[数学]必备基本知识

必备基础知识

函数

对数函数

  1. $ y=log_a^x(a>0,a\neq 1),是y=a^x的反函数$
  2. 单调性：\(当a>1时，y=log_a^x单调增加，当0<a<1时，y=log_a^x单调减小\)
  3. 常用的对数函数：\(y=lnx(自然对数，lnx=log_e^x，e=2.71828...)\)
  4. 特殊函数值：\(log_a^1=0,\ log_a^a=1,\ ln1-0,\ lne=1\)
  5. 极限：\(\lim_{x \to 0^+}lnx=-\infty,\ \lim_{x \to +\infty}lnx=+\infty\)
  6. ！！！常用公式：\(x=e^{lnx},\ u^v=e^{lnu^v}=e^{vlnu}\)

反正切函数，反余切函数

  1. 反正切函数---y=arctanx, 反余切函数---y=arccotx
  2. 性质：\(arctanx+arccotx=\frac{\pi}{2} (-\infty < x < +\infty )\)
  3. 特殊函数值：
    • \(arctan0=0,\ arctan\frac{\sqrt{3}}{3}=\frac{\pi}{6},\ arctan1=\frac{\pi}{4},\ arctan{\sqrt{3}}=\frac{\pi}{3}\)
    • \(arccot0=0,\ arccot\frac{\sqrt{3}}{3}=\frac{\pi}{6},\ arccot1=\frac{\pi}{4},\ arccot{\sqrt{3}}=\frac{\pi}{3}\)
  4. 极限：\(\lim \limits_{x \to -\infty}arctanx=-\frac{\pi}{2},\ \lim \limits_{x \to +\infty}arctanx=\frac{\pi}{2},\ \lim \limits_{x \to -\infty}arccotx=\pi,\ \lim \limits_{x \to +\infty}arccotx=0\)

三个重要的分段函数

1. 分段函数

  1. 定义：\(y=|x|=\begin{cases}x & x \geq 0\\ b & -x < 0\end{cases}称为“绝对值函数”\)
  2. 性质：
    1. 该函数在x=0处连续（没有间断），但是不可导（有折点，不光滑）。后面会看到，这个看起来不起眼的函数，会多次在我们判别似是而非的概念时给我们援手。
    2. 绝对值函数和最大、最小值函数有某种亲密关系，如下：
      设f(x)与g(x)为连续函数，如果令\(U=max\{f(x), g(x)\}, \ V=min\{f(x), g(x)\},\)则：
      \(U = max\{f(x), g(x)\} = \frac{1}{2}[f(x)+g(x)+|f(x)-g(x)|] = \begin{cases}f(x) & f(x) \geq g(x) \\ g(x) & f(x)<g(x) \end{cases}\)
      \(V = min\{f(x), g(x)\} = \frac{1}{2}[f(x)+g(x)-|f(x)-g(x)|] = \begin{cases}g(x) & f(x) \geq g(x) \\ f(x) & f(x)<g(x) \end{cases}\)
      即：\(U+V = f(x)+g(x),\ U-V = |f(x)-g(x)|,\ UV=f(x)g(x)\)

2. 符号函数

  • 定义：

\[y=sgnx=\begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases}, 称为符号函数，对于任何实数x，有x=|x|shnx \]

3. 取整函数

  1. 定义：y=[x]称为取整函数，先给出定义，设x为任一实数，不超过x的最大整数称为x的整数部分，记作[x]。如[0.99]=0, \([\pi]\)=3, [-1]=-1, [-1.99]=-2, 因此，取整函数y=[x]的定义域为R，值域为Z，在x为整数值处发生跳跃
  2. 注意点：
    1. \(x-1\leq[x],\ [x+n]=[x],\ n[x]\leq nx,\ [x]+[y]\leq[x+y]\)
    2. \(\lim_{x \to 0^+}[x]=0,\ \lim_{x \to 0^-}[x]=-1\)
    3. 考得最多：\(x-1<[x]\leq x\)

