高数题,求解答,谢谢

2025-05-18 07:36:43
推荐回答(1个)
回答(1):

1)lim(y→0+)F(x,y)

= lim(y→0+)∫[-y,y]f(x+t)dt/(2y) (0/0)
= lim(y→0+)[f(x+y)-f(x-y)(-1)]/2
= f(x);
2)DF/Dx
= (D/Dx){∫[-y,y]f(x+t)dt/(2y)}
= ∫[-y,y]f'(x+t)dt/(2y)

= [f(x+y)-f(x-y)]/(2y);
3)lim(y→0+)(DF/Dx)
= lim(y→0+)[f(x+y)-f(x-y)]/(2y) (0/0)
= lim(y→0+)[f'(x+y)-f'(x-y)(-1)]/2

= f'(x)。