利用微分的线性近似求下列各数的近似值(1)1.02开五次方(2)e^0.035(3)ln1.002(4)ln0.99(5)1.02开三次方

Question

利用微分的线性近似求下列各数的近似值
(1)1.02开五次方
(2)e^0.035
(3)ln1.002
(4)ln0.99
(5)1.02开三次方

Answer 1

f(x+u)约等于f(x)+f'(x)*u 其中u是一个小量
1)f(x)=x^(1/5),f'(x)=(1/5)x^(-4/5)
f(1.02)=f(1+0.02)=f(1)+f'(1)*0.02=1+0.2*0.02=1.004
2)f(x)=e^x,f'(x)=e^x
f(0.035)=f(0+0.035)=f(0)+f'(0)*0.035=1+1*0.035=1.035
3)f(x)=lnx,f'(x)=1/x
f(1.002)=f(1+0.002)=f(1)+f'(1)*0.002=0+1*0.002=0.002
4)f(0.99)=f(1-0.01)=f(1)+f'(1)*(-0.01)=0+1*(-0.01)= -0.01
5)f(x)=x^(1/3),f'(x)=(1/3)*x^(-2/3)
f(1.02)=f(1+0.02)=f(1)+f'(1)*0.02=1+(1/3)*0.02=1.0067