2024年成考专升本《高等数学二》每日一练试题06月15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、下列结论正确的是（）.</p><ul><li>A：若A＋B＝<img src='https://img2.meite.com/questions/202212/086391543a56a8d.png' />,则A，B互为对立事件</li><li>B：若A，B为互不相容事件，则A，B互为对立事件</li><li>C：若A，B为互不相容事件，则<img src='https://img2.meite.com/questions/202212/086391544dda179.png' />也互不相容</li><li>D：若A，B为互不相容事件，则A－B＝A</li></ul><p>答 案：D</p><p>解 析：A，B为互不相容事件是指A，B不可能同时发生；A，B为对立事件是指两者必有一件发生，即A发生则B不发生，B发生则A不发生.可知若A，B为互不相容事件，则A－B＝A.</p><p>2、下列函数在x＝0处切线斜率不存在的是（）.</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/06638f0498d6044.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/06638f04a1cd752.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/06638f04a9d3386.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/06638f04b4a46b0.png' /></li></ul><p>答 案：D</p><p>解 析：D项，<img src="https://img2.meite.com/questions/202212/06638f04c5a6bcb.png" />，当x→0时，<img src="https://img2.meite.com/questions/202212/06638f04d6baacb.png" />即切线斜率不存在.A项，<img src="https://img2.meite.com/questions/202212/06638f04e9c107f.png" />，<img src="https://img2.meite.com/questions/202212/06638f04f58566e.png" />，即函数在x＝0处切线斜率为1；B项，<img src="https://img2.meite.com/questions/202212/06638f0504b0091.png" />，即函数在x＝0处切线斜率为1；<br />C项，<img src="https://img2.meite.com/questions/202212/06638f0516a5894.png" />，即函数在x＝0处切线斜率为<img src="https://img2.meite.com/questions/202212/06638f05283271d.png" />.</p><p class="introTit">主观题</p><p>1、设事件A与B相互独立，<img src="https://img2.meite.com/questions/202212/0863917baf10c70.png" />，<img src="https://img2.meite.com/questions/202212/0863917bba46883.png" />，<img src="https://img2.meite.com/questions/202212/0863917bc65a5bd.png" />，求q．</p><p>答 案：解：因为事件A与B相互独立，故<img src="https://img2.meite.com/questions/202212/0863917c1276717.png" />，<img src="https://img2.meite.com/questions/202212/0863917c219eabb.png" />，即<img src="https://img2.meite.com/questions/202212/0863917c331e4b2.png" />，解得<img src="https://img2.meite.com/questions/202212/0863917c4c3ff46.png" />＝<img src="https://img2.meite.com/questions/202212/0863917c58641ed.png" />．</p><p>2、求一个正弦曲线与x轴所围成图形的面积（只计算一个周期的面积）．</p><p>答 案：解：取从0～2π的正弦曲线如图<img src="https://img2.meite.com/questions/202212/0763903d20af21a.png" />，设所围图形面积为S，则<img src="https://img2.meite.com/questions/202212/0763903cddd57ab.png" /><img src="https://img2.meite.com/questions/202212/0763903ce9b5e60.png" /><img src="https://img2.meite.com/questions/202212/0763903cf62e7d0.png" /><img src="https://img2.meite.com/questions/202212/0763903d056e610.png" />注意到图形面积是对称的，可直接得出<img src="https://img2.meite.com/questions/202212/0763903d341c7a8.png" />。</p><p class="introTit">填空题</p><p>1、<img src="https://img2.meite.com/questions/202212/05638d5317c56ee.png" />（）．</p><p>答 案：<img src="https://img2.meite.com/questions/202212/05638d5324d14f5.png" />＋C</p><p>解 析：<img src="https://img2.meite.com/questions/202212/05638d5331ee077.png" />．</p><p>2、<img src="https://img2.meite.com/questions/202212/076390355451b14.png" />()．</p><p>答 案：2－<img src="https://img2.meite.com/questions/202212/0763903561662ca.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202212/076390357574d22.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、已知曲线<img src="https://img2.meite.com/questions/202303/216419246048f3b.png" />在点(1,2)处有水平切线，且原点为该曲线的拐点，求a,b,c的值，并写出此曲线的方程。  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/2164192515baa08.png" /><img src="https://img2.meite.com/questions/202303/2164192520cbb96.png" /> 由已知条件得： <img src="https://img2.meite.com/questions/202303/2164192541ec9d6.png" /> 故b=0,a=-1，c=3，次曲线的方程为<img src="https://img2.meite.com/questions/202303/21641925650ce9b.png" />  </p>