第3章 3.18

(1)

$(1.004)^7 = (1+0.004)^7 = (1+x)^7 \quad (x = 0.004)$

$f = (1+x)^7$ とおくと

$f' = 7(1+x)^6 \qquad f'_{(0)} = 7$

$f'' = 42(1+x)^5 \qquad f''_{(0)} = 42$

$f''' = 210(1+x)^4 \qquad f'''_{(0)} = 210$

これから

$f \cong 1 + 7x + \frac{42}{2!}x^2 = 1 + 7\times 0.004 + 21\times 0.004^2 = 1.0283$

(2)

$\frac{1}{5.03} = \frac{1}{5}\cdot\frac{1}{(1+0.006)} = \frac{1}{5}\cdot\frac{1}{(1+x)} \quad (x=0.006)$

$f = \dfrac{1}{5}\cdot\dfrac{1}{1+x}$ とおくと

$f' = \frac{1}{5}\cdot\frac{-1}{(1+x)^2} \qquad f'_{(0)} = \frac{-1}{5}$

$f'' = \frac{1}{5}\cdot\frac{2}{(1+x)^3} \qquad f''_{(0)} = \frac{2}{5}$

$f \cong \frac{1}{5}\cdot(1-x+x^2) = \frac{1}{5}\cdot(1-0.006+0.006^2) = 0.1988$

(3)

$\sqrt[3]{4} = \frac{3}{2}\sqrt[3]{1+\frac{5}{27}} = \frac{3}{2}\sqrt[3]{1+x} \qquad x=\frac{5}{27}$

$f = \dfrac{3}{2}\sqrt[3]{1+x}$ とおくと

$f' = \frac{3}{2}\cdot\frac{1}{3(1+x)^{\frac{2}{3}}} = \frac{1}{2(1+x)^{\frac{2}{3}}} \qquad f'_{(0)} = \frac{1}{2}$

$f'' = -\frac{2}{3}\cdot\frac{1}{2(1+x)^{\frac{5}{3}}} = -\frac{1}{3(1+x)^{\frac{5}{3}}} \qquad f''_{(0)} = -\frac{1}{3}$

$f''' = \frac{5}{3}\cdot\frac{1}{3(1+x)^{\frac{8}{3}}} = \frac{5}{9(1+x)^{\frac{8}{3}}} \qquad f'''_{(0)} = \frac{5}{9}$

これから

$f \cong \frac{3}{2} + \frac{1}{2}x - \frac{1}{6}x^2 + \frac{5}{54}x^3 = \frac{3}{2} + \frac{1}{2}\cdot\frac{5}{27} - \frac{1}{6}\cdot\left(\frac{5}{27}\right)^2 + \frac{5}{54}\cdot\left(\frac{5}{27}\right)^3 = 1.5874$

(4)

$\sin 10° = \sin\frac{\pi}{18} = \sin x \qquad \left(x=\frac{\pi}{18}\right)$

$f = \sin x$ とおくと

$f \cong x - \frac{1}{3!}x^3 = \frac{\pi}{18} - \frac{1}{6}\cdot\left(\frac{\pi}{18}\right)^3 = 0.1736$