### 数学ノート (2017/04/12)

アインシュタイン曰く

というわけで、よく使われるが、すぐ忘れてしまう公式などを覚えないで済むよう、メモとして残す。

#### 式の計算

$\frac {x - 25} {x^{2} + 5x - 24}$ $= \frac {x - 25} {(x - 3)(x + 8)}$ $= \frac {A} {x - 3} + \frac {B} {x + 8}$ $= \frac {A(x + 8)+B(x - 3)} {(x - 3)(x + 8)}$ $= \frac {Ax + 8A + Bx - 3B} {(x - 3)(x + 8)}$ $A + B = 1$ $8A - 3B = -25$ $8A - 3B + 3A + 3B = -25 + 3$ $11A = -22$ $A = -2$ $-2 + B = 1$ $B = 1 + 2$ $B = 3$ $\frac {x - 25} {x^{2} + 5x - 24} = \frac {-2} {x - 3} + \frac {3} {x + 8}$

#### 図形

$h = \sqrt {\frac {2} {3}} a$

$\triangle ABC, \angle DAB = \angle DAC \Rightarrow \frac {AB} {BD} = \frac {AC} {CD}$

$S = 4\pi r^{2}$

$焦点: (-8, -1)$ $准線: y = -4$ $焦点との距離: \sqrt{(x + 8)^{2} + (y + 1)^{2}}$ $准線との距離: \sqrt{(y + 4)^{2}}$ $\sqrt{(y + 4)^{2}} = \sqrt{(x + 8)^{2} + (y + 1)^{2}}$ $(y + 4)^{2} = (x + 8)^{2} + (y + 1)^{2}$ $y^{2} + 8y + 16 = (x + 8)^{2} + y^{2} + 2y + 1$ $y^{2} -y^{2} + 8y - 2y = (x + 8)^{2} + 1 - 16$ $6y = (x + 8)^{2} - 15$ $y = \frac {(x + 8)^{2}} {6} - \frac{15} {6}$ $y = \frac {(x + 8)^{2}} {6} - \frac{5} {2}$

$\frac {x^{2}} {a^{2}} + \frac {y^{2}} {b^{2}} = 1 \ (a > b > 0)$ $焦点: (\pm \sqrt{a^{2} - b^{2}}, 0)$

$\frac {x^{2}} {a^{2}} + \frac {y^{2}} {b^{2}} = 1 \ (b > a > 0)$ $焦点: (0, \pm \sqrt{b^{2} - a^{2}})$

$\frac {(x - p)^{2}} {a^{2}} + \frac {(y - q)^{2}} {b^{2}} = 1 \ (a > b > 0)$ $焦点: (\pm \sqrt{a^{2} - b^{2}} + p, q)$

$\frac {x^{2}} {a^{2}} - \frac {y^{2}} {b^{2}} = 1$ $焦点: (\pm \sqrt{a^{2} + b^{2}}, 0)$ $\frac {y^{2}} {a^{2}} - \frac {x^{2}} {b^{2}} = 1$ $焦点: (0, \pm \sqrt{a^{2} + b^{2}})$

#### 論理

イプシロン-デルタ論法

$\forall \varepsilon \gt 0, \ \exists N \in \mathbb{N}, \ \mbox{s.t.} \ n > N \Rightarrow |a_n - L| \lt \varepsilon$

#### 対数

$\log_b(a) = \frac {\log_x(a)} {\log_x(b)}$

#### 三角関数

$\csc{\theta} = \frac {1} {\sin{\theta}}$ $\cot{\theta} = \frac {1} {\tan{\theta}}$

$\sin{\theta} = \sin{(180^{\circ}-\theta)}$

$\sin{\theta} = \sin{(-180^{\circ}-\theta)}$

$\cos{\theta} = \cos{(-\theta)}$

$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

$a^2 = b^2 + c^2 - 2bc\cos A$

ラジアン

$\frac {180 ^ {\circ}} {\pi} = 1rad$

シヌソイドの中間線

$f(x) = a\sin(bx + c) + d$

$f(x) = a\cos(bx + c) + d$

#### 数列

$S_n = \frac {a_1(1 - r ^ n)} {1 - r}$

$a_n = S_{n} - S_{n-1}$

$b < c, \ \frac {1} {(ax+b)(ax+c)} = \frac {1} {(c-b)(ax+b)} - \frac {1} {(c-b)(ax+c)}$

