必備基礎知識
函數
對數函數
- $ y=log_a^x(a>0,a\neq 1),是y=a^x的反函數$
- 單調性：\(當a>1時，y=log_a^x單調增加，當0<a<1時，y=log_a^x單調減小\)
- 常用的對數函數：\(y=lnx(自然對數，lnx=log_e^x，e=2.71828...)\)
- 特殊函數值：\(log_a^1=0,\ log_a^a=1,\ ln1-0,\ lne=1\)
- 極限：\(\lim_{x \to 0^+}lnx=-\infty,\ \lim_{x \to +\infty}lnx=+\infty\)
- ！！！常用公式：\(x=e^{lnx},\ u^v=e^{lnu^v}=e^{vlnu}\)
反正切函數，反余切函數
- 反正切函數---y=arctanx, 反余切函數---y=arccotx
- 性質：\(arctanx+arccotx=\frac{\pi}{2} (-\infty < x < +\infty )\)
- 特殊函數值：
- \(arctan0=0,\ arctan\frac{\sqrt{3}}{3}=\frac{\pi}{6},\ arctan1=\frac{\pi}{4},\ arctan{\sqrt{3}}=\frac{\pi}{3}\)
- \(arccot0=0,\ arccot\frac{\sqrt{3}}{3}=\frac{\pi}{6},\ arccot1=\frac{\pi}{4},\ arccot{\sqrt{3}}=\frac{\pi}{3}\)
- 極限：\(\lim \limits_{x \to -\infty}arctanx=-\frac{\pi}{2},\ \lim \limits_{x \to +\infty}arctanx=\frac{\pi}{2},\ \lim \limits_{x \to -\infty}arccotx=\pi,\ \lim \limits_{x \to +\infty}arccotx=0\)
三個重要的分段函數
1. 分段函數
- 定義：\(y=|x|=\begin{cases}x & x \geq 0\\ b & -x < 0\end{cases}稱為“絕對值函數”\)
- 性質：
- 該函數在x=0處連續（沒有間斷），但是不可導（有折點，不光滑）。后面會看到，這個看起來不起眼的函數，會多次在我們判別似是而非的概念時給我們援手。
- 絕對值函數和最大、最小值函數有某種親密關系，如下：
設f(x)與g(x)為連續函數，如果令\(U=max\{f(x), g(x)\}, \ V=min\{f(x), g(x)\},\)則：
\(U = max\{f(x), g(x)\} = \frac{1}{2}[f(x)+g(x)+|f(x)-g(x)|] = \begin{cases}f(x) & f(x) \geq g(x) \\ g(x) & f(x)<g(x) \end{cases}\)
\(V = min\{f(x), g(x)\} = \frac{1}{2}[f(x)+g(x)-|f(x)-g(x)|] = \begin{cases}g(x) & f(x) \geq g(x) \\ f(x) & f(x)<g(x) \end{cases}\)
即：\(U+V = f(x)+g(x),\ U-V = |f(x)-g(x)|,\ UV=f(x)g(x)\)
2. 符號函數
\[y=sgnx=\begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases}, 稱為符號函數，對於任何實數x，有x=|x|shnx \]
3. 取整函數
- 定義：y=[x]稱為取整函數，先給出定義，設x為任一實數，不超過x的最大整數稱為x的整數部分，記作[x]。如[0.99]=0, \([\pi]\)=3, [-1]=-1, [-1.99]=-2, 因此，取整函數y=[x]的定義域為R，值域為Z，在x為整數值處發生跳躍
- 注意點：
- \(x-1\leq[x],\ [x+n]=[x],\ n[x]\leq nx,\ [x]+[y]\leq[x+y]\)
- \(\lim_{x \to 0^+}[x]=0,\ \lim_{x \to 0^-}[x]=-1\)
- 考得最多：\(x-1<[x]\leq x\)
常用基礎知識
數列基礎
- 等差數列：
- 通項公式：\(a_{n} = a_{1} + (n - 1)d\)
- 前n項的和：\(S_{n} = \frac{n}{2}[2a_{1} + (n - 1)d] = \frac{n}{2}(a_{1} + a_{n})\)
- 等比數列：
- 通項公式：\(a_{n}=a_{1}r^{n-1}\)
- 前n項的和：\(S_{n}=\frac{a_{1}(1 - r^{n})}{1 - r} (r \neq 1)\)
- 常用：\(1 + r + r^{2} + … + r^{n - 1} = \frac{1 - r^{n}}{1 - r} (r \neq 1)\)
- 一些數列前n項的和：
- \(\sum_{k=1}^nk = 1 + 2 + 3 + … + n = \frac{n(n + 1)}{2}\)
- \(\sum_{k=1}^n(2k - 1) = 1 + 3 + 5 + … + (2n - 1) = n^{2}\)
