2024年成考专升本《高等数学一》每日一练试题09月28日

<p class="introTit">单选题</p><p>1、设曲线<img src="https://img2.meite.com/questions/202211/296385ce71aec04.png" />上某点处的切线方程为y＝mx，则m的值可能是（）。</p><ul><li>A：0</li><li>B：1</li><li>C：2</li><li>D：3</li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202211/296385ce8746438.png" />又曲线<img src="https://img2.meite.com/questions/202211/296385ce94a4774.png" />上某点处的切线方程为y＝mx，设该点为<img src="https://img2.meite.com/questions/202211/296385ceb05f9b5.png" />，则有<img src="https://img2.meite.com/questions/202211/296385cebf00a66.png" />，解得m=1或5。</p><p>2、下列命题中正确的有（）。  </p><ul><li>A：<img src='https://img2.meite.com/questions/202408/1666beee5dbb517.png' /></li><li>B：<img src='https://img2.meite.com/questions/202408/1666beee61969bf.png' /></li><li>C：<img src='https://img2.meite.com/questions/202408/1666beee6512209.png' /></li><li>D：<img src='https://img2.meite.com/questions/202408/1666beee686db8d.png' /></li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666beee6dec07d.png" /> <img src="https://img2.meite.com/questions/202408/1666beee7183d66.png" />  </p><p>3、<img src="https://img2.meite.com/questions/202211/30638710d23a9f9.png" />＝（）。</p><ul><li>A：0</li><li>B：<img src='https://img2.meite.com/questions/202211/30638710e00224d.png' /></li><li>C：arctanx</li><li>D：<img src='https://img2.meite.com/questions/202211/30638710ea87c2f.png' /></li></ul><p>答 案：A</p><p>解 析：当定积分<img src="https://img2.meite.com/questions/202211/30638710f9ad3eb.png" />存在时，它表示一个常数值，常数的导数等于零。</p><p class="introTit">主观题</p><p>1、求微分方程<img src="https://img2.meite.com/questions/202212/03638ac41f0fd3e.png" />的通解。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638ac42ad5381.png" />的特征值方程为<img src="https://img2.meite.com/questions/202212/03638ac43c4b5ff.png" />，则<img src="https://img2.meite.com/questions/202212/03638ac449853b1.png" />；故齐次微分方程的通解为<img src="https://img2.meite.com/questions/202212/03638ac45952791.png" />。由题意设原微分方程的特解为<img src="https://img2.meite.com/questions/202212/03638ac46ccee9d.png" />，则有<img src="https://img2.meite.com/questions/202212/03638ac47aa72ad.png" />，得<img src="https://img2.meite.com/questions/202212/03638ac4860b1ed.png" />。即微分方程的通解为<img src="https://img2.meite.com/questions/202212/03638ac4946b4d5.png" />。</p><p>2、计算二重积分<img src="https://img2.meite.com/questions/202212/03638af10c1bd16.png" />，其中D是x²＋y²≤1，x≥0，y≥0所围的平面区域．</p><p>答 案：解：D的图形见下图中阴影部分。<img src="https://img2.meite.com/questions/202212/03638af134a3609.png" />在极坐标系下D满足0≤<img src="https://img2.meite.com/questions/202212/03638af14d96d51.png" />≤<img src="https://img2.meite.com/questions/202212/03638af15506ed3.png" />，0≤r≤1，且x²＋y²＝（rcos<img src="https://img2.meite.com/questions/202212/03638af14d96d51.png" />）²＋（rsin<img src="https://img2.meite.com/questions/202212/03638af14d96d51.png" />）²＝r²，故<img src="https://img2.meite.com/questions/202212/03638af19233229.png" />。</p><p>3、计算<img src="https://img2.meite.com/questions/202211/176375daf19ee29.png" /></p><p>答 案：解：利用洛必达法则，得<img src="https://img2.meite.com/questions/202211/176375db05a5eb7.png" /></p><p class="introTit">填空题</p><p>1、二阶常系数齐次微分方程<img src="https://img2.meite.com/questions/202408/1666bf0ccf57b7a.png" />的通解为_____。  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bf0cd475f26.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666bf0cd8a41b0.png" /><img src="https://img2.meite.com/questions/202408/1666bf0cdbb3e63.png" /> <img src="https://img2.meite.com/questions/202408/1666bf0cdfa481c.png" />  </p><p>2、交换二次积分的积分次序，<img src="https://img2.meite.com/questions/202212/0163884aa14b346.png" />（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202212/0163884ab4284fb.png" /></p><p>解 析：由题设有<img src="https://img2.meite.com/questions/202212/0163884ac5b442b.png" />从而<img src="https://img2.meite.com/questions/202212/0163884adca8d6c.png" />故交换次序后二次积分为<img src="https://img2.meite.com/questions/202212/0163884aebe3b55.png" />。</p><p>3、<img src="https://img2.meite.com/questions/202408/1666bf05bb0e62f.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bf05bf72309.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666bf05c405097.png" /></p><p class="introTit">简答题</p><p>1、求函数f(x)=<img src="https://img2.meite.com/questions/202303/036401a047d0835.png" />的单调区间。  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/036401af525c8e7.png" /> <img src="https://img2.meite.com/questions/202303/036401af6400cd0.png" /><img src="https://img2.meite.com/questions/202303/036401af706ad1e.png" /> <img src="https://img2.meite.com/questions/202303/036401af8094320.png" /><img src="https://img2.meite.com/questions/202303/036401af8b285df.png" /></p>