设f(x)可导,F(x)=f(x)(1+|sinx|)则f(0)=0是F(x)在x=0处可导的充分必要条件
来源:学生作业帮 编辑:百度作业网作业帮 分类:数学作业 时间:2024/05/21 15:41:50
设f(x)可导,F(x)=f(x)(1+|sinx|)则f(0)=0是F(x)在x=0处可导的充分必要条件
①充分性: f(0)=0 ,则:
F'(0)
=lim(x->0) [f(x)(1+|sinx|)-f(0)(1+|sin0|]/x
=lim(x->0) f(x)(1+|sinx|)/x
=lim(x->0) [f(x)-f(0)]/x* (1+|sinx|)
= f'(0)*1
= f'(0)
②必要性:F(x)在x=0处可导,则:
F'(0+0)=F'(0-0)
由导数极限定理【此处也可改为极限式计算】:
F'(0+0)
=lim(x->0+) [f(x)(1+sinx)]'
=lim(x->0+) [f'(x)(1+sinx)+f(x)*cosx]
=f'(0)+f(0)
F'(0-0)
=lim(x->0+) [f(x)(1-sinx)]'
=lim(x->0+) [f'(x)(1-sinx)-f(x)*cosx]
=f'(0)-f(0)
∵F'(0+0)=F'(0-0)
∴f'(0)+f(0)=f'(0)-f(0)
∴ f(0)=0
F'(0)
=lim(x->0) [f(x)(1+|sinx|)-f(0)(1+|sin0|]/x
=lim(x->0) f(x)(1+|sinx|)/x
=lim(x->0) [f(x)-f(0)]/x* (1+|sinx|)
= f'(0)*1
= f'(0)
②必要性:F(x)在x=0处可导,则:
F'(0+0)=F'(0-0)
由导数极限定理【此处也可改为极限式计算】:
F'(0+0)
=lim(x->0+) [f(x)(1+sinx)]'
=lim(x->0+) [f'(x)(1+sinx)+f(x)*cosx]
=f'(0)+f(0)
F'(0-0)
=lim(x->0+) [f(x)(1-sinx)]'
=lim(x->0+) [f'(x)(1-sinx)-f(x)*cosx]
=f'(0)-f(0)
∵F'(0+0)=F'(0-0)
∴f'(0)+f(0)=f'(0)-f(0)
∴ f(0)=0
设f(x)可导,F(x)=f(x)(1+|sinx|),若F(X)在点x=0处可导,则必有(?)
设f(x)可导且f(x)=0,证明:F(X)=f(x)(1+/sinx/)在x=0点可导,并求F(0)的导数
1.设函数f(x)在x=0处某邻域内有定义,且f(0)=0,则f(x)在x=0处可导的充分必要条件为()
设f(x)可导,且f(0)=0,证明F(X)=f(x)(1+/SINX/)在x=0处可导
设f(x)=sinx-∫(0~t)(x-t)f(t)dt,f为连续函数,求f(x).
设f(x)为可导函数,且满足条件lim(x->0)[f(1)-f(1-x)]/2x=1,则曲线y=f(x)在(1,f(x
设函数f(X)定义在(0,+∞)上,f(1)=0,导数f'(x)=1/x,g(x)=f(x)+f'(x) .
设f(x)可导.且f(x)导数>0,f(0)=0,f(a)=b,g(x)是f(X)的反函数,求∫f(x)dx(上a下o)
设F(x)是f(x)的一个原函数,且F(0)=1,f(x)/F(x)=3x,求F(x)和f(x)
设f(x)可导,且满足lim(x→0)f(1)-f(1-x)/2x=-1,求曲线y=f(x)在点(1,f(1))出的切线
f(x)是R上的函数 f(x+3)=-f(x) 当0≤X≤1 f(x)=x 则f(9.5)等于?
高数题.导数设F(X)=f(x)g(x),x=a是g(x)的跳跃间断点.f'(x)存在,则f(x)=f'(x)=0是F(