2024年成考专升本《高等数学二》每日一练试题04月11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、下列极限正确的是（）.</p><ul><li>A：<img src='https://img2.meite.com/questions/202212/06638ea6eb215bf.png' /></li><li>B：<img src='https://img2.meite.com/questions/202212/06638ea6f55490b.png' /></li><li>C：<img src='https://img2.meite.com/questions/202212/06638ea7022646e.png' /></li><li>D：<img src='https://img2.meite.com/questions/202212/06638ea70ff3676.png' /></li></ul><p>答 案：D</p><p>解 析：A项，<img src="https://img2.meite.com/questions/202212/06638ea71e5de41.png" />；B项，<img src="https://img2.meite.com/questions/202212/06638ea729730e2.png" />；C项，因为当<img src="https://img2.meite.com/questions/202212/06638ea7397db9d.png" />时，x为无穷小量，又<img src="https://img2.meite.com/questions/202212/06638ea74eda788.png" />为有界量，所以<img src="https://img2.meite.com/questions/202212/06638ea75f770f7.png" />为0；D项，<img src="https://img2.meite.com/questions/202212/06638ea76fe915a.png" />.</p><p>2、设在(a，b)内有<img src="https://img2.meite.com/questions/202303/206417f894b9fb1.png" />则在(a，b)内必定有（）</p><ul><li>A：f(x)-g(x)=0</li><li>B：f(x)-g(x)=C</li><li>C：<img src='https://img2.meite.com/questions/202303/206417f8d2ead70.png' /></li><li>D：f(x)dx=g(x)dx</li></ul><p>答 案：B</p><p>解 析：<img src="https://img2.meite.com/questions/202303/206417f8f37f223.png" /><img src="https://img2.meite.com/questions/202303/206417f8fe734f1.png" /><img src="https://img2.meite.com/questions/202303/206417f906645af.png" /><img src="https://img2.meite.com/questions/202303/206417f914ceff5.png" /> 故f(x)-g(x)-C=0,所以f(x)-g(x)=C  </p><p class="introTit">主观题</p><p>1、求<img src="https://img2.meite.com/questions/202212/07639038ae5e4b0.png" />．</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/07639039c813814.png" /></p><p>2、设随机变量ξ的分布列为<img src="https://img2.meite.com/questions/202212/0863917f7936b4d.png" />求E（ξ）和D（ξ）．</p><p>答 案：解：E（ξ）＝-1×0.2＋0×0.1＋1×0.4＋2×0.3＝0.8．D（ξ）＝（-1－0.8）²×0.2＋（0－0.8）²×0.1＋（1－0.8）²×0.4＋（2－0.8）×0.3＝1.16．</p><p class="introTit">填空题</p><p>1、己知y＝x³-ax在x＝1的切线平行于直线5x-ｙ+1＝0，则ａ＝（）  </p><p>答 案：-2</p><p>解 析：<img src="https://img2.meite.com/questions/202212/06638eea5b9374a.png" />a=-2  </p><p>2、<img src="https://img2.meite.com/questions/202212/05638d67add2a78.png" />（）．</p><p>答 案：x＋arctanx</p><p>解 析：<img src="https://img2.meite.com/questions/202212/05638d67c1c0d2d.png" />．</p><p class="introTit">简答题</p><p>1、求函数<img src="https://img2.meite.com/questions/202212/06638eb487c7926.png" />的倒数。  </p><p>答 案：等式两边同时取对数得<img src="https://img2.meite.com/questions/202212/06638eb4a7e893e.png" /> 方程两边同时对x求导有<img src="https://img2.meite.com/questions/202212/06638eb4bc75baf.png" /> 故<img src="https://img2.meite.com/questions/202212/06638eb4c8db194.png" /><img src="https://img2.meite.com/questions/202212/06638eb4ce8d960.png" /></p><p>2、证明：<img src="https://img2.meite.com/questions/202303/206417fa32611db.png" /></p><p>答 案：令<img src="https://img2.meite.com/questions/202303/206417fa5d92774.png" />则<img src="https://img2.meite.com/questions/202303/206417fa64709ff.png" />由于此式不便判定符号，故再求出<img src="https://img2.meite.com/questions/202303/206417fa8f7819c.png" />又因<img src="https://img2.meite.com/questions/202303/206417fa9f5cd03.png" /><img src="https://img2.meite.com/questions/202303/206417faa4b9537.png" />所以f'(x)单调增加，故f'(x)>f'(4)=<img src="https://img2.meite.com/questions/202303/206417facca9272.png" />-8=8(2ln2-1)=8(ln4-1)>0, 得到f(x)单调增加，故f(x)>f(4),即<img src="https://img2.meite.com/questions/202303/206417fb29350f8.png" />因此<img src="https://img2.meite.com/questions/202303/206417fb3703866.png" /></p>