y=0</annotation></semantics></math></span> 的通解为 ____<a class="anchor" href="#15-hahahugoshortcode7s19hbhb-%e7%9a%84%e9%80%9a%e8%a7%a3%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：y=C₁e^{2x}+C₂e^{3x}</summary> (r−2)(r−3)=0 </details> <h2 id="16-hahahugoshortcode7s20hbhb-的单调减区间为-____">16. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">y=x^3-3x</annotation></semantics></math></span> 的单调减区间为 ____<a class="anchor" href="#16-hahahugoshortcode7s20hbhb-%e7%9a%84%e5%8d%95%e8%b0%83%e5%87%8f%e5%8c%ba%e9%97%b4%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：(−1, 1)</summary> f'=3(x²−1)<0 </details> <h2 id="17-设-hahahugoshortcode7s21hbhb-收敛则-hahahugoshortcode7s22hbhb-____敛散">17. 设 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><msub><mi>a</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\sum a_n</annotation></semantics></math></span> 收敛，则 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mn>2026</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum(a_n+2026)</annotation></semantics></math></span> ____（敛/散）<a class="anchor" href="#17-%e8%ae%be-hahahugoshortcode7s21hbhb-%e6%94%b6%e6%95%9b%e5%88%99-hahahugoshortcode7s22hbhb-____%e6%95%9b%e6%95%a3" aria-label="Link"></a></h2> <details><summary>答案：发散</summary> 通项 →2026 ≠ 0（必要条件秒杀，禁术 18） </details> <h2 id="18-hahahugoshortcode7s23hbhb--____">18. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_0^1(2x+1)\,dx</annotation></semantics></math></span> = ____<a class="anchor" href="#18-hahahugoshortcode7s23hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：2</summary> [x²+x]₀¹ = 2 </details> <h2 id="19-hahahugoshortcode7s24hbhb则-hahahugoshortcode7s25hbhb--____">19. <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z=\ln(x^2+y^2)</annotation></semantics></math></span>，则 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">z_x</annotation></semantics></math></span> = ____<a class="anchor" href="#19-hahahugoshortcode7s24hbhb%e5%88%99-hahahugoshortcode7s25hbhb--____" aria-label="Link"></a></h2> <details><summary>答案：2x/(x²+y²)</summary> 链式：外层 1/u × 内层 2x </details> <h2 id="20-曲线-hahahugoshortcode7s26hbhb-的水平渐近线为-____">20. 曲线 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">y=e^x</annotation></semantics></math></span> 的水平渐近线为 ____<a class="anchor" href="#20-%e6%9b%b2%e7%ba%bf-hahahugoshortcode7s26hbhb-%e7%9a%84%e6%b0%b4%e5%b9%b3%e6%b8%90%e8%bf%91%e7%ba%bf%e4%b8%ba-____" aria-label="Link"></a></h2> <details><summary>答案：y=0（x→−∞ 方向）</summary> x→+∞ 时 →+∞ 无水平；x→−∞ → 0 </details>https://one.snvedu.cn/exam/2021/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2021/<h1 id="2021-年浙江专升本真题">2021 年浙江专升本真题<a class="anchor" href="#2021-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/gaoshu/indefinite-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/indefinite-integrals/<blockquote class=''> <p>🎯 浙江考纲定位：一元微积分板块。计算题必出 2~3 道积分，凑微分与分部积分是主战场。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解原函数与不定积分的概念</li> <li>掌握基本积分公式表</li> <li>掌握第一类换元法（凑微分）、第二类换元法</li> <li>掌握分部积分法</li> </ul> <h2 id="-核心公式库">📖 核心公式库<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e5%85%ac%e5%bc%8f%e5%ba%93" aria-label="Link"></a></h2> <h3 id="1-基本积分表反向的导数表必须滚瓜烂熟">1. 基本积分表（反向的导数表，必须滚瓜烂熟）<a class="anchor" href="#1-%e5%9f%ba%e6%9c%ac%e7%a7%af%e5%88%86%e8%a1%a8%e5%8f%8d%e5%90%91%e7%9a%84%e5%af%bc%e6%95%b0%e8%a1%a8%e5%bf%85%e9%a1%bb%e6%bb%9a%e7%93%9c%e7%83%82%e7%86%9f" aria-label="Link"></a></h3> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><msup><mi>x</mi><mi>a</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><msup><mi>x</mi><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>C</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>a</mi><mo mathvariant="normal">≠</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="2em"/><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mi>ln</mi><mo>⁡</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mi>C</mi><mo separator="true">,</mo><mspace width="2em"/><mo>∫</mo><msup><mi>e</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\int x^a dx=\frac{x^{a+1}}{a+1}+C\ (a\neq-1),\qquad \int\frac{dx}{x}=\ln|x|+C,\qquad \int e^x dx=e^x+C</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>+</mo><mi>C</mi><mo separator="true">,</mo><mspace width="2em"/><mo>∫</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo>+</mo><mi>C</mi><mo separator="true">,</mo><mspace width="2em"/><mo>∫</mo><msup><mrow><mi>sec</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\int\cos x\,dx=\sin x+C,\qquad \int\sin x\,dx=-\cos x+C,\qquad \int\sec^2x\,dx=\tan x+C</annotation></semantics></math></span><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>arctan</mi><mo>⁡</mo><mi>x</mi><mo>+</mo><mi>C</mi><mo separator="true">,</mo><mspace width="2em"/><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo>=</mo><mi>arcsin</mi><mo>⁡</mo><mi>x</mi><mo>+</mo><mi>C</mi><mo separator="true">,</mo><mspace width="2em"/><mo>∫</mo><msup><mi>a</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><msup><mi>a</mi><mi>x</mi></msup><mrow><mi>ln</mi><mo>⁡</mo><mi>a</mi></mrow></mfrac><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\int\frac{dx}{1+x^2}=\arctan x+C,\qquad \int\frac{dx}{\sqrt{1-x^2}}=\arcsin x+C,\qquad \int a^x dx=\frac{a^x}{\ln a}+C</annotation></semantics></math></span><h3 id="2-凑微分对照表本章最核心资产">2. 凑微分对照表（本章最核心资产）<a class="anchor" href="#2-%e5%87%91%e5%be%ae%e5%88%86%e5%af%b9%e7%85%a7%e8%a1%a8%e6%9c%ac%e7%ab%a0%e6%9c%80%e6%a0%b8%e5%bf%83%e8%b5%84%e4%ba%a7" aria-label="Link"></a></h3> <table> <thead> <tr> <th>看见</th> <th>凑成</th> <th>看见</th> <th>凑成</th> </tr> </thead> <tbody> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x\,dx</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>d</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac12 d(x^2)</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{dx}{x}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>ln</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(\ln x)</annotation></semantics></math></span></td> </tr> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">x^2dx</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>d</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac13d(x^3)</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{dx}{x^2}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(-\frac1x)</annotation></semantics></math></span></td> </tr> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msqrt><mi>x</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\frac{dx}{\sqrt{x}}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>d</mi><mo stretchy="false">(</mo><msqrt><mi>x</mi></msqrt><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2d(\sqrt{x})</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">e^xdx</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(e^x)</annotation></semantics></math></span></td> </tr> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin x\,dx</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi>d</mi><mo stretchy="false">(</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-d(\cos x)</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\cos x\,dx</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(\sin x)</annotation></semantics></math></span></td> </tr> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dx}{1+x^2}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>arctan</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(\arctan x)</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\frac{dx}{\sqrt{1-x^2}}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>arcsin</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(\arcsin x)</annotation></semantics></math></span></td> </tr> </tbody> </table> <h3 id="3-三角代换选择表">3. 三角代换选择表<a class="anchor" href="#3-%e4%b8%89%e8%a7%92%e4%bb%a3%e6%8d%a2%e9%80%89%e6%8b%a9%e8%a1%a8" aria-label="Link"></a></h3> <table> <thead> <tr> <th>含有</th> <th>换元</th> </tr> </thead> <tbody> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{a^2-x^2}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mi>sin</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">x=a\sin t</annotation></semantics></math></span></td> </tr> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{a^2+x^2}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mi>tan</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">x=a\tan t</annotation></semantics></math></span></td> </tr> <tr> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{x^2-a^2}</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mi>sec</mi><mo>⁡</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">x=a\sec t</annotation></semantics></math></span></td> </tr> </tbody> </table> <h2 id="-题型库历年考法全覆盖完整解法版">🧩 题型库：历年考法全覆盖（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1基本公式直接积分送分别丢">题型 1：基本公式直接积分（送分，别丢）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e5%9f%ba%e6%9c%ac%e5%85%ac%e5%bc%8f%e7%9b%b4%e6%8e%a5%e7%a7%af%e5%88%86%e9%80%81%e5%88%86%e5%88%ab%e4%b8%a2" aria-label="Link"></a></h3> <p>例：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mrow><mo fence="true">(</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>2</mn><mi>x</mi></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo fence="true">)</mo></mrow><mi>d</mi><mi>x</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>2</mn><mi>ln</mi><mo>⁡</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\int\left(3x^2-\frac2x+e^x\right)dx=x^3-2\ln|x|+e^x+C</annotation></semantics></math></span> ✅</p>https://one.snvedu.cn/docs/english/cloze/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/cloze/<h2 id="-模块说明">📌 模块说明<a class="anchor" href="#-%e6%a8%a1%e5%9d%97%e8%af%b4%e6%98%8e" aria-label="Link"></a></h2> <p>完形考「<strong>上下文逻辑 + 词义辨析 + 固定搭配</strong>」的综合。核心原则：<strong>先完其意，再完其形</strong>——先读懂文章，再填空。</p> <h2 id="-题型库五种空格类型完整解法版">🧩 题型库：五种空格类型（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e4%ba%94%e7%a7%8d%e7%a9%ba%e6%a0%bc%e7%b1%bb%e5%9e%8b%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1逻辑关系词题">题型 1：逻辑关系词题<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e9%80%bb%e8%be%91%e5%85%b3%e7%b3%bb%e8%af%8d%e9%a2%98" aria-label="Link"></a></h3> <p>特征：空格处需要 but / however / therefore / because 等衔接词。 解法：判断<strong>空格前后两句的关系</strong>（转折/因果/递进/让步）。 例：He worked very hard, ____ he failed the exam. → 前后语义相反 → <strong>but / however</strong> ✅</p>https://one.snvedu.cn/docs/english-basics/04-present-simple/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/04-present-simple/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>一般现在时讲什么（习惯、事实）</li> <li><strong>第三人称单数 + s</strong>（专升本必考）</li> <li>否定 don&rsquo;t / doesn&rsquo;t 和疑问 Do / Does</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-什么时候用一般现在时">1. 什么时候用一般现在时<a class="anchor" href="#1-%e4%bb%80%e4%b9%88%e6%97%b6%e5%80%99%e7%94%a8%e4%b8%80%e8%88%ac%e7%8e%b0%e5%9c%a8%e6%97%b6" aria-label="Link"></a></h3> <p>讲<strong>习惯</strong>和<strong>事实</strong>——现在是这样，平时也这样：</p>https://one.snvedu.cn/bank/calc/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/calc/<blockquote class=''> <p>大题按<strong>步骤给分</strong>——解析里的每一步都是答题卷上该写的。每题限时 10~12 分钟。</p> </blockquote><h2 id="大题-1极限">大题 1（极限）<a class="anchor" href="#%e5%a4%a7%e9%a2%98-1%e6%9e%81%e9%99%90" aria-label="Link"></a></h2> <p>计算 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\dfrac{\tan x-\sin x}{x^3}</annotation></semantics></math></span></p> <details><summary>完整解答</summary> <p>提取公因式化加减为乘除（深坑 A1/避坑 1）：</p> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>=</mo><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>x</mi><mo>⋅</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\tan x-\sin x=\tan x(1-\cos x)\sim x\cdot\frac{x^2}{2}=\frac{x^3}{2}</annotation></semantics></math></span><p>所以原式 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mi>lim</mi><mo>⁡</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mstyle><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">=\lim\dfrac{x^3/2}{x^3}=\dfrac12</annotation></semantics></math></span>。</p> </details>https://one.snvedu.cn/docs/english-words/topics/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/topics/<blockquote class=''> <p>🎯 写作和翻译是<strong>按主题出题</strong>的——把每个主题的核心词背熟，作文翻译直接调用。</p> </blockquote><h2 id="一校园学习作文最高频主题">一、校园学习（作文最高频主题）<a class="anchor" href="#%e4%b8%80%e6%a0%a1%e5%9b%ad%e5%ad%a6%e4%b9%a0%e4%bd%9c%e6%96%87%e6%9c%80%e9%ab%98%e9%a2%91%e4%b8%bb%e9%a2%98" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>中文</th> <th>词</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>education</td> <td>教育</td> <td>knowledge</td> <td>知识</td> </tr> <tr> <td>exam / test</td> <td>考试</td> <td>course</td> <td>课程</td> </tr> <tr> <td>major</td> <td>专业</td> <td>degree</td> <td>学位</td> </tr> <tr> <td>campus</td> <td>校园</td> <td>library</td> <td>图书馆</td> </tr> <tr> <td>teacher</td> <td>老师</td> <td>classmate</td> <td>同学</td> </tr> <tr> <td>study habit</td> <td>学习习惯</td> <td>review</td> <td>复习</td> </tr> <tr> <td>concentrate</td> <td>专心</td> <td>progress</td> <td>进步</td> </tr> <tr> <td>achieve</td> <td>取得（成就）</td> <td>effort</td> <td>努力</td> </tr> <tr> <td>confidence</td> <td>自信</td> <td>motivation</td> <td>动力</td> </tr> <tr> <td>graduate</td> <td>毕业</td> <td>scholarship</td> <td>奖学金</td> </tr> </tbody> </table> <p><strong>常用搭配</strong>：make progress 取得进步 / take an exam 参加考试 / get a degree 获得学位 / pay attention to 集中注意于</p>https://one.snvedu.cn/exam/2022/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2022/<h1 id="2022-年浙江专升本真题">2022 年浙江专升本真题<a class="anchor" href="#2022-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/gaoshu/definite-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/definite-integrals/<blockquote class=''> <p>🎯 浙江考纲定位：一元微积分板块。计算题常考定积分计算，<strong>综合题常考面积/旋转体</strong>。