$\sum_{n=0}^\infty ar^n = \frac {a} {1 - r}$

テイラー展開

$f(x) = \sum_{n=0}^\infty \frac {f^{(n)}(a)} {n!}(x - a)^n$

マクローリン展開

$f(x) = \sum_{n=0}^\infty \frac {f^{(n)}(0)} {n!}x^n$

$$\sin{(x)}$$のテイラー展開

$\sin(x) = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + ...$

$$\cos{(x)}$$のテイラー展開

$\cos(x) = \sum_{n=0}^\infty \frac {(-1)^n} {(2n)!} x^{2n}$

$\cos(x) = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + ...$

$$\tan{(x)}$$のテイラー展開

$\tan(x) = x + \frac {x^3} {3} + \frac {2x^5} {15} + ...$

$$\arctan{(x)}$$のテイラー展開

$\arctan(x) = x - \frac {x^3} {3} + \frac {x^5} {5} - \frac {x^7} {7}...$

$$e^{x}$$のテイラー展開

$e^x = \sum_{n=0}^\infty \frac {x^n} {n!}$

#### 極限

$\lim_{x \to 0} \frac {\cos(x) - 1} {x} = 0$

ロピタルの定理

$\lim_{x \to a} \frac {f(x)} {g(x)} = \lim_{x \to a} \frac {f'(x)} {g'(x)}$

$$\lim_{x \to \infty} x^{\frac {1} {x}}$$の計算例

$\lim_{x \to \infty} x^{\frac {1} {x}}$ $y = x^{\frac {1} {x}}$ $\ln{(y)} = \ln{(x^{\frac {1} {x}})}$ $\ln{(y)} = \frac {\ln{(x)}} {x}$ $\ln{(y)} = \frac {(\ln{(x)})'} {(x)'}$ $\ln{(y)} = \frac {\frac {1} {x}} {1}$ $\ln{(y)} = \frac {1} {x}$ $\lim_{x \to \infty} \frac {1} {x} = 0$ $\ln{(y)} = 0$ $e^{\ln{(y)}} = e^{0}$ $y = 1$ $\lim_{x \to \infty} x^{\frac {1} {x}} = 1$

#### 微分

$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$

$\left( \frac {f(x)} {g(x)} \right)' = \frac {f'(x)g(x) - f(x)g'(x)} {(g(x))^{2}}$

$(f(g(x)))' = f'(g(x))g'(x)$

$$\tan{(x)}$$の微分

$(\tan(x))' = \frac {1} {\cos^2(x)}$

$$\sec{(x)}$$の微分

$(\sec(x))' = \sec(x)\tan(x)$

$$\csc{(x)}$$の微分

$(\csc(x))' = -\cot(x)\csc(x)$

$$\cot{(x)}$$の微分

$(\cot(x))' = -\csc^2(x)$

$$\arcsin{(x)}$$の微分

$(\arcsin(x))' = \frac {1} {\sqrt{1 - x^2}}$

$$\arccos{(x)}$$の微分

$(\arccos(x))' = -\frac {1} {\sqrt{1 - x^2}}$

$$\arctan{(x)}$$の微分

$(\arctan(x))' = \frac {1} {1 + x^2}$

$$a^{x}$$の微分

$(a^{x})' = a^{x} \ln(a)$

$$\log_{a}{(x)}$$の微分

$(\log_{a}(x))' = \frac {1} {x\ln(a)}$

$$\ln{(x)}$$の微分

$(\ln(x))' = \frac {1} {x}$

$$e^{ax}$$の微分

$(e^{ax})' = ae^{ax}$

$$e^{x^{2}}$$の微分

$(e^{x^{2}})' = 2xe^{x^{2}}$

$h(x) = f^{-1}(x)$ $h'(x) = \frac {1} {f'(h(x))}$

$$x^{x}$$の微分の計算例

$y = x^{x}$ $\ln{(y)} = \ln{(x^{x})}$ $\ln{(y)} = x\ln{(x)}$ $(\ln{(y)})' = (x\ln{(x)})'$ $\frac {1} {y} \frac {dy} {dx} = 1 * \ln{(x)} + x * \frac {1} {x}$ $\frac {1} {y} \frac {dy} {dx} = \ln{(x)} + 1$ $\frac {dy} {dx} = y(\ln{(x)} + 1)$ $\frac {dy} {dx} = x^{x}(\ln{(x)} + 1)$

$$x^{\ln{(x)}}$$の微分の計算例

$y = x^{\ln{(x)}}$ $\ln{(y)} = \ln{(x^{\ln{(x)}})}$ $\ln{(y)} = \ln{(x)} * \ln{(x)}$ $\frac {1} {y} \frac {dy} {dx} = \frac {1} {x} * \ln{(x)} + \ln{(x)} * \frac {1} {x}$ $\frac {1} {y} \frac {dy} {dx} = 2 * \frac {1} {x} * \ln{(x)}$ $\frac {dy} {dx} = 2 * \frac {1} {x} * \ln{(x)} * y$ $\frac {dy} {dx} = 2 * \frac {1} {x} * \ln{(x)} * x^{\ln{(x)}}$ $\frac {dy} {dx} = 2 * \ln{(x)} * x^{\ln{(x)}} * \frac {1} {x}$ $\frac {dy} {dx} = 2 * \ln{(x)} * x^{\ln{(x)}} * x^{-1}$ $\frac {dy} {dx} = 2 * \ln{(x)} * x^{\ln{(x)}-1}$

