2024年成考专升本《高等数学一》每日一练试题05月13日

<p class="introTit">单选题</p><p>1、设f（x）＝<img src="https://img2.meite.com/questions/202211/29638564a7b9a5d.png" />在<img src="https://img2.meite.com/questions/202211/29638564b6ea729.png" />上连续，且<img src="https://img2.meite.com/questions/202211/29638564ca6d938.png" />，则常数a，b满足（）。</p><ul><li>A：a＜0，b≤0</li><li>B：a＞0，b＞0</li><li>C：a＜0，b＜0</li><li>D：a≥0，b＜0</li></ul><p>答 案：D</p><p>解 析：因为<img src="https://img2.meite.com/questions/202211/29638564e44f0ae.png" />在<img src="https://img2.meite.com/questions/202211/29638564fd5e758.png" />上连续，所以<img src="https://img2.meite.com/questions/202211/296385650bc10f1.png" />因<img src="https://img2.meite.com/questions/202211/296385652266523.png" />则a≥0，又因为<img src="https://img2.meite.com/questions/202211/296385653c413a3.png" />所以<img src="https://img2.meite.com/questions/202211/296385654a65f73.png" />时，必有<img src="https://img2.meite.com/questions/202211/296385655769208.png" />因此应有b＜0。</p><p>2、级数<img src="https://img2.meite.com/questions/202303/0364019dcba4672.png" />的收敛半径为（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/0364019dee2b35c.png' /></li><li>B：1</li><li>C：<img src='https://img2.meite.com/questions/202303/0364019df7861a9.png' /></li><li>D：2</li></ul><p>答 案：B</p><p>解 析：由题可知<img src="https://img2.meite.com/questions/202303/036401ac6ceeef1.png" />因此级数的收敛半径为<img src="https://img2.meite.com/questions/202303/036401ac8805bbb.png" /></p><p>3、设函数<img src="https://img2.meite.com/questions/202212/03638ae5dc2fec1.png" />，则f(x)的导数f'(x)=（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/03638ae60b3d035.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/03638ae61c595c3.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/03638ae627263a9.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/03638ae632bc237.png' /></li></ul><p>答 案：C</p><p>解 析：由可变限积分求导公式<img src="https://img2.meite.com/questions/202212/03638ae645024ba.png" />可知<img src="https://img2.meite.com/questions/202212/03638ae65923189.png" /></p><p class="introTit">主观题</p><p>1、将<img src="https://img2.meite.com/questions/202212/01638861372e412.png" />展开为x的幂级数。</p><p>答 案：解：因为<img src="https://img2.meite.com/questions/202212/0163886156cd240.png" />，<img src="https://img2.meite.com/questions/202212/016388616850530.png" />，所以<img src="https://img2.meite.com/questions/202212/016388617f85441.png" /></p><p>2、计算<img src="https://img2.meite.com/questions/202212/03638af03054207.png" />．</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638af04212449.png" />从而有<img src="https://img2.meite.com/questions/202212/03638af04f52136.png" />，所以<img src="https://img2.meite.com/questions/202212/03638af05e7d49e.png" /></p><p>3、设函数f（x）由<img src="https://img2.meite.com/questions/202211/176375a9a462105.png" />所确定，求<img src="https://img2.meite.com/questions/202211/176375a9b68f239.png" /></p><p>答 案：解：方法一：方程两边同时对x求导，得<img src="https://img2.meite.com/questions/202211/176375a9cd89294.png" />即<img src="https://img2.meite.com/questions/202211/176375a9de829b7.png" />故<img src="https://img2.meite.com/questions/202211/176375a9ec6e2cb.png" /><br />方法二：设<img src="https://img2.meite.com/questions/202211/176375a9fdad664.png" />，<br />则<img src="https://img2.meite.com/questions/202211/176375aa1004c84.png" /><img src="https://img2.meite.com/questions/202211/176375aa1f17e53.png" /></p><p class="introTit">填空题</p><p>1、幂级数<img src="https://img2.meite.com/questions/202211/16637458e6bd9e4.png" />的收敛半径R＝（）。</p><p>答 案：1</p><p>解 析：对于级数<img src="https://img2.meite.com/questions/202211/16637458f013377.png" />，<img src="https://img2.meite.com/questions/202211/16637458f607186.png" />，<img src="https://img2.meite.com/questions/202211/16637458fb797e9.png" />。</p><p>2、<img src="https://img2.meite.com/questions/202211/3063872295a7417.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/306387229feed93.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/30638722b02d14e.png" />。</p><p>3、微分方程<img src="https://img2.meite.com/questions/202303/17641404d969882.png" />的通解为（）</p><p>答 案：<img src="https://img2.meite.com/questions/202303/17641404f46d77d.png" /></p><p>解 析：微分方程<img src="https://img2.meite.com/questions/202303/17641404ffe684d.png" />的特征方程为<img src="https://img2.meite.com/questions/202303/176414051194161.png" />特征根为<img src="https://img2.meite.com/questions/202303/176414052db0eda.png" />所以微分方程的通解为<img src="https://img2.meite.com/questions/202303/1764140544a698a.png" /></p><p class="introTit">简答题</p><p>1、设f(x)=<img src="https://img2.meite.com/questions/202303/1764140cba28156.png" />在x=0连续，试确定A,B.</p><p>答 案：<img src="https://img2.meite.com/questions/202303/1764140ce7d0fb9.png" /> <img src="https://img2.meite.com/questions/202303/1764140cf387f0c.png" /> 欲使f(x)在x=0处连续，应有2A=4=B+1，所以A=2,B=3.  </p>