常用基础知识

数列基础

  1. 等差数列：
    • 通项公式：\(a_{n} = a_{1} + (n - 1)d\)
    • 前n项的和：\(S_{n} = \frac{n}{2}[2a_{1} + (n - 1)d] = \frac{n}{2}(a_{1} + a_{n})\)
  2. 等比数列：
    • 通项公式：\(a_{n}=a_{1}r^{n-1}\)
    • 前n项的和：\(S_{n}=\frac{a_{1}(1 - r^{n})}{1 - r} (r \neq 1)\)
    • 常用：\(1 + r + r^{2} + … + r^{n - 1} = \frac{1 - r^{n}}{1 - r} (r \neq 1)\)
  3. 一些数列前n项的和：
    1. \(\sum_{k=1}^nk = 1 + 2 + 3 + … + n = \frac{n(n + 1)}{2}\)
    2. \(\sum_{k=1}^n(2k - 1) = 1 + 3 + 5 + … + (2n - 1) = n^{2}\)
    3. \(\sum_{k=1}^nk^{2} = 1^{2} + 2^{2} + 3^{2} + … + n^{2} = \frac{n(n + 1)(2n + 1)}{6}\)
    4. \(\sum_{k=1}^nk^{3} = 1^{3} + 2^{3} + 3^{3} + … + n^{3} = [\frac{n(n + 1)}{2}]^{2} = (\sum_1^nk)^{2}\)
    5. \(\sum_{k=1}^nk(k + 1) = 1 * 2 + 2 * 3 + 3 * 4 + … + n(n + 1) = \frac{n(n + 1)(n + 2)}{3}\)
    6. \(\sum_{k=1}^n\frac{1}{k(k + 1)} = \frac{1}{1 * 2} + \frac{1}{2 * 3} + \frac{1}{3 * 4} + … + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\)

三角函数基础

1. 三角函数基本关系

  1. \(\sin\alpha \csc\alpha = 1\)
  2. \(\csc\alpha = \frac{1}{\sin\alpha}\)
  3. \(\cos\alpha \sec\alpha = 1\)
  4. \(\sec\alpha = \frac{1}{\cos\alpha}\)
  5. \(\tan\alpha \cot\alpha = 1\)
  6. \(\cot\alpha = \frac{1}{\tan\alpha}\)
  7. \(\tan\alpha = \frac{\sin\alpha}{\cos\alpha}\)
  8. \(\cot\alpha = \frac{\cos\alpha}{\sin\alpha}\)
  9. \(\sin^{2}\alpha + \cos^{2}\alpha = 1\)
  10. \(1 - \sin^{2}\alpha = \cos^{2}\alpha\)
  11. \(1 - \cos^{2}\alpha = \sin^{2}\alpha\)
  12. \(\sec^{2}\alpha - \tan^{2}\alpha = 1\)
  13. \(1 + \tan^{2}\alpha = \sec^{2}\alpha\)
  14. \(\sec^{2}\alpha - 1 = \tan^{2}\alpha\)
  15. \(\csc^{2}\alpha - \cot^{2}\alpha = 1\)
  16. \(1 + \cot^{2}\alpha = \csc^{2}\alpha\)
  17. \(\csc^{2}\alpha - 1 = \cot^{2}\alpha\)

