2024年成考专升本《高等数学二》每日一练试题08月22日

<p class="introTit">判断题</p><p>1、若<img src="https://img2.meite.com/questions/202307/1364af5780970e6.png" />，则<img src="https://img2.meite.com/questions/202307/1364af578624075.png" />。（）  </p><p>答 案：错</p><p>解 析：<img src="https://img2.meite.com/questions/202212/06638ef8852e1bd.png" />所以<img src="https://img2.meite.com/questions/202212/06638ef88b66bc1.png" />  </p><p class="introTit">单选题</p><p>1、曲线<img src="https://img2.meite.com/questions/202303/206417d3330e6ef.png" />的凸区间为（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/206417d340c2a1a.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/206417d3450d088.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/206417d350c708e.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/206417d34c4a740.png' /></li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202303/206417d365bfbf6.png" />，<img src="https://img2.meite.com/questions/202303/206417d37603ca8.png" />显然当x>1时，<img src="https://img2.meite.com/questions/202303/206417d387245e6.png" />而当x<1时，<img src="https://img2.meite.com/questions/202303/206417d39b34957.png" />在<img src="https://img2.meite.com/questions/202303/206417d3b173d57.png" />内曲线为凸弧</p><p>2、<img src="https://img2.meite.com/questions/202408/1666bf154d9fcf1.png" />  </p><ul><li>A：0</li><li>B：e-1</li><li>C：2（e-1）</li><li>D：<img src='https://img2.meite.com/questions/202408/1666bf15518f984.png' /></li></ul><p>答 案：C</p><p>解 析：本题考查的知识点是奇、偶函数在对称区间上的定积分计算。 注意到被积函数是偶函数的特性，可知<img src="https://img2.meite.com/questions/202408/1666bf1556b22cc.png" />所以选C。  </p><p class="introTit">主观题</p><p>1、设平面图形是由曲线y＝<img src="https://img2.meite.com/questions/202212/0763903d43141de.png" />和x＋y＝4围成的．（1）求此平面图形的面积S．<br />（2）求此平面图形绕x轴旋转而成的旋转体的体积V_x．</p><p>答 案：解：曲线y＝<img src="https://img2.meite.com/questions/202212/0763903d43141de.png" />和x＋y＝4围成的图形如图阴影部分所示<img src="https://img2.meite.com/questions/202212/0763903d80ab61a.png" />．求两条曲线的交点，解方程<img src="https://img2.meite.com/questions/202212/0763903d916112e.png" />得交点（1，3）与（3，1）．（1）面积<img src="https://img2.meite.com/questions/202212/0763903da21dfd4.png" />；<br />（2）旋转体体积<img src="https://img2.meite.com/questions/202212/0763903db5b56ca.png" /></p><p>2、计算<img src="https://img2.meite.com/questions/202212/06638e9ae957f4e.png" /></p><p>答 案：解：由洛必达法则得<img src="https://img2.meite.com/questions/202212/06638e9af6dd03c.png" /></p><p class="introTit">填空题</p><p>1、二人独立破译一种密码，他们能独立译出的概率分别为0.3和0.4，则此密码能被破译的概率为（）．</p><p>答 案：0.58</p><p>解 析：<img src="https://img2.meite.com/questions/202212/086391601e59217.png" />．</p><p>2、<img src="https://img2.meite.com/questions/202408/1666bf158926dec.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202408/1666bf158dc91ea.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202408/1666bf159121439.png" /> <img src="https://img2.meite.com/questions/202408/1666bf15a5af508.png" />  </p><p class="introTit">简答题</p><p>1、已知函数f(x)=ax³-bx²+cx在区间<img src="https://img2.meite.com/questions/202212/07639002a0c63f6.png" />内是奇函数，且当x＝1时，f(x)有极小值<img src="https://img2.meite.com/questions/202212/07639002ba72fe4.png" />，求另一个极值及此曲线的拐点．  </p><p>答 案：f(x)=ax³-bx²+cx，<img src="https://img2.meite.com/questions/202212/07639002da708ed.png" /> 由于f(x)是奇函数，则必有x²的系数为0，即b=0. <img src="https://img2.meite.com/questions/202212/07639003148fbf6.png" />即a+c=<img src="https://img2.meite.com/questions/202212/076390032089cb2.png" />，<img src="https://img2.meite.com/questions/202212/0763900324b36b3.png" />得3a+c=0．解得a=<img src="https://img2.meite.com/questions/202212/076390033fe0499.png" />c=<img src="https://img2.meite.com/questions/202212/0763900347cfddd.png" /> 此时<img src="https://img2.meite.com/questions/202212/07639003591bd3b.png" /> 令<img src="https://img2.meite.com/questions/202212/0763900361d0796.png" />得<img src="https://img2.meite.com/questions/202212/07639003674342f.png" /><img src="https://img2.meite.com/questions/202212/076390036dd8f49.png" />所以<img src="https://img2.meite.com/questions/202212/076390037794ea3.png" />为极大值，<img src="https://img2.meite.com/questions/202212/07639003826209a.png" />得x=0，x<0时，<img src="https://img2.meite.com/questions/202212/0763900396d70f8.png" /> 所以（0，0）为曲线的拐点．</p><p>2、计算<img src="https://img2.meite.com/questions/202303/076406da7743065.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/076406e36bd4a94.png" /></p>