2024年成考专升本《高等数学一》每日一练试题04月18日

<p class="introTit">单选题</p><p>1、设z＝x^3y,则<img src="https://img2.meite.com/questions/202212/01638841c6daa34.png" />＝（）.</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/01638841d11f555.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/01638841db06fde.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/01638841e614a34.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/01638841f115e39.png' /></li></ul><p>答 案：D</p><p>解 析：将x看为常数，因此z为y的指数函数，可知<img src="https://img2.meite.com/questions/202212/01638842059f002.png" />。</p><p>2、设<img src="https://img2.meite.com/questions/202211/176375987592f2d.png" />，则y'＝（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/176375988687ba4.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/1763759893ae533.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/176375989ed230a.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/17637598ab8c549.png' /></li></ul><p>答 案：C</p><p>解 析：y＝x⁴，则<img src="https://img2.meite.com/questions/202211/17637598c296ca6.png" />。</p><p>3、微分方程y'＝2y的通解为y＝（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/1763759f3d985f7.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/1763759f4a76a0e.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/1763759f5740e62.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/1763759f63867f3.png' /></li></ul><p>答 案：A</p><p>解 析：将方程y'＝2y分离变量得，<img src="https://img2.meite.com/questions/202211/1763759f959aacb.png" />。</p><p class="introTit">主观题</p><p>1、将<img src="https://img2.meite.com/questions/202211/186376ef6c190a5.png" />展开为x的幂级数．</p><p>答 案：解：因为<img src="https://img2.meite.com/questions/202211/186376ef8075916.png" />，所以<img src="https://img2.meite.com/questions/202211/186376ef9acef33.png" /><img src="https://img2.meite.com/questions/202211/186376efa5a3a71.png" /></p><p>2、求微分方程<img src="https://img2.meite.com/questions/202212/03638b01839ca7b.png" />的通解．</p><p>答 案：解：对应齐次微分方程的特征方程为<img src="https://img2.meite.com/questions/202212/03638b019490467.png" />，解得r₁＝3，r₂＝－2.所以齐次通解为<img src="https://img2.meite.com/questions/202212/03638b01b0cf47a.png" />。设方程的特解设为y*＝（Ax＋B）e^x，代入原微分方程可解得，A＝<img src="https://img2.meite.com/questions/202212/03638b01cb92f6c.png" />，B＝<img src="https://img2.meite.com/questions/202212/03638b01d9339ef.png" />．即非齐次微分方程特解为<img src="https://img2.meite.com/questions/202212/03638b01ea50130.png" />。所以微分方程<img src="https://img2.meite.com/questions/202212/03638b01f95ca36.png" />的通解为<img src="https://img2.meite.com/questions/202212/03638b0209e7294.png" />。</p><p>3、求函数<img src="https://img2.meite.com/questions/202211/30638706e12f5a2.png" />的极值及凹凸区间和拐点。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/30638706ed5abd3.png" />（2）令y'＝0，得x₁＝0，x₂＝2。令y''＝0，得<img src="https://img2.meite.com/questions/202211/3063870717e6086.png" />。<br />（3）列表如下：<br /><img src="https://img2.meite.com/questions/202211/30638707319f4aa.png" /><br />函数<img src="https://img2.meite.com/questions/202211/306387074267a52.png" />的极小值为y（0）＝0，极大值为<img src="https://img2.meite.com/questions/202211/3063870757924c9.png" />函数<img src="https://img2.meite.com/questions/202211/3063870769cdfb8.png" />的凹区间为<img src="https://img2.meite.com/questions/202211/306387078489c52.png" />函数<img src="https://img2.meite.com/questions/202211/306387079b6df7b.png" />的凸区间为<img src="https://img2.meite.com/questions/202211/30638707ae8b225.png" />函数<img src="https://img2.meite.com/questions/202211/30638707c0c95ac.png" />的拐点为<img src="https://img2.meite.com/questions/202211/30638707d78bac6.png" />与<img src="https://img2.meite.com/questions/202211/30638707f184046.png" /></p><p class="introTit">填空题</p><p>1、设<img src="https://img2.meite.com/questions/202303/176414039357643.png" />连续，<img src="https://img2.meite.com/questions/202303/176414039f427ba.png" />则<img src="https://img2.meite.com/questions/202303/17641403a690592.png" />=（）</p><p>答 案：<img src="https://img2.meite.com/questions/202303/17641403c2952e9.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202303/17641403e5b9944.png" /><img src="https://img2.meite.com/questions/202303/17641403f6d79a1.png" /><img src="https://img2.meite.com/questions/202303/17641404065059a.png" /><img src="https://img2.meite.com/questions/202303/1764140411cd478.png" /></p><p>2、<img src="https://img2.meite.com/questions/202211/29638568587fe7c.png" />的间断点为（）。</p><p>答 案：x＝－3</p><p>解 析：x＝－3时，<img src="https://img2.meite.com/questions/202211/2963856869b6ff9.png" />没有定义，因此x＝-3为间断点。</p><p>3、<img src="https://img2.meite.com/questions/202211/30638721b56b1bc.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/30638721c1b356f.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/30638721cc39719.png" />。</p><p class="introTit">简答题</p><p>1、求<img src="https://img2.meite.com/questions/202303/036401a0a3e00d2.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/036401afab6aba9.png" /></p>