2024年成考专升本《高等数学一》每日一练试题12月12日

<p class="introTit">单选题</p><p>1、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>2、<img src="https://img2.meite.com/questions/202212/03638ae390d4700.png" />（）。</p><ul><li>A：0</li><li>B：1</li><li>C：<img src='https://img2.meite.com/questions/202212/03638ae39c54e94.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/03638ae3a5e5de9.png' /></li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202212/03638ae3b60d066.png" />。</p><p>3、设<img src="https://img2.meite.com/questions/202211/176375affc9e8fd.png" />，则<img src="https://img2.meite.com/questions/202211/176375b0105d066.png" />（）。</p><ul><li>A：2xy＋y²8.x²＋2xy<br /></li><li>C：4xy</li><li>D：x²＋y²</li></ul><p>答 案：A</p><p>解 析：对二元函数z，求<img src="https://img2.meite.com/questions/202211/176375b03781605.png" />时，将y看作常量，则<img src="https://img2.meite.com/questions/202211/176375b047cade5.png" />。</p><p class="introTit">主观题</p><p>1、将函数<img src="https://img2.meite.com/questions/202211/176375acbfe6f24.png" />展开成x的幂级数，并指出其收敛区间</p><p>答 案：解：因为<img src="https://img2.meite.com/questions/202211/176375acd243164.png" />所以<img src="https://img2.meite.com/questions/202211/176375ace684eb8.png" />其中5x∈（-1，1），得收敛区间<img src="https://img2.meite.com/questions/202211/176375ad137c097.png" /></p><p>2、在曲线<img src="https://img2.meite.com/questions/202212/01638815804b4d3.png" />上求一点M₀，使得如图中阴影部分的面积S₁与S₂之和S最小。<img src="https://img2.meite.com/questions/202212/01638815b132ea0.png" /></p><p>答 案：解：设点M₀的横坐标为x₀，则有<img src="https://img2.meite.com/questions/202212/01638815de92308.png" />则<img src="https://img2.meite.com/questions/202212/01638815f1da54d.png" />S为x₀的函数，将上式对x₀求导得<img src="https://img2.meite.com/questions/202212/016388161216b07.png" />令S'=0,得<img src="https://img2.meite.com/questions/202212/01638816367b2e9.png" />，所以<img src="https://img2.meite.com/questions/202212/016388164ebc9e7.png" />由于只有唯一的驻点，所以<img src="https://img2.meite.com/questions/202212/0163881664ad3a9.png" />则点M₀的坐标为<img src="https://img2.meite.com/questions/202212/016388167cc9784.png" />为所求。</p><p>3、计算<img src="https://img2.meite.com/questions/202212/0163880f6fcc230.png" />dx。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/0163880f85e1928.png" /><img src="https://img2.meite.com/questions/202212/0163880f93955c3.png" /></p><p class="introTit">填空题</p><p>1、微分方程<img src="https://img2.meite.com/questions/202212/0163886e0a7762f.png" />的通解是（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/0163886e15023dd.png" /></p><p>解 析：分离变量，得<img src="https://img2.meite.com/questions/202212/0163886e2383f9f.png" />，两边同时积分，有<img src="https://img2.meite.com/questions/202212/0163886e32a54ae.png" />。</p><p>2、<img src="https://img2.meite.com/questions/202408/1666beb0c50b9a6.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666beb0c91a5df.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666beb0cd76b63.png" /></p><p>3、设<img src="https://img2.meite.com/questions/202211/306386b23bb2946.png" />则y''＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/306386b24d5ce88.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/306386b2615b5e4.png" /><img src="https://img2.meite.com/questions/202211/306386b367aad23.png" /></p><p class="introTit">简答题</p><p>1、<img src="https://img2.meite.com/questions/202408/1666bf0652aea6c.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bf0657783e6.png" /></p>