2024年成考专升本《高等数学一》每日一练试题06月11日

<p class="introTit">单选题</p><p>1、<img src="https://img2.meite.com/questions/202405/166645bbbd3cecb.png" />（）  </p><ul><li>A：sinx+C</li><li>B：-sinx+C</li><li>C：cosx+C</li><li>D：-cosx+C</li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202405/166645bbc52708b.png" /></p><p>2、<img src="https://img2.meite.com/questions/202212/03638ae588c44f4.png" />（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/03638ae5946a99f.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/03638ae5a0a586c.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/03638ae5ab2c9de.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/03638ae5b7db878.png' /></li></ul><p>答 案：A</p><p>解 析：<img src="https://img2.meite.com/questions/202212/03638ae5cad7d82.png" />。</p><p>3、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 class="introTit">主观题</p><p>1、求微分方程<img src="https://img2.meite.com/questions/202211/166374add1c6145.png" />的通解．</p><p>答 案：解：原方程对应的齐次微分方程为<img src="https://img2.meite.com/questions/202211/166374ade8112f4.png" />特征方程为<img src="https://img2.meite.com/questions/202211/166374adf9e847c.png" />特征根为x₁＝-1，x₂＝3，<br />齐次方程的通解为<img src="https://img2.meite.com/questions/202211/166374ae1b2107f.png" /><br />设原方程的特解为<img src="https://img2.meite.com/questions/202211/166374ae3374b9e.png" />＝A，代入原方程可得<img src="https://img2.meite.com/questions/202211/166374ae434e613.png" />＝-1。<br />所以原方程的通解为<img src="https://img2.meite.com/questions/202211/166374ae5b9e010.png" />（C₁，C₂为任意常数）</p><p>2、求曲线y＝x²在点（a，a²）（a＜1）的一条切线，使由该切线与x＝0、x＝1和y＝x²所围图形的面积最小。</p><p>答 案：解：设所求切线的切点为（a，b），见下图，<img src="https://img2.meite.com/questions/202212/01638814257361c.png" />则b＝a²，<img src="https://img2.meite.com/questions/202212/016388143d3fc16.png" />，切线方程为y－b＝2a（x－a），y＝2ax－2a²＋b＝2ax－a²。设对应图形面积为A，则<img src="https://img2.meite.com/questions/202212/01638814689e85a.png" /><br />令<img src="https://img2.meite.com/questions/202212/0163881476e033e.png" />，则<img src="https://img2.meite.com/questions/202212/01638814851e37c.png" />，令<img src="https://img2.meite.com/questions/202212/0163881492513cb.png" />。当a＜<img src="https://img2.meite.com/questions/202212/01638814a227100.png" />时，f'(a)＜0；当a＞<img src="https://img2.meite.com/questions/202212/01638814a227100.png" />时，f'(a)＞0，故<img src="https://img2.meite.com/questions/202212/0163881536b8640.png" />为f(a)的最小值点，切线方程为：y＝x－<img src="https://img2.meite.com/questions/202212/01638815675a0c3.png" />。</p><p>3、计算<img src="https://img2.meite.com/questions/202211/16637484703e391.png" />，其中积分区域D由直线y＝x，x＝1及x轴围成．</p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/166374847ba8f6e.png" /></p><p class="introTit">填空题</p><p>1、过坐标原点且与平面2x－y＋z＋1＝0平行的平行方程为（）。</p><p>答 案：2x－y＋z＝0</p><p>解 析：已知平面的法线向量为(2，-1，1)，所求平面与已知平面平行<img src="https://img2.meite.com/questions/202211/1663745885379d6.png" />，因此平面方程可设为<img src="https://img2.meite.com/questions/202211/166374589042355.png" />，又平面过原点，故D＝0，即所求平面方程为2x－y＋z＝0。</p><p>2、设f（x）＝3^x，g（x）＝x³，则<img src="https://img2.meite.com/questions/202212/03638afbe35c739.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/03638afbf104955.png" />·1n3</p><p>解 析：g（x）＝x³，g'（x）＝3x²，则<img src="https://img2.meite.com/questions/202212/03638afc2482958.png" />＝f'（3x²），注意等号右端的含义为f（<img src="https://img2.meite.com/questions/202212/03638afc46bb9ed.png" />）在<img src="https://img2.meite.com/questions/202212/03638afc5095882.png" />＝3x²处的导数，而f（x）＝3^x，即f（<img src="https://img2.meite.com/questions/202212/03638afc46bb9ed.png" />）＝<img src="https://img2.meite.com/questions/202212/03638afc82ae020.png" />，则<img src="https://img2.meite.com/questions/202212/03638afc910e655.png" />＝<img src="https://img2.meite.com/questions/202212/03638afc82ae020.png" />ln3，所以<img src="https://img2.meite.com/questions/202212/03638afcab92f99.png" /></p><p>3、设<img src="https://img2.meite.com/questions/202211/186376e1cbd80d4.png" />则<img src="https://img2.meite.com/questions/202211/186376e1d9e0698.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/186376e1e5e45e8.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/186376e1f5c9fca.png" />将x看作常量，则<img src="https://img2.meite.com/questions/202211/186376e2089ad12.png" /></p><p class="introTit">简答题</p><p>1、给定曲线<img src="https://img2.meite.com/questions/202303/1764140be29ac3d.png" />与直线y=px-q（其中p>0），求p与q为关系时，直线y=px-q<img src="https://img2.meite.com/questions/202303/1764140be29ac3d.png" />的切线。</p><p>答 案：由题意知，再切点处有<img src="https://img2.meite.com/questions/202303/1764140c40e9e08.png" />两边对x求导得<img src="https://img2.meite.com/questions/202303/1764140c4f7bf80.png" /><img src="https://img2.meite.com/questions/202303/1764140c558ef84.png" /></p>