第3章 3.19

(1)

$f = e^x \qquad (x = 1)$

$f' = e^x \qquad x = 1 のとき \qquad f' = e$

$f'' = e^x \qquad x = 1 のとき \qquad f'' = e$

これから

$f \cong e + e(x-1) + \frac{e}{2}(x-1)^2$

(2)

$f = \log x \qquad (x = e)$

$f' = \dfrac{1}{x} \qquad x = e のとき \qquad f' = \dfrac{1}{e}$

$f'' = -\dfrac{1}{x^2} \qquad x = e のとき \qquad f'' = -\dfrac{1}{e^2}$

これから

$f \cong 1 + \frac{1}{e}(x-e) - \frac{1}{2e^2}(x-e)^2$

(3)

$f = \sin x \qquad (x = \pi)$

$f' = \cos x \qquad x = \pi のとき \qquad f' = -1$

$f'' = -\sin x \qquad x = \pi のとき \qquad f'' = 0$

$f''' = -\cos x \qquad x = \pi のとき \qquad f''' = 1$

$f^{(4)} = \sin x \qquad x = \pi のとき \qquad f^{(4)} = 0$

$f^{(5)} = \cos x \qquad x = \pi のとき \qquad f^{(5)} = -1$

これから

$f \cong -(x-\pi) + \frac{1}{3!}(x-\pi)^3 - \frac{1}{5!}(x-\pi)^5 = -(x-\pi) + \frac{1}{6}(x-\pi)^3 - \frac{1}{120}(x-\pi)^5$

(4)

$f = \cos x \qquad \left(x = \frac{\pi}{4}\right)$

$f' = -\sin x \qquad x = \frac{\pi}{4}\text{のとき} \qquad f' = -\frac{1}{\sqrt{2}}$

$f'' = -\cos x \qquad x = \frac{\pi}{4}\text{のとき} \qquad f' = -\frac{1}{\sqrt{2}}$

これから

$f \cong \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\left(x - \frac{\pi}{4}\right) - \frac{1}{2\sqrt{2}}\left(x - \frac{\pi}{4}\right)^2$