2024年成考专升本《高等数学一》每日一练试题04月03日

<p class="introTit">单选题</p><p>1、如果级数<img src="https://img2.meite.com/questions/202212/016388565aa2005.png" />收敛，那么以下级数收敛的是（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/01638856693a12a.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/01638856729a2d8.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/016388567c54e3e.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/0163885686f3f8b.png' /></li></ul><p>答 案：A</p><p>解 析：A项。级数<img src="https://img2.meite.com/questions/202212/016388569477790.png" />收敛，则<img src="https://img2.meite.com/questions/202212/01638856a1d6cfc.png" />收敛；由极限收敛的必要条件可知，<img src="https://img2.meite.com/questions/202212/01638856b353526.png" />＝0，则B项，<img src="https://img2.meite.com/questions/202212/01638856c5ef172.png" />＝1；C项，<img src="https://img2.meite.com/questions/202212/01638856d39abc7.png" />；D项，<img src="https://img2.meite.com/questions/202212/01638856e95c74d.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、当x→0时，<img src="https://img2.meite.com/questions/202211/2863848346e0955.png" />与1-cosx比较，可得（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/286384835ef41b5.png' />是较1-cosx高阶的无穷小量</li><li>B：<img src='https://img2.meite.com/questions/202211/286384836049dd1.png' />是较1-cosx低阶的无穷小量</li><li>C：<img src='https://img2.meite.com/questions/202211/2863848361ac933.png' />与1-cosx是同阶无穷小量，但不是等价无穷小量</li><li>D：<img src='https://img2.meite.com/questions/202211/2863848363a9f99.png' />与1-cosx是等价无穷小量</li></ul><p>答 案：B</p><p>解 析：因为<img src="https://img2.meite.com/questions/202211/28638483a1af7dc.png" />，所以<img src="https://img2.meite.com/questions/202211/28638483ac91a76.png" />是较1-cosx的低阶无穷小量。</p><p class="introTit">主观题</p><p>1、试证：当x＞0时，有不等式<img src="https://img2.meite.com/questions/202212/03638affc6aa0f3.png" /></p><p>答 案：证：先证x＞sinx（x＞0）。设f（x）＝x－sinx，则f（x）＝1－cosx≥0（x＞0），所以f（x）为单调递增函数，于是对x＞0有f（x）＞f（0）＝0，即x－sinx＞0，亦即x＞sinx（x＞0）。再证<img src="https://img2.meite.com/questions/202212/03638affee811dd.png" /><br />令<img src="https://img2.meite.com/questions/202212/03638afffc44f70.png" /><br />则<img src="https://img2.meite.com/questions/202212/03638b0009b99c4.png" />，所以g'(x)单调递增，又g'(x)＝0，可知g'(x)＞g'(0)＝0（x＞0），那么有g（x）单调递增，又g（0）＝0，可知g（x）＞g（0）＝0（x＞0），所以<img src="https://img2.meite.com/questions/202212/03638b0054d6d06.png" />即<img src="https://img2.meite.com/questions/202212/03638b0063d9343.png" /><br />综上可得：当x＞0时，<img src="https://img2.meite.com/questions/202212/03638b007c4f31d.png" />。</p><p>2、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>3、设<img src="https://img2.meite.com/questions/202211/2963856c2de4338.png" />求C的值。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/2963856c7163364.png" />则<img src="https://img2.meite.com/questions/202211/2963856cb44e6d7.png" />，有<img src="https://img2.meite.com/questions/202211/2963856cc277361.png" />，<img src="https://img2.meite.com/questions/202211/2963856cd1025c5.png" />。</p><p class="introTit">填空题</p><p>1、<img src="https://img2.meite.com/questions/202303/1764141bea38c4f.png" />（）</p><p>答 案：<img src="https://img2.meite.com/questions/202303/1764141c03d308e.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202303/1764141c185c3e5.png" /><img src="https://img2.meite.com/questions/202303/1764141c1fe101b.png" /></p><p>2、微分方程y"＋2y'＋y＝0满足初始条件<img src="https://img2.meite.com/questions/202212/02638958b6bd20a.png" />，<img src="https://img2.meite.com/questions/202212/02638958c4c632d.png" />的特解是（）。</p><p>答 案：（2＋5x）e^-x</p><p>解 析：微分方程的特征方程为<img src="https://img2.meite.com/questions/202212/02638958dd1b432.png" />，得<img src="https://img2.meite.com/questions/202212/02638958ea972f8.png" />，微分方程的通解为<img src="https://img2.meite.com/questions/202212/02638958f9950e9.png" />.将<img src="https://img2.meite.com/questions/202212/0263895905dec75.png" />，<img src="https://img2.meite.com/questions/202212/0263895913c0ae4.png" />代入得<img src="https://img2.meite.com/questions/202212/0263895923131b1.png" />，<img src="https://img2.meite.com/questions/202212/026389592f3ee81.png" />，则<img src="https://img2.meite.com/questions/202212/026389593e3bb42.png" />．故微分方程通解为<img src="https://img2.meite.com/questions/202212/026389594fa6b48.png" />。</p><p>3、若级数<img src="https://img2.meite.com/questions/202212/0163885fc36d9dd.png" />条件收敛（其中k＞0为常数），则k的取值范围是（）。</p><p>答 案：0＜k≤l</p><p>解 析：k＞1时，级数各项取绝对值，得正项级数<img src="https://img2.meite.com/questions/202212/0163885fe0df37f.png" />，是收敛的p级数，从而原级数绝对收敛；当0＜k≤l时，由莱布尼茨交错级数收敛性条件可判明原级数条件收敛，因此应有0＜k≤1。</p><p class="introTit">简答题</p><p>1、<img src="https://img2.meite.com/questions/202303/17641426bb3e6fd.png" /></p><p>答 案：<img src="https://img2.meite.com/questions/202303/17641426cdbf5a7.png" /></p>