2024年成考专升本《高等数学一》每日一练试题04月09日

<p class="introTit">单选题</p><p>1、设y^(n-2)=sinx，则y⁽ⁿ⁾=()  </p><ul><li>A：cosx</li><li>B：-cosx</li><li>C：sinx</li><li>D：-sinx</li></ul><p>答 案：D</p><p>解 析：<img src="https://img2.meite.com/questions/202303/036401aacb205c4.png" />因此<img src="https://img2.meite.com/questions/202303/036401aad58b2c6.png" /></p><p>2、<img src="https://img2.meite.com/questions/202211/176375ff5de2068.png" />（）。</p><ul><li>A：<img src='https://img2.meite.com/questions/202211/176375ff69711d4.png' /></li><li>B：<img src='https://img2.meite.com/questions/202211/176375ff7573db3.png' /></li><li>C：<img src='https://img2.meite.com/questions/202211/176375ff810ebb6.png' /></li><li>D：<img src='https://img2.meite.com/questions/202211/176375ff9317f47.png' /></li></ul><p>答 案：D</p><p>解 析：由不定积分的基本积分公式可得，<img src="https://img2.meite.com/questions/202211/176375ffa253b48.png" />。</p><p>3、已知f(xy,x-y)=<img src="https://img2.meite.com/questions/202303/17641400eb570c7.png" />则<img src="https://img2.meite.com/questions/202303/17641400fc3e15f.png" />等于（）</p><ul><li>A：2</li><li>B：2x</li><li>C：2y</li><li>D：2x+2y</li></ul><p>答 案：A</p><p>解 析：因f(xy,x-y)=<img src="https://img2.meite.com/questions/202303/17641400eb570c7.png" />=<img src="https://img2.meite.com/questions/202303/176414014ddbd40.png" />故<img src="https://img2.meite.com/questions/202303/17641401605bb19.png" />从而<img src="https://img2.meite.com/questions/202303/176414016972068.png" /></p><p class="introTit">主观题</p><p>1、设f(x)是以T为周期的连续函数，a为任意常数，证明：<img src="https://img2.meite.com/questions/202212/01638810a04c178.png" />。</p><p>答 案：证：因为<img src="https://img2.meite.com/questions/202212/01638810b105867.png" />令x＝T＋t，做变量替换得<img src="https://img2.meite.com/questions/202212/01638810cade25f.png" />故<img src="https://img2.meite.com/questions/202212/01638810de0e205.png" /></p><p>2、求微分方程<img src="https://img2.meite.com/questions/202212/03638abf9c40386.png" />的通解。</p><p>答 案：解：<img src="https://img2.meite.com/questions/202212/03638abfadd756c.png" />为一阶线性微分方程，则<img src="https://img2.meite.com/questions/202212/03638abfbbaff67.png" /></p><p>3、求微分方程<img src="https://img2.meite.com/questions/202212/03638b01839ca7b.png" />的通解．</p><p>答 案：解：对应齐次微分方程的特征方程为<img src="https://img2.meite.com/questions/202212/03638b019490467.png" />，解得r₁＝3，r₂＝－2.所以齐次通解为<img src="https://img2.meite.com/questions/202212/03638b01b0cf47a.png" />。设方程的特解设为y*＝（Ax＋B）e^x，代入原微分方程可解得，A＝<img src="https://img2.meite.com/questions/202212/03638b01cb92f6c.png" />，B＝<img src="https://img2.meite.com/questions/202212/03638b01d9339ef.png" />．即非齐次微分方程特解为<img src="https://img2.meite.com/questions/202212/03638b01ea50130.png" />。所以微分方程<img src="https://img2.meite.com/questions/202212/03638b01f95ca36.png" />的通解为<img src="https://img2.meite.com/questions/202212/03638b0209e7294.png" />。</p><p class="introTit">填空题</p><p>1、函数<img src="https://img2.meite.com/questions/202211/306386fad35365a.png" />在[1，2]上符合拉格朗日中值定理的<img src="https://img2.meite.com/questions/202211/306386fdd7622ab.png" />＝_。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/306386fde85c879.png" /></p><p>解 析：由拉格朗日中值定理有<img src="https://img2.meite.com/questions/202211/306386fe2edc9ca.png" />解得<img src="https://img2.meite.com/questions/202211/306386fe47ab869.png" />，其中<img src="https://img2.meite.com/questions/202211/306386fdd7622ab.png" />＝－<img src="https://img2.meite.com/questions/202211/306386fde85c879.png" />（舍），得<img src="https://img2.meite.com/questions/202211/306386fdd7622ab.png" />＝<img src="https://img2.meite.com/questions/202211/306386fde85c879.png" />。</p><p>2、<img src="https://img2.meite.com/questions/202211/3063872295a7417.png" />＝（）。</p><p>答 案：<img src="https://img2.meite.com/questions/202211/306387229feed93.png" /></p><p>解 析：<img src="https://img2.meite.com/questions/202211/30638722b02d14e.png" />。</p><p>3、<img src="https://img2.meite.com/questions/202211/186376e093361ff.png" />＝（）。</p><p>答 案：</p><p>解 析：所给求极限的表达式为分式，x＝1时分母不为零，可将x＝1直接代入函数求得极限<img src="https://img2.meite.com/questions/202211/186376e0b072c89.png" /></p><p class="introTit">简答题</p><p>1、求<img src="https://img2.meite.com/questions/202303/036401a0a3e00d2.png" />  </p><p>答 案：<img src="https://img2.meite.com/questions/202303/036401afab6aba9.png" /></p>