- \(\sum_{k=1}^nk^{2} = 1^{2} + 2^{2} + 3^{2} + … + n^{2} = \frac{n(n + 1)(2n + 1)}{6}\)
- \(\sum_{k=1}^nk^{3} = 1^{3} + 2^{3} + 3^{3} + … + n^{3} = [\frac{n(n + 1)}{2}]^{2} = (\sum_1^nk)^{2}\)
- \(\sum_{k=1}^nk(k + 1) = 1 * 2 + 2 * 3 + 3 * 4 + … + n(n + 1) = \frac{n(n + 1)(n + 2)}{3}\)
- \(\sum_{k=1}^n\frac{1}{k(k + 1)} = \frac{1}{1 * 2} + \frac{1}{2 * 3} + \frac{1}{3 * 4} + … + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\)
三角函數基礎
1. 三角函數基本關系
- \(\sin\alpha \csc\alpha = 1\)
- \(\csc\alpha = \frac{1}{\sin\alpha}\)
- \(\cos\alpha \sec\alpha = 1\)
- \(\sec\alpha = \frac{1}{\cos\alpha}\)
- \(\tan\alpha \cot\alpha = 1\)
- \(\cot\alpha = \frac{1}{\tan\alpha}\)
- \(\tan\alpha = \frac{\sin\alpha}{\cos\alpha}\)
- \(\cot\alpha = \frac{\cos\alpha}{\sin\alpha}\)
- \(\sin^{2}\alpha + \cos^{2}\alpha = 1\)
- \(1 - \sin^{2}\alpha = \cos^{2}\alpha\)
- \(1 - \cos^{2}\alpha = \sin^{2}\alpha\)
- \(\sec^{2}\alpha - \tan^{2}\alpha = 1\)
- \(1 + \tan^{2}\alpha = \sec^{2}\alpha\)
- \(\sec^{2}\alpha - 1 = \tan^{2}\alpha\)
- \(\csc^{2}\alpha - \cot^{2}\alpha = 1\)
- \(1 + \cot^{2}\alpha = \csc^{2}\alpha\)
- \(\csc^{2}\alpha - 1 = \cot^{2}\alpha\)
2. 誘導公式
| \(函數/角\theta\) |
\(\frac{\pi}{2} - \alpha\) |
\(\frac{\pi}{2} + \alpha\) |
\(\pi - \alpha\) |
\(\pi + \alpha\) |
\(\frac{3\pi}{2} - \alpha\) |
\(\frac{3\pi}{2} + \alpha\) |
\(2\pi - \alpha\) |
| \(函數/角\theta\) |
\(90°- a\) |
\(90°+ a\) |
\(80°- a\) |
\(180°+ a\) |
\(270°- a\) |
\(270°+ a\) |
\(360°- a\) |
| \(\sin\theta\) |
\(\cos\alpha\) |
\(\cos\alpha\) |
\(\sin\alpha\) |
-\(\sin\alpha\) |
\(-\cos\alpha\) |
\(-\cos\alpha\) |
\(-\sin\alpha\) |
| \(\cos\theta\) |
\(\sin\alpha\) |
-\(\sin\alpha\) |
\(-\cos\alpha\) |
\(-\cos\alpha\) |
\(-\sin\alpha\) |
\(\sin\alpha\) |
\(\cos\alpha\) |
| \(\tan\theta\) |
\(\cot\alpha\) |
-\(\cot\alpha\) |
\(-\tan\alpha\) |
\(\tan\alpha\) |
\(\cot\alpha\) |
\(-\cot\alpha\) |
\(-\tan\alpha\) |
| \(\cot\theta\) |
\(\tan\alpha\) |
-\(\tan\alpha\) |
\(-\cot\alpha\) |
\(\cot\alpha\) |
\(\tan\alpha\) |
\(-\tan\alpha\) |
\(-\cot\alpha\) |
| 口訣：奇變偶不變，符號看象限 |
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| \(函數/角\theta 所在象限\) |
\(第一象限\) |
\(第二象限\) |
\(第三象\) |
\(第四象限\) |
| \(\sin\theta\) |
\(+\) |
\(+\) |
\(-\) |
\(-\) |
| \(\cos\theta\) |
\(+\) |
\(-\) |
\(-\) |
\(+\) |
| \(\tan\theta\) |
\(+\) |
\(-\) |
\(+\) |
\(-\) |
| \(\cot\theta\) |
\(+\) |
\(-\) |
\(+\) |
\(-\) |
| 口訣：全STC |