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解定积分的概念与几何意义</li> <li>掌握牛顿—莱布尼茨公式</li> <li>掌握定积分的换元法与分部积分法</li> <li>理解变上限积分函数，会求其导数</li> <li>会用定积分求平面图形面积、旋转体体积</li> </ul> <h2 id="-核心公式库">📖 核心公式库<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e5%85%ac%e5%bc%8f%e5%ba%93" aria-label="Link"></a></h2> <ul> <li>牛顿—莱布尼茨：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_a^b f(x)dx=F(b)-F(a)</annotation></semantics></math></span></li> <li>变上限求导：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mo>∫</mo><mi>a</mi><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>φ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\int_a^{\varphi(x)}f(t)dt=f(\varphi(x))\cdot\varphi&#x27;(x)</annotation></semantics></math></span></li> <li>面积：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mo stretchy="false">[</mo><mtext>上</mtext><mo>−</mo><mtext>下</mtext><mo stretchy="false">]</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">S=\int_a^b[\text{上}-\text{下}]dx</annotation></semantics></math></span></li> <li>旋转体（绕 x 轴）：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mi>π</mi><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><msup><mi>f</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">V=\pi\int_a^b f^2(x)dx</annotation></semantics></math></span>（环状：外平方 − 内平方）</li> </ul> <h3 id="-快速解法秒杀技巧">⚡ 快速解法（秒杀技巧）<a class="anchor" href="#-%e5%bf%ab%e9%80%9f%e8%a7%a3%e6%b3%95%e7%a7%92%e6%9d%80%e6%8a%80%e5%b7%a7" aria-label="Link"></a></h3> <ul> <li><strong>对称区间先想奇偶</strong>：奇函数 = 0，偶函数 = 2 倍半区间</li> <li><strong>周期性</strong>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mi>T</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>T</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_a^{a+T}f(x)dx=\int_0^T f(x)dx</annotation></semantics></math></span></li> <li><strong>华里士公式（若考纲涉及）</strong>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>π</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msubsup><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mi>n</mi></msup><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_0^{\pi/2}\sin^nx\,dx</annotation></semantics></math></span> 双阶乘速算</li> </ul> <h2 id="-题型库历年考法全覆盖完整解法版">🧩 题型库：历年考法全覆盖（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1直接计算送分">题型 1：直接计算（送分）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e7%9b%b4%e6%8e%a5%e8%ae%a1%e7%ae%97%e9%80%81%e5%88%86" aria-label="Link"></a></h3> <p>例：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo stretchy="false">(</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mo stretchy="false">[</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><msubsup><mo stretchy="false">]</mo><mn>0</mn><mn>1</mn></msubsup><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\int_0^1(3x^2+2x)dx=[x^3+x^2]_0^1=2</annotation></semantics></math></span> ✅</p>https://one.snvedu.cn/docs/english/passage-questions/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/passage-questions/<blockquote class=''> <p>🎯 浙江题型定位：与完形同板块，5 题 10 分。</p> </blockquote><h2 id="-模块说明">📌 模块说明<a class="anchor" href="#-%e6%a8%a1%e5%9d%97%e8%af%b4%e6%98%8e" aria-label="Link"></a></h2> <p>读短文回答 5 个问题——<strong>答案基本都能在原文找到</strong>，考「定位 + 提炼 + 简洁作答」。这是主观题里最容易公式化拿分的题型。</p>https://one.snvedu.cn/docs/english-words/days-31-40/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/days-31-40/<blockquote class=''> <p>第二期开始：J~Z + 高频补漏。用法不变：遮中文自测，动词 ⭐ 优先，△ 标记遗忘。</p> </blockquote><h2 id="day-31jk">Day 31（J~K）<a class="anchor" href="#day-31jk" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>jet</td> <td>n.</td> <td>喷气式飞机</td> </tr> <tr> <td>jewel</td> <td>n.</td> <td>宝石</td> </tr> <tr> <td>jog</td> <td>v.</td> <td>慢跑</td> </tr> <tr> <td>join ⭐</td> <td>v.</td> <td>加入；连接</td> </tr> <tr> <td>journey</td> <td>n.</td> <td>旅程</td> </tr> <tr> <td>jungle</td> <td>n.</td> <td>丛林</td> </tr> <tr> <td>junior</td> <td>adj./n.</td> <td>初级的；晚辈</td> </tr> <tr> <td>junk</td> <td>n.</td> <td>废物</td> </tr> <tr> <td>justify</td> <td>v.</td> <td>为…辩护</td> </tr> <tr> <td>keyboard</td> <td>n.</td> <td>键盘</td> </tr> <tr> <td>kid</td> <td>n./v.</td> <td>小孩；开玩笑</td> </tr> <tr> <td>kill</td> <td>v.</td> <td>杀死</td> </tr> <tr> <td>kindergarten</td> <td>n.</td> <td>幼儿园</td> </tr> <tr> <td>kingdom</td> <td>n.</td> <td>王国</td> </tr> <tr> <td>kite</td> <td>n.</td> <td>风筝</td> </tr> <tr> <td>knee</td> <td>n.</td> <td>膝盖</td> </tr> <tr> <td>kneel</td> <td>v.</td> <td>跪下</td> </tr> <tr> <td>knock</td> <td>v.</td> <td>敲；碰撞</td> </tr> <tr> <td>knot</td> <td>n.</td> <td>结</td> </tr> <tr> <td>knowledgeable</td> <td>adj.</td> <td>知识渊博的</td> </tr> <tr> <td>label</td> <td>n./v.</td> <td>标签；标注</td> </tr> <tr> <td>labor</td> <td>n.</td> <td>劳动</td> </tr> <tr> <td>lack ⭐</td> <td>v./n.</td> <td>缺乏</td> </tr> <tr> <td>ladder</td> <td>n.</td> <td>梯子</td> </tr> <tr> <td>lag</td> <td>v.</td> <td>落后</td> </tr> </tbody> </table> <h2 id="day-32l">Day 32（L）<a class="anchor" href="#day-32l" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>lamb</td> <td>n.</td> <td>羔羊</td> </tr> <tr> <td>lamp</td> <td>n.</td> <td>台灯</td> </tr> <tr> <td>lane</td> <td>n.</td> <td>小巷；车道</td> </tr> <tr> <td>lap</td> <td>n.</td> <td>膝上；一圈</td> </tr> <tr> <td>largely</td> <td>adv.</td> <td>主要地</td> </tr> <tr> <td>laser</td> <td>n.</td> <td>激光</td> </tr> <tr> <td>latter</td> <td>adj.</td> <td>后者的</td> </tr> <tr> <td>lawyer</td> <td>n.</td> <td>律师</td> </tr> <tr> <td>lazy</td> <td>adj.</td> <td>懒的</td> </tr> <tr> <td>leaf</td> <td>n.</td> <td>叶子</td> </tr> <tr> <td>league</td> <td>n.</td> <td>联盟</td> </tr> <tr> <td>leak</td> <td>v./n.</td> <td>泄漏</td> </tr> <tr> <td>lean</td> <td>v.</td> <td>倚靠</td> </tr> <tr> <td>leap</td> <td>v./n.</td> <td>跳跃</td> </tr> <tr> <td>leather</td> <td>n.</td> <td>皮革</td> </tr> <tr> <td>lecture</td> <td>n.</td> <td>讲座</td> </tr> <tr> <td>legal</td> <td>adj.</td> <td>合法的</td> </tr> <tr> <td>legend</td> <td>n.</td> <td>传说</td> </tr> <tr> <td>leisure</td> <td>n.</td> <td>闲暇</td> </tr> <tr> <td>lend ⭐</td> <td>v.</td> <td>借出</td> </tr> <tr> <td>length</td> <td>n.</td> <td>长度</td> </tr> <tr> <td>lesson</td> <td>n.</td> <td>课；教训</td> </tr> <tr> <td>let ⭐</td> <td>v.</td> <td>让</td> </tr> <tr> <td>level</td> <td>n.</td> <td>水平</td> </tr> <tr> <td>liable</td> <td>adj.</td> <td>易于…的</td> </tr> </tbody> </table> <h2 id="day-33l">Day 33（L）<a class="anchor" href="#day-33l" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>liberty</td> <td>n.</td> <td>自由</td> </tr> <tr> <td>librarian</td> <td>n.</td> <td>图书管理员</td> </tr> <tr> <td>license</td> <td>n.</td> <td>执照</td> </tr> <tr> <td>lick</td> <td>v.</td> <td>舔</td> </tr> <tr> <td>lid</td> <td>n.</td> <td>盖子</td> </tr> <tr> <td>lighten</td> <td>v.</td> <td>减轻；照亮</td> </tr> <tr> <td>likely</td> <td>adj./adv.</td> <td>很可能的</td> </tr> <tr> <td>limit ⭐</td> <td>n./v.</td> <td>限制</td> </tr> <tr> <td>line</td> <td>n.</td> <td>线；排</td> </tr> <tr> <td>link</td> <td>v./n.</td> <td>链接</td> </tr> <tr> <td>lip</td> <td>n.</td> <td>嘴唇</td> </tr> <tr> <td>liquid</td> <td>n./adj.</td> <td>液体</td> </tr> <tr> <td>list</td> <td>n./v.</td> <td>列表</td> </tr> <tr> <td>literature</td> <td>n.</td> <td>文学</td> </tr> <tr> <td>lively</td> <td>adj.</td> <td>活泼的</td> </tr> <tr> <td>living</td> <td>adj./n.</td> <td>活着的；生计</td> </tr> <tr> <td>load</td> <td>n./v.</td> <td>装载</td> </tr> <tr> <td>loaf</td> <td>n.</td> <td>一条（面包）</td> </tr> <tr> <td>local</td> <td>adj.</td> <td>当地的</td> </tr> <tr> <td>locate</td> <td>v.</td> <td>位于</td> </tr> <tr> <td>lock</td> <td>v./n.</td> <td>锁</td> </tr> <tr> <td>lodge</td> <td>v.</td> <td>暂住</td> </tr> <tr> <td>log</td> <td>n./v.</td> <td>原木；记录</td> </tr> <tr> <td>logic</td> <td>n.</td> <td>逻辑</td> </tr> <tr> <td>lonely</td> <td>adj.</td> <td>孤独的</td> </tr> </tbody> </table> <h2 id="day-34lm">Day 34（L~M）<a class="anchor" href="#day-34lm" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>long ⭐</td> <td>adj./v.</td> <td>长的；渴望</td> </tr> <tr> <td>loose</td> <td>adj.</td> <td>松的</td> </tr> <tr> <td>lord</td> <td>n.</td> <td>主人；勋爵</td> </tr> <tr> <td>lorry</td> <td>n.</td> <td>卡车</td> </tr> <tr> <td>lose ⭐</td> <td>v.</td> <td>丢失</td> </tr> <tr> <td>loss</td> <td>n.</td> <td>损失</td> </tr> <tr> <td>lot</td> <td>n.</td> <td>许多；地块</td> </tr> <tr> <td>loud</td> <td>adj.</td> <td>大声的</td> </tr> <tr> <td>lounge</td> <td>n.</td> <td>休息室</td> </tr> <tr> <td>love ⭐</td> <td>v./n.</td> <td>爱</td> </tr> <tr> <td>loyal</td> <td>adj.</td> <td>忠诚的</td> </tr> <tr> <td>luck</td> <td>n.</td> <td>运气</td> </tr> <tr> <td>luggage</td> <td>n.</td> <td>行李</td> </tr> <tr> <td>lump</td> <td>n.</td> <td>团；肿块</td> </tr> <tr> <td>lunch</td> <td>n.</td> <td>午餐</td> </tr> <tr> <td>lung</td> <td>n.</td> <td>肺</td> </tr> <tr> <td>machine</td> <td>n.</td> <td>机器</td> </tr> <tr> <td>magic</td> <td>n./adj.</td> <td>魔法</td> </tr> <tr> <td>magnet</td> <td>n.</td> <td>磁铁</td> </tr> <tr> <td>magnificent</td> <td>adj.</td> <td>壮丽的</td> </tr> <tr> <td>maid</td> <td>n.</td> <td>女仆</td> </tr> <tr> <td>mail</td> <td>n./v.</td> <td>邮件</td> </tr> <tr> <td>main</td> <td>adj.</td> <td>主要的</td> </tr> <tr> <td>mainland</td> <td>n.</td> <td>大陆</td> </tr> <tr> <td>maintain ⭐</td> <td>v.</td> <td>维持</td> </tr> </tbody> </table> <h2 id="day-35m">Day 35（M）<a class="anchor" href="#day-35m" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>major</td> <td>adj./n.</td> <td>主要的；专业</td> </tr> <tr> <td>majority</td> <td>n.</td> <td>大多数</td> </tr> <tr> <td>make ⭐</td> <td>v.</td> <td>制作</td> </tr> <tr> <td>male</td> <td>adj./n.</td> <td>男性的</td> </tr> <tr> <td>manage ⭐</td> <td>v.</td> <td>管理；设法做到</td> </tr> <tr> <td>manner</td> <td>n.</td> <td>方式；礼貌</td> </tr> <tr> <td>manufacture</td> <td>v.</td> <td>制造</td> </tr> <tr> <td>many</td> <td>adj.</td> <td>许多</td> </tr> <tr> <td>map</td> <td>n.</td> <td>地图</td> </tr> <tr> <td>marble</td> <td>n.</td> <td>大理石</td> </tr> <tr> <td>march</td> <td>v./n.</td> <td>行军；三月</td> </tr> <tr> <td>margin</td> <td>n.</td> <td>边缘；余地</td> </tr> <tr> <td>mark</td> <td>n./v.</td> <td>标记</td> </tr> <tr> <td>market</td> <td>n.</td> <td>市场</td> </tr> <tr> <td>marriage</td> <td>n.</td> <td>婚姻</td> </tr> <tr> <td>marry</td> <td>v.</td> <td>结婚</td> </tr> <tr> <td>marvelous</td> <td>adj.</td> <td>了不起的</td> </tr> <tr> <td>mask</td> <td>n.</td> <td>面罩</td> </tr> <tr> <td>mass</td> <td>n.</td> <td>大量；群众</td> </tr> <tr> <td>master</td> <td>n./v.</td> <td>大师；掌握</td> </tr> <tr> <td>mat</td> <td>n.</td> <td>垫子</td> </tr> <tr> <td>match</td> <td>n./v.</td> <td>比赛；匹配</td> </tr> <tr> <td>material</td> <td>n.</td> <td>材料</td> </tr> <tr> <td>mathematics</td> <td>n.</td> <td>数学</td> </tr> <tr> <td>matter</td> <td>n./v.</td> <td>事情；要紧</td> </tr> </tbody> </table>https://one.snvedu.cn/bank/creative/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/creative/<blockquote class=''> <p>考场不止一种题面。这一页练三种「不常规」题型：<strong>找错题</strong>（给你错误解法，指出错哪一步）、<strong>变式链</strong>（同一考点连续变形）、<strong>应用创新题</strong>。做不了常规题的人做不了它们；做得懂它们的人，常规题闭眼做。</p> </blockquote>https://one.snvedu.cn/docs/english-basics/05-past-simple/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/05-past-simple/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>一般过去时讲什么（过去发生、已结束）</li> <li>规则动词 + ed 和它的拼写规则</li> <li>专升本必背的 20 个不规则动词</li> <li>否定 didn&rsquo;t 和疑问 Did</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-什么时候用一般过去时">1. 什么时候用一般过去时<a class="anchor" href="#1-%e4%bb%80%e4%b9%88%e6%97%b6%e5%80%99%e7%94%a8%e4%b8%80%e8%88%ac%e8%bf%87%e5%8e%bb%e6%97%b6" aria-label="Link"></a></h3> <p>事情发生在<strong>过去某个时间</strong>，现在已经结束：</p>https://one.snvedu.cn/exam/2023/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2023/<h1 id="2023-年浙江专升本真题">2023 年浙江专升本真题<a class="anchor" href="#2023-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/english/translation/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/translation/<blockquote class=''> <p>🎯 浙江题型定位：共 30 分——<strong>汉译英 5 题 15 分 + 英译汉 5 题 15 分</strong>，两个方向策略完全不同。</p> </blockquote><h2 id="一英译汉题型库完整解法版">一、英译汉：题型库（完整解法版）<a class="anchor" href="#%e4%b8%80%e8%8b%b1%e8%af%91%e6%b1%89%e9%a2%98%e5%9e%8b%e5%ba%93%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1定语从句最高频">题型 1：定语从句（最高频）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e5%ae%9a%e8%af%ad%e4%bb%8e%e5%8f%a5%e6%9c%80%e9%ab%98%e9%a2%91" aria-label="Link"></a></h3> <p>例：The book <strong>which I bought yesterday</strong> is very interesting. 解法：从句短 → <strong>前置</strong>成「……的」；从句长 → 独立成句。 译：我昨天买的那本书非常有趣。✅ 长从句版：He made a plan which helped us save a lot of time and money, which surprised everyone. → 「他制定了一个帮我们省下大量时间和金钱的计划，这让大家很惊讶。」（第二个 which 独立成句译成「这」）</p>https://one.snvedu.cn/docs/gaoshu/ordinary-differential-equations/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/ordinary-differential-equations/<blockquote class=''> <p>🎯 浙江考纲定位：独立板块之一，计算题稳定出 1 道，难度不大，<strong>必须拿满分的一章</strong>。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解微分方程的阶、解、通解、特解概念</li> <li>掌握可分离变量方程的解法</li> <li>掌握一阶线性微分方程的解法（公式法 / 常数变易法）</li> <li>掌握二阶常系数线性齐次方程的特征根法</li> <li>了解二阶常系数非齐次方程特解的设法</li> </ul> <h2 id="-核心公式库">📖 核心公式库<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e5%85%ac%e5%bc%8f%e5%ba%93" aria-label="Link"></a></h2> <h3 id="1-一阶线性-y--pxy--qx-通解">1. 一阶线性 y′ + P(x)y = Q(x) 通解<a class="anchor" href="#1-%e4%b8%80%e9%98%b6%e7%ba%bf%e6%80%a7-y--pxy--qx-%e9%80%9a%e8%a7%a3" aria-label="Link"></a></h3> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></msup><mrow><mo fence="true">(</mo><mo>∫</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext><msup><mi>e</mi><mrow><mo>∫</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></msup><mtext> </mtext><mi>d</mi><mi>x</mi><mo>+</mo><mi>C</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">y=e^{-\int P(x)dx}\left(\int Q(x)\,e^{\int P(x)dx}\,dx+C\right)</annotation></semantics></math></span><h3 id="2-二阶齐次特征根三分支r--pr--q--0">2. 二阶齐次特征根三分支（r² + pr + q = 0）<a class="anchor" href="#2-%e4%ba%8c%e9%98%b6%e9%bd%90%e6%ac%a1%e7%89%b9%e5%be%81%e6%a0%b9%e4%b8%89%e5%88%86%e6%94%afr--pr--q--0" aria-label="Link"></a></h3> <table> <thead> <tr> <th>特征根</th> <th>通解</th> </tr> </thead> <tbody> <tr> <td>两不等实根 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo mathvariant="normal">≠</mo><msub><mi>r</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">r_1\neq r_2</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><msup><mi>e</mi><mrow><msub><mi>r</mi><mn>1</mn></msub><mi>x</mi></mrow></msup><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><msup><mi>e</mi><mrow><msub><mi>r</mi><mn>2</mn></msub><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">y=C_1e^{r_1x}+C_2e^{r_2x}</annotation></semantics></math></span></td> </tr> <tr> <td>二重根 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mi>r</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">y=(C_1+C_2x)e^{rx}</annotation></semantics></math></span></td> </tr> <tr> <td>共轭复根 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>±</mo><mi>β</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">\alpha\pm\beta i</annotation></semantics></math></span></td> <td><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>e</mi><mrow><mi>α</mi><mi>x</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mi>cos</mi><mo>⁡</mo><mi>β</mi><mi>x</mi><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mi>sin</mi><mo>⁡</mo><mi>β</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)</annotation></semantics></math></span></td> </tr> </tbody> </table> <h3 id="-快速解法秒杀技巧">⚡ 快速解法（秒杀技巧）<a class="anchor" href="#-%e5%bf%ab%e9%80%9f%e8%a7%a3%e6%b3%95%e7%a7%92%e6%9d%80%e6%8a%80%e5%b7%a7" aria-label="Link"></a></h3> <ul> <li><strong>通解常数个数 = 阶数</strong></li> <li><strong>非齐次特解速设</strong>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mi>λ</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">P_m(x)e^{\lambda x}</annotation></semantics></math></span> → <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>k</mi></msup><msub><mi>Q</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mi>λ</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">x^kQ_m(x)e^{\lambda x}</annotation></semantics></math></span>，<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span> = λ 作为特征根的重数（非根 0 / 单根 1 / 重根 2）</li> </ul> <h2 id="-题型库历年考法全覆盖完整解法版">🧩 题型库：历年考法全覆盖（完整解法版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96%e5%ae%8c%e6%95%b4%e8%a7%a3%e6%b3%95%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1可分离变量计算题常客">题型 1：可分离变量（计算题常客）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e5%8f%af%e5%88%86%e7%a6%bb%e5%8f%98%e9%87%8f%e8%ae%a1%e7%ae%97%e9%a2%98%e5%b8%b8%e5%ae%a2" aria-label="Link"></a></h3> <p>例：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">\frac{dy}{dx}=xy</annotation></semantics></math></span>：</p>https://one.snvedu.cn/docs/english-words/days-41-50/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/days-41-50/<blockquote class=''> <p>用法同前。