2024年成考专升本《高等数学一》每日一练试题06月04日

<p class="introTit">单选题</p><p>1、微分方程<img src="https://img2.meite.com/questions/202212/0163886a137ea79.png" />的通解为（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/0163886a1ea5b29.png' /></li><li>B：y＝<img src='https://img2.meite.com/questions/202212/0163886a26e68ce.png' /></li><li>C：y＝<img src='https://img2.meite.com/questions/202212/0163886a301f771.png' /></li><li>D：y＝<img src='https://img2.meite.com/questions/202212/0163886a39edf14.png' /></li></ul><p>答 案：D</p><p>解 析：由一阶线性微分方程的通解公式可知<img src="https://img2.meite.com/questions/202212/0163886a4b5803d.png" />的通解为<img src="https://img2.meite.com/questions/202212/0163886a5b4915b.png" /></p><p>2、方程x²＋2y²－z²＝0表示的曲面是（）。</p><ul><li>A：椭球面</li><li>B：锥面</li><li>C：柱面</li><li>D：平面</li></ul><p>答 案：B</p><p>解 析：二次锥面的标准方程为：<img src="https://img2.meite.com/questions/202212/0163881b95e7949.png" />，故所给曲面为锥面。</p><p>3、当x→0时，下列函数以零为极限的是（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/28638483f35f4ab.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/28638483fc4275c.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/2863848407d6c9f.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/286384841540851.png' /></li></ul><p>答 案：C</p><p>解 析：A项，<img src="https://img2.meite.com/questions/202211/28638484209200a.png" />；B项，<img src="https://img2.meite.com/questions/202211/286384842e99e0d.png" />；C项，<img src="https://img2.meite.com/questions/202211/286384843b1594e.png" />；D项，<img src="https://img2.meite.com/questions/202211/2863848445c88a3.png" />不存在。</p><p class="introTit">主观题</p><p>1、求y＝<img src="https://img2.meite.com/questions/202211/306387018466ce4.png" />的一阶导数y'。</p><p>答 案：解：两边取对数得<img src="https://img2.meite.com/questions/202211/306387019d12b78.png" /><img src="https://img2.meite.com/questions/202211/30638701ab3b7a1.png" />两边对x求导得<img src="https://img2.meite.com/questions/202211/30638701d789e30.png" />故<img src="https://img2.meite.com/questions/202211/306387020ad880a.png" /></p><p>2、求微分方程<img src="https://img2.meite.com/questions/202212/03638ac38e5fd44.png" />的通解。</p><p>答 案：解：对应的齐次方程为<img src="https://img2.meite.com/questions/202212/03638ac39bb126c.png" />。特征方程<img src="https://img2.meite.com/questions/202212/03638ac3abef614.png" />，特征根<img src="https://img2.meite.com/questions/202212/03638ac3bb18486.png" />齐次方程通解为<img src="https://img2.meite.com/questions/202212/03638ac3c81e1f0.png" />原方程特解为<img src="https://img2.meite.com/questions/202212/03638ac3d7442b3.png" />，代入原方程可得<img src="https://img2.meite.com/questions/202212/03638ac3ea0899f.png" />，因此<img src="https://img2.meite.com/questions/202212/03638ac3fe7cb5d.png" />。<br />方程通解为<img src="https://img2.meite.com/questions/202212/03638ac40c27e0d.png" /></p><p>3、设<img src="https://img2.meite.com/questions/202212/03638aff4f737d0.png" />存在且<img src="https://img2.meite.com/questions/202212/03638aff59d37e0.png" />，求<img src="https://img2.meite.com/questions/202212/03638aff6554eac.png" /></p><p>答 案：解：设<img src="https://img2.meite.com/questions/202212/03638aff7078ded.png" />对<img src="https://img2.meite.com/questions/202212/03638aff830ad7a.png" />两边同时求极限，得<img src="https://img2.meite.com/questions/202212/03638affa03127b.png" />，即<img src="https://img2.meite.com/questions/202212/03638affaad78d4.png" />，得<img src="https://img2.meite.com/questions/202212/03638affb8a0ed1.png" />。</p><p class="introTit">填空题</p><p>1、微分方程dy＋xdx＝0的通解为（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/176375d75458c80.png" /></p><p>解 析：所给方程为可分离变量方程，分离变量得，dy＝-xdx，等式两边分别积分<img src="https://img2.meite.com/questions/202211/176375d772333c4.png" /></p><p>2、设y＝sin（x＋2），则y'＝（）。</p><p>答 案：cos（x＋2）</p><p>解 析：<img src="https://img2.meite.com/questions/202211/166374963d23a47.png" /></p><p>3、<img src="https://img2.meite.com/questions/202211/16637457808ddc2.png" />＝（）。</p><p>答 案：2e</p><p>解 析：<img src="https://img2.meite.com/questions/202211/166374578574bd6.png" /></p><p class="introTit">简答题</p><p>1、<img src="https://img2.meite.com/questions/202303/1764140b5eb19d9.png" /></p><p>答 案：<img src="https://img2.meite.com/questions/202303/1764140b998c053.png" /></p>