$$\frac {dy} {dx}$$の計算例

$2xy + x^3 - 3y^2 = 5$ $\frac {d} {dx} (2xy + x^3 - 3y^2) = \frac {d} {dx} (5)$ $\frac {d} {dx} (2xy) + \frac {d} {dx} (x^3) - \frac {d} {dx} (3y^2) = \frac {d} {dx} (5)$ $\frac {d} {dx} (2xy) + 3x^2 - \frac {d} {dx} (3y^2) = 0$ $2\frac {d} {dx} (xy) + 3x^2 - 3\frac {d} {dx} (y^2) = 0$ $2(\frac {d} {dx} (x) * \frac {d} {dx} (y)) + 3x^2 - 3\frac {d} {dx} (y^2) = 0$ $2(1 * y + x * 1 \frac {dy} {dx}) + 3x^2 - 3\frac {d} {dx} (y^2) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3\frac {d} {dx} (y^2) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3\frac {d} {dx} (1 * y^2) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3\frac {d} {dx} (x^0 * y^2) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3(\frac {d} {dx} (x^0) * \frac {d} {dx} (y^2)) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3((0 * y^2) + (x^0 * 2y \frac {dy} {dx})) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3((0 * y^2) + (1 * 2y \frac {dy} {dx})) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 3(0 + 2y \frac {dy} {dx}) = 0$ $2y + 2x \frac {dy} {dx} + 3x^2 - 6y \frac {dy} {dx} = 0$ $2y + 3x^2 + 2x \frac {dy} {dx} - 6y \frac {dy} {dx} = 0$ $2y + 3x^2 + (2x - 6y) \frac {dy} {dx} = 0$ $(2x - 6y) \frac {dy} {dx} = -(2y + 3x^2)$ $\frac {dy} {dx} = -\frac {2y + 3x^2} {2x - 6y}$

$\left( \int_{0}^{x ^ {2}} \sin(t) dt \right)'$ $= \frac {d} {dx} \int_{0} ^ {x ^ {2}} \sin(t) dt$ $= \sin(x ^ {2}) \cdot \frac {d} {dx} (x ^ {2})$ $= \sin(x ^ {2}) \cdot 2x$ $= 2x \sin(x ^ {2})$

オイラー法

$y_{n+1}= y_{n} + \Delta y_{n}$

$\frac {f(b) -f(a)} {b - a} = f'(c)$

$x = f(t), \ y = g(t)$ $\frac {d^{2}y} {dx^{2}} = \frac {f'(t)g''(t) - f''(t)g'(t)} {(f'(t))^{3}}$

#### 積分

$$\sin{(ax)}$$の積分

$\int \sin(ax) dx = -\frac {1} {a} \cos(ax) + C$

$$\sec^{2}{(ax)}$$の積分

$\int \sec^2(ax) dx = \frac {1} {a} \tan(ax) + C$

$$\csc^{2}{(x)}$$の積分

$\int \csc^2(x) dx = -\cot(x) + C$

$$e^{ax}$$の積分

$\int e^{ax} dx = \frac {e^{ax}} {a} + C$

$$\frac {1} {1 + x ^ {2}}$$の積分

$\int \frac {1} {1 + x ^ {2}} dx = \tan ^ {-1}(x) + C$

$L = \int_a^b \sqrt{1 + (f'(x))^2} dx$

ｙ軸周りの回転体の体積（バームクーヘン積分）

$V = 2\pi\int_a^b x f(x) dx$

$\frac {1} {b - a} \int_a^b f(x) dx$

$\int \frac {f'(x)} {f(x)} dx = \log |f(x)| + C$

$\int \frac {a} {b(x + c)} dx = \frac {a \ln |x + c|} {b} + C$

$A = \int_\alpha^\beta \frac {1} {2} r^2 d\theta$

リーマン積分

$\int_a^b \sin(x) dx = \lim_{n \to \infty} \sum_{i = 1}^{n} \sin \left(\frac {b - a} {n} i\right)\cdot \frac {b - a} {n}$

$\int \frac {x} {\sqrt{16 - x ^ 2}} dx$ $u = 16 - x ^ 2$ $\frac {du} {dx} = -2x$ $-\frac {du} {2x} = dx$ $\int \frac {x} {\sqrt{u}} \cdot (-\frac {du} {2x})$ $= -\int \frac {1} {2\sqrt{u}} du$ $= -\int \frac {1} {2} u ^ {-\frac {1} {2}} du$ $= -u ^ {\frac {1} {2}} + C$ $= -\sqrt{u} + C$ $= -\sqrt{16 - x ^ 2} + C$

