证明:若f在[a,+∞)上一致连续,且收敛,则
证明:若f在[a,+∞)上一致连续,且<img src='https://img2.soutiyun.com/ask/2021-02-05/98138680072271.png' />收敛,则<img src='https://img2.soutiyun.com/ask/2021-02-05/981386807293086.png' />
时间:2024-02-29 07:25:40
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由莱布尼兹公式可知:若函数f(x)在[a,b]上连续,且存在原函数,则f在区间[a,b]上可积。()
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设{f<sub>n</sub>}在D上一致收敛于f,{g<sub>n</sub>}在D上一致收敛于g,证明{f<sub>n</sub>±g<sub>n</sub>}在D上一致收敛于f±g.
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证明:若函数f(x)在[a,b]连续、非负,且使f(x0)>0,则
证明:若函数f(x)在[a,b]连续、非负,且<img src='https://img2.soutiyun.com/ask/2020-11-12/97406224084526.png' />使f(x0)>0,则
<img src='https://img2.soutiyun.com/ask/2020-11-12/974062250390806.png' />
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证明:若函数f(x)在a连续,则函数在a都连续.
证明:若函数f(x)在a连续,则函数
<img src='https://img2.soutiyun.com/ask/2020-11-11/973957882099598.png' />
在a都连续.
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若f(z)在周线C内部除有一个一阶级点外解析,且连续到C,在C上|f(z)|=1.证明:f(z)=a(a| >1) 在C内部恰好有一个根. 提示用辐角原理证明N(f(z)-a,C)-P(f(z)-a,C)=0.
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设函数f(x)在[a,b]上连续,且f(x)>0,证明:在(a,b)内存在一个ξ,使得
设函数f(x)在[a,b]上连续,且f(x)>0,证明:在(a,b)内存在一个ξ,使得
<img src='https://img2.soutiyun.com/ask/2021-01-14/979465674691464.png' />
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证明:若函数f(x)在区间[a,+∞)上连续且有极限则(x)在区间[a,+∞)上是有界的.
证明:若函数f(x)在区间[a,+∞)上连续且有极限<img src='https://img2.soutiyun.com/ask/2020-12-13/976732708656138.png' />则(x)在区间[a,+∞)上是有界的.
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设(x)在[a,b]上连续,在(a,b)内可导且f'(x)≤0,证明在(a,b)内有F'(x)≤0.
设(x)在[a,b]上连续,在(a,b)内可导且f'(x)≤0,
<img src='https://img2.soutiyun.com/ask/2020-08-06/965576645302938.png' />
证明在(a,b)内有F'(x)≤0.
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设f(x)在[a, b]上连续,在(a, b)内可导且f'(x)≤0,证明:在(a, b)内有F'(a)≤0
设f(x)在[a, b]上连续,在(a, b)内可导且f'(x)≤0,
<img src='https://img2.soutiyun.com/ask/2020-12-14/976805726019948.png' />
证明:在(a, b)内有F'(a)≤0
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证明:若函数f(x)与φ(x)在[a,b]连续,则
证明:若函数f(x)与φ(x)在[a,b]连续,则
<img src='https://img2.soutiyun.com/ask/2020-11-12/974060778329609.png' />
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证明:若函数f(x)在R有任意阶导函数,且函数列{f<sup>(n)</sup>(x)}在R一致收敛于极限函数φ(x),则φ(x)=ce<sup>x</sup>,其中c是常数.
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证明:若函数f(x)在(a,+∞)连续,且则f(x)在(a,+∞)有界.
证明:若函数f(x)在(a,+∞)连续,且<img src='https://img2.soutiyun.com/ask/2020-11-11/97395757022676.png' />则f(x)在(a,+∞)有界.
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证明:若函数y=f(x)在[a,b]严格增加,且连续则反丽数x=f<sup>-1</sup>(y)在点a=f(a)右连续,即
证明:若函数y=f(x)在[a,b]严格增加,且连续则反丽数x=f<sup>-1</sup>(y)在点a=f(a)右连续,即
<img src='https://img2.soutiyun.com/ask/2020-11-11/973957297199144.png' />
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证明:若f(x),g(x)在任何区间[a,A]可积,又设f<sup>2</sup>(x),g<sup>2</sup>(x)在[a,+∞)积分收敛,那末[f(x)+g(x)]<sup>2</sup>和|f(x)·g(x)|在[a,+∞)上皆可积.
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证明:若无穷积分绝对收敛,函数φ(x)在[a,+∞)单调有界,则无穷积分收敛.
证明:若无穷积分<img src='https://img2.soutiyun.com/ask/2020-11-13/97414103002922.jpg' />绝对收敛,函数φ(x)在[a,+∞)单调有界,则无穷积分<img src='https://img2.soutiyun.com/ask/2020-11-13/974141041163856.jpg' />收敛.
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证明点列{fn}按题2中距离收敛于f∈C<sup>∞</sup>[a,b]的充要条件为f的各阶导数在[a,b]上一致收敛于f的各阶导数.
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判断下列复级数的敛散性,若收敛指明条件收敛还是绝对收敛. 设D是一个有界区域,其边界为aD,若fn()+… 在 上一致收敛.
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证明:若可积函数列f<sub>n</sub>(x)(n=1,2,...)在区间[a,b]上一致收敛于可积函数f(x),则它也平均收敛于f(x)[相反的结论不成立].
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设f(x,y)在[a,+∞;c,d]连续,对[c,d)上每一个收敛,但积分在y= d发散.证明这积分在[c,d]非一致收
设f(x,y)在[a,+∞;c,d]连续,对[c,d)上每一个<img src='https://img2.soutiyun.com/ask/2021-01-27/98058817504593.png' />收敛,但积分在y= d发散.证明这积分在[c,d]非一致收敛。
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设I为有限区间.证明:若f在I上一致连续,则f在I上有界,举例说明此结论当I为无限区间不一定成立.
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设f(x,y)在[a,b;c,∞)上连续,且保持同一符号,y)dy在[a,b]上连续,证明:
设f(x,y)在[a,b;c,∞)上连续,且保持同一符号,<img src='https://img2.soutiyun.com/ask/2021-01-06/978797314201327.png' />y)dy在[a,b]上连续,证明:
<img src='https://img2.soutiyun.com/ask/2021-01-06/978797330365252.png' />
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证明:(1)若函数f在[a,b]上可导,且f'(x)≥m,则(2)若函数f在[a,b]上可导,且(3)对任意实数x<sub>1
证明:(1)若函数f在[a,b]上可导,且f'(x)≥m,则
<img src='https://img2.soutiyun.com/ask/2021-02-04/98128598322409.png' />
(2)若函数f在[a,b]上可导,且
<img src='https://img2.soutiyun.com/ask/2021-02-04/981285989538451.png' />
(3)对任意实数x<sub>1</sub>,x<sub>2</sub>,都有
<img src='https://img2.soutiyun.com/ask/2021-02-04/981286001647143.png' />
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设f(x)在[a,b]上连续,在(a,b)内可导且f'(x)≤0,证明在(a,b)内有F'(x)<0.
设f(x)在[a,b]上连续,在(a,b)内可导且f'(x)≤0,
<img src='https://img2.soutiyun.com/ask/2020-12-04/975925572077622.png' />
证明在(a,b)内有F'(x)<0.
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设f在(a,b)内连续,且.证明:f在(a.b)内有最大值或最小值.
设f在(a,b)内连续,且<img src='https://img2.soutiyun.com/ask/2020-11-30/975588322827546.png' />.证明:f在(a.b)内有最大值或最小值.