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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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若函数f在[a,b]上的黎曼和的极限存在,则函数f在 [a,b] 上可积.
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如果函数 y=f(x) 在闭区间[ a,b ]内连续,且 f(a) 和 f(b) 符号相反,即 f(a)·f(b)<0 ,那么存在某个 ξ∈(a,b) ,使得 ( )
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若函数f(x)在区间【a,b】上连续,则它在这个区间上可能不存在原函数
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证明:若函数f,g在区间[a,b]上可导,且f'(x)>g'(x),f(a)=g(a),则在(a,b]内有f(x)>g(x).
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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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(a)证明:若x(t)是偶函数,即x(t)=x(—t),则X(s)=X(—s).(b)证明:若x(t)是奇函数,即x(t)=—x(—t),
(a)证明:若x(t)是偶函数,即x(t)=x(—t),则X(s)=X(—s).
(b)证明:若x(t)是奇函数,即x(t)=—x(—t),则X(s)= —X(—s).
(c)对于图9-24所示的零-极点图,判断有无与一个偶时间函数相对应的零-极点图?若有,对这些图指出所需的收敛域。
<img src='https://img2.soutiyun.com/ask/2020-09-16/969094317052211.png' />
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证明:若函数f(x)在[a,b]可积,则函数[f(x)]<sup>2</sup>在[a,b]也可积.
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设函数f(x)在[a,b]上二阶可导,且f(A)= f(b)=0,令F(x)=(x-(A)f(x),证明:在(a,b) 内至少存在一点ξ,使得F"(ξ)=0.
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设f(x)为[-a,a]上的连续函数,证明:(1)若f(x)是偶函数,则是[-a,a]上的奇函数;(2)若f(x)是奇函数
设f(x)为[-a,a]上的连续函数,证明:
(1)若f(x)是偶函数,则<img src='https://img2.soutiyun.com/ask/2020-10-12/971368555517115.png' />是[-a,a]上的奇函数;
(2)若f(x)是奇函数,则<img src='https://img2.soutiyun.com/ask/2020-10-12/971368586317876.png' />是[-a,a]上的偶函数。
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证明.若函数f(x)在区间[-π,π]可积,且a<sub>k</sub>,b<sub>k</sub>,是函数f(x)的傅里叶系数,则有不等式后者称
证明.若函数f(x)在区间[-π,π]可积,且a<sub>k</sub>,b<sub>k</sub>,是函数f(x)的傅里叶系数,则<img src='https://img2.soutiyun.com/ask/2020-11-13/974121166578095.jpg' />有不等式
<img src='https://img2.soutiyun.com/ask/2020-11-13/974121183156043.png' />
后者称为贝塞尔①不等式.(证明1),讨论积分<img src='https://img2.soutiyun.com/ask/2020-11-13/974121199972005.png' />
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证明:若函数f(x)>0,在[a,b]可积,令则
证明:若函数f(x)>0,在[a,b]可积,令<img src='https://img2.soutiyun.com/ask/2020-11-13/974108578848118.png' />则
<img src='https://img2.soutiyun.com/ask/2020-11-13/974108619490443.png' />
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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,y)在R(a<sub>1</sub>≤x≤b<sub>1</sub>,a<sub>2</sub>≤y≤b<sub>2</sub>)连续,
证明:若函数f(x,y)在R(a<sub>1</sub>≤x≤b<sub>1</sub>,a<sub>2</sub>≤y≤b<sub>2</sub>)连续,
<img src='https://img2.soutiyun.com/ask/2020-11-14/974189428787492.png' />
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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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设f(x)是[a,b]上的有限函数,若存在M>0,使对任何ε>0都有则f(x)是[a,b]上有界差函数.
设f(x)是[a,b]上的有限函数,若存在M>0,使对任何ε>0都有<img src='https://img2.soutiyun.com/ask/2020-08-13/966171531356788.png' /><img src='https://img2.soutiyun.com/shangxueba/ask/50574001-50577000/50574553/spacer.gif' />
则f(x)是[a,b]上有界差函数.
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设函数f在[a,b]上可导.证明:存在∈(a,b),使得
设函数f在[a,b]上可导.证明:存在<img src='https://img2.soutiyun.com/ask/2020-11-29/975510230161692.png' />∈(a,b),使得
<img src='https://img2.soutiyun.com/ask/2020-11-29/975511152857467.png' />
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若函数f(x)在[a,b]上可积,证明存在折线函数列
若函数f(x)在[a,b]上可积,证明存在折线函数列<img src='https://img2.soutiyun.com/ask/2021-01-22/98018065819823.png' />
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证明:若函数f(x)在[a,b]单调增加,则
证明:若函数f(x)在[a,b]单调增加,则
<img src='https://img2.soutiyun.com/ask/2020-11-12/974060207561962.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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证明:若函数f(x)与g(x)在[a,b]可积,则φ(x)=max{f(x),g(x)}与φ(x)=min{f(x),g(x)}在[a,b]都可积.
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证明:若函数f(x,u)在矩形域R(a≤x≤b,a≤u≤β)连续,而函数a(u)与b(u)在区间[a,β]也连续,且有a≤a(u
证明:若函数f(x,u)在矩形域R(a≤x≤b,a≤u≤β)连续,而函数a(u)与b(u)在区间[a,β]也连续,且<img src='https://img2.soutiyun.com/ask/2020-11-13/974144402448111.png' />有
a≤a(u)≤b,a≤b(u)≤b,
则函数<img src='https://img2.soutiyun.com/ask/2020-11-13/97414442394134.png' />在区间[a,β]连续.