1.\(x≒0\)のとき、次の式の近似式を求めなさい。
(1)\(e^x\)
\(f(x)=e^x\)とおくと、
\(\displaystyle f'(x)=e^x\)
\(\displaystyle f(0)=1,f'(0)=1\)
よって、
\(\displaystyle e^x≒1+x\)
(2)\(\log(1+x)\)
\(f(x)=\log(1+x)\)とおくと、
\(\displaystyle f'(x)=\frac{1}{1+x}\)
\(\displaystyle f(0)=0,f'(0)=1\)
よって、
\(\displaystyle \log(1+x)≒x\)
(3)\(\displaystyle \frac{1}{1+x}\)
\(\displaystyle f(x)=\frac{1}{1+x}\)とおくと、
\(\displaystyle f'(x)=-\frac{1}{(1+x)^2}\)
\(\displaystyle f(0)=1,f'(0)=-1\)
よって、
\(\displaystyle \frac{1}{1+x}≒1-x\)
2.次の式の近似値を求めなさい。
(1)\(\sqrt[4]{1.03}\)
\(x≒0\)のとき、\(\displaystyle \sqrt[4]{1+x}≒1+\frac{1}{4}x\)
よって、
\(\displaystyle \sqrt[4]{1+0.03}≒1+\frac{1}{4}・0.03=1.0075\)
(2)\(\log1.01\)
\(x≒0\)のとき、\(\displaystyle \log(1+x)≒x\)
よって、
\(\displaystyle \log(1+0.01)≒0.01\)
(3)\(\displaystyle \frac{1}{0.998}\)
\(x≒0\)のとき、\(\displaystyle \frac{1}{1+x}≒1-x\)
よって、
\(\displaystyle \frac{1}{1+(-0.002)}≒1.002\)
(4)\(e^{0.01}\)
\(x≒0\)のとき、\(\displaystyle e^x≒1+x\)
よって、
\(\displaystyle e^{0.01}≒1.01\)