第二期注意：遇到第一期背过的词（画 △ 的），当作「重逢」重点巩固。</p> </blockquote><h2 id="day-41m">Day 41（M）<a class="anchor" href="#day-41m" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>mature</td> <td>adj./v.</td> <td>成熟的</td> </tr> <tr> <td>maximum</td> <td>n./adj.</td> <td>最大值</td> </tr> <tr> <td>meal</td> <td>n.</td> <td>一餐</td> </tr> <tr> <td>meaning</td> <td>n.</td> <td>意义</td> </tr> <tr> <td>meanwhile</td> <td>adv.</td> <td>同时</td> </tr> <tr> <td>measure</td> <td>v./n.</td> <td>测量；措施</td> </tr> <tr> <td>media</td> <td>n.</td> <td>媒体</td> </tr> <tr> <td>medium</td> <td>adj./n.</td> <td>中等的；媒介</td> </tr> <tr> <td>melt</td> <td>v.</td> <td>融化</td> </tr> <tr> <td>member</td> <td>n.</td> <td>成员</td> </tr> <tr> <td>memorial</td> <td>n./adj.</td> <td>纪念碑；纪念的</td> </tr> <tr> <td>memory</td> <td>n.</td> <td>记忆</td> </tr> <tr> <td>mend</td> <td>v.</td> <td>修补</td> </tr> <tr> <td>mental</td> <td>adj.</td> <td>精神的</td> </tr> <tr> <td>mention</td> <td>v.</td> <td>提到</td> </tr> <tr> <td>menu</td> <td>n.</td> <td>菜单</td> </tr> <tr> <td>merchant</td> <td>n.</td> <td>商人</td> </tr> <tr> <td>mercy</td> <td>n.</td> <td>仁慈</td> </tr> <tr> <td>mere</td> <td>adj.</td> <td>仅仅的</td> </tr> <tr> <td>merry</td> <td>adj.</td> <td>欢乐的</td> </tr> <tr> <td>mess</td> <td>n.</td> <td>混乱</td> </tr> <tr> <td>message</td> <td>n.</td> <td>消息</td> </tr> <tr> <td>metal</td> <td>n.</td> <td>金属</td> </tr> <tr> <td>method</td> <td>n.</td> <td>方法</td> </tr> <tr> <td>midnight</td> <td>n.</td> <td>午夜</td> </tr> </tbody> </table> <h2 id="day-42m">Day 42（M）<a class="anchor" href="#day-42m" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>mild</td> <td>adj.</td> <td>温和的</td> </tr> <tr> <td>military</td> <td>adj.</td> <td>军事的</td> </tr> <tr> <td>mill</td> <td>n.</td> <td>磨坊；工厂</td> </tr> <tr> <td>million</td> <td>num.</td> <td>百万</td> </tr> <tr> <td>mineral</td> <td>n.</td> <td>矿物</td> </tr> <tr> <td>minimum</td> <td>n./adj.</td> <td>最小值</td> </tr> <tr> <td>minister</td> <td>n.</td> <td>部长</td> </tr> <tr> <td>minor</td> <td>adj.</td> <td>次要的</td> </tr> <tr> <td>minority</td> <td>n.</td> <td>少数派</td> </tr> <tr> <td>minute</td> <td>n.</td> <td>分钟；片刻</td> </tr> <tr> <td>miracle</td> <td>n.</td> <td>奇迹</td> </tr> <tr> <td>mirror</td> <td>n.</td> <td>镜子</td> </tr> <tr> <td>miserable</td> <td>adj.</td> <td>痛苦的</td> </tr> <tr> <td>miss ⭐</td> <td>v.</td> <td>错过；想念</td> </tr> <tr> <td>missile</td> <td>n.</td> <td>导弹</td> </tr> <tr> <td>mist</td> <td>n.</td> <td>薄雾</td> </tr> <tr> <td>mistake</td> <td>n./v.</td> <td>错误</td> </tr> <tr> <td>mix ⭐</td> <td>v.</td> <td>混合</td> </tr> <tr> <td>mixture</td> <td>n.</td> <td>混合物</td> </tr> <tr> <td>mobile</td> <td>adj.</td> <td>移动的</td> </tr> <tr> <td>model</td> <td>n.</td> <td>模型；模特</td> </tr> <tr> <td>moderate</td> <td>adj.</td> <td>适度的</td> </tr> <tr> <td>modern</td> <td>adj.</td> <td>现代的</td> </tr> <tr> <td>modest</td> <td>adj.</td> <td>谦虚的</td> </tr> <tr> <td>moment</td> <td>n.</td> <td>时刻</td> </tr> </tbody> </table> <h2 id="day-43mn">Day 43（M~N）<a class="anchor" href="#day-43mn" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>monitor</td> <td>n./v.</td> <td>监控</td> </tr> <tr> <td>monkey</td> <td>n.</td> <td>猴子</td> </tr> <tr> <td>month</td> <td>n.</td> <td>月</td> </tr> <tr> <td>mood</td> <td>n.</td> <td>心情</td> </tr> <tr> <td>moral</td> <td>adj./n.</td> <td>道德的</td> </tr> <tr> <td>more ⭐</td> <td>adj./adv.</td> <td>更多</td> </tr> <tr> <td>moreover</td> <td>adv.</td> <td>而且</td> </tr> <tr> <td>mosquito</td> <td>n.</td> <td>蚊子</td> </tr> <tr> <td>mostly</td> <td>adv.</td> <td>主要地</td> </tr> <tr> <td>mother</td> <td>n.</td> <td>母亲</td> </tr> <tr> <td>motion</td> <td>n.</td> <td>运动；提议</td> </tr> <tr> <td>motivate</td> <td>v.</td> <td>激励</td> </tr> <tr> <td>mount</td> <td>v./n.</td> <td>登上；山</td> </tr> <tr> <td>mountain</td> <td>n.</td> <td>山</td> </tr> <tr> <td>mouse</td> <td>n.</td> <td>老鼠</td> </tr> <tr> <td>movement</td> <td>n.</td> <td>运动</td> </tr> <tr> <td>nail</td> <td>n.</td> <td>钉子；指甲</td> </tr> <tr> <td>naked</td> <td>adj.</td> <td>赤裸的</td> </tr> <tr> <td>name</td> <td>n./v.</td> <td>名字；命名</td> </tr> <tr> <td>narrow</td> <td>adj.</td> <td>狭窄的</td> </tr> <tr> <td>nation</td> <td>n.</td> <td>国家；民族</td> </tr> <tr> <td>nationality</td> <td>n.</td> <td>国籍</td> </tr> <tr> <td>native</td> <td>adj.</td> <td>本土的</td> </tr> <tr> <td>natural</td> <td>adj.</td> <td>自然的</td> </tr> <tr> <td>nature</td> <td>n.</td> <td>自然</td> </tr> </tbody> </table> <h2 id="day-44n">Day 44（N）<a class="anchor" href="#day-44n" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>naval</td> <td>adj.</td> <td>海军的</td> </tr> <tr> <td>navigate</td> <td>v.</td> <td>导航</td> </tr> <tr> <td>near</td> <td>prep./adj.</td> <td>靠近</td> </tr> <tr> <td>neat</td> <td>adj.</td> <td>整洁的</td> </tr> <tr> <td>necessarily</td> <td>adv.</td> <td>必然地</td> </tr> <tr> <td>necessary</td> <td>adj.</td> <td>必要的</td> </tr> <tr> <td>necessity</td> <td>n.</td> <td>必需品</td> </tr> <tr> <td>neck</td> <td>n.</td> <td>脖子</td> </tr> <tr> <td>negative</td> <td>adj.</td> <td>消极的</td> </tr> <tr> <td>neglect</td> <td>v.</td> <td>忽视</td> </tr> <tr> <td>negotiate</td> <td>v.</td> <td>谈判</td> </tr> <tr> <td>neighbor</td> <td>n.</td> <td>邻居</td> </tr> <tr> <td>neighborhood</td> <td>n.</td> <td>街区</td> </tr> <tr> <td>neither</td> <td>conj./adj.</td> <td>两者都不</td> </tr> <tr> <td>nephew</td> <td>n.</td> <td>侄子</td> </tr> <tr> <td>nerve</td> <td>n.</td> <td>神经</td> </tr> <tr> <td>nervous</td> <td>adj.</td> <td>紧张的</td> </tr> <tr> <td>nest</td> <td>n.</td> <td>巢</td> </tr> <tr> <td>net</td> <td>n.</td> <td>网</td> </tr> <tr> <td>network</td> <td>n.</td> <td>网络</td> </tr> <tr> <td>neutral</td> <td>adj.</td> <td>中立的</td> </tr> <tr> <td>never</td> <td>adv.</td> <td>从不</td> </tr> <tr> <td>nevertheless</td> <td>adv.</td> <td>然而</td> </tr> <tr> <td>niece</td> <td>n.</td> <td>侄女</td> </tr> <tr> <td>normal</td> <td>adj.</td> <td>正常的</td> </tr> </tbody> </table> <h2 id="day-45no">Day 45（N~O）<a class="anchor" href="#day-45no" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>note</td> <td>n./v.</td> <td>笔记；注意到</td> </tr> <tr> <td>novel</td> <td>n./adj.</td> <td>小说；新颖的</td> </tr> <tr> <td>now</td> <td>adv.</td> <td>现在</td> </tr> <tr> <td>nowhere</td> <td>adv.</td> <td>无处</td> </tr> <tr> <td>nuclear</td> <td>adj.</td> <td>核的</td> </tr> <tr> <td>number</td> <td>n.</td> <td>数字</td> </tr> <tr> <td>numerous</td> <td>adj.</td> <td>众多的</td> </tr> <tr> <td>nurse</td> <td>n.</td> <td>护士</td> </tr> <tr> <td>object</td> <td>n./v.</td> <td>物体；反对</td> </tr> <tr> <td>objection</td> <td>n.</td> <td>反对</td> </tr> <tr> <td>objective</td> <td>adj./n.</td> <td>客观的；目标</td> </tr> <tr> <td>oblige</td> <td>v.</td> <td>迫使</td> </tr> <tr> <td>observe</td> <td>v.</td> <td>观察</td> </tr> <tr> <td>obstacle</td> <td>n.</td> <td>障碍</td> </tr> <tr> <td>obtain</td> <td>v.</td> <td>获得</td> </tr> <tr> <td>obvious</td> <td>adj.</td> <td>明显的</td> </tr> <tr> <td>occasion</td> <td>n.</td> <td>场合</td> </tr> <tr> <td>occupy</td> <td>v.</td> <td>占据</td> </tr> <tr> <td>occur</td> <td>v.</td> <td>发生</td> </tr> <tr> <td>ocean</td> <td>n.</td> <td>海洋</td> </tr> <tr> <td>odd</td> <td>adj.</td> <td>奇怪的</td> </tr> <tr> <td>offend</td> <td>v.</td> <td>冒犯</td> </tr> <tr> <td>offer ⭐</td> <td>v./n.</td> <td>提供</td> </tr> <tr> <td>office</td> <td>n.</td> <td>办公室</td> </tr> <tr> <td>official</td> <td>adj./n.</td> <td>官方的</td> </tr> </tbody> </table>https://one.snvedu.cn/docs/english-basics/06-future/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/06-future/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>will 的用法（临时决定 / 预测 / 承诺）</li> <li>be going to 的用法（计划打算）</li> <li>否定 won&rsquo;t 和疑问 Will&hellip;?</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-will人人通用零变形">1. will：人人通用，零变形<a class="anchor" href="#1-will%e4%ba%ba%e4%ba%ba%e9%80%9a%e7%94%a8%e9%9b%b6%e5%8f%98%e5%bd%a2" aria-label="Link"></a></h3> <p><strong>will + 动词原形</strong>——好消息：will 不分人称，所有人都是它。</p>https://one.snvedu.cn/bank/mock-1/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/mock-1/<blockquote class=''> <p><strong>满分 150 分，限时 150 分钟</strong>。建议打印式作答（草稿本+手写），全程不翻资料。做完再打开卷末答案自评。题目标注分值，按步骤给分。</p> </blockquote><h2 id="一选择题5--4--20-分">一、选择题（5 × 4 = 20 分）<a class="anchor" href="#%e4%b8%80%e9%80%89%e6%8b%a9%e9%a2%985--4--20-%e5%88%86" aria-label="Link"></a></h2> <p><strong>M1.</strong> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>sin</mi><mo>⁡</mo><mn>3</mn><mi>x</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\dfrac{e^{2x}-1}{\sin 3x}</annotation></semantics></math></span><link rel="stylesheet" href="https://one.snvedu.cn/katex/katex.min.css" /> = ? A. 2/3　B. 3/2　C. 1　D. 0</p>https://one.snvedu.cn/exam/2024/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2024/<h1 id="2024-年浙江专升本真题">2024 年浙江专升本真题<a class="anchor" href="#2024-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/gaoshu/infinite-series/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/infinite-series/<blockquote class=''> <p>🎯 浙江考纲定位：四大板块之一（占比小于一元微积分，多为选择/填空题量级）。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解数项级数收敛、发散及收敛和的概念</li> <li>掌握正项级数的比较判别法、比值判别法</li> <li>掌握交错级数的莱布尼茨判别法</li> <li>理解绝对收敛与条件收敛</li> <li>会求幂级数的收敛半径与收敛区间</li> </ul> <h2 id="-核心考点">📖 核心考点<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e8%80%83%e7%82%b9" aria-label="Link"></a></h2> <h3 id="1-必背参照级数">1. 必背参照级数<a class="anchor" href="#1-%e5%bf%85%e8%83%8c%e5%8f%82%e7%85%a7%e7%ba%a7%e6%95%b0" aria-label="Link"></a></h3> <ul> <li><strong>等比级数</strong> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mi>a</mi><msup><mi>q</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\sum aq^n</annotation></semantics></math></span><link rel="stylesheet" href="https://one.snvedu.cn/katex/katex.min.css" />：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>q</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|q|&lt;1</annotation></semantics></math></span> 收敛（和 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo><mfrac><mi>a</mi><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">=\frac{a}{1-q}</annotation></semantics></math></span>），<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>q</mi><mi mathvariant="normal">∣</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|q|\ge1</annotation></semantics></math></span> 发散</li> <li><strong>p 级数</strong> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mfrac><mn>1</mn><msup><mi>n</mi><mi>p</mi></msup></mfrac></mrow><annotation encoding="application/x-tex">\sum\frac{1}{n^p}</annotation></semantics></math></span>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p&gt;1</annotation></semantics></math></span> 收敛，<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p\le1</annotation></semantics></math></span> 发散（调和级数 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p=1</annotation></semantics></math></span> 发散）</li> </ul> <h3 id="2-判别法选择流程">2. 判别法选择流程<a class="anchor" href="#2-%e5%88%a4%e5%88%ab%e6%b3%95%e9%80%89%e6%8b%a9%e6%b5%81%e7%a8%8b" aria-label="Link"></a></h3> <ol> <li><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u_n \to 0</annotation></semantics></math></span>？不趋零<strong>必发散</strong>（先秒杀）</li> <li>正项级数：含 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">n!</annotation></semantics></math></span> / <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>n</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">n^n</annotation></semantics></math></span> / 指数式 → <strong>比值判别法</strong>；与 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><msup><mi>n</mi><mi>p</mi></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{n^p}</annotation></semantics></math></span> 同形 → <strong>比较判别法</strong></li> <li>交错级数 → <strong>莱布尼茨</strong>（<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span> 单调减 + 趋零）</li> </ol> <h3 id="-快速解法秒杀技巧">⚡ 快速解法（秒杀技巧）<a class="anchor" href="#-%e5%bf%ab%e9%80%9f%e8%a7%a3%e6%b3%95%e7%a7%92%e6%9d%80%e6%8a%80%e5%b7%a7" aria-label="Link"></a></h3> <ul> <li>含 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">n!</annotation></semantics></math></span> → 直接比值判别法（阶乘连乘消干净）</li> <li>收敛半径：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>=</mo><mi>lim</mi><mo>⁡</mo><mrow><mo fence="true">∣</mo><mfrac><msub><mi>a</mi><mi>n</mi></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac><mo fence="true">∣</mo></mrow></mrow><annotation encoding="application/x-tex">R=\lim\left|\frac{a_n}{a_{n+1}}\right|</annotation></semantics></math></span>（注意方向，别写反）</li> </ul> <h2 id="-题型库历年考法全覆盖">🧩 题型库：历年考法全覆盖<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e5%8e%86%e5%b9%b4%e8%80%83%e6%b3%95%e5%85%a8%e8%a6%86%e7%9b%96" aria-label="Link"></a></h2> <h3 id="题型-1通项极限秒杀发散选择常客">题型 1：通项极限秒杀发散（选择常客）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e9%80%9a%e9%a1%b9%e6%9e%81%e9%99%90%e7%a7%92%e6%9d%80%e5%8f%91%e6%95%a3%e9%80%89%e6%8b%a9%e5%b8%b8%e5%ae%a2" aria-label="Link"></a></h3> <p>例：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\sum\frac{n}{n+1}</annotation></semantics></math></span>：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>→</mo><mn>1</mn><mo mathvariant="normal">≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u_n\to1\neq0</annotation></semantics></math></span> → <strong>发散</strong> ✅</p>https://one.snvedu.cn/docs/english/writing-templates/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english/writing-templates/<blockquote class=''> <p>🎯 浙江题型定位：写作 30 分。<strong>性价比最高</strong>的模块：两大文体模板背熟 + 考前每周写一篇并修改。</p> </blockquote><h2 id="-题型库两大文体完整示范版">🧩 题型库：两大文体（完整示范版）<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e4%b8%a4%e5%a4%a7%e6%96%87%e4%bd%93%e5%ae%8c%e6%95%b4%e7%a4%ba%e8%8c%83%e7%89%88" aria-label="Link"></a></h2> <h3 id="题型-1议论文观点--论述--总结">题型 1：议论文（观点 + 论述 + 总结）<a class="anchor" href="#%e9%a2%98%e5%9e%8b-1%e8%ae%ae%e8%ae%ba%e6%96%87%e8%a7%82%e7%82%b9--%e8%ae%ba%e8%bf%b0--%e6%80%bb%e7%bb%93" aria-label="Link"></a></h3> <p><strong>模板骨架：</strong></p>https://one.snvedu.cn/docs/english-words/days-51-60/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-words/days-51-60/<blockquote class=''> <p>最后十天的安排：Day 51<del>57 覆盖 O</del>Z 字母段，Day 58<del>60 是「高频补漏」——按词性集中补第一期 A</del>I 段漏掉的高频词。</p> </blockquote><h2 id="day-51op">Day 51（O~P）<a class="anchor" href="#day-51op" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>operate</td> <td>v.</td> <td>操作；手术</td> </tr> <tr> <td>opinion</td> <td>n.</td> <td>观点</td> </tr> <tr> <td>opponent</td> <td>n.</td> <td>对手</td> </tr> <tr> <td>opportunity</td> <td>n.</td> <td>机会</td> </tr> <tr> <td>oppose</td> <td>v.</td> <td>反对</td> </tr> <tr> <td>opposite</td> <td>adj.</td> <td>相反的</td> </tr> <tr> <td>optimistic</td> <td>adj.</td> <td>乐观的</td> </tr> <tr> <td>option</td> <td>n.</td> <td>选项</td> </tr> <tr> <td>order</td> <td>n./v.</td> <td>顺序；命令</td> </tr> <tr> <td>ordinary</td> <td>adj.