2024年成考高起点《数学(理)》每日一练试题03月20日
聚题库
03/20
<p class="introTit">单选题</p><p>1、函数<img src="https://img2.meite.com/questions/202303/156411640a2cd90.png" />的定义域是（）</p><ul><li>A：{x|-3＜x＜-1}</li><li>B：{x|x＜-3或x＞-1}</li><li>C：{x|1＜x＜3}</li><li>D：{x|x＜1或x＞3}</li></ul><p>答 案：D</p><p>解 析：由对数函数的性质可知<img src="https://img2.meite.com/questions/202303/15641167889eb44.png" />，解得x＞3或x＜1，因此函数的定义域为{x|x＜1或x＞3}</p><p>2、设0<a<b<1,则下列正确的是
</p><ul><li>A：<img src='https://img2.meite.com/questions/202303/2864225b1e4a590.png' /></li><li>B：<img src='https://img2.meite.com/questions/202303/2864225b29000a7.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/2864225b2e590ed.png' /></li><li>D：<img src='https://img2.meite.com/questions/202303/2864225b34e87f0.png' /></li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202303/2864225e9994bba.png" /><img src="https://img2.meite.com/questions/202303/2864225ea183366.png" />
</p><p>3、设集合A={0,1}，B={0,1,2}，则A∩B=（）
</p><ul><li>A：{1,2}</li><li>B：{0,2}</li><li>C：{0,1}</li><li>D：{0,1,2}</li></ul><p>答 案：C</p><p>解 析：<img src="https://img2.meite.com/questions/202303/286422489c9882b.png" /></p><p>4、在△ABC中，若b=<img src="https://img2.meite.com/questions/202303/2864228792c7c59.png" />，c=<img src="https://img2.meite.com/questions/202303/286422879bef613.png" /><img src="https://img2.meite.com/questions/202303/28642287a3c143e.png" />则a等于（）</p><ul><li>A：2</li><li>B：<img src='https://img2.meite.com/questions/202303/28642287b3e4835.png' /></li><li>C：<img src='https://img2.meite.com/questions/202303/28642287b8ca84c.png' /></li><li>D：无解</li></ul><p>答 案：B</p><p>解 析：此题是已知两边和其中一边的对角,解三角形时,会出现一解、两解、无解的情况,要注意这一点.用余弦定理<img src="https://img2.meite.com/questions/202303/286422888d98016.png" />可得<img src="https://img2.meite.com/questions/202303/28642288b870fd3.png" /><img src="https://img2.meite.com/questions/202303/28642288bcdcd26.png" /><img src="https://img2.meite.com/questions/202303/28642288cb0279a.png" /><img src="https://img2.meite.com/questions/202303/28642288d358a96.png" /><img src="https://img2.meite.com/questions/202303/28642288e0d440f.png" />解出<img src="https://img2.meite.com/questions/202303/28642288f8ec133.png" /><img src="https://img2.meite.com/questions/202303/286422890103ec2.png" /><img src="https://img2.meite.com/questions/202303/286422890874181.png" /><img src="https://img2.meite.com/questions/202303/2864228911473af.png" /></p><p class="introTit">主观题</p><p>1、设函数f(x)=<img src="https://img2.meite.com/questions/202303/28642286431b211.png" />
(Ⅰ)求f(x)的单调区间;
(Ⅱ)求 f(x)的极值</p><p>答 案：(Ⅰ)函数的定义域为<img src="https://img2.meite.com/questions/202303/28642286bee9cc3.png" /> <img src="https://img2.meite.com/questions/202303/28642286c7d68a9.png" />
(Ⅱ)<img src="https://img2.meite.com/questions/202303/28642286d3444c8.png" />
</p><p>2、某工厂每月生产x台游戏机的收入为R(x)=<img src="https://img2.meite.com/questions/202303/28642259676bd4d.png" />+130x-206(百元)，成本函数为C(x)=50x+100(百元)，当每月生产多少台时,获利润最大?最大利润为多少?
</p><p>答 案：利润 =收入-成本， L(x)=R(x)-C(x)=<img src="https://img2.meite.com/questions/202303/28642259c06a284.png" />+130x-206-(50x+100)=<img src="https://img2.meite.com/questions/202303/28642259e2077b7.png" />+80x-306
法一：用二次函数<img src="https://img2.meite.com/questions/202303/28642259fb36916.png" />当a<0时有最大值
<img src="https://img2.meite.com/questions/202303/2864225a14ed5aa.png" />是开口向下的抛物线，有最大值
<img src="https://img2.meite.com/questions/202303/2864225a2c75330.png" />
法二：用导数来求解
<img src="https://img2.meite.com/questions/202303/2864225a43988b8.png" />
因为x=90是函数在定义域内唯一驻点
所以x=90是函数的极大值点，也是函数的最大值点，其最大值为L（90）=3294
</p><p>3、设函数f(x)=xlnx+x.(I）求曲线y=f(x）在点（(1,f(1))处的切线方程；<br />(II）求f（x）的极值．</p><p>答 案：(I）f(1)=1,f'(x)=2+lnx，故f'(1)=2.所以曲线y=f(x）在点（1,f(1))处的切线方程为y=2x-1.(II）令f'(x)=0，解得<img src="https://img2.meite.com/questions/202303/1564116d2d14a94.png" />当<img src="https://img2.meite.com/questions/202303/1564116d3d33026.png" />时，f'(x)<O；当<img src="https://img2.meite.com/questions/202303/1564116d6f6aec3.png" />时，f'(x)＞O.故f(x）在区间<img src="https://img2.meite.com/questions/202303/1564116db9a0764.png" />单调递减，在区间<img src="https://img2.meite.com/questions/202303/1564116dc99fc91.png" />单调递增．因此f(x）在<img src="https://img2.meite.com/questions/202303/1564116ddb842d0.png" />时取得极小值<img src="https://img2.meite.com/questions/202303/1564116de4f1b79.png" /></p><p>4、建筑一个容积为8000<img src="https://img2.meite.com/questions/202303/2864224b406cbf6.png" />,深为6m的长方体蓄水池，池壁每<img src="https://img2.meite.com/questions/202303/2864224b5cac16d.png" />的造价为15元，池底每<img src="https://img2.meite.com/questions/202303/2864224b60ac28e.png" />的造价为30元。（I）把总造价y(元)表示为长x(m)的函数；（Ⅱ）求函数的定义域
</p><p>答 案：<img src="https://img2.meite.com/questions/202303/2864224be4311f4.png" /><img src="https://img2.meite.com/questions/202303/2864224bee67713.png" /></p><p class="introTit">填空题</p><p>1、函数<img src="https://img2.meite.com/questions/202303/2864225857f0d64.png" />的图像与坐标轴的交点共有（）
</p><p>答 案：2</p><p>解 析：当x=0时，y=<img src="https://img2.meite.com/questions/202303/286422587f2a68b.png" />-2=-1，故函数与y轴交于(0,-1)点，令y=0,则有<img src="https://img2.meite.com/questions/202303/28642258b372965.png" />故函数与x轴交于(1,0) 点,因此函数 <img src="https://img2.meite.com/questions/202303/28642258c6c51c8.png" />与坐标轴的交点共有 2个.</p><p>2、若平面向量a=(x,1),b=(1,-2),且a//b,则x=（）
</p><p>答 案：<img src="https://img2.meite.com/questions/202303/286422508f0554f.png" /></p><p>解 析：由于a//b，故<img src="https://img2.meite.com/questions/202303/286422509cd5c14.png" /></p>