2. 诱导公式

\(函数/角\theta\) \(\frac{\pi}{2} - \alpha\) \(\frac{\pi}{2} + \alpha\) \(\pi - \alpha\) \(\pi + \alpha\) \(\frac{3\pi}{2} - \alpha\) \(\frac{3\pi}{2} + \alpha\) \(2\pi - \alpha\)
\(函数/角\theta\) \(90°- a\) \(90°+ a\) \(80°- a\) \(180°+ a\) \(270°- a\) \(270°+ a\) \(360°- a\)
\(\sin\theta\) \(\cos\alpha\) \(\cos\alpha\) \(\sin\alpha\) -\(\sin\alpha\) \(-\cos\alpha\) \(-\cos\alpha\) \(-\sin\alpha\)
\(\cos\theta\) \(\sin\alpha\) -\(\sin\alpha\) \(-\cos\alpha\) \(-\cos\alpha\) \(-\sin\alpha\) \(\sin\alpha\) \(\cos\alpha\)
\(\tan\theta\) \(\cot\alpha\) -\(\cot\alpha\) \(-\tan\alpha\) \(\tan\alpha\) \(\cot\alpha\) \(-\cot\alpha\) \(-\tan\alpha\)
\(\cot\theta\) \(\tan\alpha\) -\(\tan\alpha\) \(-\cot\alpha\) \(\cot\alpha\) \(\tan\alpha\) \(-\tan\alpha\) \(-\cot\alpha\)
口诀：奇变偶不变，符号看象限
\(函数/角\theta 所在象限\) \(第一象限\) \(第二象限\) \(第三象\) \(第四象限\)
\(\sin\theta\) \(+\) \(+\) \(-\) \(-\)
\(\cos\theta\) \(+\) \(-\) \(-\) \(+\)
\(\tan\theta\) \(+\) \(-\) \(+\) \(-\)
\(\cot\theta\) \(+\) \(-\) \(+\) \(-\)
口诀：全STC
\(\alpha\) \(0°\) \(30°\) \(45°\) \(60°\) \(90\) \(120°\) \(135°\) \(150°\) \(180°\) \(270°\) \(360°\)
\(\alpha\) \(0\) \(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{3}\) \(\frac{\pi}{2}\) \(\frac{2\pi}{3}\) \(\frac{3\pi}{4}\) \(\frac{5\pi}{6}\) \(\pi\) \(\frac{3\pi}{2}\) \(2\pi\)
\(\sin\alpha\) \(0\) \(\frac{1}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{3}}{2}\) \(1\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{1}{2}\) \(0\) \(-1\) \(0\)
\(\cos\alpha\) \(1\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{1}{2}\) \(0\) \(-\frac{1}{2}\) \(-\frac{\sqrt{2}}{2}\) \(-\frac{\sqrt{3}}{2}\) \(-1\) \(0\) \(1\)
\(\tan\alpha\) \(0\) \(\frac{\sqrt{3}}{3}\) \(1\) \(\sqrt{3}\) \(\infty\) \(-\sqrt{3}\) \(-1\) \(-\frac{\sqrt{3}}{3}\) \(0\) \(\infty\) \(0\)
\(\cot\alpha\) \(\infty\) \(\sqrt{3}\) \(1\) \(\frac{\sqrt{3}}{3}\) \(0\) \(-\frac{\sqrt{3}}{3}\) \(-1\) \(-\sqrt{3}\) \(\infty\) \(0\) \(\infty\)

3. 倍角公式

  1. \(\sin2\alpha=2\sin\alpha\cos\alpha\)
  2. \(\cos2\alpha=cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1\)
  3. \(\sin3\alpha=-4\sin^3\alpha+3\sin\alpha\)
  4. \(\cos3\alpha=4\cos^3\alpha-3\cos\alpha\)
  5. \(\sin^2\alpha=\frac{1}{2}(1-\cos2\alpha)\)
  6. \(\cos^2\alpha=\frac{1}{2}(1+\cos2\alpha)\)
  7. \(\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2}\)
  8. \(\cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}\)

4. 半角公式

  1. \(\sin^2\frac{\alpha}{2}=\frac{1}{2}(1-\cos\alpha)\)
  2. \(\cos^2\frac{\alpha}{2}=\frac{1}{2}(1+\cos\alpha)\)
  3. \(\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}\)
  4. \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}\)
  5. \(\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}=\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}\)
  6. \(\cot\frac{\alpha}{2}=\frac{\sin\alpha}{1-\cos\alpha}=\frac{1+\cos\alpha}{\sin\alpha}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}\)

5. 和差公式

  1. \(\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\)
  2. \(\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta\)
  3. \(\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}\)
  4. \(\cot(\alpha\pm\beta）=\frac{\cot\alpha\cot\beta\mp1}{\cot\beta\pm\cot\alpha}\)

