任何可导函数 \(f(x)=\sum a_{n}x^{n}\)，\(x\rightarrow 0\)时
\(sinx=x-\frac{1}{6}x^{3}+o(x^{3})\)\(arcsinx=x+\frac{1}{6}x^{^{3}}+o(x^{^{3}})\)\(tanx=x+\frac{1}{3}x^{3}+o(x^{3})\)\(arctanx=x-\frac{1}{3}x^{3}+o(x^{3})\)\(cosx=1-\frac{1}{2}x^{2}+\frac{1}{24}x^{4}+o(x^{4})\)\(ln(1+x)=x-\frac{1}{2}x^{2}+\frac{1}{3}x^{3}-\frac{1}{4}x^{4}+o(x^{4})\)\(e^{x}=1+x+\frac{1}{2!}x^{2}+\frac{1}{3!}x^{3}+o(x^{4})\)\(\frac{1}{1-x}=1+x+x^{2}+x^{3}+o(x^{3})(\left | x \leq 1\right |)\)Back to TOC基本微分公式\(({x^{n}})'=nx^{n-1}\)
\({(a^{x})}'=a^{x}lna\)
\({(e^{x})}'=e^{x}\)
\({(lnx)}'=\frac{1}{x}\)
\({(sinx)}'=cosx\)
\({(cosx)}'=-sinx\)
\({(tanx)}'=sec^{2}x\)
\({(cotx)}'=-csc^{2}x\)
\({(secx)}'=secxtanx\)
\({(cscx)}'=-cscxcotx\)
\({(arcsinx)}'=\frac{1}{\sqrt{1-x^{2}}}\)
\({(arccosx)}'=-\frac{1}{\sqrt{1-x^{2}}}\)
\({(arctanx)}'=\frac{1}{1+x^{2}}\)
\({(arccotx)}'=-\frac{1}{1+x^{2}}\)
\({(ln(x+\sqrt{x^{2}+1})})'=\frac{1}{\sqrt{x^{2}+1}}\)
\(({ln(x+\sqrt{x^{2}-1})})'=\frac{1}{\sqrt{x^{2}-1}}\)
Back to TOC常用等价无穷小\(x \rightarrow 0\)\(sin x \sim x\)\(arcsin x \sim x\)\(tan x \sim x\)\(arctan x \sim x\)\(e^{x} - 1 \sim x\)\(ln(1 + x) \sim x\)\((1 + x)^{\alpha } - 1 \sim \alpha x\)\(1 - cos x \sim \frac{1}{2} x^{2}\)Back to TOC函数极限定义\(\lim \limits_{x \to x0}f(x)=A \Leftrightarrow \forall \epsilon > 0, \exists \delta >0, 当 00, \exists N>0, 当 n>N时，有 |x_{0}-A|0, \delta>0,当00\)\(若\lim \limits_{x \rightarrow x_{0}}f(x)=A0,使得|f(x)| \leq K, \forall x \in[a,b]\)
Back to TOC最值定理设f(x)在[a,b]连续，则：\(当m\leq \mu \leq M时，其中m,M分别为f(x)在[a,b]上的最小最大值\)
Back to TOC介值定理设f(x)在[a,b]连续，则：\(当m\leq \mu \leq M时，则\exists \xi \in (a,b),使得f(\xi)=0\)
Back to TOC零点定理设f(x)在[a,b]连续，则：\(当f(a) \cdot f(b)0 \Rightarrow 极小值；若f(x)在x=x_{0}处二阶可导，{f}'(x_{0})=0，{f}''(x_{0}) f(\frac{x_1+x_2}{2}) \Rightarrow f(x)， 是凹曲线 \\ \frac{f(x_1)+f(x_2)}{2} < f(\frac{x_1+x_2}{2}) \Rightarrow f(x)， 是凸曲线 \end{cases}\)
Back to TOC函数拐点连续曲线凹凸弧的分界点
Back to TOC拐点判别法设f(x)在I上二阶可导
\(\begin{cases}若{f}''(x_0)>0，\forall x\in I \Rightarrow f(x)是凹的\\ 若{f}''(x_0)0 \Rightarrow \lambda_1\neq\lambda_2 \Rightarrow y=C_1e^{\lambda_1x}+C_2e^{\lambda_2x}\\ \triangle=0 \Rightarrow \lambda_1=\lambda_2=\lambda \Rightarrow y(C_1+C_2x)e^{kx} \\ \triangle0,a\neq 1),是y=a^x的反函数$单调性：\(当a>1时，y=log_a^x单调增加，当0