2024年成考专升本《高等数学一》每日一练试题07月31日

<p class="introTit">单选题</p><p>1、设<img src="https://img2.meite.com/questions/202405/166645bb843eba8.png" />,则y'＝（）。  </p><ul><li>A：<img src='https://img2.meite.com/questions/202405/166645bb8aac47c.png' /></li><li>B：<img src='https://img2.meite.com/questions/202405/166645bb8f51282.png' /></li><li>C：<img src='https://img2.meite.com/questions/202405/166645bb945a2f8.png' /></li><li>D：<img src='https://img2.meite.com/questions/202405/166645bb9a0f96a.png' /></li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202405/166645bb9f7a368.png" /></p><p>2、下列四项中，正确的是（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/3063871d2161e9c.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/3063871d2cdb389.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/3063871d39050ff.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/3063871d44a6079.png' /></li></ul><p>答 案：C</p><p>解 析：A项，<img src="https://img2.meite.com/questions/202211/3063871d503b11a.png" />在（-1,1）不连续；B项，<img src="https://img2.meite.com/questions/202211/3063871de19d022.png" />不存在；C项，<img src="https://img2.meite.com/questions/202211/3063871decd1627.png" />在（-1,1）为奇函数，所以<img src="https://img2.meite.com/questions/202211/3063871e0b4f3d9.png" />；D项，<img src="https://img2.meite.com/questions/202211/3063871e15c6603.png" />也不存在。</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、求过点M₀（0，2，4），且与两个平面π1，π2都平行的直线方程，其中<img src="https://img2.meite.com/questions/202212/03638af2a68be37.png" /></p><p>答 案：解：如果直线l平行于π1，则平面π1的法线向量n1必定垂直于直线l的方向向量s．同理，直线l平行于π2，则平面π2的法线向量n2必定满足n2⊥s．由向量积的定义可知，取<img src="https://img2.meite.com/questions/202212/03638af2cfcec2f.png" />由于直线l过点M₀（0，2，4），由直线的标准方程可知<img src="https://img2.meite.com/questions/202212/03638af2eb2ecba.png" />为所求直线方程。</p><p>2、求<img src="https://img2.meite.com/questions/202212/0163880ec16943e.png" />。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/0163880ed358bb0.png" /></p><p>3、设z＝（x，y）由<img src="https://img2.meite.com/questions/202212/0163884f384ccd0.png" />所确定，求dz。</p><p>答 案：解：设F（x，y，z）＝<img src="https://img2.meite.com/questions/202212/0163884f5cb8369.png" />，则<img src="https://img2.meite.com/questions/202212/0163884f77d14e4.png" /></p><p class="introTit">填空题</p><p>1、微分方程y'-2y=3的通解为＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/0163886f26ac016.png" /></p><p>解 析：分离变量<img src="https://img2.meite.com/questions/202212/0163886f3ac7856.png" />两边分别积分<img src="https://img2.meite.com/questions/202212/0163886f5084bb4.png" /><img src="https://img2.meite.com/questions/202212/0163886f5ca216e.png" />方程的通解为<img src="https://img2.meite.com/questions/202212/0163886f6f7619a.png" /></p><p>2、微分方程y'＝x＋1的通解为y＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/176375a42256bd4.png" /></p><p>解 析：方程为可分离变量方程，<img src="https://img2.meite.com/questions/202211/176375a439da652.png" />，等式两边分别积分<img src="https://img2.meite.com/questions/202211/176375a44b27a58.png" /></p><p>3、<img src="https://img2.meite.com/questions/202211/296385663fd47de.png" />（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/296385664b18199.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/296385665d40f4a.png" /></p><p class="introTit">简答题</p><p>1、设<img src="https://img2.meite.com/questions/202303/1764140acc4272d.png" />求常数a，b</p><p>答 案：<img src="https://img2.meite.com/questions/202303/1764140aef77cdc.png" /> 由此积分收敛知，应有b-a=0，即b=a， <img src="https://img2.meite.com/questions/202303/1764140b178b135.png" />  </p>