2024年成考专升本《高等数学二》每日一练试题04月02日

<p class="introTit">判断题</p><p>1、若<img src="https://img2.meite.com/questions/202307/1364af5780970e6.png" />，则<img src="https://img2.meite.com/questions/202307/1364af578624075.png" />。（）  </p><p>答 案：错</p><p>解 析：<img src="https://img2.meite.com/questions/202212/06638ef8852e1bd.png" />所以<img src="https://img2.meite.com/questions/202212/06638ef88b66bc1.png" />  </p><p class="introTit">单选题</p><p>1、下列变量在给定的变化过程中是无穷小量的是（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/206417ce8f2c3b1.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/206417cec99c4a4.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/206417ced214f2c.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/206417ceee8d55f.png' /></li></ul><p>答 案：D</p><p>解 析：由<img src="https://img2.meite.com/questions/202303/206417cf23b6e0f.png" />故由无穷小量知应选D，<img src="https://img2.meite.com/questions/202303/206417cf4bbe3bd.png" /></p><p>2、设f'(x)在－闭区间[0，1]上连续，则曲线y＝f(x)与直线x＝0，x＝1和y＝0所围成的平面图形的面积等于（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/0763902c77052f0.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/0763902c82b2b13.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/0763902c8fc1abf.png' /></li><li>D：不确定</li></ul><p>答 案：C</p><p>解 析：由定积分的几何意义可知，当在区间[a，b]上<img src="https://img2.meite.com/questions/202212/0763902ca72050e.png" />时，<img src="https://img2.meite.com/questions/202212/0763902dca58e8c.png" />表示曲线y＝f（x）与两条直线x=a,x=b以及x轴所围成的曲边梯形的面积；当在区间[a，b]上<img src="https://img2.meite.com/questions/202212/0763902e65ec6bd.png" />时，<img src="https://img2.meite.com/questions/202212/0763902e72890cd.png" />表示曲线y＝f（x）与两条直线x=a,x=b以及x轴所围成的曲边梯形面积的负值</p><p class="introTit">主观题</p><p>1、袋中有10个乒乓球．其中6个白球、4个黄球，随机地抽取两次，每次取1个，不放回．设A＝{第一次取到白球），B＝{第二次取到白球}，求P（B|A）．</p><p>答 案：解：因为样本空间的基本事件有<img src="https://img2.meite.com/questions/202212/08639181d5ba642.png" />个．而AB表示第一次取白球且第二次也取白球，故引起事件AB的基本事件有<img src="https://img2.meite.com/questions/202212/08639181e48c844.png" />个，所以<img src="https://img2.meite.com/questions/202212/08639181f538b34.png" />而<img src="https://img2.meite.com/questions/202212/086391820301908.png" />；所以<img src="https://img2.meite.com/questions/202212/0863918212099fd.png" />；</p><p>2、设事件A与B相互独立，<img src="https://img2.meite.com/questions/202212/0863917baf10c70.png" />，<img src="https://img2.meite.com/questions/202212/0863917bba46883.png" />，<img src="https://img2.meite.com/questions/202212/0863917bc65a5bd.png" />，求q．</p><p>答 案：解：因为事件A与B相互独立，故<img src="https://img2.meite.com/questions/202212/0863917c1276717.png" />，<img src="https://img2.meite.com/questions/202212/0863917c219eabb.png" />，即<img src="https://img2.meite.com/questions/202212/0863917c331e4b2.png" />，解得<img src="https://img2.meite.com/questions/202212/0863917c4c3ff46.png" />＝<img src="https://img2.meite.com/questions/202212/0863917c58641ed.png" />．</p><p class="introTit">填空题</p><p>1、<img src="https://img2.meite.com/questions/202212/0863914040da2e2.png" />（）</p><p>答 案：<img src="https://img2.meite.com/questions/202212/08639140519a78c.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202212/08639140602075c.png" /></p><p>2、已知<img src="https://img2.meite.com/questions/202212/076390322a88dff.png" />，且f(x)在[a,b]连续，则由曲线y=f(x)，x＝a、x＝b及x轴围成的平面图形的面积A＝()．</p><p>答 案：<img src="https://img2.meite.com/questions/202212/076390324a9cdf2.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202212/0763903255a4c9d.png" />，则有<img src="https://img2.meite.com/questions/202212/076390326234373.png" /></p><p class="introTit">简答题</p><p>1、求函数<img src="https://img2.meite.com/questions/202212/06638ed915483af.png" />的单调区间、极值及凹凸区间．  </p><p>答 案：函数定义域为<img src="https://img2.meite.com/questions/202212/06638ed949685f2.png" /> 求导得<img src="https://img2.meite.com/questions/202212/06638ed953f3056.png" />令<img src="https://img2.meite.com/questions/202212/06638ed95e80a90.png" />得<img src="https://img2.meite.com/questions/202212/06638ed966b2b31.png" /> 列表得<img src="https://img2.meite.com/questions/202212/06638ed978b781e.png" /> 函数<img src="https://img2.meite.com/questions/202212/06638ed9856cfaa.png" />的单调增加区间为<img src="https://img2.meite.com/questions/202212/06638ed9b10a910.png" />单调减少区间为<img src="https://img2.meite.com/questions/202212/06638ed9bd94c42.png" /><img src="https://img2.meite.com/questions/202212/06638ed9c35852d.png" />为极大值，<img src="https://img2.meite.com/questions/202212/06638ed9cf63ef2.png" />极小值；凸区间为<img src="https://img2.meite.com/questions/202212/06638ed9ee9bb53.png" />凹区间为<img src="https://img2.meite.com/questions/202212/06638eda01249f0.png" />。</p><p>2、已知函数f(x)=ax³-bx²+cx在区间<img src="https://img2.meite.com/questions/202212/07639002a0c63f6.png" />内是奇函数，且当x＝1时，f(x)有极小值<img src="https://img2.meite.com/questions/202212/07639002ba72fe4.png" />，求另一个极值及此曲线的拐点．  </p><p>答 案：f(x)=ax³-bx²+cx，<img src="https://img2.meite.com/questions/202212/07639002da708ed.png" /> 由于f(x)是奇函数，则必有x²的系数为0，即b=0. <img src="https://img2.meite.com/questions/202212/07639003148fbf6.png" />即a+c=<img src="https://img2.meite.com/questions/202212/076390032089cb2.png" />，<img src="https://img2.meite.com/questions/202212/0763900324b36b3.png" />得3a+c=0．解得a=<img src="https://img2.meite.com/questions/202212/076390033fe0499.png" />c=<img src="https://img2.meite.com/questions/202212/0763900347cfddd.png" /> 此时<img src="https://img2.meite.com/questions/202212/07639003591bd3b.png" /> 令<img src="https://img2.meite.com/questions/202212/0763900361d0796.png" />得<img src="https://img2.meite.com/questions/202212/07639003674342f.png" /><img src="https://img2.meite.com/questions/202212/076390036dd8f49.png" />所以<img src="https://img2.meite.com/questions/202212/076390037794ea3.png" />为极大值，<img src="https://img2.meite.com/questions/202212/07639003826209a.png" />得x=0，x<0时，<img src="https://img2.meite.com/questions/202212/0763900396d70f8.png" /> 所以（0，0）为曲线的拐点．</p>