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| \(\alpha\) |
\(0°\) |
\(30°\) |
\(45°\) |
\(60°\) |
\(90\) |
\(120°\) |
\(135°\) |
\(150°\) |
\(180°\) |
\(270°\) |
\(360°\) |
| \(\alpha\) |
\(0\) |
\(\frac{\pi}{6}\) |
\(\frac{\pi}{4}\) |
\(\frac{\pi}{3}\) |
\(\frac{\pi}{2}\) |
\(\frac{2\pi}{3}\) |
\(\frac{3\pi}{4}\) |
\(\frac{5\pi}{6}\) |
\(\pi\) |
\(\frac{3\pi}{2}\) |
\(2\pi\) |
| \(\sin\alpha\) |
\(0\) |
\(\frac{1}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{\sqrt{3}}{2}\) |
\(1\) |
\(\frac{\sqrt{3}}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{1}{2}\) |
\(0\) |
\(-1\) |
\(0\) |
| \(\cos\alpha\) |
\(1\) |
\(\frac{\sqrt{3}}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{1}{2}\) |
\(0\) |
\(-\frac{1}{2}\) |
\(-\frac{\sqrt{2}}{2}\) |
\(-\frac{\sqrt{3}}{2}\) |
\(-1\) |
\(0\) |
\(1\) |
| \(\tan\alpha\) |
\(0\) |
\(\frac{\sqrt{3}}{3}\) |
\(1\) |
\(\sqrt{3}\) |
\(\infty\) |
\(-\sqrt{3}\) |
\(-1\) |
\(-\frac{\sqrt{3}}{3}\) |
\(0\) |
\(\infty\) |
\(0\) |
| \(\cot\alpha\) |
\(\infty\) |
\(\sqrt{3}\) |
\(1\) |
\(\frac{\sqrt{3}}{3}\) |
\(0\) |
\(-\frac{\sqrt{3}}{3}\) |
\(-1\) |
\(-\sqrt{3}\) |
\(\infty\) |
\(0\) |
\(\infty\) |
3. 倍角公式
- \(\sin2\alpha=2\sin\alpha\cos\alpha\)
- \(\cos2\alpha=cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1\)
- \(\sin3\alpha=-4\sin^3\alpha+3\sin\alpha\)
- \(\cos3\alpha=4\cos^3\alpha-3\cos\alpha\)
- \(\sin^2\alpha=\frac{1}{2}(1-\cos2\alpha)\)
- \(\cos^2\alpha=\frac{1}{2}(1+\cos2\alpha)\)
- \(\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2}\)
- \(\cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}\)
4. 半角公式
- \(\sin^2\frac{\alpha}{2}=\frac{1}{2}(1-\cos\alpha)\)
- \(\cos^2\frac{\alpha}{2}=\frac{1}{2}(1+\cos\alpha)\)
- \(\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}\)
- \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}\)
- \(\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}=\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}\)
- \(\cot\frac{\alpha}{2}=\frac{\sin\alpha}{1-\cos\alpha}=\frac{1+\cos\alpha}{\sin\alpha}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}\)
5. 和差公式
- \(\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\)
- \(\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta\)
- \(\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}\)
- \(\cot(\alpha\pm\beta）=\frac{\cot\alpha\cot\beta\mp1}{\cot\beta\pm\cot\alpha}\)
