2024年成考专升本《高等数学一》每日一练试题12月04日

<p class="introTit">单选题</p><p>1、设y₁、y₂是二阶常系数线性齐次方程<img src="https://img2.meite.com/questions/202212/0163886ae3f2fde.png" />的两个特解，C₁、C₂为两个任意常数，则下列命题中正确的是（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/0163886b0926a4b.png' />为该方程的通解</li><li>B：<img src='https://img2.meite.com/questions/202212/0163886b1b6e4b2.png' />不可能是该方程的通解</li><li>C：<img src='https://img2.meite.com/questions/202212/0163886b25e2368.png' />为该方程的解</li><li>D：<img src='https://img2.meite.com/questions/202212/0163886b306966b.png' />不是该方程的解</li></ul><p>答 案：C</p><p>解 析：由线性方程解的结构定理知<img src="https://img2.meite.com/questions/202212/0163886b3e67058.png" />为该方程的解，题中没说明y₁、y₂是否线性无关，无法判断<img src="https://img2.meite.com/questions/202212/0163886b5f7957c.png" />是否为通解。</p><p>2、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>3、<img src="https://img2.meite.com/questions/202408/1666bef8e4666d7.png" />（）。  </p><ul><li>A：<img src='https://img2.meite.com/questions/202408/1666bef8e7c7ad9.png' /></li><li>B：<img src='https://img2.meite.com/questions/202408/1666bef8eb8a209.png' /></li><li>C：π</li><li>D：2π</li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666bef8f981d9b.png" /><img src="https://img2.meite.com/questions/202408/1666bef8fdd1f8e.png" /></p><p class="introTit">主观题</p><p>1、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>2、设<img src="https://img2.meite.com/questions/202211/166374afc293c27.png" />，求<img src="https://img2.meite.com/questions/202211/166374afcf46756.png" />。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/166374afe188e20.png" /></p><p>3、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 class="introTit">填空题</p><p>1、函数y=cosx在<img src="https://img2.meite.com/questions/202303/17641418a57013d.png" />上满足罗尔定理，则<img src="https://img2.meite.com/questions/202303/17641418ba67da0.png" />=（）  </p><p>答 案：π</p><p>解 析：cos2π-cos0=<img src="https://img2.meite.com/questions/202303/17641418f1e8eeb.png" />即<img src="https://img2.meite.com/questions/202303/17641418fc6f1fa.png" />所以<img src="https://img2.meite.com/questions/202303/1764141903b7030.png" /></p><p>2、<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>3、定积分<img src="https://img2.meite.com/questions/202211/30638723be8da0b.png" />dx＝（）。</p><p>答 案：</p><p>解 析：因为<img src="https://img2.meite.com/questions/202211/30638723cfd41f5.png" />是奇函数，所以定积分<img src="https://img2.meite.com/questions/202211/30638723ef3e7de.png" />。</p><p class="introTit">简答题</p><p>1、<img src="https://img2.meite.com/questions/202408/1666bf065d31bbe.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bf0661f3d75.png" /></p>