2024年成考专升本《高等数学二》每日一练试题06月02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、甲、乙两人独立地对同一目标射击一次，其命中率分别0.6和0.5，现已知目标被命中，是甲射中的概率为（）</p><ul><li>A：0.6</li><li>B：0.75</li><li>C：0.85</li><li>D：0.9</li></ul><p>答 案：B</p><p>解 析：设A1={甲射中目标}，A2={乙射中目标}，B={目标被命中}。由题意，P(A1)=0.6,P=(A2)=0.5,<img src="https://img2.meite.com/questions/202303/2164194b35c9583.png" /><img src="https://img2.meite.com/questions/202303/2164194b3ca2cb4.png" />=1-(1-0.6)(1-0.5)=0.8;故所求概率为<img src="https://img2.meite.com/questions/202303/2164194b7a98b93.png" /></p><p>2、设f(x)的一个原函数是arctanx，则f(x)的导函数是（）</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/07639005f6525f4.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/07639006047c034.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/076390060db60f4.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/07639006177e267.png' /></li></ul><p>答 案：D</p><p>解 析：根据原函数的定义可知<img src="https://img2.meite.com/questions/202212/076390062c6731c.png" />，则<img src="https://img2.meite.com/questions/202212/076390063e2abef.png" /></p><p class="introTit">主观题</p><p>1、设函数<img src="https://img2.meite.com/questions/202212/0863914d993661e.png" />，其中<img src="https://img2.meite.com/questions/202212/0863914da44a2d3.png" />有二阶偏导数．</p><p>答 案：证明：<img src="https://img2.meite.com/questions/202212/0863914e2bb08ab.png" />证：对x求导，<img src="https://img2.meite.com/questions/202212/0863914e39cfe7e.png" />再对x求导，得<img src="https://img2.meite.com/questions/202212/0863914e4aa3cca.png" />；<img src="https://img2.meite.com/questions/202212/0863914e56aca9c.png" />对y求导，得<img src="https://img2.meite.com/questions/202212/0863914e643ec28.png" />类似可得，<img src="https://img2.meite.com/questions/202212/0863914e7553f04.png" />；所以<img src="https://img2.meite.com/questions/202212/0863914e87881ba.png" /><img src="https://img2.meite.com/questions/202212/0863914e949f865.png" /></p><p>2、证明：当x＞1时，x＞1＋lnx．</p><p>答 案：证：设f(x)＝x－1－lnx，则f'(x)＝<img src="https://img2.meite.com/questions/202212/05638da08c22129.png" />．当x＞1时，f'(x)＞0，则f(x)单调上升．所以当x＞1时，f(x)＞f(1)＝0，即x－1－lnx＞0,得x＞1＋lnx．</p><p class="introTit">填空题</p><p>1、<img src="https://img2.meite.com/questions/202212/05638d67cfaf891.png" />（）．</p><p>答 案：<img src="https://img2.meite.com/questions/202212/05638d67d98bd4e.png" /></p><p>解 析：因为积分区间关于原点对称，<img src="https://img2.meite.com/questions/202212/05638d67e53bc93.png" />是奇函数，故<img src="https://img2.meite.com/questions/202212/05638d67f510a1a.png" />，则<img src="https://img2.meite.com/questions/202212/05638d68008bbef.png" /></p><p>2、设函数y＝ln（1＋x），则y''＝（）．</p><p>答 案：<img src="https://img2.meite.com/questions/202212/05638d98a46e891.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202212/05638d98b010d11.png" />．</p><p class="introTit">简答题</p><p>1、设随机变量X的概率分布为：<img src="https://img2.meite.com/questions/202212/06638eda161b9ed.png" /> 求X的期望、方差以及标准差．</p><p>答 案：<img src="https://img2.meite.com/questions/202212/06638eda2e62e11.png" /> <img src="https://img2.meite.com/questions/202212/06638eda375c467.png" /></p><p>2、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>