</td> <td>普通的</td> </tr> <tr> <td>organ</td> <td>n.</td> <td>器官</td> </tr> <tr> <td>organize</td> <td>v.</td> <td>组织</td> </tr> <tr> <td>origin</td> <td>n.</td> <td>起源</td> </tr> <tr> <td>original</td> <td>adj.</td> <td>原创的</td> </tr> <tr> <td>otherwise</td> <td>adv.</td> <td>否则</td> </tr> <tr> <td>outcome</td> <td>n.</td> <td>结果</td> </tr> <tr> <td>outdoor</td> <td>adj.</td> <td>户外的</td> </tr> <tr> <td>outline</td> <td>n.</td> <td>大纲</td> </tr> <tr> <td>output</td> <td>n.</td> <td>产量</td> </tr> <tr> <td>overcome</td> <td>v.</td> <td>克服</td> </tr> <tr> <td>overhead</td> <td>adv.</td> <td>在头顶上</td> </tr> <tr> <td>overlook</td> <td>v.</td> <td>忽略；俯瞰</td> </tr> <tr> <td>own</td> <td>adj./v.</td> <td>自己的；拥有</td> </tr> <tr> <td>owner</td> <td>n.</td> <td>主人</td> </tr> <tr> <td>pack</td> <td>v./n.</td> <td>打包</td> </tr> </tbody> </table> <h2 id="day-52p">Day 52（P）<a class="anchor" href="#day-52p" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>packet</td> <td>n.</td> <td>小包</td> </tr> <tr> <td>painful</td> <td>adj.</td> <td>痛苦的</td> </tr> <tr> <td>paint</td> <td>v./n.</td> <td>油漆</td> </tr> <tr> <td>painter</td> <td>n.</td> <td>画家</td> </tr> <tr> <td>pale</td> <td>adj.</td> <td>苍白的</td> </tr> <tr> <td>palm</td> <td>n.</td> <td>手掌</td> </tr> <tr> <td>paragraph</td> <td>n.</td> <td>段落</td> </tr> <tr> <td>parallel</td> <td>adj.</td> <td>平行的</td> </tr> <tr> <td>parcel</td> <td>n.</td> <td>包裹</td> </tr> <tr> <td>pardon</td> <td>v./n.</td> <td>原谅</td> </tr> <tr> <td>parent</td> <td>n.</td> <td>父母</td> </tr> <tr> <td>park</td> <td>n./v.</td> <td>公园；停车</td> </tr> <tr> <td>parliament</td> <td>n.</td> <td>议会</td> </tr> <tr> <td>part</td> <td>n.</td> <td>部分</td> </tr> <tr> <td>participate</td> <td>v.</td> <td>参加</td> </tr> <tr> <td>particular</td> <td>adj.</td> <td>特别的</td> </tr> <tr> <td>partner</td> <td>n.</td> <td>伙伴</td> </tr> <tr> <td>party</td> <td>n.</td> <td>聚会；政党</td> </tr> <tr> <td>passage</td> <td>n.</td> <td>段落；通道</td> </tr> <tr> <td>passenger</td> <td>n.</td> <td>乘客</td> </tr> <tr> <td>passion</td> <td>n.</td> <td>激情</td> </tr> <tr> <td>passive</td> <td>adj.</td> <td>被动的</td> </tr> <tr> <td>passport</td> <td>n.</td> <td>护照</td> </tr> <tr> <td>past</td> <td>n./prep.</td> <td>过去</td> </tr> <tr> <td>paste</td> <td>v.</td> <td>粘贴</td> </tr> </tbody> </table> <h2 id="day-53p">Day 53（P）<a class="anchor" href="#day-53p" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>path</td> <td>n.</td> <td>小路</td> </tr> <tr> <td>patience</td> <td>n.</td> <td>耐心</td> </tr> <tr> <td>patient</td> <td>n./adj.</td> <td>病人；耐心的</td> </tr> <tr> <td>pattern</td> <td>n.</td> <td>模式</td> </tr> <tr> <td>pause</td> <td>v./n.</td> <td>暂停</td> </tr> <tr> <td>pay</td> <td>v.</td> <td>支付</td> </tr> <tr> <td>payment</td> <td>n.</td> <td>付款</td> </tr> <tr> <td>peace</td> <td>n.</td> <td>和平</td> </tr> <tr> <td>peaceful</td> <td>adj.</td> <td>和平的</td> </tr> <tr> <td>peak</td> <td>n.</td> <td>顶峰</td> </tr> <tr> <td>pearl</td> <td>n.</td> <td>珍珠</td> </tr> <tr> <td>peculiar</td> <td>adj.</td> <td>奇怪的</td> </tr> <tr> <td>pen</td> <td>n.</td> <td>钢笔</td> </tr> <tr> <td>percentage</td> <td>n.</td> <td>百分比</td> </tr> <tr> <td>perfect</td> <td>adj.</td> <td>完美的</td> </tr> <tr> <td>perform</td> <td>v.</td> <td>表演；执行</td> </tr> <tr> <td>performance</td> <td>n.</td> <td>表现</td> </tr> <tr> <td>perhaps</td> <td>adv.</td> <td>也许</td> </tr> <tr> <td>period</td> <td>n.</td> <td>时期</td> </tr> <tr> <td>permanent</td> <td>adj.</td> <td>永久的</td> </tr> <tr> <td>permit</td> <td>v./n.</td> <td>允许；许可证</td> </tr> <tr> <td>persist</td> <td>v.</td> <td>坚持</td> </tr> <tr> <td>person</td> <td>n.</td> <td>人</td> </tr> <tr> <td>personal</td> <td>adj.</td> <td>个人的</td> </tr> <tr> <td>perspective</td> <td>n.</td> <td>视角</td> </tr> </tbody> </table> <h2 id="day-54pq">Day 54（P~Q）<a class="anchor" href="#day-54pq" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>persuade</td> <td>v.</td> <td>说服</td> </tr> <tr> <td>pessimistic</td> <td>adj.</td> <td>悲观的</td> </tr> <tr> <td>photograph</td> <td>n.</td> <td>照片</td> </tr> <tr> <td>phrase</td> <td>n.</td> <td>短语</td> </tr> <tr> <td>physical</td> <td>adj.</td> <td>身体的；物理的</td> </tr> <tr> <td>physician</td> <td>n.</td> <td>内科医生</td> </tr> <tr> <td>piano</td> <td>n.</td> <td>钢琴</td> </tr> <tr> <td>picture</td> <td>n.</td> <td>图片</td> </tr> <tr> <td>pie</td> <td>n.</td> <td>馅饼</td> </tr> <tr> <td>piece</td> <td>n.</td> <td>片</td> </tr> <tr> <td>pile</td> <td>n./v.</td> <td>堆</td> </tr> <tr> <td>pilgrim</td> <td>n.</td> <td>朝圣者</td> </tr> <tr> <td>pilot</td> <td>n.</td> <td>飞行员</td> </tr> <tr> <td>pin</td> <td>n./v.</td> <td>别针</td> </tr> <tr> <td>pinch</td> <td>v.</td> <td>捏</td> </tr> <tr> <td>pink</td> <td>adj.</td> <td>粉色的</td> </tr> <tr> <td>pioneer</td> <td>n.</td> <td>先锋</td> </tr> <tr> <td>pipe</td> <td>n.</td> <td>管子</td> </tr> <tr> <td>pity</td> <td>n.</td> <td>遗憾</td> </tr> <tr> <td>plain</td> <td>adj./n.</td> <td>简单的；平原</td> </tr> <tr> <td>plan</td> <td>n./v.</td> <td>计划</td> </tr> <tr> <td>plane</td> <td>n.</td> <td>飞机</td> </tr> <tr> <td>planet</td> <td>n.</td> <td>行星</td> </tr> <tr> <td>plant</td> <td>n./v.</td> <td>植物；种植</td> </tr> <tr> <td>plastic</td> <td>n./adj.</td> <td>塑料</td> </tr> </tbody> </table> <h2 id="day-55qr">Day 55（Q~R）<a class="anchor" href="#day-55qr" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>platform</td> <td>n.</td> <td>站台；平台</td> </tr> <tr> <td>pleasant</td> <td>adj.</td> <td>令人愉快的</td> </tr> <tr> <td>plot</td> <td>n.</td> <td>情节；阴谋</td> </tr> <tr> <td>plug</td> <td>n./v.</td> <td>插头</td> </tr> <tr> <td>pocket</td> <td>n.</td> <td>口袋</td> </tr> <tr> <td>poem</td> <td>n.</td> <td>诗</td> </tr> <tr> <td>poet</td> <td>n.</td> <td>诗人</td> </tr> <tr> <td>poetry</td> <td>n.</td> <td>诗歌（总称）</td> </tr> <tr> <td>point</td> <td>n./v.</td> <td>点；指出</td> </tr> <tr> <td>poison</td> <td>n./v.</td> <td>毒</td> </tr> <tr> <td>policy</td> <td>n.</td> <td>政策</td> </tr> <tr> <td>polish</td> <td>v.</td> <td>擦亮</td> </tr> <tr> <td>polite</td> <td>adj.</td> <td>礼貌的</td> </tr> <tr> <td>political</td> <td>adj.</td> <td>政治的</td> </tr> <tr> <td>pond</td> <td>n.</td> <td>池塘</td> </tr> <tr> <td>pool</td> <td>n.</td> <td>水池</td> </tr> <tr> <td>popular</td> <td>adj.</td> <td>流行的</td> </tr> <tr> <td>population</td> <td>n.</td> <td>人口</td> </tr> <tr> <td>port</td> <td>n.</td> <td>港口</td> </tr> <tr> <td>portion</td> <td>n.</td> <td>一部分</td> </tr> <tr> <td>portrait</td> <td>n.</td> <td>肖像</td> </tr> <tr> <td>position</td> <td>n.</td> <td>位置</td> </tr> <tr> <td>positive</td> <td>adj.</td> <td>积极的</td> </tr> <tr> <td>possess</td> <td>v.</td> <td>拥有</td> </tr> <tr> <td>possibility</td> <td>n.</td> <td>可能性</td> </tr> </tbody> </table> <h2 id="day-56r">Day 56（R）<a class="anchor" href="#day-56r" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>post</td> <td>n./v.</td> <td>邮寄；帖子</td> </tr> <tr> <td>postpone</td> <td>v.</td> <td>推迟</td> </tr> <tr> <td>potential</td> <td>adj./n.</td> <td>潜在的</td> </tr> <tr> <td>pour</td> <td>v.</td> <td>倾倒</td> </tr> <tr> <td>poverty</td> <td>n.</td> <td>贫穷</td> </tr> <tr> <td>powder</td> <td>n.</td> <td>粉末</td> </tr> <tr> <td>power</td> <td>n.</td> <td>力量；权力</td> </tr> <tr> <td>practical</td> <td>adj.</td> <td>实用的</td> </tr> <tr> <td>practice</td> <td>n./v.</td> <td>练习</td> </tr> <tr> <td>praise</td> <td>v./n.</td> <td>表扬</td> </tr> <tr> <td>pray</td> <td>v.</td> <td>祈祷</td> </tr> <tr> <td>precious</td> <td>adj.</td> <td>珍贵的</td> </tr> <tr> <td>precise</td> <td>adj.</td> <td>精确的</td> </tr> <tr> <td>predict</td> <td>v.</td> <td>预测</td> </tr> <tr> <td>prefer</td> <td>v.</td> <td>更喜欢</td> </tr> <tr> <td>pregnant</td> <td>adj.</td> <td>怀孕的</td> </tr> <tr> <td>prejudice</td> <td>n.</td> <td>偏见</td> </tr> <tr> <td>prepare</td> <td>v.</td> <td>准备</td> </tr> <tr> <td>presence</td> <td>n.</td> <td>在场</td> </tr> <tr> <td>present</td> <td>adj./n./v.</td> <td>现在的；礼物；呈现</td> </tr> <tr> <td>preserve</td> <td>v.</td> <td>保存</td> </tr> <tr> <td>president</td> <td>n.</td> <td>总统；校长</td> </tr> <tr> <td>press</td> <td>v./n.</td> <td>按；新闻界</td> </tr> <tr> <td>pressure</td> <td>n.</td> <td>压力</td> </tr> <tr> <td>pretend</td> <td>v.</td> <td>假装</td> </tr> </tbody> </table> <h2 id="day-57r">Day 57（R）<a class="anchor" href="#day-57r" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>pretty</td> <td>adv./adj.</td> <td>相当；漂亮的</td> </tr> <tr> <td>prevent</td> <td>v.</td> <td>防止</td> </tr> <tr> <td>previous</td> <td>adj.</td> <td>以前的</td> </tr> <tr> <td>price</td> <td>n.</td> <td>价格</td> </tr> <tr> <td>pride</td> <td>n.</td> <td>自豪</td> </tr> <tr> <td>primary</td> <td>adj.</td> <td>主要的；初级的</td> </tr> <tr> <td>prime</td> <td>adj.</td> <td>首要的</td> </tr> <tr> <td>primitive</td> <td>adj.</td> <td>原始的</td> </tr> <tr> <td>prince</td> <td>n.</td> <td>王子</td> </tr> <tr> <td>princess</td> <td>n.</td> <td>公主</td> </tr> <tr> <td>principle</td> <td>n.</td> <td>原则</td> </tr> <tr> <td>print</td> <td>v./n.</td> <td>打印</td> </tr> <tr> <td>prior</td> <td>adj.</td> <td>优先的</td> </tr> <tr> <td>priority</td> <td>n.</td> <td>优先事项</td> </tr> <tr> <td>prison</td> <td>n.</td> <td>监狱</td> </tr> <tr> <td>private</td> <td>adj.</td> <td>私人的</td> </tr> <tr> <td>prize</td> <td>n.</td> <td>奖品</td> </tr> <tr> <td>probable</td> <td>adj.</td> <td>很可能的</td> </tr> <tr> <td>problem</td> <td>n.</td> <td>问题</td> </tr> <tr> <td>procedure</td> <td>n.</td> <td>程序</td> </tr> <tr> <td>proceed</td> <td>v.</td> <td>继续进行</td> </tr> <tr> <td>process</td> <td>n./v.</td> <td>过程</td> </tr> <tr> <td>produce</td> <td>v.</td> <td>生产</td> </tr> <tr> <td>product</td> <td>n.</td> <td>产品</td> </tr> <tr> <td>profession</td> <td>n.</td> <td>职业</td> </tr> </tbody> </table> <h2 id="day-58rs--高频补漏动词">Day 58（R~S + 高频补漏：动词）<a class="anchor" href="#day-58rs--%e9%ab%98%e9%a2%91%e8%a1%a5%e6%bc%8f%e5%8a%a8%e8%af%8d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>profit</td> <td>n./v.</td> <td>利润</td> </tr> <tr> <td>program</td> <td>n.</td> <td>节目；程序</td> </tr> <tr> <td>progress</td> <td>n./v.</td> <td>进步</td> </tr> <tr> <td>prohibit</td> <td>v.</td> <td>禁止</td> </tr> <tr> <td>project</td> <td>n.</td> <td>项目</td> </tr> <tr> <td>promise</td> <td>n./v.</td> <td>承诺</td> </tr> <tr> <td>promote</td> <td>v.</td> <td>促进；晋升</td> </tr> <tr> <td>proof</td> <td>n.</td> <td>证据</td> </tr> <tr> <td>proper</td> <td>adj.</td> <td>恰当的</td> </tr> <tr> <td>property</td> <td>n.</td> <td>财产</td> </tr> <tr> <td>proposal</td> <td>n.</td> <td>提议</td> </tr> <tr> <td>protect</td> <td>v.</td> <td>保护</td> </tr> <tr> <td>protest</td> <td>v./n.</td> <td>抗议</td> </tr> <tr> <td>proud</td> <td>adj.</td> <td>骄傲的</td> </tr> <tr> <td>prove</td> <td>v.</td> <td>证明</td> </tr> <tr> <td>provide</td> <td>v.</td> <td>提供</td> </tr> <tr> <td>publish</td> <td>v.</td> <td>出版</td> </tr> <tr> <td>pull</td> <td>v.</td> <td>拉</td> </tr> <tr> <td>punish</td> <td>v.</td> <td>惩罚</td> </tr> <tr> <td>purchase</td> <td>v./n.</td> <td>购买</td> </tr> <tr> <td>pure</td> <td>adj.</td> <td>纯的</td> </tr> <tr> <td>purpose</td> <td>n.</td> <td>目的</td> </tr> <tr> <td>pursue</td> <td>v.</td> <td>追求</td> </tr> <tr> <td>push</td> <td>v.</td> <td>推</td> </tr> <tr> <td>puzzle</td> <td>n./v.</td> <td>谜；使困惑</td> </tr> </tbody> </table> <h2 id="day-59s--高频补漏形容词">Day 59（S + 高频补漏：形容词）<a class="anchor" href="#day-59s--%e9%ab%98%e9%a2%91%e8%a1%a5%e6%bc%8f%e5%bd%a2%e5%ae%b9%e8%af%8d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>qualify</td> <td>v.</td> <td>使合格</td> </tr> <tr> <td>quality</td> <td>n.</td> <td>质量</td> </tr> <tr> <td>quantity</td> <td>n.</td> <td>数量</td> </tr> <tr> <td>quarrel</td> <td>v./n.</td> <td>争吵</td> </tr> <tr> <td>quarter</td> <td>n.</td> <td>四分之一</td> </tr> <tr> <td>queue</td> <td>n.</td> <td>排队</td> </tr> <tr> <td>quit</td> <td>v.</td> <td>辞职；戒掉</td> </tr> <tr> <td>quiz</td> <td>n.</td> <td>小测验</td> </tr> <tr> <td>quote</td> <td>v./n.</td> <td>引用</td> </tr> <tr> <td>rabbit</td> <td>n.</td> <td>兔子</td> </tr> <tr> <td>racial</td> <td>adj.</td> <td>种族的</td> </tr> <tr> <td>radiation</td> <td>n.</td> <td>辐射</td> </tr> <tr> <td>rainbow</td> <td>n.</td> <td>彩虹</td> </tr> <tr> <td>raise</td> <td>v.</td> <td>举起；筹集</td> </tr> <tr> <td>random</td> <td>adj.</td> <td>随机的</td> </tr> <tr> <td>range</td> <td>n./v.</td> <td>范围</td> </tr> <tr> <td>rank</td> <td>n./v.</td> <td>排名</td> </tr> <tr> <td>rapid</td> <td>adj.</td> <td>快速的</td> </tr> <tr> <td>rare</td> <td>adj.</td> <td>稀有的</td> </tr> <tr> <td>rat</td> <td>n.</td> <td>老鼠</td> </tr> <tr> <td>rate</td> <td>n.</td> <td>比率</td> </tr> <tr> <td>rather</td> <td>adv.</td> <td>相当；宁愿</td> </tr> <tr> <td>raw</td> <td>adj.</td> <td>生的</td> </tr> <tr> <td>ray</td> <td>n.</td> <td>光线</td> </tr> <tr> <td>react</td> <td>v.</td> <td>反应</td> </tr> </tbody> </table> <h2 id="day-60sz--高频补漏连接词与副词">Day 60（S~Z + 高频补漏：连接词与副词）<a class="anchor" href="#day-60sz--%e9%ab%98%e9%a2%91%e8%a1%a5%e6%bc%8f%e8%bf%9e%e6%8e%a5%e8%af%8d%e4%b8%8e%e5%89%af%e8%af%8d" aria-label="Link"></a></h2> <table> <thead> <tr> <th>词</th> <th>词性</th> <th>中文</th> </tr> </thead> <tbody> <tr> <td>reasonable</td> <td>adj.</td> <td>合理的</td> </tr> <tr> <td>rebel</td> <td>v./n.</td> <td>反叛</td> </tr> <tr> <td>recall</td> <td>v.</td> <td>回想起</td> </tr> <tr> <td>receipt</td> <td>n.</td> <td>收据</td> </tr> <tr> <td>receive</td> <td>v.</td> <td>收到</td> </tr> <tr> <td>recent</td> <td>adj.</td> <td>近来的</td> </tr> <tr> <td>recognize</td> <td>v.</td> <td>认出</td> </tr> <tr> <td>recommend</td> <td>v.</td> <td>推荐</td> </tr> <tr> <td>record</td> <td>n./v.</td> <td>记录</td> </tr> <tr> <td>recover</td> <td>v.</td> <td>恢复</td> </tr> <tr> <td>recruit</td> <td>v.</td> <td>招募</td> </tr> <tr> <td>reduce</td> <td>v.</td> <td>减少</td> </tr> <tr> <td>refer</td> <td>v.</td> <td>提及；参考</td> </tr> <tr> <td>reflect</td> <td>v.</td> <td>反射；反思</td> </tr> <tr> <td>reform</td> <td>v./n.</td> <td>改革</td> </tr> <tr> <td>refresh</td> <td>v.</td> <td>使清新</td> </tr> <tr> <td>refuse</td> <td>v.</td> <td>拒绝</td> </tr> <tr> <td>regard</td> <td>v./n.</td> <td>看待；问候</td> </tr> <tr> <td>region</td> <td>n.</td> <td>地区</td> </tr> <tr> <td>register</td> <td>v.</td> <td>注册</td> </tr> <tr> <td>regret</td> <td>v./n.</td> <td>后悔</td> </tr> <tr> <td>reject</td> <td>v.</td> <td>拒绝</td> </tr> <tr> <td>relate</td> <td>v.</td> <td>有关</td> </tr> <tr> <td>relax</td> <td>v.</td> <td>放松</td> </tr> <tr> <td>release</td> <td>v./n.</td> <td>释放</td> </tr> </tbody> </table>https://one.snvedu.cn/docs/english-basics/07-progressive/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/07-progressive/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>现在进行时 = be + doing（正在做某事）</li> <li>doing 的三条拼写规则</li> <li>感叹句 Look! / Listen! 的提示作用</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-结构be--doing">1. 结构：be + doing<a class="anchor" href="#1-%e7%bb%93%e6%9e%84be--doing" aria-label="Link"></a></h3> <p><strong>谁在干什么（此刻）</strong>，动词前面加一个 be 动词、动词本身变 doing：</p>https://one.snvedu.cn/exam/2025/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/exam/2025/<h1 id="2025-年浙江专升本真题">2025 年浙江专升本真题<a class="anchor" href="#2025-%e5%b9%b4%e6%b5%99%e6%b1%9f%e4%b8%93%e5%8d%87%e6%9c%ac%e7%9c%9f%e9%a2%98" aria-label="Link"></a></h1> <blockquote class=''> <p>⏱ 建议完成时间：150 分钟（高数）/ 150 分钟（英语）。