2024年成考专升本《高等数学一》每日一练试题09月26日

<p class="introTit">单选题</p><p>1、若f(x)为[a，b]上的连续函数，则<img src="https://img2.meite.com/questions/202211/30638711c237f0d.png" />（）。</p><ul><li>A：小于0</li><li>B：大于0</li><li>C：等于0</li><li>D：不确定</li></ul><p>答 案：C</p><p>解 析：f（x）为[a，b]上的连续函数，故<img src="https://img2.meite.com/questions/202211/30638711d4aa408.png" />存在，它为一个确定的常数，由定积分与变量无关的性质，可知<img src="https://img2.meite.com/questions/202211/30638711eabc5eb.png" />故<img src="https://img2.meite.com/questions/202211/30638711f6c6ef5.png" />＝0。</p><p>2、级数<img src="https://img2.meite.com/questions/202212/0163885515b244e.png" />（k为非零常数）（）。</p><ul><li>A：发散</li><li>B：绝对收敛</li><li>C：条件收敛</li><li>D：收敛性与k有关</li></ul><p>答 案：C</p><p>解 析：级数各项取绝对值得级数<img src="https://img2.meite.com/questions/202212/016388552b2e6d2.png" />为发散级数；由莱布尼茨判别法可知<img src="https://img2.meite.com/questions/202212/016388553fc0466.png" />收敛，故<img src="https://img2.meite.com/questions/202212/0163885551ad5f1.png" />为条件收敛。</p><p>3、曲线<img src="https://img2.meite.com/questions/202303/176413e20ef1385.png" />与其过原点的切线及y轴所围面积为（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/176413e22131363.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/176413e228b6826.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/176413e22d7bf72.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/176413e233b0b84.png' /></li></ul><p>答 案：A</p><p>解 析：设<img src="https://img2.meite.com/questions/202303/176413e24ca4a55.png" />为切点，则切线方程为<img src="https://img2.meite.com/questions/202303/176413e25bd7507.png" />联立<img src="https://img2.meite.com/questions/202303/176413e26a09694.png" />得<img src="https://img2.meite.com/questions/202303/176413e279016e2.png" />所以切线方程为y=ex，故所求面积为<img src="https://img2.meite.com/questions/202303/176413e29f7d093.png" /></p><p class="introTit">主观题</p><p>1、已知当x→0时，<img src="https://img2.meite.com/questions/202211/2963856d1f10115.png" />是等价无穷小量，求常数a的值。</p><p>答 案：解：因为当x→0时，<img src="https://img2.meite.com/questions/202211/2963856d36e0130.png" />是等价无穷小量，所以有<img src="https://img2.meite.com/questions/202211/2963856d485d2ef.png" />则<img src="https://img2.meite.com/questions/202211/2963856d5c23a0c.png" />解得a＝2。</p><p>2、计算<img src="https://img2.meite.com/questions/202212/03638b00b3592fa.png" />。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638b00c097bc9.png" /></p><p>3、判定级数<img src="https://img2.meite.com/questions/202212/03638af1a91015d.png" />的敛散性．</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638af1ba750d3.png" />含有参数a＞0，要分情况讨论：（1）如果0＜a＜1，则<img src="https://img2.meite.com/questions/202212/03638af1ce30308.png" />，由级数收敛的必要条件可知，原级数发散。（2）如果a＞1，令<img src="https://img2.meite.com/questions/202212/03638af1f837489.png" />＝<img src="https://img2.meite.com/questions/202212/03638af1fd403f6.png" />；因为<img src="https://img2.meite.com/questions/202212/03638af20aec47c.png" />＜1，因而<img src="https://img2.meite.com/questions/202212/03638af2197e35f.png" />是收敛的，比较法：<img src="https://img2.meite.com/questions/202212/03638af22c14c44.png" /><br />所以<img src="https://img2.meite.com/questions/202212/03638af2377e88e.png" />也收敛。<br />（3）如果a＝1，则<img src="https://img2.meite.com/questions/202212/03638af24ff2071.png" />所以<img src="https://img2.meite.com/questions/202212/03638af2632cc46.png" />，由级数收敛的必要条件可知，原级数发散。所以<img src="https://img2.meite.com/questions/202212/03638af274be0c7.png" /></p><p class="introTit">填空题</p><p>1、设z＝arctanxy，则<img src="https://img2.meite.com/questions/202212/016388488057895.png" />＋<img src="https://img2.meite.com/questions/202212/016388488b08cef.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/0163884895bf306.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202212/01638848cf2d5c9.png" />，故<img src="https://img2.meite.com/questions/202212/01638848de73a53.png" />。</p><p>2、<img src="https://img2.meite.com/questions/202408/1666beee8ddfcb8.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666beee918e6cd.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666beee950ba55.png" /></p><p>3、<img src="https://img2.meite.com/questions/202211/30638722026e1bf.png" />（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/306387220ce46d3.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/306387221aad9d8.png" /></p><p class="introTit">简答题</p><p>1、求方程<img src="https://img2.meite.com/questions/202303/17641427a20cfc0.png" />的通解。  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/17641427b60f0c0.png" /></p>