6. 積化和差公式
- \(\sin\alpha\cos\beta=\frac{1}{2}\left[\sin（\alpha+\beta）+\sin（\alpha-\beta）\right]\)
- \(\cos\alpha\sin\beta=\frac{1}{2}\left[\sin（\alpha+\beta）-\sin（\alpha-\beta）\right]\)
- \(\cos\alpha\cos\beta=\frac{1}{2}\left[\cos（\alpha+\beta）+\cos（\alpha-\beta）\right]\)
- \(\sin\alpha\sin\beta=\frac{1}{2}\left[\cos（\alpha-\beta）-\cos（\alpha+\beta）\right]\)
7. 和差化積公式
- \(\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\)
- \(\sin\alpha-\sin\beta=2\sin\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}\)
- \(\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}\)
- \(\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}\)
8. 萬能公式
- \(若\mu=\tan\frac{x}{2}(-\pi < x < \pi)，則\sin\chi=\frac{2\mu}{1+\mu^2}\)
指數運算法則
- \(a^{\alpha}*a^{\beta}=a^{\alpha+\beta}\)
- \(\frac{a^{\alpha}}{a^{\beta}}=a^{\alpha-\beta}\)
- \(\left(a^{\alpha}\right)^{\beta}=a^{\alpha\beta}\)
- \(\left(ab\right)^\alpha=a^{\alpha}b^{\alpha}\)
- \(\left(\frac{a}{b}\right)^\alpha=\frac{a^{\alpha}}{b^{\alpha}}\)
\(其中，a,b是正實數，\alpha,\beta是任意實數\)
對數運算法則
- \(\log_{a}{MN}=\log_{a}{M} + \log_{a}{N}\)
- \(\log_{a}{\frac{M}{N}}=\log_{a}{M}-\log_{a}{N}\)
- \(\log_{a}{M^{n}}=n\log_{a}{M}\)
- \(\log_{a}{\sqrt[n]{M}}=\frac{1}{n}\log_{a}{M}\)
一元二次方程基礎
- 一元二次方程： \(ax^{2} + bx + c=0(a\neq0)\)
- 根的公式： \(x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
- 根與系數的關系： \(x_{1} + x_{2}=-\frac{b}{a},x_{1}x_{2}=\frac{c}{a}\)
- 判別式： \(\triangle=b^{2}-4ac:\)
- \(\triangle>0\)，方程有兩個不等的實根
- \(\triangle=0\)，方程有兩個相等的實根，
- \(\triangle<0\)，方程有兩個共軛的復根。
- 拋物線： \(y=ax^{2}+bx+c\) 的頂點：\(\left( -\frac{b}{2a},c-\frac{b^{2}}{4a}\right)\)
因式分解公式
- \((a+b)^{2}=a^{2}+2ab+b^{2}\)
- \((a-b)^{2}=a^{2}-2ab+b^{2}\)
- \((a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\)
- \((a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\)
- \((a+b)(a-b)=a^{2}-b^{2}\)
- \(a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})\)
- \(a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})\)
- \(a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})\) (n是正整數)
- n是正偶數時，\(a^{n}-b^{n}=(a+b)(a^{n-1}-a^{n-2}b+...+ab^{n-2}-b^{n-1})\)
- n是正奇數時，\(a^{n}+b^{n}=(a+b)(a^{n-1}-a^{n-2}b+...-ab^{n-2}+b^{n-1})\)
階乘與雙階乘
- \(n!= 1\times2\times3\times ... \times n，規定0!=1\)
- \((2n)!! = 2\times4\times6\times ... \times 2n=2^n\cdot n!\)
- \((2n-1)!! = 1\times3\times5\times ... \times (2n-1)\)