状态：☐ 未做 / ☐ 已归档</p> </blockquote><h2 id="高等数学">高等数学<a class="anchor" href="#%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6" aria-label="Link"></a></h2> <h3 id="高数-1考点____--对应题型-__--深坑-__">高数 1（考点：____ ｜ 对应：题型 __ / 深坑 __）<a class="anchor" href="#%e9%ab%98%e6%95%b0-1%e8%80%83%e7%82%b9____--%e5%af%b9%e5%ba%94%e9%a2%98%e5%9e%8b-__--%e6%b7%b1%e5%9d%91-__" aria-label="Link"></a></h3> <p><strong>我的解法</strong>：</p>https://one.snvedu.cn/docs/gaoshu/space-analytic-geometry/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/space-analytic-geometry/<blockquote class=''> <p>⚠️ <strong>浙江考纲定位提示</strong>：浙江专升本高数四大板块为「一元微积分、多元微积分、无穷级数、常微分方程」，本章<strong>未单列</strong>——请对照你当年的最新考纲确认，不在考纲内可直接跳过，把时间投给一元微积分。</p> </blockquote>https://one.snvedu.cn/docs/english-basics/08-perfect/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/08-perfect/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>现在完成时的结构：have / has + 过去分词</li> <li>它和一般过去时的本质区别（专升本最爱考）</li> <li>since 和 for 的区别</li> <li>标志词全家桶</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-结构have--has--过去分词">1. 结构：have / has + 过去分词<a class="anchor" href="#1-%e7%bb%93%e6%9e%84have--has--%e8%bf%87%e5%8e%bb%e5%88%86%e8%af%8d" aria-label="Link"></a></h3> <table> <thead> <tr> <th>人</th> <th>结构</th> <th>例句</th> </tr> </thead> <tbody> <tr> <td>I / you / we / they</td> <td><strong>have</strong> + 分词</td> <td>They <strong>have finished</strong> the work.（他们已经完成工作了。）</td> </tr> <tr> <td>he / she / it</td> <td><strong>has</strong> + 分词</td> <td>She <strong>has lost</strong> her keys.（她把钥匙弄丢了。）</td> </tr> </tbody> </table> <p><strong>过去分词</strong>哪来的？两种来源：</p>https://one.snvedu.cn/docs/gaoshu/multivariable-calculus/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/multivariable-calculus/<blockquote class=''> <p>🎯 浙江考纲定位：四大板块之一，题目类型固定、套路性强，<strong>学会套路就能拿分</strong>。</p> </blockquote><h2 id="-考纲要求">🎯 考纲要求<a class="anchor" href="#-%e8%80%83%e7%ba%b2%e8%a6%81%e6%b1%82" aria-label="Link"></a></h2> <ul> <li>理解二元函数的偏导数概念，会求一阶偏导数</li> <li>会求简单二元函数的全微分</li> <li>掌握多元复合函数求导法则（链式法则）</li> <li>理解二元函数极值的必要条件</li> <li>会计算简单的二重积分（直角坐标）</li> </ul> <h2 id="-核心考点">📖 核心考点<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e8%80%83%e7%82%b9" aria-label="Link"></a></h2> <h3 id="1-偏导数把别的变量当常数">1. 偏导数：把别的变量当常数<a class="anchor" href="#1-%e5%81%8f%e5%af%bc%e6%95%b0%e6%8a%8a%e5%88%ab%e7%9a%84%e5%8f%98%e9%87%8f%e5%bd%93%e5%b8%b8%e6%95%b0" aria-label="Link"></a></h3> <p><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x,y)=x^2y</annotation></semantics></math></span> → <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>x</mi></msub><mo>=</mo><mn>2</mn><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">f_x=2xy</annotation></semantics></math></span>、<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>y</mi></msub><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f_y=x^2</annotation></semantics></math></span></p>https://one.snvedu.cn/docs/english-basics/09-review-tenses/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/09-review-tenses/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>把前六课装进<strong>一张表</strong>，然后用 12 道混练题检验——这 12 题的水平，就是你现在做专升本语法题的水平。</p> <h2 id="-六大时态总表保存这一页考前就背它">📖 六大时态总表（保存这一页，考前就背它）<a class="anchor" href="#-%e5%85%ad%e5%a4%a7%e6%97%b6%e6%80%81%e6%80%bb%e8%a1%a8%e4%bf%9d%e5%ad%98%e8%bf%99%e4%b8%80%e9%a1%b5%e8%80%83%e5%89%8d%e5%b0%b1%e8%83%8c%e5%ae%83" aria-label="Link"></a></h2> <table> <thead> <tr> <th>时态</th> <th>结构</th> <th>标志词</th> <th>例句</th> </tr> </thead> <tbody> <tr> <td>一般现在时</td> <td>动词原形（三单 + s）</td> <td>always / usually / every day</td> <td>He <strong>goes</strong> to school every day.</td> </tr> <tr> <td>一般过去时</td> <td>动词过去式</td> <td>yesterday / last / &hellip;ago / in 2020</td> <td>He <strong>went</strong> to school yesterday.</td> </tr> <tr> <td>一般将来时</td> <td>will + 原形 / be going to</td> <td>tomorrow / next / soon / in 2030</td> <td>He <strong>will go</strong> to school tomorrow.</td> </tr> <tr> <td>现在进行时</td> <td>be + doing</td> <td>now / Look! / Listen!</td> <td>He <strong>is going</strong> to school now.</td> </tr> <tr> <td>现在完成时</td> <td>have / has + 分词</td> <td>since / for / already / just / yet / ever</td> <td>He <strong>has gone</strong> to school already.</td> </tr> <tr> <td>一般过去(be)</td> <td>was / were</td> <td>同一般过去时</td> <td>He <strong>was</strong> at school yesterday.</td> </tr> </tbody> </table> <h3 id="做题流程四步每题都走一遍">做题流程（四步，每题都走一遍）<a class="anchor" href="#%e5%81%9a%e9%a2%98%e6%b5%81%e7%a8%8b%e5%9b%9b%e6%ad%a5%e6%af%8f%e9%a2%98%e9%83%bd%e8%b5%b0%e4%b8%80%e9%81%8d" aria-label="Link"></a></h3> <ol> <li><strong>找标志词</strong>（now？yesterday？since？tomorrow？）</li> <li>没有标志词 → 看<strong>上下文时间</strong>（前一句是昨天的事？）</li> <li>定时态 → 按<strong>结构表</strong>写动词</li> <li>最后检查：三单？分词形式对不对？</li> </ol> <h2 id="-混练-12-题先盖住答案">📝 混练 12 题（先盖住答案）<a class="anchor" href="#-%e6%b7%b7%e7%bb%83-12-%e9%a2%98%e5%85%88%e7%9b%96%e4%bd%8f%e7%ad%94%e6%a1%88" aria-label="Link"></a></h2> <ol> <li>Look! The baby ____ (sleep).</li> <li>My father ____ (read) newspapers every morning.</li> <li>They ____ (visit) the Great Wall last summer.</li> <li>I ____ (not / see) him since last month.</li> <li>____ you ____ (finish) your work yet?</li> <li>She ____ (be) a nurse for ten years.</li> <li>Tomorrow we ____ (have) an English test.</li> <li>Listen! Someone ____ (knock) at the door.</li> <li>He ____ (buy) a new phone two days ago.</li> <li>改错：She have studied English for three years.</li> <li>改错：Did you saw the movie last night?</li> <li>改错：Look! It rains outside.</li> </ol> <details> <summary>答案与解析（对照四步流程检查）</summary> <ol> <li><strong>is sleeping</strong>（Look! → 进行时；baby 三单 → is）</li> <li><strong>reads</strong>（every day 类标志 → 一般现在；father 三单 → +s）</li> <li><strong>visited</strong>（last summer → 过去；规则动词 + ed）</li> <li><strong>haven&rsquo;t seen</strong>（since → 完成时否定；see 的分词是 seen）</li> <li><strong>Have</strong> you <strong>finished</strong>&hellip;?（yet → 完成时疑问；have 提前）</li> <li><strong>has been</strong>（for + 时间段 → 完成时；she → has；be 的分词 been）</li> <li><strong>will have / are going to have</strong>（tomorrow → 将来）</li> <li><strong>is knocking</strong>（Listen! → 进行时；someone 三单 → is）</li> <li><strong>bought</strong>（two days ago → 过去；buy 不规则 → bought）</li> <li>have ❌ → <strong>has</strong> studied（she 三单用 has）</li> <li>saw ❌ → <strong>see</strong>（Did 后面用原形）</li> <li>rains ❌ → <strong>is raining</strong>（Look! → 进行时）</li> </ol> <p><strong>错 3 题以内</strong>：合格，继续往下学。<strong>错 4 题以上</strong>：把第 4、5、8 课重读一遍再做混练，别硬闯。</p> </details>https://one.snvedu.cn/docs/gaoshu/comprehensive-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/comprehensive-proofs/<blockquote class=''> <p>🎯 浙江考纲定位：综合题 3 × 10 = <strong>30 分</strong>，是拉开分差的地方。题型高度固定，把下面几个套路练熟就是稳定得分。</p> </blockquote><h2 id="-题型库综合题四大套路">🧩 题型库：综合题四大套路<a class="anchor" href="#-%e9%a2%98%e5%9e%8b%e5%ba%93%e7%bb%bc%e5%90%88%e9%a2%98%e5%9b%9b%e5%a4%a7%e5%a5%97%e8%b7%af" aria-label="Link"></a></h2> <h3 id="套路-1方程根的存在性与唯一性零点定理--单调性">套路 1：方程根的存在性与唯一性（零点定理 + 单调性）<a class="anchor" href="#%e5%a5%97%e8%b7%af-1%e6%96%b9%e7%a8%8b%e6%a0%b9%e7%9a%84%e5%ad%98%e5%9c%a8%e6%80%a7%e4%b8%8e%e5%94%af%e4%b8%80%e6%80%a7%e9%9b%b6%e7%82%b9%e5%ae%9a%e7%90%86--%e5%8d%95%e8%b0%83%e6%80%a7" aria-label="Link"></a></h3> <p><strong>标准答题模板（背下来）：</strong></p>https://one.snvedu.cn/docs/english-basics/10-passive/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/10-passive/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>什么时候用被动（主语是被动作打到的一方）</li> <li>万能公式 <strong>be + 过去分词</strong>，六种时态的被动全推导</li> <li>情态动词的被动</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-什么时候用被动">1. 什么时候用被动？<a class="anchor" href="#1-%e4%bb%80%e4%b9%88%e6%97%b6%e5%80%99%e7%94%a8%e8%a2%ab%e5%8a%a8" aria-label="Link"></a></h3> <p>主动句：<strong>人们</strong> speak Chinese.（人们说中文——主语是动作执行者） 被动句：Chinese <strong>is spoken</strong> by people.（中文<strong>被</strong>人们说——主语是承受者）</p>https://one.snvedu.cn/bank/en-choice-1/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/en-choice-1/<blockquote class=''> <p>限时每题 45 秒（考场英语语法题均速）。先选再看解析。考点对应<a href="https://one.snvedu.cn/docs/english/vocabulary-grammar/">词汇与语法模块</a>。</p> </blockquote><h2 id="1-by-the-time-he-arrived-the-meeting-____-for-half-an-hour">1. By the time he arrived, the meeting ____ for half an hour.<a class="anchor" href="#1-by-the-time-he-arrived-the-meeting-____-for-half-an-hour" aria-label="Link"></a></h2> <p>A. has begun　B. had been on　C. had begun　D. began</p> <details><summary>答案：B</summary> begin 是瞬间动词，持续一段时间用 **be on**；by the time + 过去 → 过去完成：had been on </details> <h2 id="2-look-out-a-car-____-toward-you">2. Look out! A car ____ toward you.<a class="anchor" href="#2-look-out-a-car-____-toward-you" aria-label="Link"></a></h2> <p>A. comes　B. is coming　C. has come　D. came</p>https://one.snvedu.cn/docs/gaoshu/traps-compendium/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/traps-compendium/<blockquote class=''> <p>🎯 浙江考生<strong>考前必刷页</strong>：全考纲 60+ 个坑，按板块排列。每条都是「❌ 错做法 → ✅ 正确姿势 + 一句防错」。</p> </blockquote><h2 id="一极限与连续14-坑">一、极限与连续（14 坑）<a class="anchor" href="#%e4%b8%80%e6%9e%81%e9%99%90%e4%b8%8e%e8%bf%9e%e7%bb%ad14-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ 加减中乱用等价无穷小：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>lim</mi><mo>⁡</mo><mfrac><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\lim\frac{\tan x-\sin x}{x^3}</annotation></semantics></math></span> 当 0</td> <td>✅ 先提取公因式变乘除：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\tan x(1-\cos x)\sim\frac{x^3}{2}</annotation></semantics></math></span></td> </tr> <tr> <td>2</td> <td>❌ 两个重要极限混用</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \to 0</annotation></semantics></math></span> 配 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(1+x)^{1/x}</annotation></semantics></math></span>；<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">x \to \infty</annotation></semantics></math></span> 配 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><msup><mo stretchy="false">)</mo><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">(1+\frac1x)^x</annotation></semantics></math></span></td> </tr> <tr> <td>3</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">e^{1/x}</annotation></semantics></math></span>、<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>arctan</mi><mo>⁡</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">\arctan\frac1x</annotation></semantics></math></span> 不分左右极限</td> <td>✅ 见到这两个信号词，<strong>必分 x→0⁺ / 0⁻</strong></td> </tr> <tr> <td>4</td> <td>❌ 含 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math></span> 直接去绝对值</td> <td>✅ 绝对值 = 分段函数，分侧算</td> </tr> <tr> <td>5</td> <td>❌ 洛必达前不验型</td> <td>✅ 必须是 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac00</annotation></semantics></math></span> 或 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi mathvariant="normal">∞</mi><mi mathvariant="normal">∞</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac\infty\infty</annotation></semantics></math></span>，用一次验一次</td> </tr> <tr> <td>6</td> <td>❌ 洛必达后极限不存在 → 断言原极限不存在</td> <td>✅ 洛必达失效（振荡型）≠ 原极限不存在，换等价/夹逼/泰勒</td> </tr> <tr> <td>7</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">x \to -\infty</annotation></semantics></math></span> 时 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sqrt{x^2}=x</annotation></semantics></math></span></td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sqrt{x^2}=|x|=-x</annotation></semantics></math></span>（x→−∞ 时）</td> </tr> <tr> <td>8</td> <td>❌ 抓大头用在 x→0</td> <td>✅ 抓大头只适用 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">x \to \infty</annotation></semantics></math></span>；x→0 时抓<strong>最低次</strong>（小头）</td> </tr> <tr> <td>9</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>1</mn><mi mathvariant="normal">∞</mi></msup></mrow><annotation encoding="application/x-tex">1^\infty</annotation></semantics></math></span> 直接写 1</td> <td>✅ 用 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>lim</mi><mo>⁡</mo><mi>v</mi><mo stretchy="false">(</mo><mi>u</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">e^{\lim v(u-1)}</annotation></semantics></math></span>，结果是 e 的多少次方</td> </tr> <tr> <td>10</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>⋅</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">0\cdot\infty</annotation></semantics></math></span> 型直接写 0</td> <td>✅ 未定式！翻一个因子到分母再处理</td> </tr> <tr> <td>11</td> <td>❌ 「可导必连续」记成「连续必可导」</td> <td>✅ 单向 implication：可导 ⇒ 连续，反例 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">y=|x|</annotation></semantics></math></span></td> </tr> <tr> <td>12</td> <td>❌ 无界 = 无穷大</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mi>n</mi></mrow><annotation encoding="application/x-tex">(-1)^n n</annotation></semantics></math></span> 无界但非无穷大（不单调趋∞）</td> </tr> <tr> <td>13</td> <td>❌ 分段函数分界点只算一侧极限</td> <td>✅ 左 = 右 = 函数值，三者缺一不可</td> </tr> <tr> <td>14</td> <td>❌ 含参数连续只令左右极限相等</td> <td>✅ 还要令 <strong>极限值 = 该点函数值</strong>（参数常有两个方程）</td> </tr> </tbody> </table> <h2 id="二导数与微分10-坑">二、导数与微分（10 坑）<a class="anchor" href="#%e4%ba%8c%e5%af%bc%e6%95%b0%e4%b8%8e%e5%be%ae%e5%88%8610-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ 分段函数分界点直接代公式求导</td> <td>✅ 必须<strong>用定义算左右导数</strong></td> </tr> <tr> <td>2</td> <td>❌ 隐函数中 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">y^2</annotation></semantics></math></span> 求导得 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">2y</annotation></semantics></math></span></td> <td>✅ y 是 x 的函数：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>y</mi><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">2yy&#x27;</annotation></semantics></math></span>；<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>y</mi></msup></mrow><annotation encoding="application/x-tex">e^y</annotation></semantics></math></span> → <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mi>y</mi></msup><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">e^y y&#x27;</annotation></semantics></math></span></td> </tr> <tr> <td>3</td> <td>❌ 参数方程二阶导 = <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><msup><mi>x</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y&#x27;&#x27;(t)/x&#x27;&#x27;(t)</annotation></semantics></math></span></td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mtext> ⁣</mtext><mrow><mo fence="true">(</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo fence="true">)</mo></mrow><mi mathvariant="normal">/</mi><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{d}{dt}\!