2023年揚州大學高數(shù)期終試題A及答案

揚州大學20高等數(shù)學I（2）統(tǒng)考試卷(A)班級學號姓名得分注意事項：1.本試卷共６頁,3大題，20小題,滿分100分，考試時間120分鐘;2.請將試卷后所附的兩張空白紙所有撕下作草稿紙。題號選擇題填空題11～1213~141５～１617~1８１９～2０扣分扣分一、選擇題（每小題3分,共1５分)１.考慮二元函數(shù)的下面4條性質：①在點處連續(xù)②偏導數(shù)，存在③在點處可微④，在點處連續(xù)若“”表達由性質推出性質,則有【】Ａ.③②①Ｂ.②③①C.④②①Ｄ.④③②2．設函數(shù)為由方程所擬定的函數(shù)，其中為可導函數(shù)，為常數(shù)，則【】A.Ｂ.Ｃ.Ｄ.３．若二重積分可化為二次積分,則積分域可表達為【】Ａ．B．C.Ｄ.4.下列級數(shù)收斂的是【】Ａ.Ｂ.C.D.5.設常數(shù)，則級數(shù)【】Ａ.絕對收斂B.條件收斂C.發(fā)散D.斂散性與的取值有關扣分二、填空題（每小題３分,共1５分)6．設,則全微分.７．設,其中具有二階連續(xù)偏導數(shù),則．8.曲面在點處的切平面方程為.9.函數(shù)在點處沿該點梯度方向的方向導數(shù)為．10.設為圓周,則．三、計算題（每小題7分,共70分)扣分1１．求函數(shù)的極值．扣分1２.計算二重積分,其中是由直線,,所圍成的閉區(qū)域.?１3．求旋轉拋物面位于面上方部分的面積.扣分扣分14．計算曲線積分,其中為圓周取逆時針方向．?１５．計算三重積分，其中是由圓錐面與平面所圍成的空間閉區(qū)域.扣分扣分１6．計算曲面積分,其中為拋物面在平面下方的部分．

17.計算曲面積分,其中為上半球面的上側．扣分扣分18.求冪級數(shù)的收斂域與和函數(shù)．?19.將函數(shù)展開成的冪級數(shù)．扣分扣分20．計算,其中是由點經拋物線到點的有向曲線弧.2０期終試題(Ａ)參考答案及評分標準一、選擇題（每小題３分,共15分)1.D２.A３.C4．D5.Ｂ二、填空題(每小題3分，共15分)6.7.8.９．１0.三、計算題(每小題7分，共70分)11．,;，，.........．..．....．.．..．.．.．..............．．..．..．．...．2分由得,,解得駐點:,．.....．.．....．.．.．.....．..．．.．.....１分對于駐點，,由于，故不是極值;對于駐點，,由于，且，故是極小值..........．....．.．...．..．..．．..．．.．．．..．.．．....．．.....４分12．...．..．....．.........．............．........．.......．.５分.....．........．.．......．．．．.．..．.．．.．．....．......．....2分1３．.．..．．.．....．．...3分．.．...............．．．.．..．.......．.．．....．．...．．..．2分．.．．.............．．..．.....．..．.．.............．...．．..2分14．..．．..．．.．．......．......．．．..．.．.．．．．．..........．...．......．4分.．..．......．．...．...．.．..．..．．...．...............．...．．...．.3分1５..................．.．.............．．..．．....．.．...............................．．.．.．...．...4分......．........．.．.．.．.．.．..．．...．．.．．.....．...．..．.．..．．........．．.....．.........．．.．．.．...．．.．．...２分...........．.....．．.．....．...．........．...．．..．..．..．...．....．.....．..．..．..............．．.....．.．...1分解法二...．.......．．．.................．．..．.....．...．.............．．......．4分...．..．．......．.．...．.....．.．.．.......．．.......．．.....．..．....．．．..．.3分16...．．．.．.．．.....．．．.....．.．..．.．...．...．.．．.．..．.．．．.....．..．．...４分..................．．....．.．...．..．....．．.......．......．..．2分.........．....．．．.........．．..．.．........．.．....．.......．．..．.......．..........．．．1分17．增補平面塊,取下側.由高斯公式得:...．．....．...．......．..........．.....2分...．．....．...．．....．．...．．..............．...．．........．．.．...．．....．..．.．．.......３分..．．．..．........．........．...．.............．．....．．...．.....．.．...．.．.．..．.．．..．..2分18.（1）．令.當時,原級數(shù)成為，是發(fā)散的．故原級數(shù)的收斂域為（，）.．.．..．.......．....．．.....．......．.....．..．...．...3分(2）令，則．..．．....．2分...．....．....．...．..．...．．.2分19．．...．．．...．...．.....．..．．.．...............．.....．.1分....．.2分.....．.．.．．...............．．......．............．．２分.．.．.．.．．.．..．．.．.．．．.....．......．......．....．..．1分．.．.．.．.．.．．．．．..．........．....．.．....．.．..．.．.．．.1分20.令，,則．于是,在不包含原點的單連通區(qū)域內曲線積分與途徑無關..．..．..............．........．..．....．．．．．.....２分取途徑（從到)，則．．..．.......．.．．.....．...．.