2024年成考专升本《高等数学一》每日一练试题05月14日

<p class="introTit">单选题</p><p>1、若级数<img src="https://img2.meite.com/questions/202212/01638854cfa0e9b.png" />收敛，则<img src="https://img2.meite.com/questions/202212/01638854db021ec.png" />（）。</p><ul><li>A：发散</li><li>B：条件收敛</li><li>C：绝对收敛</li><li>D：无法判定敛散性</li></ul><p>答 案：C</p><p>解 析：级数绝对收敛的性质可知，<img src="https://img2.meite.com/questions/202212/01638854ef9f908.png" />收敛，则<img src="https://img2.meite.com/questions/202212/01638855016b01e.png" />收敛，且为绝对收敛。</p><p>2、曲线<img src="https://img2.meite.com/questions/202211/296385d4df8e71b.png" />的拐点是（）。</p><ul><li>A：（2，0）</li><li>B：（－2，0）</li><li>C：（1，0）</li><li>D：不存在</li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202211/296385d4f5b5abe.png" />，<img src="https://img2.meite.com/questions/202211/296385d5004f2d6.png" />，令<img src="https://img2.meite.com/questions/202211/296385d50b70b3e.png" />得x=-2，当x＜-2时<img src="https://img2.meite.com/questions/202211/296385d530ce548.png" />，当x＞-2时<img src="https://img2.meite.com/questions/202211/296385d54d0ae85.png" />，点（－2，0）为拐点。</p><p>3、<img src="https://img2.meite.com/questions/202211/16637453b2a966c.png" />＝（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/16637453b864f37.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/16637453bce41b0.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/16637453c14c4df.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/16637453c621372.png' /></li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202211/16637453cd1b78b.png" />。</p><p class="introTit">主观题</p><p>1、求<img src="https://img2.meite.com/questions/202211/186376ebf64f60f.png" /></p><p>答 案：解：用洛必达法则，得<img src="https://img2.meite.com/questions/202211/186376ec0b31598.png" /></p><p>2、在曲线<img src="https://img2.meite.com/questions/202212/01638815804b4d3.png" />上求一点M₀，使得如图中阴影部分的面积S₁与S₂之和S最小。<img src="https://img2.meite.com/questions/202212/01638815b132ea0.png" /></p><p>答 案：解：设点M₀的横坐标为x₀，则有<img src="https://img2.meite.com/questions/202212/01638815de92308.png" />则<img src="https://img2.meite.com/questions/202212/01638815f1da54d.png" />S为x₀的函数，将上式对x₀求导得<img src="https://img2.meite.com/questions/202212/016388161216b07.png" />令S'=0,得<img src="https://img2.meite.com/questions/202212/01638816367b2e9.png" />，所以<img src="https://img2.meite.com/questions/202212/016388164ebc9e7.png" />由于只有唯一的驻点，所以<img src="https://img2.meite.com/questions/202212/0163881664ad3a9.png" />则点M₀的坐标为<img src="https://img2.meite.com/questions/202212/016388167cc9784.png" />为所求。</p><p>3、求<img src="https://img2.meite.com/questions/202211/186376ef0f207ed.png" />其中<img src="https://img2.meite.com/questions/202211/186376ef20557ae.png" /></p><p>答 案：解：D在极坐标系下可以表示为<img src="https://img2.meite.com/questions/202211/186376ef330a6ec.png" />则<img src="https://img2.meite.com/questions/202211/186376ef4baac2c.png" /><img src="https://img2.meite.com/questions/202211/186376ef5c72ad1.png" /></p><p class="introTit">填空题</p><p>1、设y＝5＋lnx，则dy＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/16637457c2bf6b0.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/16637457c7adbbf.png" /></p><p>2、设函数z＝f（x，y）可微，（x₀，y₀）为其极值点，则<img src="https://img2.meite.com/questions/202211/176375a3a0e8b5c.png" />（）。</p><p>答 案：</p><p>解 析：由于z＝f（x，y）可微，则偏导数必定存在，再由二元函数极值的必要条件可知，若点（x₀，y₀）为z＝f（x，y）的极值点，且<img src="https://img2.meite.com/questions/202211/176375a3d3cfd7f.png" />，<img src="https://img2.meite.com/questions/202211/176375a3e9da99a.png" />在点（x₀，y₀）处存在，则必有<img src="https://img2.meite.com/questions/202211/176375a408b43ee.png" /></p><p>3、已知<img src="https://img2.meite.com/questions/202211/3063872438d1a7f.png" />，则<img src="https://img2.meite.com/questions/202211/30638724485b269.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/3063872454cd35d.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/3063872462b1da2.png" /><img src="https://img2.meite.com/questions/202211/306387246f61d8b.png" /><img src="https://img2.meite.com/questions/202211/306387247f5c8bb.png" /></p><p class="introTit">简答题</p><p>1、求微分方程<img src="https://img2.meite.com/questions/202303/036401a0b4bb252.png" />满足初值条件<img src="https://img2.meite.com/questions/202303/036401a0c888c77.png" />的特解  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/036401afc1cb58a.png" /> <img src="https://img2.meite.com/questions/202303/036401afd313124.png" />  </p>