2024年成考专升本《高等数学一》每日一练试题08月15日

<p class="introTit">单选题</p><p>1、设<img src="https://img2.meite.com/questions/202303/176413fe09102f5.png" />当<img src="https://img2.meite.com/questions/202303/176413fe1433b0d.png" />时f(x)与g(x)是（）</p><ul><li>A：等价无穷小</li><li>B：f(x)是比g(x)高阶无穷小</li><li>C：f(x)是比g(x)低阶无穷小</li><li>D：f(x)与g(x)是同阶但非等价无穷小</li></ul><p>答 案：D</p><p>解 析：由<img src="https://img2.meite.com/questions/202303/176413feb5e8982.png" /><img src="https://img2.meite.com/questions/202303/176413febdc61fc.png" />(等价无穷小代换)<img src="https://img2.meite.com/questions/202303/176413feea7ef77.png" /><img src="https://img2.meite.com/questions/202303/176413fefd27438.png" />故f(x)与g(x)是同阶但非等价无穷小</p><p>2、当x→0时，<img src="https://img2.meite.com/questions/202303/036401aa8d87bb3.png" />为x的（）  </p><ul><li>A：高阶无穷小量</li><li>B：等价无穷小量</li><li>C：同阶但不等价无穷小量</li><li>D：低阶无穷小量</li></ul><p>答 案：A</p><p>解 析：由题可知<img src="https://img2.meite.com/questions/202303/036401aa7d80577.png" />，故<img src="https://img2.meite.com/questions/202303/036401aa90e5aba.png" />是x的高阶无穷小量。</p><p>3、设函数f(x)满足<img src="https://img2.meite.com/questions/202303/176413e0e837e78.png" />且f(0)=0,则f(x)=()</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/176413e11e492e3.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/176413e113de62f.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/176413e1196224a.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/176413e10cadab6.png' /></li></ul><p>答 案：D</p><p>解 析：由<img src="https://img2.meite.com/questions/202303/176413e0e837e78.png" />知<img src="https://img2.meite.com/questions/202303/176413e16f36bbb.png" />令<img src="https://img2.meite.com/questions/202303/176413e17d276a1.png" />故<img src="https://img2.meite.com/questions/202303/176413e184a50ac.png" />所以f(u)=u-<img src="https://img2.meite.com/questions/202303/176413e1ab07ac9.png" />由f(0)=0，得C=0.所以<img src="https://img2.meite.com/questions/202303/176413e1c444863.png" /></p><p class="introTit">主观题</p><p>1、设<img src="https://img2.meite.com/questions/202212/03638aff4f737d0.png" />存在且<img src="https://img2.meite.com/questions/202212/03638aff59d37e0.png" />，求<img src="https://img2.meite.com/questions/202212/03638aff6554eac.png" /></p><p>答 案：解：设<img src="https://img2.meite.com/questions/202212/03638aff7078ded.png" />对<img src="https://img2.meite.com/questions/202212/03638aff830ad7a.png" />两边同时求极限，得<img src="https://img2.meite.com/questions/202212/03638affa03127b.png" />，即<img src="https://img2.meite.com/questions/202212/03638affaad78d4.png" />，得<img src="https://img2.meite.com/questions/202212/03638affb8a0ed1.png" />。</p><p>2、设函数f（x）＝x-lnx，求f（x）的单调增区间．</p><p>答 案：解：函数f（x）的定义域为（0，＋∞）。令y＝f（x），则<img src="https://img2.meite.com/questions/202211/166374ad22c2d4a.png" />令y'＝0，解得x＝1。当0＜x＜1时，y'＜0；当x＞1时，y'＞0。<br />因此函数f（x）的单调增区间为（1，＋∞）。</p><p>3、求<img src="https://img2.meite.com/questions/202211/2963856ce27e6d8.png" /></p><p>答 案：解：<img src="https://img2.meite.com/questions/202211/2963856cf281662.png" /><img src="https://img2.meite.com/questions/202211/2963856d0bd9b82.png" /></p><p class="introTit">填空题</p><p>1、过点M（1，2，－1）且与平面<img src="https://img2.meite.com/questions/202212/0163881f75e91dc.png" />垂直的直线方程为（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/0163881f8240a4c.png" /></p><p>解 析：由于直线与平面x－2y＋4z＝0垂直，可取直线方向向量为（1，－2，4），因此所求直线方程为<img src="https://img2.meite.com/questions/202212/0163881f8e4796b.png" /></p><p>2、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>3、积分<img src="https://img2.meite.com/questions/202211/30638722beb47e6.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/30638722c91eb86.png" /></p><p>解 析：利用分部积分进行求解，得<img src="https://img2.meite.com/questions/202211/30638722d638f8e.png" /></p><p class="introTit">简答题</p><p>1、求微分方程<img src="https://img2.meite.com/questions/202303/036401a0b4bb252.png" />满足初值条件<img src="https://img2.meite.com/questions/202303/036401a0c888c77.png" />的特解  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/036401afc1cb58a.png" /> <img src="https://img2.meite.com/questions/202303/036401afd313124.png" />  </p>