2024年成考专升本《高等数学一》每日一练试题08月23日

<p class="introTit">单选题</p><p>1、设y＝f（x）在点x₀的某邻域内可导，且<img src="https://img2.meite.com/questions/202211/296385cc977db31.png" />＝0，则点x₀一定是（）。</p><ul><li>A：极大值点</li><li>B：极小值点</li><li>C：驻点</li><li>D：拐点</li></ul><p>答 案：C</p><p>解 析：极值点是函数某段子区间的最值，一般在驻点或者不可导点取得；驻点是函数一阶导数为0的点对应的x值；拐点是凸曲线与凹曲线的连接点，当函数图像上的某点使函数的二阶导数为零，且三阶导数不为零时，这点即为函数的拐点；综上所述，点x₀为该函数的驻点。</p><p>2、<img src="https://img2.meite.com/questions/202211/176375ff5de2068.png" />（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/176375ff69711d4.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/176375ff7573db3.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/176375ff810ebb6.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/176375ff9317f47.png' /></li></ul><p>答 案：D</p><p>解 析：由不定积分的基本积分公式可得，<img src="https://img2.meite.com/questions/202211/176375ffa253b48.png" />。</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、求微分方程<img src="https://img2.meite.com/questions/202212/03638ac41f0fd3e.png" />的通解。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638ac42ad5381.png" />的特征值方程为<img src="https://img2.meite.com/questions/202212/03638ac43c4b5ff.png" />，则<img src="https://img2.meite.com/questions/202212/03638ac449853b1.png" />；故齐次微分方程的通解为<img src="https://img2.meite.com/questions/202212/03638ac45952791.png" />。由题意设原微分方程的特解为<img src="https://img2.meite.com/questions/202212/03638ac46ccee9d.png" />，则有<img src="https://img2.meite.com/questions/202212/03638ac47aa72ad.png" />，得<img src="https://img2.meite.com/questions/202212/03638ac4860b1ed.png" />。即微分方程的通解为<img src="https://img2.meite.com/questions/202212/03638ac4946b4d5.png" />。</p><p>2、求<img src="https://img2.meite.com/questions/202211/166374ad60c8c33.png" /></p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/166374ad7a47718.png" /></p><p>3、求微分方程<img src="https://img2.meite.com/questions/202211/166374add1c6145.png" />的通解．</p><p>答 案：解：原方程对应的齐次微分方程为<img src="https://img2.meite.com/questions/202211/166374ade8112f4.png" />特征方程为<img src="https://img2.meite.com/questions/202211/166374adf9e847c.png" />特征根为x₁＝-1，x₂＝3，<br />齐次方程的通解为<img src="https://img2.meite.com/questions/202211/166374ae1b2107f.png" /><br />设原方程的特解为<img src="https://img2.meite.com/questions/202211/166374ae3374b9e.png" />＝A，代入原方程可得<img src="https://img2.meite.com/questions/202211/166374ae434e613.png" />＝-1。<br />所以原方程的通解为<img src="https://img2.meite.com/questions/202211/166374ae5b9e010.png" />（C₁，C₂为任意常数）</p><p class="introTit">填空题</p><p>1、微分方程<img src="https://img2.meite.com/questions/202212/0163886e468bd4b.png" />的通解是（）。</p><p>答 案：y＝<img src="https://img2.meite.com/questions/202212/0163886e528ad3e.png" /></p><p>解 析：该方程是一阶线性方程，其中<img src="https://img2.meite.com/questions/202212/0163886e62849ed.png" />由通解公式，有<img src="https://img2.meite.com/questions/202212/0163886e7515c8d.png" />因为<img src="https://img2.meite.com/questions/202212/0163886e90e9394.png" />所以<img src="https://img2.meite.com/questions/202212/0163886ea78b114.png" /></p><p>2、<img src="https://img2.meite.com/questions/202408/1666bec73d620ab.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bec7417342e.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666bec7458fa06.png" /></p><p>3、曲线<img src="https://img2.meite.com/questions/202211/306386d334837c7.png" />的铅直渐近线方程为（）。</p><p>答 案：x＝2</p><p>解 析：因为<img src="https://img2.meite.com/questions/202211/306386d34bea9e7.png" />，故曲线的铅直线渐近线方程x＝2。</p><p class="introTit">简答题</p><p>1、<img src="https://img2.meite.com/questions/202408/1666bf0d2601008.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bf0d2a9f522.png" /></p>