\left(\frac{dy}{dx}\right)/x&#x27;(t)</annotation></semantics></math></span></td> </tr> <tr> <td>4</td> <td>❌ 复合函数漏内层导</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">e^{-x^2}</annotation></semantics></math></span> 的导数带 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">-2x</annotation></semantics></math></span>；每层都要剥</td> </tr> <tr> <td>5</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f&#x27;(x_0)</annotation></semantics></math></span> 与 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">[f(x_0)]&#x27;</annotation></semantics></math></span> 混淆</td> <td>✅ 前者是导数值；后者是常数求导 = 0</td> </tr> <tr> <td>6</td> <td>❌ 切线只想到斜率存在</td> <td>✅ 导数无穷大时切线是竖直线 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x=x_0</annotation></semantics></math></span></td> </tr> <tr> <td>7</td> <td>❌ 微分只写 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">y&#x27;</annotation></semantics></math></span></td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>y</mi><mo>=</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">dy=y&#x27;dx</annotation></semantics></math></span>，dx 不能丢</td> </tr> <tr> <td>8</td> <td>❌ 法线斜率忘负倒数</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mfrac><mn>1</mn><mi>k</mi></mfrac></mrow><annotation encoding="application/x-tex">-\frac1k</annotation></semantics></math></span></td> </tr> <tr> <td>9</td> <td>❌ 见式就硬求导</td> <td>✅ 先化简：对数先拆、根式先提幂、乘除先分离</td> </tr> <tr> <td>10</td> <td>❌ 对数求导法最后漏乘 y</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mi>y</mi></mfrac><mo>=</mo><mi>A</mi><mo>⇒</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mi>y</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\frac{y&#x27;}{y}=A \Rightarrow y&#x27;=yA</annotation></semantics></math></span>，还原这步别丢</td> </tr> </tbody> </table> <h2 id="三中值定理与导数应用9-坑">三、中值定理与导数应用（9 坑）<a class="anchor" href="#%e4%b8%89%e4%b8%ad%e5%80%bc%e5%ae%9a%e7%90%86%e4%b8%8e%e5%af%bc%e6%95%b0%e5%ba%94%e7%94%a89-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ 驻点 = 极值点</td> <td>✅ 还要<strong>变号</strong>（<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">x^3</annotation></semantics></math></span> 在 0 驻而不极）</td> </tr> <tr> <td>2</td> <td>❌ 极值只在驻点里找</td> <td>✅ <strong>不可导点</strong>也可能是极值点（<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math></span>）</td> </tr> <tr> <td>3</td> <td>❌ 用中值定理不验条件</td> <td>✅ 「闭区间连续、开区间可导」逐条写出再引用</td> </tr> <tr> <td>4</td> <td>❌ 认为拉格朗日的 ξ 唯一</td> <td>✅ 定理只保证<strong>存在</strong>，不保证唯一</td> </tr> <tr> <td>5</td> <td>❌ 渐近线只找一种</td> <td>✅ 三种按顺序查：垂直（分母零点）→ 水平 → 斜</td> </tr> <tr> <td>6</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f&#x27;&#x27;(x_0)=0</annotation></semantics></math></span> 就是拐点</td> <td>✅ 两侧 <strong>凹凸性要变号</strong>（<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">x^4</annotation></semantics></math></span> 在 0 不是拐点）</td> </tr> <tr> <td>7</td> <td>❌ 最值只比较驻点</td> <td>✅ 驻点 + 不可导点 + <strong>端点</strong>全代入</td> </tr> <tr> <td>8</td> <td>❌ 应用题不写定义域</td> <td>✅ 几何量非负、边长有限，定义域是采分点</td> </tr> <tr> <td>9</td> <td>❌ f&rsquo; 图像的零点全当极值点</td> <td>✅ 零点且<strong>变号</strong>才是极值点</td> </tr> </tbody> </table> <h2 id="四积分12-坑">四、积分（12 坑）<a class="anchor" href="#%e5%9b%9b%e7%a7%af%e5%88%8612-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ 不定积分漏 +C</td> <td>✅ 每题必写，养成肌肉记忆</td> </tr> <tr> <td>2</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mi>sin</mi><mo>⁡</mo><mn>2</mn><mi>x</mi><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">\int\sin 2x\,dx=-\cos 2x</annotation></semantics></math></span></td> <td>✅ 内层导数补倒数：<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mfrac><mrow><mi>cos</mi><mo>⁡</mo><mn>2</mn><mi>x</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">-\frac{\cos 2x}{2}</annotation></semantics></math></span></td> </tr> <tr> <td>3</td> <td>❌ 分部 u 选错越积越复杂</td> <td>✅ 「反对幂三指」选 u；复杂了立刻换向</td> </tr> <tr> <td>4</td> <td>❌ 循环积分一直积下去</td> <td>✅ 分部两次后<strong>移项解方程</strong></td> </tr> <tr> <td>5</td> <td>❌ 第二类换元不回代</td> <td>✅ 最后必须换回 x</td> </tr> <tr> <td>6</td> <td>❌ 定积分换元不换限</td> <td>✅ <strong>新变量新限</strong>，或最后用原变量原限</td> </tr> <tr> <td>7</td> <td>❌ 对称区间不看奇偶硬算</td> <td>✅ 先查区间对称性：奇函数 = 0，偶函数折半</td> </tr> <tr> <td>8</td> <td>❌ 绝对值积分不拆区间</td> <td>✅ 变号点拆开分别去绝对值</td> </tr> <tr> <td>9</td> <td>❌ 变上限被积函数含 x 直接套公式</td> <td>✅ 先把 x 移出积分或拆项，再用乘积法则</td> </tr> <tr> <td>10</td> <td>❌ 面积直接 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\int f</annotation></semantics></math></span></td> <td>✅ 「上减下」；x 轴下方区间用 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>f</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|f|</annotation></semantics></math></span></td> </tr> <tr> <td>11</td> <td>❌ 圆环体积 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mo>∫</mo><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><mi>r</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\pi\int(R-r)^2</annotation></semantics></math></span></td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi><mo>∫</mo><mo stretchy="false">(</mo><msup><mi>R</mi><mn>2</mn></msup><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi\int(R^2-r^2)</annotation></semantics></math></span>（平方后相减）</td> </tr> <tr> <td>12</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>=</mo><mi>ln</mi><mo>⁡</mo><mi mathvariant="normal">∣</mi><msup><mi>x</mi><mn>2</mn></msup><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\int\frac{dx}{x^2}=\ln|x^2|</annotation></semantics></math></span></td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">-\frac1x+C</annotation></semantics></math></span>（基本公式表要过关）</td> </tr> </tbody> </table> <h2 id="五微分方程6-坑">五、微分方程（6 坑）<a class="anchor" href="#%e4%ba%94%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b6-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ 特征方程判别式算错</td> <td>✅ 先慢后快，b² − 4ac 用笔算</td> </tr> <tr> <td>2</td> <td>❌ 复根通解漏 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>α</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">e^{\alpha x}</annotation></semantics></math></span></td> <td>✅ 完整形态 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>α</mi><mi>x</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>C</mi><mn>1</mn></msub><mi>cos</mi><mo>⁡</mo><mi>β</mi><mi>x</mi><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mi>sin</mi><mo>⁡</mo><mi>β</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)</annotation></semantics></math></span></td> </tr> <tr> <td>3</td> <td>❌ 一阶线性公式指数正负写反</td> <td>✅ 外乘负指数、内乘正指数；代 P=0 验证退化</td> </tr> <tr> <td>4</td> <td>❌ λ 是特征根仍设 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><msup><mi>e</mi><mrow><mi>λ</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">Ae^{\lambda x}</annotation></semantics></math></span></td> <td>✅ 乘 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">x^k</annotation></semantics></math></span>：单根乘 x，重根乘 x²</td> </tr> <tr> <td>5</td> <td>❌ 通解常数个数 ≠ 阶数</td> <td>✅ n 阶方程 n 个独立常数</td> </tr> <tr> <td>6</td> <td>❌ 初始条件直接往特解假设里代</td> <td>✅ 流程：齐次通解 + 特解 → 合并通解 → 再代条件</td> </tr> </tbody> </table> <h2 id="六无穷级数6-坑">六、无穷级数（6 坑）<a class="anchor" href="#%e5%85%ad%e6%97%a0%e7%a9%b7%e7%ba%a7%e6%95%b06-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u_n \to 0</annotation></semantics></math></span> 就判收敛</td> <td>✅ 必要不充分（调和级数反例）</td> </tr> <tr> <td>2</td> <td>❌ 收敛区间不验端点</td> <td>✅ 端点逐个代入判敛，区间可能闭</td> </tr> <tr> <td>3</td> <td>❌ 绝对/条件判断顺序颠倒</td> <td>✅ 先判 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><mi mathvariant="normal">∣</mi><msub><mi>u</mi><mi>n</mi></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">\sum|u_n|</annotation></semantics></math></span>，再判原级数</td> </tr> <tr> <td>4</td> <td>❌ 比较判别法放缩方向反</td> <td>✅ 证收敛找「更大的收敛级数」，证发散找「更小的发散级数」</td> </tr> <tr> <td>5</td> <td>❌ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>=</mo><mi>lim</mi><mo>⁡</mo><mfrac><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>a</mi><mi>n</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">R=\lim\frac{a_{n+1}}{a_n}</annotation></semantics></math></span> 当 R</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>ρ</mi></mrow><annotation encoding="application/x-tex">R=1/\rho</annotation></semantics></math></span> 或 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo>=</mo><mi>lim</mi><mo>⁡</mo><mfrac><msub><mi>a</mi><mi>n</mi></msub><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mrow><annotation encoding="application/x-tex">R=\lim\frac{a_n}{a_{n+1}}</annotation></semantics></math></span>（方向别反）</td> </tr> <tr> <td>6</td> <td>❌ 比值判别 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\rho=1</annotation></semantics></math></span> 仍下结论</td> <td>✅ ρ=1 时<strong>失效</strong>，换比较/积分判别</td> </tr> </tbody> </table> <h2 id="七多元微积分6-坑">七、多元微积分（6 坑）<a class="anchor" href="#%e4%b8%83%e5%a4%9a%e5%85%83%e5%be%ae%e7%a7%af%e5%88%866-%e5%9d%91" aria-label="Link"></a></h2> <table> <thead> <tr> <th>#</th> <th>坑</th> <th>防错</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>❌ 偏导时忘记其他变量当常数</td> <td>✅ 对 x 求导时 y² 是系数</td> </tr> <tr> <td>2</td> <td>❌ 全微分漏 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">dx</annotation></semantics></math></span> 或 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">dy</annotation></semantics></math></span></td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>z</mi><mo>=</mo><msub><mi>f</mi><mi>x</mi></msub><mi>d</mi><mi>x</mi><mo>+</mo><msub><mi>f</mi><mi>y</mi></msub><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">dz=f_xdx+f_ydy</annotation></semantics></math></span> 两项都要带尾巴</td> </tr> <tr> <td>3</td> <td>❌ 极坐标漏乘 r</td> <td>✅ <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>σ</mi><mo>=</mo><mi>r</mi><mtext> </mtext><mi>d</mi><mi>r</mi><mtext> </mtext><mi>d</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">d\sigma=r\,dr\,d\theta</annotation></semantics></math></span></td> </tr> <tr> <td>4</td> <td>❌ 交换次序后内层限写成常数</td> <td>✅ 内层限是<strong>外层变量的函数</strong>才对</td> </tr> <tr> <td>5</td> <td>❌ 极值判别 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>C</mi><mo>−</mo><msup><mi>B</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">AC-B^2</annotation></semantics></math></span> 用错</td> <td>✅ &gt;0 且 A&lt;0 极大、&gt;0 且 A&gt;0 极小、&lt;0 非极值、=0 失效</td> </tr> <tr> <td>6</td> <td>❌ 二重积分上下限想当然</td> <td>✅ <strong>先画区域</strong>再定限，90% 的错在限</td> </tr> </tbody> </table> <h2 id="使用建议">使用建议<a class="anchor" href="#%e4%bd%bf%e7%94%a8%e5%bb%ba%e8%ae%ae" aria-label="Link"></a></h2> <ol> <li><strong>考前一周</strong>：把这页从头到尾过两遍，每条 ❌ 在脑中过一遍错的过程</li> <li><strong>平时刷题</strong>：错题对照本页编号，记进<a href="https://one.snvedu.cn/cuoti/">错题本</a>时写上「避坑 #N」</li> <li>同一个坑错两次 → 回对应章节「题型库」重练同类题</li> </ol>https://one.snvedu.cn/docs/english-basics/11-comparison/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/11-comparison/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>形容词变「更……」（比较级）和「最……」（最高级）的规则</li> <li>than 和 as&hellip;as 的用法</li> <li>不规则变化四兄弟：good / bad / many / little</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-两个概念">1. 两个概念<a class="anchor" href="#1-%e4%b8%a4%e4%b8%aa%e6%a6%82%e5%bf%b5" aria-label="Link"></a></h3> <ul> <li>比较级 =「<strong>更</strong>……」：Tom is <strong>taller</strong> than Jack.（汤姆比杰克高）</li> <li>最高级 =「<strong>最</strong>……」（前面加 the）：Tom is <strong>the tallest</strong> in our class.（汤姆是班里最高的）</li> </ul> <h3 id="2-变化规则">2. 变化规则<a class="anchor" href="#2-%e5%8f%98%e5%8c%96%e8%a7%84%e5%88%99" aria-label="Link"></a></h3> <table> <thead> <tr> <th>规则</th> <th>比较级</th> <th>最高级</th> </tr> </thead> <tbody> <tr> <td>单音节一般 + er / est</td> <td>tall → taller</td> <td>the tallest</td> </tr> <tr> <td>以 e 结尾 + r / st</td> <td>large → larger</td> <td>the largest</td> </tr> <tr> <td>辅音 + y：y → ier / iest</td> <td>easy → easier</td> <td>the easiest</td> </tr> <tr> <td>重读闭音节双写</td> <td>big → bigger</td> <td>the biggest</td> </tr> <tr> <td><strong>多音节词：more / most + 原形</strong></td> <td>beautiful → <strong>more</strong> beautiful</td> <td><strong>the most</strong> beautiful</td> </tr> <tr> <td>部分双音节两种都行</td> <td>clever → cleverer / more clever</td> <td>—</td> </tr> </tbody> </table> <p>口诀：<strong>短词加尾，长词戴帽</strong>（两三个音节以上 → more / most）。</p>https://one.snvedu.cn/bank/en-choice-2/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/en-choice-2/<h2 id="21-i-wish-i-____-how-to-swim">21. I wish I ____ how to swim.<a class="anchor" href="#21-i-wish-i-____-how-to-swim" aria-label="Link"></a></h2> <p>A. know　B. knew　C. have known　D. will know</p> <details><summary>答案：B</summary> wish 对现在愿望 → 过去式 knew </details> <h2 id="22-without-electricity-life-____-quite-different">22. Without electricity, life ____ quite different.<a class="anchor" href="#22-without-electricity-life-____-quite-different" aria-label="Link"></a></h2> <p>A. is　B. would be　C. will be　D. was</p> <details><summary>答案：B</summary> without = 含蓄虚拟条件 → would be（18 课） </details> <h2 id="23-its-high-time-we-____-up">23. It&rsquo;s high time we ____ up.<a class="anchor" href="#23-its-high-time-we-____-up" aria-label="Link"></a></h2> <p>A. get　B. got　C. will get　D. getting</p>https://one.snvedu.cn/docs/gaoshu/hard-points/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/hard-points/<blockquote class=''> <p>🎯 浙江专升本高数里<strong>真正拉开分差的 6 个难点</strong>，每个按「为什么难 → 突破三步 → 完整例题」手把手过。</p> </blockquote><h2 id="难点-1幂指型未定式全攻略hahahugoshortcode68s0hbhb">难点 1：幂指型未定式全攻略（<span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>1</mn><mi mathvariant="normal">∞</mi></msup><mo separator="true">,</mo><mtext> </mtext><msup><mn>0</mn><mn>0</mn></msup><mo separator="true">,</mo><mtext> </mtext><msup><mi mathvariant="normal">∞</mi><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">1^\infty,\ 0^0,\ \infty^0</annotation></semantics></math></span>）<a class="anchor" href="#%e9%9a%be%e7%82%b9-1%e5%b9%82%e6%8c%87%e5%9e%8b%e6%9c%aa%e5%ae%9a%e5%bc%8f%e5%85%a8%e6%94%bb%e7%95%a5hahahugoshortcode68s0hbhb" aria-label="Link"></a></h2> <p><strong>为什么难</strong>：指数和底数同时在变，四则运算法则全失效，很多同学直接卡死。</p>https://one.snvedu.cn/docs/english-basics/12-prepositions/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/12-prepositions/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>in / on / at 的时间与地点分工（口诀化）</li> <li>专升本完形、词汇题最爱考的动词 + 介词搭配</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-时间三兄弟in--on--at从大到小">1. 时间三兄弟：in / on / at（从大到小）<a class="anchor" href="#1-%e6%97%b6%e9%97%b4%e4%b8%89%e5%85%84%e5%bc%9fin--on--at%e4%bb%8e%e5%a4%a7%e5%88%b0%e5%b0%8f" aria-label="Link"></a></h3> <table> <thead> <tr> <th>词</th> <th>范围</th> <th>例句</th> </tr> </thead> <tbody> <tr> <td><strong>in</strong></td> <td>大块时间：年 / 月 / 季节 / 早晚</td> <td>in 2026 / in May / in spring / <strong>in the morning</strong></td> </tr> <tr> <td><strong>on</strong></td> <td>具体某一天 / 某天早上</td> <td>on Monday / <strong>on May 1st</strong> / <strong>on Sunday morning</strong></td> </tr> <tr> <td><strong>at</strong></td> <td>具体时刻 / 小节点</td> <td><strong>at 7 o&rsquo;clock</strong> / at noon / at night / at that time</td> </tr> </tbody> </table> <p>口诀：<strong>年月季节用 in，具体某天用 on，几点几分用 at</strong>。</p>https://one.snvedu.cn/bank/en-comprehensive/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/bank/en-comprehensive/<blockquote class=''> <p>四种题型各一组：完形（10 空）→ 十五选十（5 空）→ 阅读（3 题）→ 翻译（6 句）。限时 40 分钟，按考场顺序做。</p> </blockquote><h2 id="一完形填空10-空">一、完形填空（10 空）<a class="anchor" href="#%e4%b8%80%e5%ae%8c%e5%bd%a2%e5%a1%ab%e7%a9%ba10-%e7%a9%ba" aria-label="Link"></a></h2> <blockquote class=''> <p>Last summer, I decided to learn swimming. At first, I was (1)____ of water and dared not go into the pool. My coach was very (2)____ and encouraged me every day. He told me that the most important thing was to (3)____ calm in the water. I practiced blowing bubbles and holding my (4)____ for a few seconds. Little by little, I made (5)<strong><strong>. One day, I suddenly found that I could (6)</strong></strong> for several meters without help. How (7)____ I was! From then on, I went to the pool (8)____ and practiced harder. Now swimming has become my favorite (9)<strong><strong>. This experience taught me a (10)</strong></strong> lesson: nothing is impossible if you keep trying.