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f(x)
=ln(1+2x) ; -1/2<x≤1
=ax+b : x>1
f(1)=f(1-) =lim(x->1-) ln(1+2x) =ln3
f(1+) =lim(x->1+) (ax+b) = a+b
a+b= ln3 (1)
f'(1-)
=lim(h->0) [ ln(1+2(1+h) ) - f(1) ]/h
=lim(h->0) [ ln(3+2h) - ln3 ]/h (0/0 分子分母分别求导)
=lim(h->0) 2/(3+2h)
=2/3
f'(1+)
=lim(h->0) [ a(1+h) +b - f(1) ]/h
=lim(h->0) [(a+b - ln3) +ah ] /h (0/0 分子分母分别求导)
=lim(h->0) ah/h
=a
=> a=2/3
from (1)
a+b= ln3
b= ln3 - 2/3
(a,b) =( 2/3 , ln3 -2/3)
=ln(1+2x) ; -1/2<x≤1
=ax+b : x>1
f(1)=f(1-) =lim(x->1-) ln(1+2x) =ln3
f(1+) =lim(x->1+) (ax+b) = a+b
a+b= ln3 (1)
f'(1-)
=lim(h->0) [ ln(1+2(1+h) ) - f(1) ]/h
=lim(h->0) [ ln(3+2h) - ln3 ]/h (0/0 分子分母分别求导)
=lim(h->0) 2/(3+2h)
=2/3
f'(1+)
=lim(h->0) [ a(1+h) +b - f(1) ]/h
=lim(h->0) [(a+b - ln3) +ah ] /h (0/0 分子分母分别求导)
=lim(h->0) ah/h
=a
=> a=2/3
from (1)
a+b= ln3
b= ln3 - 2/3
(a,b) =( 2/3 , ln3 -2/3)
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