试卷代号:7032
上海开放大学2017至2018学年第一学期
《高等数学基础》期末复习题答案
一.选择题
1.D 2.C 3.A 4.D 5.B 6. C 7. C 8.C 9.B 10. B 11.A 12.B 13.C 14.A 15.B 16.A
二.填空题
1.3?x?4 2.x??1且x?3 3.?1?x?5且x?0
4.?1,e?2? 5.y?ln2?12?x?2? 6. ?2sin2xf?(cos2x)dx7. 4x2?1 8. ?4sin2x 9. 122F(x?1)?C
10.23 11.0 12.?xcosx2
13.e?1
14. ?F(cosx)?C 15.1 三.计算题
1?2x1、求极限lim?4x?1?x????4x?1??
1?2x1?2x1?2 解:lim?4x?1?x????4x?1???lim?4x?1?2?x
x???lim?2????4x?1?x????1?4x?1??=e?4x?12、求极限lim?2x?1?x????2x?3??
?4x?1?4x?1?4x?1解:lim?2x?1?8x????2x?3???lim?2x?3?4x?????2x?3???lim?x????1?4?2x?3??=e
4x3、求极限lim?x???3x??3x?2?? 4x4x8解: lim?x???3x??2??3x?2???limx????1?3x?2???e?3
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4、求极限limsin3xx?01?4x?1
解:limsin3xx?01?4x?1?lim3x3x?0?2x??2
5、求极限limxln(1?3x2)x?01?3x3?1 解:limxln(1?3x2)x?01?3x3?1?limx??3x2x?03x3??2 26、求极限limln(1?2x)x?01?4x?1
解:limln(1?2x)?x?01?4x?1?lim2xx?02x??1 7、设函数y?x?ecosx?2x?,求dy。
3解:y?xecosx?2x2
y??x?ecosx?x?ecosx??3?1???2x2??ecosx?xsinxecosx?3x2???
dy??1??ecosx?xsinxecosx?3x2???dx8、设函数y?xcos(3x?1),求dy。
解:y???x??cos(3x?1)?x?cos(3x?1)???12xcos(3x?1)?3xsin(3x?1)dy???1?2xcos(3x?1)?3xsin(3x?1)???dx
9、设函数y?x2?ln2x?x?,求dy。
5解: y?x2ln2x?x2
5y??(x2)?ln2x?x2(ln2x)??(x2)?
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?x?2xln2x?5322x 3 dy???x?2xln2?x52?x
?2? d x ?10、设函数y?3x?1cos2x,求dy。
解:y???3x?1??cos2x??3x?1??cos2x??2?3x?1?sin2x?cos2x?2?3cos2x??cos2x?2 dy?3cos2x?2?3x?1?sin2x?cos2x?2dx
11、设函数y?2x1?e3x,求dy。
3x解:y???2x????1?e3x??2x??1?e??6xe3x?1?e3x?2?2?1?e3x???1?e3x?2
3xdy?2?1?e??6xe3x?1?e3x?2dxy?e?2x12、设函数1?x2,求dy。
?2x2?2x22?2x解:y???e????1?x??e??1?x???2?1?x?x?e?1?x2?2??1?x2?2
2?2xdy??2?1?x?x?e?1?x2?2dx13、设函数y?sin2x1?cosx,求dy。
解:y???sin2x???1?cosx??sin2x??1?cosx???1?cosx?2
?2cosx2??1cxo??ssxi??n?2?xsin?1?coxs?2
dy?2cosx2??1cxo?s?sxi?n2x?1?coxs?2ds xi n
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14、计算不定积分
?x2sinx2dx 解:x2 2x 2 0
+ — + sinx2 ?2cosx2 ?4sinx2 8cosx2
?x2sinx2xxx2dx=?2xcos2?8xsin2?16cos2?C 15、计算不定积分
?x2cosx3dx 解:x2 2x 2 0
+ — + cosx3 3sinx3 ?9coxsx3 ?27sin3 ?x2cosx2xxx3dx?3xsin3?18xcos3?54sin3?c
16、计算不定积分
?x2e?3xdx
解: x2 2x 2 0
+ — + e?3x ?1e?3x3
19e?3x ?127e?3x ?x2e?3xdx??x23e?x3?2x9e?x3?227e?x3?c
四、 应用题
1、求由抛物线y?2?x2与直线y??x所围的面积。
解:由??y?2?x2?x?y??x1??1,x2?2 S=?2(2?x2?(?x))dx??2?1?1(2?x2?x)dx ?92
yy?2?x2 x y??x 9
2、解:抛物线y?x2与直线y?2?x的交点为(?2,4),(1,1)
面积A? ???2?x?x?dx
2?219 2
3、求由抛物线y?x2?x与直线y?x所围的面积。
y?y?x2?xy?x解:由??x1?0,x2?2 ?y?xy?x2?x所围的面积S??20(x?(x?x))dx??(2x?x2)dx022x ?43
yy?x4、解:抛物线y?x2?2与直线y?x的交点为(?1,?1),(2,2) 面积A???x?(x?122?2)?dx xy?x2?2?
9 25、解:解:抛物线y?x与直线y?6?x的交点为(?3,9),(2,4) 面积A?2125? 6?x?xdx???622?3yy?6?xy?x2x
6、要做一个有底无盖的圆柱体容器,已知容器的容积为4立方米,试问如何选取底半径和高的尺寸,才能使所用材料最省。 解:设圆柱体底半径为r,高为h,
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