</p> </blockquote>https://one.snvedu.cn/docs/gaoshu/deep-traps/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/deep-traps/<blockquote class=''> <p>🎯 和<a href="https://one.snvedu.cn/docs/gaoshu/traps-compendium/">避坑大全</a>不同：这一页收的是<strong>藏得深的坑</strong>——知识点你都懂、公式都会背，但出题人在前提、条件、边界上做了手脚，做错时甚至不知道自己错了。每个坑都还原「出题人怎么埋 → 为什么难察觉 → 例题拆局」。</p> </blockquote><h2 id="a-类--概念伪装坑考定义不考计算">A 类 · 概念伪装坑（考定义，不考计算）<a class="anchor" href="#a-%e7%b1%bb--%e6%a6%82%e5%bf%b5%e4%bc%aa%e8%a3%85%e5%9d%91%e8%80%83%e5%ae%9a%e4%b9%89%e4%b8%8d%e8%80%83%e8%ae%a1%e7%ae%97" aria-label="Link"></a></h2> <h3 id="深坑-a1导数定义的系数迷局">深坑 A1：导数定义的「系数迷局」<a class="anchor" href="#%e6%b7%b1%e5%9d%91-a1%e5%af%bc%e6%95%b0%e5%ae%9a%e4%b9%89%e7%9a%84%e7%b3%bb%e6%95%b0%e8%bf%b7%e5%b1%80" aria-label="Link"></a></h3> <p><strong>出题人怎么埋</strong>：给你一个长得像导数定义但系数不对的极限。</p>https://one.snvedu.cn/docs/english-basics/13-clauses/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/13-clauses/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>从句到底是什么（一个句子降级当零件）</li> <li>三种最基础从句：宾语从句、定语从句、状语从句（各自一句口诀）</li> <li>长难句不再害怕的第一步</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-从句的本质">1. 从句的本质<a class="anchor" href="#1-%e4%bb%8e%e5%8f%a5%e7%9a%84%e6%9c%ac%e8%b4%a8" aria-label="Link"></a></h3> <p>两个短句：</p>https://one.snvedu.cn/docs/gaoshu/formula-sheet/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/formula-sheet/<blockquote class=''> <p>🎯 <strong>考前最后一晚只看这一页</strong>。每条都链回出处；卡在哪条，回对应章节重学。</p> </blockquote><h2 id="一极限">一、极限<a class="anchor" href="#%e4%b8%80%e6%9e%81%e9%99%90" aria-label="Link"></a></h2> <p><strong>两个重要极限</strong>：</p> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mspace width="2em"/><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><mo fence="true">(</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo fence="true">)</mo></mrow><mi>x</mi></msup><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\frac{\sin x}{x}=1,\qquad \lim_{x \to \infty}\left(1+\frac{1}{x}\right)^x=e</annotation></semantics></math></span><p><strong>等价无穷小（x→0）</strong>：</p>https://one.snvedu.cn/docs/english-basics/14-nonfinite/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/14-nonfinite/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>为什么需要非谓语（一个句子只准有一个「心脏」动词）</li> <li>to do / doing / done 三兄弟各自的心情</li> <li>三步判断法（专升本必考）</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-一个句子只有一个谓语">1. 一个句子只有一个谓语<a class="anchor" href="#1-%e4%b8%80%e4%b8%aa%e5%8f%a5%e5%ad%90%e5%8f%aa%e6%9c%89%e4%b8%80%e4%b8%aa%e8%b0%93%e8%af%ad" aria-label="Link"></a></h3> <p>英文句子规矩很严：<strong>一个简单句只能有一个真正的谓语动词</strong>（句子心脏）。其他动作想出场，就必须<strong>改头换面</strong>：</p>https://one.snvedu.cn/docs/gaoshu/forbidden-techniques/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/gaoshu/forbidden-techniques/<blockquote class=''> <p>🎯 浙江卷<strong>选择 20 分 + 填空 40 分 = 60 分</strong>，禁术的主战场。先说清楚：<strong>禁术 = 保命 + 提速</strong>，用于「卡壳时」「检查时」「选项排除时」——基础题仍靠实力，硬套禁术不做题的人考场上会死得很惨。每条禁术配「识别信号 → 操作 → 演示」。</p> </blockquote>https://one.snvedu.cn/docs/english-basics/15-vocabulary/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/15-vocabulary/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>从 1000 词到专升本要求词汇量的<strong>具体时间表</strong></li> <li>词根词缀法（让 1 个词变 5 个词）</li> <li>反遗忘的复习节奏（艾宾浩斯简化版）</li> <li>什么词值得背、什么词不值得背</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-时间表以每天-30-分钟为标准">1. 时间表（以每天 30 分钟为标准）<a class="anchor" href="#1-%e6%97%b6%e9%97%b4%e8%a1%a8%e4%bb%a5%e6%af%8f%e5%a4%a9-30-%e5%88%86%e9%92%9f%e4%b8%ba%e6%a0%87%e5%87%86" aria-label="Link"></a></h3> <table> <thead> <tr> <th>阶段</th> <th>时长</th> <th>任务</th> <th>目标</th> </tr> </thead> <tbody> <tr> <td>第一阶段</td> <td>第 1~4 周</td> <td>高频词 800（专升本真题里出现最多的词）</td> <td>认识 1800</td> </tr> <tr> <td>第二阶段</td> <td>第 5~10 周</td> <td>考纲核心词 1200 + 复习</td> <td>认识 3000</td> </tr> <tr> <td>第三阶段</td> <td>第 11~14 周</td> <td>真题生词回填 + 收尾</td> <td>稳定 3500</td> </tr> <tr> <td>持续动作</td> <td>每天</td> <td>早上新词 30 个 + 晚上复习 30 个</td> <td>不中断</td> </tr> </tbody> </table> <p><strong>原则</strong>：专升本词汇 3500 里，<strong>真题词 &gt; 考纲词 &gt; 生僻词</strong>。优先背你在真题和这套课里见过的词。</p>https://one.snvedu.cn/docs/english-basics/16-modals/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/16-modals/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>常用情态动词的含义分工：can / could / may / might / must / should</li> <li>情态动词 + have done 的「推测过去」（考试高频）</li> <li>must 和 have to 的区别、各情态动词的否定含义</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-两条铁律先记住">1. 两条铁律先记住<a class="anchor" href="#1-%e4%b8%a4%e6%9d%a1%e9%93%81%e5%be%8b%e5%85%88%e8%ae%b0%e4%bd%8f" aria-label="Link"></a></h3> <ol> <li>情态动词后面<strong>永远跟动词原形</strong>：He <strong>can swim</strong>.（❌ can swims）</li> <li>情态动词<strong>没有人称变化</strong>：He can / I can / They can——永远 can</li> </ol> <h3 id="2-含义分工表">2. 含义分工表<a class="anchor" href="#2-%e5%90%ab%e4%b9%89%e5%88%86%e5%b7%a5%e8%a1%a8" aria-label="Link"></a></h3> <table> <thead> <tr> <th>情态动词</th> <th>核心意思</th> <th>例句</th> </tr> </thead> <tbody> <tr> <td>can</td> <td>能力；许可；可能</td> <td>She <strong>can</strong> speak three languages. / Can I come in?</td> </tr> <tr> <td>could</td> <td>can 的过去；更委婉的请求</td> <td><strong>Could</strong> you help me?（比 Can you 礼貌）</td> </tr> <tr> <td>may</td> <td>许可（正式）；可能</td> <td><strong>May</strong> I ask a question? / He <strong>may</strong> be at home.</td> </tr> <tr> <td>might</td> <td>可能（比 may 把握更小）</td> <td>She <strong>might</strong> come later.（不太确定）</td> </tr> <tr> <td>must</td> <td>必须；肯定推测</td> <td>You <strong>must</strong> finish it today. / He <strong>must</strong> be tired.（肯定很累）</td> </tr> <tr> <td>should</td> <td>应该；建议</td> <td>You <strong>should</strong> practice every day.</td> </tr> <tr> <td>will / would</td> <td>意愿；请求</td> <td><strong>Would</strong> you please open the window?</td> </tr> <tr> <td>need</td> <td>需要（否定 needn&rsquo;t = 不必）</td> <td>You <strong>needn&rsquo;t</strong> worry.（你不必担心）</td> </tr> </tbody> </table> <h3 id="3-把握程度光谱推测题就按这个排">3. 把握程度光谱（推测题就按这个排）<a class="anchor" href="#3-%e6%8a%8a%e6%8f%a1%e7%a8%8b%e5%ba%a6%e5%85%89%e8%b0%b1%e6%8e%a8%e6%b5%8b%e9%a2%98%e5%b0%b1%e6%8c%89%e8%bf%99%e4%b8%aa%e6%8e%92" aria-label="Link"></a></h3> <pre tabindex="0"><code>must（90% 肯定）&gt; may（50%）&gt; might（20~30%）&gt; can&#39;t（90% 否定）</code></pre><ul> <li>The light is on. He <strong>must</strong> be home.（灯亮着，他肯定在家）</li> <li>He <strong>may/might</strong> be home.（可能在家）</li> <li>He <strong>can&rsquo;t</strong> be home—the door is locked.（<strong>不可能</strong>在家——can&rsquo;t 表否定推测！）</li> </ul> <h3 id="4-情态动词--have-done对过去的推测高频考点">4. 情态动词 + have done：对过去的推测（高频考点）<a class="anchor" href="#4-%e6%83%85%e6%80%81%e5%8a%a8%e8%af%8d--have-done%e5%af%b9%e8%bf%87%e5%8e%bb%e7%9a%84%e6%8e%a8%e6%b5%8b%e9%ab%98%e9%a2%91%e8%80%83%e7%82%b9" aria-label="Link"></a></h3> <p><strong>must have done</strong>：过去肯定做了</p>https://one.snvedu.cn/docs/english-basics/17-sentence-types/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/17-sentence-types/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>四种高频特殊句式：there be 存在句、感叹句、祈使句、反意疑问句——每个都有固定套路，套对就拿分。</p> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-there-be有的存在句">1. there be：「有」的存在句<a class="anchor" href="#1-there-be%e6%9c%89%e7%9a%84%e5%ad%98%e5%9c%a8%e5%8f%a5" aria-label="Link"></a></h3> <p>结构：<strong>There be + 名词 + 地点/时间</strong>。</p>https://one.snvedu.cn/docs/english-basics/18-subjunctive/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/18-subjunctive/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>虚拟语气全量知识：if 三态、wish 系列、suggest 系列、would rather、It&rsquo;s time、含蓄条件、错综条件——专升本语法的最高频大考点。</p>https://one.snvedu.cn/docs/english-basics/19-relative-clauses-2/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/19-relative-clauses-2/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>第 13 课只学了 who/which/that 的皮毛，这课拿下定语从句全部考点：whose、介词 + which、只用 that / 只用 which 的场合、非限制性从句。</p> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-whose的的从句版">1. whose：「……的」的从句版<a class="anchor" href="#1-whose%e7%9a%84%e7%9a%84%e4%bb%8e%e5%8f%a5%e7%89%88" aria-label="Link"></a></h3> <p>先行词（人或物）与从句主语是<strong>所属关系</strong> → 用 <strong>whose</strong>：</p>https://one.snvedu.cn/docs/english-basics/20-noun-clauses/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/20-noun-clauses/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>四大家族全拿下：主语从句、宾语从句、表语从句、同位语从句——外加连接词选择的总算法和两个必考陷阱。</p> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-四大家族--从句当名词用">1. 四大家族 = 从句当名词用<a class="anchor" href="#1-%e5%9b%9b%e5%a4%a7%e5%ae%b6%e6%97%8f--%e4%bb%8e%e5%8f%a5%e5%bd%93%e5%90%8d%e8%af%8d%e7%94%a8" aria-label="Link"></a></h3> <table> <thead> <tr> <th>家族</th> <th>位置</th> <th>例句</th> </tr> </thead> <tbody> <tr> <td><strong>主语</strong>从句</td> <td>句首当主语</td> <td><strong>What he said</strong> is true. / <strong>That he failed</strong> surprised us.</td> </tr> <tr> <td><strong>宾语</strong>从句</td> <td>动词/介词后</td> <td>I know <strong>where he lives</strong>. / It depends on <strong>whether you come</strong>.</td> </tr> <tr> <td><strong>表语</strong>从句</td> <td>be 动词后</td> <td>The problem is <strong>that we lack money</strong>.</td> </tr> <tr> <td><strong>同位语</strong>从句</td> <td>抽象名词后，解释内容</td> <td>The news <strong>that our team won</strong> excited us all.</td> </tr> </tbody> </table> <p>常接同位语从句的抽象名词：<strong>news / fact / idea / truth / hope / promise / suggestion / problem</strong>。</p>https://one.snvedu.cn/docs/english-basics/21-adverbial-clauses/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/21-adverbial-clauses/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>九类状语从句的连接词全家福 + 五个高频陷阱（主将从现、although/but、because/so、时间从句的时态、no sooner&hellip;than）。</p>https://one.snvedu.cn/docs/english-basics/22-inversion-emphasis/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/22-inversion-emphasis/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>部分倒装的全部触发条件（一张清单）</li> <li>完全倒装的两种场合</li> <li>强调句 It is&hellip;that 的判断法和陷阱</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-部分倒装只把助动词情态动词be提到主语前">1. 部分倒装：只把「助动词/情态动词/be」提到主语前<a class="anchor" href="#1-%e9%83%a8%e5%88%86%e5%80%92%e8%a3%85%e5%8f%aa%e6%8a%8a%e5%8a%a9%e5%8a%a8%e8%af%8d%e6%83%85%e6%80%81%e5%8a%a8%e8%af%8dbe%e6%8f%90%e5%88%b0%e4%b8%bb%e8%af%ad%e5%89%8d" aria-label="Link"></a></h3> <p><strong>触发条件清单（看到这些开头 → 倒装）</strong>：</p>https://one.snvedu.cn/docs/english-basics/23-agreement/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/23-agreement/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>主谓一致的三大原则（语法一致 / 意义一致 / 就近原则）+ 十个高频判断场景——选择填空的送分点，也是最容易粗心的地方。</p> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-三大原则">1. 三大原则<a class="anchor" href="#1-%e4%b8%89%e5%a4%a7%e5%8e%9f%e5%88%99" aria-label="Link"></a></h3> <ul> <li><strong>语法一致</strong>：形式上单复数对应（he plays / they play）</li> <li><strong>意义一致</strong>：看意思不看形式（集合名词、不可数）</li> <li><strong>就近原则</strong>：谓语跟<strong>最近的主语</strong>一致（there be、or、not only&hellip;but also）</li> </ul> <h3 id="2-十个高频场景速查">2. 十个高频场景速查<a class="anchor" href="#2-%e5%8d%81%e4%b8%aa%e9%ab%98%e9%a2%91%e5%9c%ba%e6%99%af%e9%80%9f%e6%9f%a5" aria-label="Link"></a></h3> <table> <thead> <tr> <th>场景</th> <th>规则</th> <th>例</th> </tr> </thead> <tbody> <tr> <td>① 集体名词 family / class / team / group</td> <td>看意思：整体 → 单；成员 → 复</td> <td>My family <strong>is</strong> big. / My family <strong>are</strong> all early risers.</td> </tr> <tr> <td>② the + 形容词（一类人）</td> <td>复数</td> <td>The rich <strong>are</strong> not always happy.</td> </tr> <tr> <td>③ each / every / either / neither + 单数名词</td> <td>单数</td> <td>Each of the students <strong>has</strong> a book. / Every boy and girl <strong>is</strong> invited.（every 修饰并列单数仍单数）</td> </tr> <tr> <td>④ a number of vs the number of</td> <td>a number of + 复数名词 → <strong>复</strong>；the number of → <strong>单</strong></td> <td>A number of students <strong>are</strong> here. / The number of students <strong>is</strong> growing.</td> </tr> <tr> <td>⑤ there be + 并列主语</td> <td>就近</td> <td>There <strong>is</strong> a pen and two books.</td> </tr> <tr> <td>⑥ either&hellip;or / neither&hellip;nor / not only&hellip;but also</td> <td>就近</td> <td>Neither you nor he <strong>is</strong> right.</td> </tr> <tr> <td>⑦ 主语 + as well as / with / along with / together with</td> <td><strong>就前</strong>（跟前面主语）</td> <td>The teacher as well as the students <strong>likes</strong> music.</td> </tr> <tr> <td>⑧ and 连接两个主语</td> <td>通常复数；指<strong>同一人/物/概念</strong> → 单</td> <td>Bread and butter <strong>is</strong> his favorite.（黄油面包=一样东西）</td> </tr> <tr> <td>⑨ 分数 / 百分数 / a lot of / plenty of + of 短语</td> <td>看 <strong>of 后面的名词</strong></td> <td>Two thirds of the water <strong>is</strong> polluted. / Two thirds of the students <strong>are</strong> here.</td> </tr> <tr> <td>⑩ many a + 单数名词</td> <td>形复意单 → <strong>单数</strong></td> <td>Many a student <strong>has</strong> made this mistake.</td> </tr> </tbody> </table> <h3 id="3-三个易漏点">3. 三个易漏点<a class="anchor" href="#3-%e4%b8%89%e4%b8%aa%e6%98%93%e6%bc%8f%e7%82%b9" aria-label="Link"></a></h3> <ol> <li><strong>时间 / 金钱 / 距离</strong>作主语当整体 → 单数： <blockquote class=''> <p>Ten years <strong>has</strong> passed. / Five dollars <strong>is</strong> enough.</p> </blockquote></li> <li><strong>不定代词</strong>（something, everyone, nobody, each&hellip;）→ 单数： <blockquote class=''> <p>Everyone <strong>is</strong> here. / Nothing <strong>was</strong> said.</p> </blockquote></li> <li><strong>疑问词 what 引导的主语从句</strong>：看表语—— <blockquote class=''> <p>What we need <strong>is</strong> more time.（need 的东西=时间，不可数→单） What we need <strong>are</strong> more teachers.（可数→复）</p> </blockquote></li> </ol>https://one.snvedu.cn/docs/english-basics/24-clause-comparison/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/24-clause-comparison/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>三大从句（定语 / 名词性 / 状语）学完了，这课专门解决「<strong>同一个连接词在不同从句里的不同用法</strong>」——这是专升本辨析题的核心套路。</p> <h2 id="-核心辨析">📖 核心辨析<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e8%be%a8%e6%9e%90" aria-label="Link"></a></h2> <h3 id="辨析一同一个-when-的三种身份">辨析一：同一个 when 的三种身份<a class="anchor" href="#%e8%be%a8%e6%9e%90%e4%b8%80%e5%90%8c%e4%b8%80%e4%b8%aa-when-%e7%9a%84%e4%b8%89%e7%a7%8d%e8%ba%ab%e4%bb%bd" aria-label="Link"></a></h3> <table> <thead> <tr> <th>从句类型</th> <th>例句</th> <th>判断依据</th> </tr> </thead> <tbody> <tr> <td>状语从句（时间）</td> <td>I&rsquo;ll go <strong>when he comes</strong>.</td> <td>when 后是完整句，整句修饰主句动作</td> </tr> <tr> <td>宾语从句</td> <td>I don&rsquo;t know <strong>when he will come</strong>.</td> <td>when 后完整句，作 know 的宾语</td> </tr> <tr> <td>定语从句</td> <td>I&rsquo;ll never forget the day <strong>when we met</strong>.</td> <td>when 前有先行词（the day），when 在从句中作状语</td> </tr> </tbody> </table> <p><strong>鉴别流程</strong>：① 前面有先行词？→ 定语从句。② 跟在 know/tell 等动词后？→ 宾语从句。③ 否则 → 状语从句。