Newton's method
works (very)well
If |f'| not small and |f''| not too big
MEAN VALUE THEOREM (MVT)
If you go from Boston to LA
(3000 mi) in 6 hours,then at sometimes you are going at ave speed 500 mi/hor
THEOREM(MVT) (*)
for some c,a<c<b
provided f is differntiable in a<x<b and continious in
PROOF OF MVT
one bad point ruins the proof:
we need f'(x) to exist at all x,a<x<b
Applications to grahing
1.If f'>0 ,then f is increasing
2.If f'<0 ,then f is decreasing
3.If f'=0 ,then f is constant
PROOF REWRITE(*)
f(b) -f(a) = f'(c).(b-a)
f(b) = f(a)+f'(c).(b-a)
1.f'(c)>0
f(b)>f(a)
2.f'(c)>0
f(b)<f(a)
3.f'(c)=0
f(b)=f(a)
comparison linear approx :
b near a (b-a =
)
some c between a and b:
ave spped
initial speed
来源：CSDN
作者：KuFun人工智能
链接：https://blog.csdn.net/fsdaewrq/article/details/103628627