高数笔记 P00a：基础知识补充

1 基础

• 一元二次方程的根 \(x_{1,2} = \cfrac {-b \pm \sqrt{b^2 - 4ac}}{2a}\)，并且\(x_1 + x_2 = -\cfrac ba, \ \ x_1 x_2 = \cfrac ca\)
• \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)，\((a-b)^3 = a^3 -3a^2b+3ab^2-b^3\)
• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)， \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
• \((a+b)^n = \displaystyle \sum_{k=0}^{n} C_n^k a^k b^{n-k}\)

2 对数

• \(\log _a (MN) = \log_a M + \log_a N\)
• \(\log_a \cfrac MN = \log_a M - \log_a N\)
• \(\log_a M^n = n \log_a M\)

3 三角函数

• \(\csc x = \cfrac 1{\sin x}, \ \sec x = \cfrac 1{\cos x}, \ \cot x = \cfrac 1{\tan x}\)
• \(\sin^2 x + \cos^2 x = 1, \ 1 + \tan^2 x = \sec^2 x, \ 1 + \cot^2 x = \csc^2 x\)
• \(\sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \sin \beta \cos \alpha\)
• \(\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\)
• \(\tan (\alpha \pm \beta) = \cfrac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}\)
• \(\sin 2\alpha = 2\sin \alpha \cos \alpha\)
• \(\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha\)
• \(\sin^2 \alpha = \cfrac 12 (1 - \cos 2\alpha), \cos ^2 \alpha = \cfrac 12 (1 + \cos 2\alpha)\)
• \(a\sin \alpha + b\cos \alpha = \sqrt{a^2 + b^2} \sin (\alpha + \varphi), \ \varphi = \arctan \cfrac ba\)
• 积化和差
  • \(\sin \alpha \sin \beta = -\cfrac 12 [\cos (\alpha + \beta) - \cos (\alpha -\beta)]\)
  • \(\sin \alpha \cos \beta = \cfrac 12 [\sin (\alpha + \beta) + \sin (\alpha - \beta)]\)
  • \(\cos \alpha \cos \beta = \cfrac 12 [\cos (\alpha + \beta) + \cos (\alpha -\beta)]\)
• 和差化积
  • \(\sin \alpha + \sin \beta = 2\sin \cfrac {\alpha + \beta}2 \cos \cfrac {\alpha -\beta}2\)
  • \(\cos \alpha + \cos \beta = 2 \cos \cfrac {\alpha + \beta}2 \cos \cfrac {\alpha -\beta}2\)
  • \(\sin \alpha - \sin \beta = 2 \sin \cfrac {\alpha -\beta}2 \cos \cfrac {\alpha + \beta}2\)
  • \(\cos \alpha -\cos \beta = -2 \sin \cfrac {\alpha + \beta}2 \sin \cfrac {\alpha - \beta}2\)

4 不等式

• \(2|ab| \le a^2 + b^2\)
• \(|a \pm b| \le |a| + |b|\)
• \(|a_1 \pm a_2 \pm \cdots \pm a_n| \le |a_1| + |a_2| + \cdots + |a_n|\)
• \(|a - b| \ge |a| - |b|\)
• \(\cfrac {a_1 + a_2 + \cdots + a_n}n \ge \sqrt[n]{a_1 a_2 \cdots a_n}\)
• \(\left|\cfrac {a_1 + a_2 + \cdots + a_n}n \right| \le \sqrt{\cfrac {a_1^2 + a_2^2 + \cdots + a_n^2}n}\)
• \(x,y,p,q \gt 0, \cfrac 1p + \cfrac 1q = 1\)，则\(xy \le \cfrac {x^p}p + \cfrac {y^q}q\)
• \((a^2 + b^2)(c^2 + d^2) \ge (ac+bd)^2\)

5 数列

• 等差数列 \(a_n = a_1 + (n-1)d\)：\(S_n = \cfrac n2 (a_1 + a_n)\)
• 等比数列 \(a_n = a_1 q^{n-1}\)：\(S_n = \cfrac {a_1(1-q^n)}{1-q}\)

常用数列的和：

• \(1 + 2 + 3 + \cdots + n = \cfrac {n(n+1)}2\)
• \(1 + 3 + 5 + \cdots + (2n-1) = n^2\)
• \(1^2 + 2^2 + 3^2 + \cdots + n^2 = \cfrac {n(n+1)(2n+1)}6\)
• \(1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[ \cfrac {n(n+1)}2 \right]^2\)

6 排列组合

• \(A_n^m = \cfrac {n!}{(n-m)!}, C_n^m = \cfrac {n!}{m! (n-m)!}\)

• 特别的，规定\(0! = 1\)。
• \(C_n^m = C_n^{n-m},\ \ C_{n+1}^m = C_n^m + C_n^{m-1}\)

7 曲线和曲面

• 圆锥体积\(V = \cfrac 13 sh\)，球的体积\(V = \cfrac 43 \pi R^3\)，球表面积\(S = 4 \pi R^2\)

• 椭圆 \(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} = 1\)，双曲线 \(\cfrac {x^2}{a^2} - \cfrac {y^2}{b^2} = 1\)，抛物线 \(y^2 = 2px\)
• 椭球面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} + \cfrac {z^2}{c^2} = 1\)，二次锥面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 0\)
• 单页双曲面\(\cfrac {x^2}{a^2} + \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 1\)，双叶双曲面\(\cfrac {x^2}{a^2} - \cfrac {y^2}{b^2} - \cfrac {z^2}{c^2} = 1\)
• 椭圆抛物面\(\cfrac {x^2}{2p} + \cfrac {y^2}{2q} = z\)，双曲抛物面/马鞍面\(\cfrac {x^2}{2p} - \cfrac {y^2}{2q} = z\)

8 极坐标

• 弧长 \(l = r \theta\)
• 扇形面积 \(S = \frac 12 r^2 \theta\)
• 极坐标换直角坐标 \(\begin{cases} x = r(\theta) \cos \theta \\ y = r(\theta) \sin \theta \end{cases}\)
• 圆心在\(x\)轴\(\cfrac a2\)处的圆 \(\rho = a \cos \theta\)，圆心在\(y\)轴\(\cfrac a2\)处的圆 \(\rho = a \sin \theta\)
• 摆线\(\begin{cases} x = a(\theta - \sin \theta) \\ y = a(1 - \cos \theta) \\ \end{cases}\)
• 阿基米德螺线 \(\rho = a \theta, \ (a \gt 0)\)