概要

数学Ⅱで使用する公式のTex記法を下記に示す。

TeX記法数学Ⅱ

微分と積分

微分係数
\lim_{h \to 0}\frac{f(a + h) - f(a)}{h} = f'(a)

\lim_{h \to 0}\frac{f(a + h) - f(a)}{h} = f'(a)

微分
nは自然数、cは定数とする
(x^n)' = n x^{n-1}
(c)' = 0
\{c f(x)\}' = cf'(x)
\{c f(x) + g(x)\}' = cf'(x) + g'(x)

(x^n)' = n x^{n-1} \\
(c)' = 0 \\
\{c f(x)\}' = cf'(x) \\
\{c f(x) + g(x)\}' = cf'(x) + g'(x)

接線の方程式
y = f(x)上の点A(a, f(a))を通る接線の式は
y - f(a) = f'(a)(x - a)

y - f(a) = f'(a)(x - a)

法線の方程式
y = f(x)上の点A(a, f(a))を通る法線の式は
y = -\frac{1}{f'(a)}(x - a) + f(a)

y = -\frac{1}{f'(a)}(x - a) + f(a)

不定積分
\int F'(x)dx = F(x) + C
\int x^{n}dx = \frac{x^{n + 1}}{n + 1} + C

\int F'(x)dx = F(x) + C \\
\int x^{n}dx = \frac{x^{n + 1}}{n + 1} + C

定積分
\int_{a}^{b} F'(x)dx = [F(x)]_{a}^{b} = F(b) - F(a)
\int_{a}^{b} k f(x)dx = k\int_{a}^{b} f(x)dx
\int_{a}^{b} \{f(x) + g(x)\}dx = \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx
\int_{a}^{b} f(x)dx = -\int_{b}^{a} f(x)dx

\int_{a}^{b} F'(x)dx = [F(x)]_{a}^{b} = F(b) - F(a) \\
\int_{a}^{b} k f(x)dx = k\int_{a}^{b} f(x)dx \\
\int_{a}^{b} \{f(x) + g(x)\}dx = \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx\\
\int_{a}^{b} f(x)dx = -\int_{b}^{a} f(x)dx

1/6の公式
\int_{\alpha}^{\beta} (x - \alpha)(x - \beta)dx = -\frac{1}{6}(\beta - \alpha)^3

\int_{\alpha}^{\beta} (x - \alpha)(x - \beta)dx = -\frac{1}{6}(\beta - \alpha)^3