【笔记】高数常用计算公式

最新推荐文章于 2024-09-19 11:20:17 发布

冰临天下

最新推荐文章于 2024-09-19 11:20:17 发布

阅读量1.1k

点赞数

分类专栏：笔记

本文链接：https://blog.youkuaiyun.com/songyuwen0808/article/details/105960503

版权

笔记专栏收录该内容

2 篇文章

订阅专栏

极限：

$\underset{n->\infty }{limit}\ q^{n} = \left\{\begin{matrix} 1 & q = 1\\ 0 & |q| < 1\\ \infty & |q| > 1\\ none & q = -1 \end{matrix}\right.$

$\underset{n->\infty }{limit}\ \frac {1}{n^{k}}= 0,\underset{n->\infty }{limit}\ n^{k}= \infty(k > 0)$

$\underset{n->\infty }{limit} \sqrt[n]{a} = 1(a > 0),\underset{n->\infty }{limit} \sqrt[n]{n} = 1$

$\underset{n->\infty }{limit}\ \frac {a^{n}}{n!} = 0(a > 0),\underset{n->\infty }{limit}\ \frac {n^{k}}{a^{n}} = 0(a > 1, k > 0)$

$\underset{n->\infty }{limit}\ \frac {n!}{n^{n}} = 0,\underset{n->\infty }{limit}\ \frac {1}{\sqrt [n]{n}} = 0$

$\underset{n->\infty }{limit}(1 + \frac {1}{2} + ... + \frac {1}{n}) = \infty$

$\underset{n->\infty }{limit}\ (1 + \frac {1}{n})^{n} = e,(e = 2.718281...)$

$\underset{x -> \infty }{limit}\ (1 + \frac {1}{x})^{x} = e, \underset{x -> 0 }{limit}(1 + x)^{\frac {1}{x}}=e$

$\underset{x -> \infty }{limit} (\frac {x + a}{x + b})^{x} = \frac {e^{a}}{e^{b}} = e^{a - b}$

$\underset{x -> 0 }{limit}\ \frac {sinx}{x} = 1$

$\underset{x -> 0 }{limit}\ \frac {ln(1 + x)}{x} = 1$

$\underset{x -> 0 }{limit}\ \frac {\sqrt {1 + x} - 1}{x} = \frac {1}{2}$

$\underset{x -> \infty }{limit}\ \frac {a_{0}x^{m} + a_{1}x^{m - 1} + ... + a_{m - 1}x + a_{m}}{b0x^{n} + b_{1}x^{n - 1} + ... + b_{n - 1}x + b_{n}} = \left\{\begin{matrix} \frac {a_{0}}{b_{0}} & m = n\\ 0 & m < n\\ \infty & m > n \end{matrix}\right.$

等价无穷小

当 $x\rightarrow 0$ 时，存在下记等价无穷小：

$\\sinx \sim x \ \ \ \ \ \ \ tanx\sim x\\ e^{x} - 1\sim x \ \ \ \ \ \ \ ln(1 + x)\sim x\\ (1 + x)^{a} - 1 \sim ax \ \ \ \ \ \ \ arcsinx \sim x \\ arctanx \sim x \ \ \ \ \ \ \ a^{x} - 1 \sim xlna(a > 0, a \neq 1)\\ x^{m} + x^{k} \sim x^{m}(k > m > 0) \ \ \ \ \ \ \ 1 - cosx \sim \frac {1}{2}x^{2}\\ \sqrt {1 + x} - 1 \sim \frac {x}{2} \ \ \ \ \ \ \ x - sinx \sim \frac {x^{3}}{6}\\ tanx - x \sim \frac {x^{3}}{3} \ \ \ \ \ \ \ tanx - sinx \sim \frac {x^{2}}{2}\\ arcsinx - x \sim \frac {x^3}{6} \ \ \ \ \ \ \ x - arctanx \sim \frac {x^{3}}{3}\\ x - ln(1 + x) \sim \frac {x^{2}}{2}$

导数

$\\(x^{a})' = ax^{a - 1} \ \ \ \ \ \ \ (a^{x})' = a^{x}lna(a > 0, a \neq 1)\\ (e^{x})' = e^{x} \ \ \ \ \ \ \ (log_{a}x)' = \frac {1}{xlna}(a > 0, a \neq 1)\\ (lnx)' = \frac {1}{x} \ \ \ \ \ \ \ (sinx)' = cosx\\ (cosx)' = -sinx \ \ \ \ \ \ \ (tanx)' = sec^{2}x\\ (cotx)' = -csc^{2}x \ \ \ \ \ \ \ (secx)' = secx\ tanx\\ (cscx)' = -cscx\ cotx \ \ \ \ \ \ \ (arcsinx)' = \frac {1}{\sqrt {1 - x^{2}}}\\ (arccosx)' = -\frac {1}{1 - x^{2}} \ \ \ \ \ \ \ (arctanx)' = \frac {1}{1 + x^{2}}\\ (arccotx)' = - \frac {1}{1 + x^{2}}$

麦克劳林公式

$e^{x} = 1 + x + \frac {1}{2!}x^{2} + ... + \frac {1}{n!}x^{n} + o(x^{n})$

$sinx = x - \frac {1}{3!}x^{3} + ... + \frac {(-1)^{n}}{(2n + 1)!}x^{2n + 1} + o(x^{2n + 2})$

$cosx = 1 - \frac {1}{2!}x^{2} + ... + \frac {(-1)n}{(2n)!}x^{2n} + o(x^{2n + 1})$

$ln(1 + x) = x - \frac {x^{2}}{2} + \frac {x^{3}}{3} - ... + (-1)^{n - 1}\frac {x^{n}}{n} + o(x^{n})$

$(1 + x)^{m} = 1 + mx + \frac {m(m - 1)}{2!}x^{2} + ... + \frac {m(m - 1)...(m - n + 1)}{n!}x^{n} + o(x^{n})$

$arcsinx = x + \frac {1}{2} \frac {x^{3}}{3} + \frac {1}{2}\frac {3}{4}\frac {x^{5}}{5} + ... + \frac{(2n - 3)!!}{(2n - 2)!!}\frac {x^{2n - 1}}{2n - 1} + o(x^{2n})$