6. 积化和差公式

  1. \(\sin\alpha\cos\beta=\frac{1}{2}\left[\sin（\alpha+\beta）+\sin（\alpha-\beta）\right]\)
  2. \(\cos\alpha\sin\beta=\frac{1}{2}\left[\sin（\alpha+\beta）-\sin（\alpha-\beta）\right]\)
  3. \(\cos\alpha\cos\beta=\frac{1}{2}\left[\cos（\alpha+\beta）+\cos（\alpha-\beta）\right]\)
  4. \(\sin\alpha\sin\beta=\frac{1}{2}\left[\cos（\alpha-\beta）-\cos（\alpha+\beta）\right]\)

7. 和差化积公式

  1. \(\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\)
  2. \(\sin\alpha-\sin\beta=2\sin\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}\)
  3. \(\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\)
  4. \(\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}\)

8. 万能公式

  1. \(若\mu=\tan\frac{x}{2}(-\pi < x < \pi)，则\sin\chi=\frac{2\mu}{1+\mu^2}\)

指数运算法则

  1. \(a^{\alpha}*a^{\beta}=a^{\alpha+\beta}\)
  2. \(\frac{a^{\alpha}}{a^{\beta}}=a^{\alpha-\beta}\)
  3. \(\left(a^{\alpha}\right)^{\beta}=a^{\alpha\beta}\)
  4. \(\left(ab\right)^\alpha=a^{\alpha}b^{\alpha}\)
  5. \(\left(\frac{a}{b}\right)^\alpha=\frac{a^{\alpha}}{b^{\alpha}}\)

\(其中，a,b是正实数，\alpha,\beta是任意实数\)

对数运算法则

  1. \(\log_{a}{MN}=\log_{a}{M} + \log_{a}{N}\)
  2. \(\log_{a}{\frac{M}{N}}=\log_{a}{M}-\log_{a}{N}\)
  3. \(\log_{a}{M^{n}}=n\log_{a}{M}\)
  4. \(\log_{a}{\sqrt[n]{M}}=\frac{1}{n}\log_{a}{M}\)

一元二次方程基础

  • 一元二次方程： \(ax^{2} + bx + c=0(a\neq0)\)
  1. 根的公式： \(x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
  2. 根与系数的关系： \(x_{1} + x_{2}=-\frac{b}{a},x_{1}x_{2}=\frac{c}{a}\)
  3. 判别式： \(\triangle=b^{2}-4ac:\)
    1. \(\triangle>0\)，方程有两个不等的实根
    2. \(\triangle=0\)，方程有两个相等的实根，
    3. \(\triangle<0\)，方程有两个共轭的复根。
  4. 抛物线： \(y=ax^{2}+bx+c\) 的顶点：\(\left( -\frac{b}{2a},c-\frac{b^{2}}{4a}\right)\)

因式分解公式

  1. \((a+b)^{2}=a^{2}+2ab+b^{2}\)
  2. \((a-b)^{2}=a^{2}-2ab+b^{2}\)
  3. \((a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\)
  4. \((a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\)
  5. \((a+b)(a-b)=a^{2}-b^{2}\)
  6. \(a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})\)
  7. \(a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})\)
  8. \(a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})\) (n是正整数)
  9. n是正偶数时，\(a^{n}-b^{n}=(a+b)(a^{n-1}-a^{n-2}b+...+ab^{n-2}-b^{n-1})\)
  10. n是正奇数时，\(a^{n}+b^{n}=(a+b)(a^{n-1}-a^{n-2}b+...-ab^{n-2}+b^{n-1})\)

阶乘与双阶乘

  1. \(n!= 1\times2\times3\times ... \times n，规定0!=1\)
  2. \((2n)!! = 2\times4\times6\times ... \times 2n=2^n\cdot n!\)
  3. \((2n-1)!! = 1\times3\times5\times ... \times (2n-1)\)
posted @ 2021-09-23 20:35  Xu_Lin  阅读(716)  评论(0)  编辑  收藏  举报