$\int x \cos(\pi x) dx$ $\int u(x)v'(x)dx = u(x)v(x) - \int u'(x)v(x) dx$ $\int u dv = uv - \int v du$ $u = x, dv = \cos(\pi x) dx$ $du = dx, v = \int \cos(\pi x) dx = \frac {\sin(\pi x)} {\pi}$ $\int x \cos(\pi x) dx$ $= \frac {x \sin(\pi x)} {\pi} - \int \frac {\sin(\pi x)} {\pi} dx$ $= \frac {x \sin(\pi x)} {\pi} - \frac {1} {\pi} \left( -\frac {\cos(\pi x )} {\pi} \right) + C$ $= \frac {x \sin(\pi x)} {\pi} + \frac {\cos(\pi x)} {\pi ^ 2} + C$

#### 微分方程式

$\frac {dy} {dt} = 3y$ $\frac {dy} {y} = 3 dt$ $\int \frac {dy} {y} = \int 3 dt$ $\int \frac {1} {y} dy = \int 3 dt$ $\ln |y| = 3t + C$

$\frac {dy} {dx} = g(x) \cdot h(y)$ $\frac {1} {h(y)} dy = g(x) dx$

#### 複素数

$|z| = \sqrt{a^{2} + b^{2}}$

$\theta = \arctan \left ( \frac {b} {a} \right )\$

$a = |z|\cos{\theta}$

$b = |z|\sin{\theta}$

$\tan{\theta} = \frac {Im(z)} {Re(z)}$

$z_{1} = r_{1}[\cos{(\theta_{1})} + i \sin{(\theta_{1})}], \ z_{2} = r_{2}[\cos{(\theta_{2})} + i \sin{(\theta_{2})}]$ $z_{1} \cdot z_{2} = r_{1}r_{2}[\cos{(\theta_{1} + \theta_{2})} + i \sin{(\theta_{1} + \theta_{2})}]$

$z_{1} = r_{1}[\cos{(\theta_{1})} + i \sin{(\theta_{1})}], \ z_{2} = r_{2}[\cos{(\theta_{2})} + i \sin{(\theta_{2})}]$ $\frac {z_{1}} {z_{2}} = \frac {r_{1}} {r_{2}}[\cos{(\theta_{1} - \theta_{2})} + i \sin{(\theta_{1} - \theta_{2})}]$

$z^{3} = -512, \ 270^{\circ} \le \theta \le 360^{\circ}$ $z^{3} = r^{3}[\cos{(3 \cdot \theta)} + i \sin{(3 \cdot \theta)}]$ $r^{3}[\cos{(3 \cdot \theta)} + i \sin{(3 \cdot \theta)}] = 512[\cos{(180^{\circ} + k \cdot 360^{\circ})} + i \sin{(180^{\circ} + k \cdot 360^{\circ})}]$ $r^{3} = 512$ $r = 8$ $3 \cdot \theta = 180^{\circ} + k \cdot 360^{\circ}$ $\theta = 60^{\circ} + k \cdot 120^{\circ}$ $\ 270^{\circ} \le \theta \le 360^{\circ}$ $\theta = 300^{\circ}$ $z = r[\cos{(\theta)} + i \cdot \sin{(\theta)}]$ $z = 8[\cos{(300^{\circ})} + i \cdot \sin{(300^{\circ})}]$ $z = 8\cos{(300^{\circ})} + 8\sin{(300^{\circ})} \cdot i$ $z = 4 - 6.928i$

$z^{5} = -7776i, \ 270^{\circ} \le \theta \le 360^{\circ}$ $z^{5} = r^{5}[\cos{(5 \cdot \theta)} + i \sin{(5 \cdot \theta)}]$ $r^{5}[\cos{(5 \cdot \theta)} + i \sin{(5 \cdot \theta)}] = 7776[\cos{(270^{\circ} + k \cdot 360^{\circ})} + i \sin{(270^{\circ} + k \cdot 360^{\circ})}]$ $r^{5} = 7776$ $r = 6$ $5 \cdot \theta = 270^{\circ} + k \cdot 360^{\circ}$ $\theta = 54^{\circ} + k \cdot 72^{\circ}$ $\ 270^{\circ} \le \theta \le 360^{\circ}$ $\theta = 342^{\circ}$ $z = r[\cos{(\theta)} + i \cdot \sin{(\theta)}]$ $z = 6[\cos{(342^{\circ})} + i \cdot \sin{(342^{\circ})}]$ $z = 6\cos{(342^{\circ})} + 6\sin{(342^{\circ})} \cdot i$ $z = 5.706 - 1.854i$

#### 行列

2×2行列の逆行列

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ $A^{-1} = \frac {1} {ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$