?
成立??11?2x2?bx?a?2???1dx?ln(1?r)dr??x(2x?a)?0??
46.计算??cos
Dx?y
dxdy,其中D是由直线x?y?1与两坐标轴围成的三角形区域x?y47.设一平面过原点和点(6,?3,2),且与平面4x?y?2z?8垂直,求该平面方程48.设函数f(x)在区间[a,b]上非负可积,求证:1b??1b?
(1)当0???1时,?f(x)dx??f(x)dx??aab?ab?a??1b??1b?(2)当??1或??0时,?f(x)dx?f(x)dx???aab?a?b?a?
xsinxarccot(2017x)
dx
1?cos4x
x2?y2?z2在(x?y)2?z2?1条件下的极值12k2??49.计算定积分I?
????50.求函数u(x,y,z)?
51.计算lim
n???arctan
k?1n52.计算?
x0??
sinx,0?x???2f(t)g(x?t)dt(x?0),其中x?0时f(x)?x,而g(x)??
??0,x??2?
4?2x?dx?x2?lnx
f???dx?ln2?f???
1?2x?x?x2?x53.设f(x)为连续函数,求证:?
4154.设函数f(x)是连续函数,且满足f(x)?x?x
62?
x20f(x2?t)dt???f(xy)dxdy,其中D
D是以(?1,?1),(1,?1),(1,1)为顶点的三角形,f(1)?0,求?
10f(x)dx
1?sin2nx255.计算limdxn??lnn?0sinx134
x?2x(0?x?1)绕直线L2:y?x旋转所生成的旋转曲面的面3356.求由曲线L1:y?积。57.设函数f(x)在x?0时具有连续导数且f(1)?2,在右半平面(x?0)内存在可微函数u?u(x,y)使得du?4x3ydx?xf(x)dy,求函数f(x)和u(x,y)
58.设锐角三角形ABC的外接圆半径是一定值,且?A、?B、?C所对的边长分别是a、b、c,求证:59.设曲线Cn:x60.(1)记An?
dadbdc???0cosAcosBcosC2n?y2n?1(n?N?),记Ln为Cn的长,求证:limLn?8
n??nnn
??????,求A,使得limAn?A2222n??n?1n?4n?nn??(2)记Bn?n(An?A),求B,使得limBn?B(3)记Cn?n(Bn?B),求C,使得limCn?C
n??61.设函数f(x)在区间[0,1]上连续,求证:lim
n??0?
1nxnf(x)dx?f(1)
22262.设函数f(x,y)在区域D?(x,y)x?y?a上具有一阶连续偏导数,且满足22?????f?f??22?,其中a?0,求证:?a,以及max??????a???(x,y)?D?x?y????????
??f(x,y)x2?y2?a2??
D4
f(x,y)dxdy??a43763.计算?333dx(1?e
x?1x)(1?x2)
64.设函数f(x)在[a,b]上连续,且在(a,b)可微(0?a?b),f(a)?f(b)?0,?
baf(x)dx?1,证明:?x2[f?(x)]2dx?
2ab14u(x)x0?65.设函数f??(x)连续,f??(x)?0,f(0)?0,f?(0)?0,求lim?
x?0f(t)dt
0f(t)dt
。其中u(x)
是曲线y?f(x)上点P(x,f(x))处的切线在x轴上的截距。66.设函数f(x)是[0,1]上的可积函数且满足?
102求证:f(x)dx??xf(x)dx?1,?f(x)?4
001167.设数列?xn?为x1?收敛并求极限3,x2?3?3,xn?2?3?3?xn(n?1,2,?),求证:数列?xn?68.设f(x)在[0,??)上可导,f(0)?0,其反函数为g(x),若求f(x)
?
x?f(x)xg(t?x)dt?x2ex,69.求过点(?2,0,0)和(0,?2,0)且与锥面x?y?z的交线为抛物线的平面方程。22270.设函数f(x)在闭区间[0,1]上具有连续导数,且f(0)?f(1)?0,求证:2?1xf(x)dx??11[f?(x)]2dx取等号的条件是当且仅当f(x)?A(x3?x)时成立,其中????0?45?0A为常数22271.设函数f(x,y,z)在区域??(x,y,z)x?y?z?1上具有二阶连续偏导数,且满足????f?f?f?
?x?y?zdiv(grad(f(x,y,z)))?x2?y2?z2计算I???????x?dV?y?z???
72.已知平面区域D是(x,y)|x?y?1,L是D的边界正向一周,求证:8?22?xesinydy?ye?sinxdx?I???
2017x2?2018y21009L73.设函数f(x)是连续函数,且满足f(x)?1?
111f(y)f(y?x)dy,求f(x)dx??x0274.设函数f(x)在闭区间[a,b]上二阶可导,f(a)?0,f(b)?0,且对任意的x?[a,b],有f?(x)?0,f??(x)?0,又对于数列?xn?,其满足xn?1?xn?(1)求证:方程f(x)?0在(a,b)内恰有一个根?(2)求证:数列?xn?收敛于?f(xn)
,n?0,1,2,...,x0?b。f?(xn)75.设s?0,计算In?
?
??0xne?sxdx
1176.设函数f(x)在[0,1]上连续,且?
0xf(x)?0(k?0,1,...,n?1),?xnf(x)dx?1,求证:0k?x0?[0,1],使得f(x0)?(n?1)?2n222?x?y?z?1??77.设x,y,z?R,求方程组?的解333??7x?14y?21z?6
78.设an?0,Sn?
?a
k?1nk?an?1an(1)当a1?1且limSn??时,判断级数?和?的敛散性2n??n?1SnlnSnn?1SnlnSn?(2)当??0时,求证:级数an收敛??n?2SnSn?1?79.设I(r)?
(y?1)dx?(x?1)dy222C,其中曲线为,取正向。(x?1)?(y?1)?r22??(x?y?2x?2y?2)C求极限limI(r)
r?0980.设函数f(x)在[0,1]上可积,且0?m?f(x)?M,求证:f(x)dx
?
10?
10dx(m?M)2?f(x)4mM81.设函数f(x)在[0,1]上具有二阶连续导数,且有f(0)?f(1)?0,f??(x)?A。求证:对任意的x?[0,1],有f?(x)?
A
282.设??1,求证:级数n
收敛?????1?2?3?????nn?1?83.设f(x,y)为具有二阶连续偏导数的二次齐次函数,即对任意的x、y、t,都有f(tx,ty)?t2f(x,y).?????(1)求证:?x?yf(x,y)?2f(x,y)??x??y??
(2)设D是由L:x?y?4正向一周所围成的闭区域,求证:222?f(x,y)ds???div(gradf(x,y))dxdy
LD84.设函数f(x)在区间[a,b]上连续,f??(a)?1,g(x)在[a,b]上正值连续,如果?
xaf(t)g(t)dt?f(?(x))?g(t)dt,?(x)?(a,x),求lim?ax?ax?(x)?a
x?a85.已知平面区域D是(x,y)|x?y?1,L是D的边界正向一周,求证:?22?xesinydy?ye?sinxdx4I????22x?xy?2y5L86.设函数f(x)在[0,1]上具有一阶连续导数,且f(0)?0,f(1)?1,求证:对任意的x?[0,1],有?f?(x)?f(x)dx?
011e
x2?y287.求证:对任意的x?0,y?0,有?ex?y?2410
专升本必做120道基础习题
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