</p>https://one.snvedu.cn/docs/english-basics/25-irregular-verbs-1/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/25-irregular-verbs-1/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>不规则动词的<strong>分组记忆法</strong>（不是死背 200 个散词，而是按变化模式分四组）+ 前半份核心表（A~L）。这张表是现在完成时、被动语态、非谓语的地基。</p>https://one.snvedu.cn/docs/english-basics/26-irregular-verbs-2/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/26-irregular-verbs-2/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>表的后半份（M~Z）+ 三对<strong>最易混淆动词</strong>的终极辨析（lie 之争每年必考）。</p> <h2 id="-核心表下mz">📖 核心表（下：M~Z）<a class="anchor" href="#-%e6%a0%b8%e5%bf%83%e8%a1%a8%e4%b8%8bmz" aria-label="Link"></a></h2> <p><strong>ABB 组（14 个）</strong>：</p> <table> <thead> <tr> <th>原形</th> <th>过去式</th> <th>分词</th> <th>含义</th> </tr> </thead> <tbody> <tr> <td>mean</td> <td>meant</td> <td>meant</td> <td>意指</td> </tr> <tr> <td>lead</td> <td>led</td> <td>led</td> <td>带领</td> </tr> <tr> <td>feed</td> <td>fed</td> <td>fed</td> <td>喂</td> </tr> <tr> <td>send</td> <td>sent</td> <td>sent</td> <td>送</td> </tr> <tr> <td>lend</td> <td>lent</td> <td>lent</td> <td>借出</td> </tr> <tr> <td>win</td> <td>won</td> <td>won</td> <td>赢</td> </tr> <tr> <td>stand</td> <td>stood</td> <td>stood</td> <td>站</td> </tr> <tr> <td>understand</td> <td>understood</td> <td>understood</td> <td>明白</td> </tr> <tr> <td>shine</td> <td>shone</td> <td>shone</td> <td>照耀</td> </tr> <tr> <td>lay</td> <td>laid</td> <td>laid</td> <td>放置</td> </tr> <tr> <td>hide</td> <td>hid</td> <td>hidden</td> <td>藏</td> </tr> <tr> <td>dig</td> <td>dug</td> <td>dug</td> <td>挖</td> </tr> <tr> <td>stick</td> <td>stuck</td> <td>stuck</td> <td>粘</td> </tr> <tr> <td>strike</td> <td>struck</td> <td>struck</td> <td>击打</td> </tr> </tbody> </table> <p><strong>ABC 组（M~Z，18 个）</strong>：</p>https://one.snvedu.cn/docs/english-basics/27-phrasal-verbs/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/27-phrasal-verbs/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>八大高频动词的短语家族（look / get / take / put / turn / give / set / come + 其他）——完形填空与词汇题的半壁江山。按「动词家族」记，每个家族一次记全。</p>https://one.snvedu.cn/docs/english-basics/28-collocations/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/28-collocations/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>三张搭配表：<strong>be + 形容词 + 介词</strong>、<strong>动词/名词 + 介词</strong>、<strong>易混搭配对比</strong>——词汇题和完形的最后一块拼图。</p> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-be--形容词--介词30-组黄金组合">1. be + 形容词 + 介词（30 组黄金组合）<a class="anchor" href="#1-be--%e5%bd%a2%e5%ae%b9%e8%af%8d--%e4%bb%8b%e8%af%8d30-%e7%bb%84%e9%bb%84%e9%87%91%e7%bb%84%e5%90%88" aria-label="Link"></a></h3> <table> <thead> <tr> <th>搭配</th> <th>意思</th> <th>搭配</th> <th>意思</th> </tr> </thead> <tbody> <tr> <td>be good <strong>at</strong></td> <td>擅长</td> <td>be bad <strong>at</strong></td> <td>不擅长</td> </tr> <tr> <td>be interested <strong>in</strong></td> <td>对…感兴趣</td> <td>be worried <strong>about</strong></td> <td>担心</td> </tr> <tr> <td>be afraid <strong>of</strong></td> <td>害怕</td> <td>be proud <strong>of</strong></td> <td>自豪</td> </tr> <tr> <td>be full <strong>of</strong></td> <td>充满</td> <td>be tired <strong>of</strong></td> <td>厌烦</td> </tr> <tr> <td>be famous <strong>for</strong>（因…）</td> <td>出名</td> <td>be famous <strong>as</strong>（作为…）</td> <td>以身份出名</td> </tr> <tr> <td>be angry <strong>with</strong> sb / <strong>at</strong> sth</td> <td>生…的气</td> <td>be pleased <strong>with</strong></td> <td>对…满意</td> </tr> <tr> <td>be strict <strong>with</strong> sb / <strong>in</strong> sth</td> <td>严格</td> <td>be popular <strong>with</strong></td> <td>受…欢迎</td> </tr> <tr> <td>be late <strong>for</strong></td> <td>迟到</td> <td>be sorry <strong>for</strong></td> <td>抱歉</td> </tr> <tr> <td>be different <strong>from</strong></td> <td>与…不同</td> <td>be similar <strong>to</strong></td> <td>与…相似</td> </tr> <tr> <td>be ready <strong>for</strong></td> <td>准备好</td> <td>be fit <strong>for</strong></td> <td>适合</td> </tr> <tr> <td>be interested&hellip; 上文</td> <td>—</td> <td>be surprised <strong>at</strong></td> <td>对…惊讶</td> </tr> <tr> <td>be excited <strong>about</strong></td> <td>对…兴奋</td> <td>be nervous <strong>about</strong></td> <td>紧张</td> </tr> <tr> <td>be busy <strong>with</strong> / doing</td> <td>忙于</td> <td>be satisfied <strong>with</strong></td> <td>满意</td> </tr> <tr> <td>be kind <strong>to</strong></td> <td>对…友善</td> <td>be polite <strong>to</strong></td> <td>礼貌</td> </tr> <tr> <td>be used <strong>to</strong> doing</td> <td>习惯于</td> <td>be made <strong>of/from</strong></td> <td>由…制成</td> </tr> <tr> <td>be covered <strong>with</strong></td> <td>覆盖着</td> <td>be filled <strong>with</strong></td> <td>装满</td> </tr> </tbody> </table> <h3 id="2-动词--名词--介词20-组">2. 动词 / 名词 + 介词（20 组）<a class="anchor" href="#2-%e5%8a%a8%e8%af%8d--%e5%90%8d%e8%af%8d--%e4%bb%8b%e8%af%8d20-%e7%bb%84" aria-label="Link"></a></h3> <table> <thead> <tr> <th>搭配</th> <th>意思</th> <th>搭配</th> <th>意思</th> </tr> </thead> <tbody> <tr> <td>listen <strong>to</strong></td> <td>听</td> <td>belong <strong>to</strong></td> <td>属于</td> </tr> <tr> <td>wait <strong>for</strong></td> <td>等</td> <td>depend <strong>on</strong></td> <td>依靠；取决于</td> </tr> <tr> <td>agree <strong>with</strong> sb</td> <td>同意某人</td> <td>agree <strong>to</strong> sth</td> <td>同意某事</td> </tr> <tr> <td>laugh <strong>at</strong></td> <td>嘲笑</td> <td>worry <strong>about</strong></td> <td>担忧</td> </tr> <tr> <td>spend&hellip;<strong>on</strong> sth</td> <td>花费</td> <td>congratulate sb <strong>on</strong> sth</td> <td>祝贺</td> </tr> <tr> <td>thank sb <strong>for</strong></td> <td>感谢</td> <td>help sb <strong>with</strong></td> <td>帮助</td> </tr> <tr> <td>stop sb <strong>from</strong> doing</td> <td>阻止</td> <td>protect&hellip;<strong>from</strong></td> <td>保护…免受</td> </tr> <tr> <td>the key <strong>to</strong></td> <td>…的钥匙/关键</td> <td>the answer <strong>to</strong></td> <td>…的答案</td> </tr> <tr> <td>a visit <strong>to</strong></td> <td>对…的访问</td> <td>the entrance <strong>to</strong></td> <td>…的入口</td> </tr> <tr> <td>pay <strong>for</strong></td> <td>付款</td> <td>insist <strong>on</strong></td> <td>坚持</td> </tr> </tbody> </table> <h3 id="3-高频易混对比每年都考的两组">3. 高频易混对比（每年都考的两组）<a class="anchor" href="#3-%e9%ab%98%e9%a2%91%e6%98%93%e6%b7%b7%e5%af%b9%e6%af%94%e6%af%8f%e5%b9%b4%e9%83%bd%e8%80%83%e7%9a%84%e4%b8%a4%e7%bb%84" aria-label="Link"></a></h3> <p><strong>① arrive / reach / get to</strong>（到达）</p>https://one.snvedu.cn/docs/english-basics/29-word-formation/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/29-word-formation/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <p>构词法全量：<strong>前缀表（改意思）+ 后缀表（改词性）+ 常用词根</strong>——这是从 1000 词到 3500+ 词的杠杆，也是阅读「同义改写」的解码器。</p> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-前缀改变词义不改变词性">1. 前缀（改变词义，不改变词性）<a class="anchor" href="#1-%e5%89%8d%e7%bc%80%e6%94%b9%e5%8f%98%e8%af%8d%e4%b9%89%e4%b8%8d%e6%94%b9%e5%8f%98%e8%af%8d%e6%80%a7" aria-label="Link"></a></h3> <table> <thead> <tr> <th>前缀</th> <th>意思</th> <th>例子</th> </tr> </thead> <tbody> <tr> <td>un- / in- / im- / ir- / il-</td> <td>否定</td> <td>happy→unhappy; possible→impossible（p 前 im）; regular→irregular; legal→illegal</td> </tr> <tr> <td>dis-</td> <td>相反</td> <td>agree→disagree; like→dislike; honest→dishonest</td> </tr> <tr> <td>mis-</td> <td>错误</td> <td>understand→misunderstand; lead→mislead</td> </tr> <tr> <td>re-</td> <td>再次</td> <td>tell→retell; write→rewrite; build→rebuild</td> </tr> <tr> <td>pre-</td> <td>前</td> <td>view→preview; pay→prepay</td> </tr> <tr> <td>post-</td> <td>后</td> <td>graduate→postgraduate</td> </tr> <tr> <td>over-</td> <td>过度</td> <td>sleep→oversleep; work→overwork</td> </tr> <tr> <td>under-</td> <td>不足</td> <td>stand→understand(本义); pay→underpaid 低薪</td> </tr> <tr> <td>sub-</td> <td>下、次</td> <td>way→subway; title→subtitle</td> </tr> <tr> <td>inter-</td> <td>之间</td> <td>national→international; view→interview</td> </tr> <tr> <td>trans-</td> <td>跨越、转换</td> <td>port→transport; form→transform</td> </tr> <tr> <td>anti-</td> <td>反对</td> <td>war→antiwar; social→antisocial</td> </tr> <tr> <td>auto-</td> <td>自动、自己</td> <td>mobile→automobile; graph→autograph</td> </tr> <tr> <td>multi-</td> <td>多</td> <td>national→multinational; media→multimedia</td> </tr> <tr> <td>self-</td> <td>自我</td> <td>confident→self-confident</td> </tr> </tbody> </table> <p>（记忆要点：im- 常在 p/b/m 前，il- 在 l 前，ir- 在 r 前——拼写规律）</p>https://one.snvedu.cn/docs/english-basics/30-sentence-analysis/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/30-sentence-analysis/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>五大句子成分是什么、怎么认</li> <li>找谓语的三步法（长难句的「心脏定位术」）</li> <li>长难句拆解五步法 + 12 个实战例句（从专升本真题风格由浅到深）</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-五大成分一个句子的零件清单">1. 五大成分（一个句子的零件清单）<a class="anchor" href="#1-%e4%ba%94%e5%a4%a7%e6%88%90%e5%88%86%e4%b8%80%e4%b8%aa%e5%8f%a5%e5%ad%90%e7%9a%84%e9%9b%b6%e4%bb%b6%e6%b8%85%e5%8d%95" aria-label="Link"></a></h3> <table> <thead> <tr> <th>成分</th> <th>回答的问题</th> <th>常见形态</th> </tr> </thead> <tbody> <tr> <td>主语</td> <td>谁/什么</td> <td>名词、代词、doing、to do、从句</td> </tr> <tr> <td>谓语</td> <td>干什么/是什么</td> <td>动词（含助动词）——<strong>全句唯一的核心</strong></td> </tr> <tr> <td>宾语</td> <td>对谁/什么</td> <td>名词、代词、doing、从句</td> </tr> <tr> <td>表语</td> <td>主语是什么样</td> <td>be/系动词后的名词、形容词、从句</td> </tr> <tr> <td>修饰成分</td> <td>什么样的？何时何地为何？</td> <td><strong>定语</strong>（修饰名词：形/介短/从句）、<strong>状语</strong>（修饰动词/整句：副词/介短/从句）</td> </tr> </tbody> </table> <p><strong>一句话心法</strong>：长难句 = 简单句 + 一堆修饰零件。砍掉修饰，主干就一句大白话。</p>https://one.snvedu.cn/docs/english-basics/31-discourse-graduation/Mon, 01 Jan 0001 00:00:00 +0000https://one.snvedu.cn/docs/english-basics/31-discourse-graduation/<h2 id="-这节课你会学到">🎯 这节课你会学到<a class="anchor" href="#-%e8%bf%99%e8%8a%82%e8%af%be%e4%bd%a0%e4%bc%9a%e5%ad%a6%e5%88%b0" aria-label="Link"></a></h2> <ol> <li>语篇衔接词全表（阅读猜词、七选五、写作三用的底层能力）</li> <li>三种篇章结构（读文章、写作文通用）</li> <li><strong>毕业总测 20 题</strong>——通关后正式进入<a href="https://one.snvedu.cn/docs/english/">英语备考模块</a>。</li> </ol> <h2 id="-讲解">📖 讲解<a class="anchor" href="#-%e8%ae%b2%e8%a7%a3" aria-label="Link"></a></h2> <h3 id="1-衔接词全表按逻辑分类背">1. 衔接词全表（按逻辑分类背）<a class="anchor" href="#1-%e8%a1%94%e6%8e%a5%e8%af%8d%e5%85%a8%e8%a1%a8%e6%8c%89%e9%80%bb%e8%be%91%e5%88%86%e7%b1%bb%e8%83%8c" aria-label="Link"></a></h3> <table> <thead> <tr> <th>逻辑</th> <th>衔接词</th> </tr> </thead> <tbody> <tr> <td>增补</td> <td>and, also, besides, in addition, moreover, what&rsquo;s more, furthermore</td> </tr> <tr> <td>转折</td> <td>but, yet, however, while, nevertheless, on the contrary</td> </tr> <tr> <td>因果</td> <td>because, since, therefore, thus, as a result, consequently, so</td> </tr> <tr> <td>顺序</td> <td>first(ly), second(ly), finally, then, next, afterwards</td> </tr> <tr> <td>举例</td> <td>for example, for instance, such as, namely, that is</td> </tr> <tr> <td>总结</td> <td>in short, in a word, in conclusion, to sum up, on the whole</td> </tr> <tr> <td>对比</td> <td>while, whereas, on the one hand&hellip;on the other hand, instead</td> </tr> <tr> <td>强调</td> <td>indeed, certainly, above all, especially</td> </tr> <tr> <td>让步</td> <td>although, though, even if, despite, in spite of, after all</td> </tr> </tbody> </table> <p><strong>三用</strong>：①阅读里它们是「作者态度路标」；②七选五里它们是「空格答案依据」；③写作里它们是「提分粘合剂」。</p>https://one.snvedu.cn/daka/2026-11-08/Sun, 08 Nov 2026 00:00:00 +0000https://one.snvedu.cn/daka/2026-11-08/<h2 id="今日完成">今日完成<a class="anchor" href="#%e4%bb%8a%e6%97%a5%e5%ae%8c%e6%88%90" aria-label="Link"></a></h2> <ul> <li><input checked="" disabled="" type="checkbox"> 高数：极限章题型 1~6 过完一遍（对应<a href="https://one.snvedu.cn/docs/gaoshu/functions-limits-continuity/">题型库</a>）</li> <li><input checked="" disabled="" type="checkbox"> 英语：词汇语法题 20 道 + 错词记录</li> </ul> <h2 id="用时">用时<a class="anchor" href="#%e7%94%a8%e6%97%b6" aria-label="Link"></a></h2> <p>约 2.5 小时</p> <h2 id="一句话心得">一句话心得<a class="anchor" href="#%e4%b8%80%e5%8f%a5%e8%af%9d%e5%bf%83%e5%be%97" aria-label="Link"></a></h2> <p>等价无穷小加减不能乱替换——今天这道错题值 4 分。</p>https://one.snvedu.cn/cuoti/2026-11-08-limit-trap/Sun, 08 Nov 2026 00:00:00 +0000https://one.snvedu.cn/cuoti/2026-11-08-limit-trap/<h2 id="原题">原题<a class="anchor" href="#%e5%8e%9f%e9%a2%98" aria-label="Link"></a></h2> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\lim_{x \to 0}\frac{\tan x-\sin x}{x^3}</annotation></semantics></math></span><h2 id="我的错解">我的错解<a class="anchor" href="#%e6%88%91%e7%9a%84%e9%94%99%e8%a7%a3" aria-label="Link"></a></h2> <p>把 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\tan x</annotation></semantics></math></span> 和 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>sin</mi><mo>⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin x</annotation></semantics></math></span> 都替换成 <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span>，分子变 0，得出极限 = 0。❌</p> <h2 id="正确解法">正确解法<a class="anchor" href="#%e6%ad%a3%e7%a1%ae%e8%a7%a3%e6%b3%95" aria-label="Link"></a></h2> <p>先提取公因式把加减变乘除：</p> <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo>−</mo><mi>sin</mi><mo>⁡</mo><mi>x</mi><mo>=</mo><mi>tan</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mo>⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>x</mi><mo>⋅</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\tan x-\sin x=\tan x(1-\cos x)\sim x\cdot\frac{x^2}{2}=\frac{x^3}{2}</annotation></semantics></math></span><p>所以极限 = 1/2。</p>https://one.snvedu.cn/posts/study-plan/Sat, 07 Nov 2026 00:00:00 +0000https://one.snvedu.cn/posts/study-plan/<h2 id="-考试信息">📅 考试信息<a class="anchor" href="#-%e8%80%83%e8%af%95%e4%bf%a1%e6%81%af" aria-label="Link"></a></h2> <ul> <li>考试时间：____（填这里，倒计时挂心里）</li> <li>目标分数：高数 ____ / 英语 ____</li> </ul> <h2 id="-三轮复习规划">🗓️ 三轮复习规划<a class="anchor" href="#-%e4%b8%89%e8%bd%ae%e5%a4%8d%e4%b9%a0%e8%a7%84%e5%88%92" aria-label="Link"></a></h2> <table> <thead> <tr> <th>轮次</th> <th>时间段</th> <th>目标</th> <th>对应本站动作</th> </tr> </thead> <tbody> <tr> <td>第一轮 · 基础</td> <td>__月__日 ~ __月__日</td> <td>过考纲，建立知识框架</td> <td>填完各章「核心考点」</td> </tr> <tr> <td>第二轮 · 强化</td> <td>__月__日 ~ __月__日</td> <td>真题分章刷，吃透每类题</td> <td>错题写进各章「易错点」</td> </tr> <tr> <td>第三轮 · 冲刺</td> <td>__月__日 ~ 考前</td> <td>套卷 + 背模板</td> <td>只看「易错点」+「真题演练」</td> </tr> </tbody> </table> <h2 id="-每日节奏">⏰ 每日节奏<a class="anchor" href="#-%e6%af%8f%e6%97%a5%e8%8a%82%e5%a5%8f" aria-label="Link"></a></h2> <table> <thead> <tr> <th>时间段</th> <th>内容</th> </tr> </thead> <tbody> <tr> <td>早上</td> <td>英语词汇 30 个（滚动复习昨天的）</td> </tr> <tr> <td>上午</td> <td>高数一个专题 + 例题</td> </tr> <tr> <td>晚上</td> <td>整理当日错题 → 写进对应章节「易错点」</td> </tr> <tr> <td>睡前</td> <td>复盘今天学了啥，一句话写进<a href="https://one.snvedu.cn/posts/">备考日记</a></td> </tr> </tbody> </table> <h2 id="-每周复盘">📌 每周复盘<a class="anchor" href="#-%e6%af%8f%e5%91%a8%e5%a4%8d%e7%9b%98" aria-label="Link"></a></h2> <p>每周日问自己三个问题：</p>https://one.snvedu.cn/posts/how-i-learn/Sat, 07 Nov 2026 00:00:00 +0000https://one.snvedu.cn/posts/how-i-learn/<p>费曼技巧四步：</p> <ol> <li><strong>选一个概念</strong>（比如「洛必达法则」），写在纸的最上面</li> <li><strong>假装讲给完全不懂的人听</strong>——这个站就是讲台，写成文章</li> <li><strong>卡壳的地方回去重学</strong>，直到能流畅讲出来</li> <li><strong>简化</strong>：把比喻和例子留下，把术语废话删掉</li> </ol> <h2 id="错题三问">错题三问<a class="anchor" href="#%e9%94%99%e9%a2%98%e4%b8%89%e9%97%ae" aria-label="Link"></a></h2> <p>每道错题除了抄进对应章节的「易错点」，还问自己三个问题：</p>