2024年成考专升本《高等数学一》每日一练试题05月29日

<p class="introTit">单选题</p><p>1、微分方程<img src="https://img2.meite.com/questions/202303/176413fd1db62a4.png" />的通解为（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/176413fd3002785.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/176413fd3b07929.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/176413fd42006ba.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/176413fd477c552.png' /></li></ul><p>答 案：C</p><p>解 析：设<img src="https://img2.meite.com/questions/202303/176413fd6e81f66.png" />代入有<img src="https://img2.meite.com/questions/202303/176413fd7d72bde.png" />所以<img src="https://img2.meite.com/questions/202303/176413fd90182b3.png" /><img src="https://img2.meite.com/questions/202303/176413fda798e44.png" />原方程的通解为<img src="https://img2.meite.com/questions/202303/176413fdb8328b7.png" /></p><p>2、若y＝a^x（a＞0且a≠1），则<img src="https://img2.meite.com/questions/202211/296385bed009377.png" />等于（）。</p><ul><li>A：lnⁿa</li><li>B：a^xlnⁿa</li><li>C：<img src='https://img2.meite.com/questions/202211/296385bef0bc6ad.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/296385befa7445a.png' /></li></ul><p>答 案：A</p><p>解 析：因为<img src="https://img2.meite.com/questions/202211/296385bf0d7ab44.png" />，故<img src="https://img2.meite.com/questions/202211/296385bf1e28d19.png" />。</p><p>3、设y＝f（x）为分段函数，x₀为其分段点，且函数在x₀处连续，则下列命题（）正确。</p><ul><li>A：f（x）在点x₀处必定可导</li><li>B：f（x）在点x₀处必定可微</li><li>C：<img src='https://img2.meite.com/questions/202211/296385ca8b64899.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/296385ca9c078cb.png' /></li></ul><p>答 案：C</p><p>解 析：函数在x₀处连续，即在x₀处f(x)的左右极限存在且相等，所以<img src="https://img2.meite.com/questions/202211/296385cac3c5a82.png" />。</p><p class="introTit">主观题</p><p>1、计算<img src="https://img2.meite.com/questions/202212/03638b00b3592fa.png" />。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638b00c097bc9.png" /></p><p>2、求过原点且与直线<img src="https://img2.meite.com/questions/202212/03638b00cedc248.png" />平行的直线的方程．</p><p>答 案：解：直线<img src="https://img2.meite.com/questions/202212/03638b00ddc6c33.png" />的方向向量为<img src="https://img2.meite.com/questions/202212/03638b00ed97e2a.png" />因所求直线与已知直线平行，所以所求直线的方向向量也为s．所求直线过原点．故由标准式可得所求直线的方程为<img src="https://img2.meite.com/questions/202212/03638b01053a5fd.png" /></p><p>3、求二元函数<img src="https://img2.meite.com/questions/202212/01638851cc53e6f.png" />的极值。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/01638851e4935b5.png" />则由<img src="https://img2.meite.com/questions/202212/01638851f7adb87.png" /><img src="https://img2.meite.com/questions/202212/01638852088c159.png" />点P（－1，1）为唯一驻点，<img src="https://img2.meite.com/questions/202212/016388528d5cc07.png" />因此点（－1，－1）为z的极小值点，极小值为－1。</p><p class="introTit">填空题</p><p>1、<img src="https://img2.meite.com/questions/202211/186376e05e7dfa5.png" />＝（）。</p><p>答 案：e^-1</p><p>解 析：<img src="https://img2.meite.com/questions/202211/186376e07bb2733.png" /></p><p>2、<img src="https://img2.meite.com/questions/202211/30638721dc0b38c.png" />（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/30638721e77f5eb.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/30638721f6571b1.png" /></p><p>3、设z=e^xy，则dz=（）  </p><p>答 案：e^xy(ydx+xdy)</p><p>解 析：<img src="https://img2.meite.com/questions/202405/166645bdee37852.png" /></p><p class="introTit">简答题</p><p>1、设f(x)=<img src="https://img2.meite.com/questions/202303/1764140cba28156.png" />在x=0连续，试确定A,B.</p><p>答 案：<img src="https://img2.meite.com/questions/202303/1764140ce7d0fb9.png" /> <img src="https://img2.meite.com/questions/202303/1764140cf387f0c.png" /> 欲使f(x)在x=0处连续，应有2A=4=B+1，所以A=2,B=3.  </p>