2024年成考专升本《高等数学一》每日一练试题07月16日

<p class="introTit">单选题</p><p>1、极限<img src="https://img2.meite.com/questions/202307/2464bdf1755ed09.png" />等于（）。</p><ul><li>A：5</li><li>B：<img src='https://img2.meite.com/questions/202211/2863847c806bcbe.png' /></li><li>C：3</li><li>D：0</li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202211/2863847c92afe5e.png" />。</p><p>2、曲线<img src="https://img2.meite.com/questions/202211/296385d4df8e71b.png" />的拐点是（）。</p><ul><li>A：（2，0）</li><li>B：（－2，0）</li><li>C：（1，0）</li><li>D：不存在</li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202211/296385d4f5b5abe.png" />，<img src="https://img2.meite.com/questions/202211/296385d5004f2d6.png" />，令<img src="https://img2.meite.com/questions/202211/296385d50b70b3e.png" />得x=-2，当x＜-2时<img src="https://img2.meite.com/questions/202211/296385d530ce548.png" />，当x＞-2时<img src="https://img2.meite.com/questions/202211/296385d54d0ae85.png" />，点（－2，0）为拐点。</p><p>3、下列级数中绝对收敛的是（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/016388576f03337.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/0163885779c5ce7.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/016388578377773.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/016388578e33b50.png' /></li></ul><p>答 案：C</p><p>解 析：本题考查绝对收敛的定义.A项，<img src="https://img2.meite.com/questions/202212/01638857a312410.png" />发散；B项，<img src="https://img2.meite.com/questions/202212/01638857b475ae0.png" />发散，即<img src="https://img2.meite.com/questions/202212/01638857c637a0a.png" />条件收敛；C项，<img src="https://img2.meite.com/questions/202212/01638857d6187ff.png" />收敛；D项，<img src="https://img2.meite.com/questions/202212/01638857ecd2e95.png" />发散。</p><p class="introTit">主观题</p><p>1、判断级数<img src="https://img2.meite.com/questions/202212/016388675422a30.png" />的敛散性。</p><p>答 案：解：令<img src="https://img2.meite.com/questions/202212/0163886766018da.png" />，则<img src="https://img2.meite.com/questions/202212/0163886777ec373.png" />，由于<img src="https://img2.meite.com/questions/202212/0163886785bde3b.png" />故有当<img src="https://img2.meite.com/questions/202212/016388678ff3e3f.png" />＜1，即a＞e时，该级数收敛；当<img src="https://img2.meite.com/questions/202212/016388678ff3e3f.png" />＞1，即a＜e时，该级数发散。</p><p>2、计算<img src="https://img2.meite.com/questions/202211/176375daf19ee29.png" /></p><p>答 案：解：利用洛必达法则，得<img src="https://img2.meite.com/questions/202211/176375db05a5eb7.png" /></p><p>3、求<img src="https://img2.meite.com/questions/202211/2963856ce27e6d8.png" /></p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/2963856cf281662.png" /><img src="https://img2.meite.com/questions/202211/2963856d0bd9b82.png" /></p><p class="introTit">填空题</p><p>1、设区域<img src="https://img2.meite.com/questions/202211/1663749b17373d3.png" />，则<img src="https://img2.meite.com/questions/202211/1663749b265759c.png" />（）。</p><p>答 案：3π</p><p>解 析：积分区域D为半径为1的圆域，其面积为π，因此<img src="https://img2.meite.com/questions/202211/1663749b4c66dfa.png" />。</p><p>2、微分方程xy'+y＝0满足y（1）＝1的解为y＝（）  </p><p>答 案：<img src="https://img2.meite.com/questions/202405/166645be1779835.png" /></p><p>解 析：由xy'+y＝0得<img src="https://img2.meite.com/questions/202405/166645be1dd3ade.png" />，通解为<img src="https://img2.meite.com/questions/202405/166645be235c986.png" />，将y（1）＝1代入通解，得C＝1,故所求的解为<img src="https://img2.meite.com/questions/202405/166645be2993b0d.png" /></p><p>3、幂级数<img src="https://img2.meite.com/questions/202212/0163885e81d16c9.png" />的收敛半径是（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/0163885e8d29a3a.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202212/0163885e99c41cd.png" /><img src="https://img2.meite.com/questions/202212/0163885ea7f2ed3.png" />，当<img src="https://img2.meite.com/questions/202212/0163885eb6f2f57.png" />时，级数收敛，故收敛区间为<img src="https://img2.meite.com/questions/202212/0163885ec622071.png" />，收敛半径<img src="https://img2.meite.com/questions/202212/0163885eceec350.png" />。</p><p class="introTit">简答题</p><p>1、证明：当x>0时<img src="https://img2.meite.com/questions/202303/036401a13497bbc.png" />>1+x.  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/036401aff003c5f.png" /></p>