求解下面两道高数题QAQ
利用(1+x)∧k≈1+kx求以下两个函数在x→0时的近似值1）f(x)=(1-x)∧62)f(x)=[1-(1/(2+x)]∧(2/3)...
利用(1+x)∧k≈1+kx求以下两个函数在x→0时的近似值
1）f(x)=(1-x)∧6
2) f(x)=[1-(1/(2+x)]∧(2/3) 展开
1）f(x)=(1-x)∧6
2) f(x)=[1-(1/(2+x)]∧(2/3) 展开
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(1)
f(x) = (1-x)^6≈1-6x
(2)
[1- 1/(2+x) ]∧(2/3)
≈[1-(2+x)^(-1) ]^(2/3)
={1- (1/2)[1+(1/2)x]^(-1) }^(2/3)
≈{ 1- (1/2)[ 1-(1/2)x ] }^(2/3)
=[1/2 + (1/4)x ]^(2/3)
=(1/2)^(2/3) . ( 1 + (1/2)x )^(2/3)
≈(1/2)^(2/3) . ( 1 + (1/3)x )
f(x) = (1-x)^6≈1-6x
(2)
[1- 1/(2+x) ]∧(2/3)
≈[1-(2+x)^(-1) ]^(2/3)
={1- (1/2)[1+(1/2)x]^(-1) }^(2/3)
≈{ 1- (1/2)[ 1-(1/2)x ] }^(2/3)
=[1/2 + (1/4)x ]^(2/3)
=(1/2)^(2/3) . ( 1 + (1/2)x )^(2/3)
≈(1/2)^(2/3) . ( 1 + (1/3)x )
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