【高校数学Ⅲ】4-1-7 近似式｜問題集

\(f(x)=e^x\)とおくと、 \(\displaystyle f'(x)=e^x\)
\(\displaystyle f(0)=1,f'(0)=1\)
よって、
\(\displaystyle e^x\approx1+x\)

\(f(x)=\log(1+x)\)とおくと、 \(\displaystyle f'(x)=\frac{1}{1+x}\)
\(\displaystyle f(0)=0,f'(0)=1\)
よって、
\(\displaystyle \log(1+x)\approx x\)

\(\displaystyle f(x)=\frac{1}{1+x}\)とおくと、 \(\displaystyle f'(x)=-\frac{1}{(1+x)^2}\)
\(\displaystyle f(0)=1,f'(0)=-1\)
よって、
\(\displaystyle \frac{1}{1+x}\approx1-x\)

\(x\approx0\)のとき、\(\displaystyle \sqrt[4]{1+x}\approx1+\frac{1}{4}x\)
よって、
\(\displaystyle \sqrt[4]{1+0.03}\approx1+\frac{1}{4}・0.03=1.0075\)

\(x\approx0\)のとき、\(\displaystyle \log(1+x)\approx x\)
よって、
\(\displaystyle \log(1+0.01)\approx0.01\)

\(x\approx0\)のとき、\(\displaystyle \frac{1}{1+x}\approx1-x\)
よって、
\(\displaystyle \frac{1}{1+(-0.002)}\approx1.002\)

\(x\approx0\)のとき、\(\displaystyle e^x\approx1+x\)
よって、
\(\displaystyle e^{0.01}\approx1.01\)