中国数学考研 Graduate Math Exams

中国”考研”究生:

考题难, 重视理论基础, 不是技巧。计算量大, 时间(3小时)不够。

国家 “及格” 底线 : 58~ 90分 (总分 : 150 分) – 根据 理工 / 经管系 , 不同重点大学, 底线各异。

http://www.bilibili.com/mobile/video/av2261356.html

[例子] p (x) = a + bx+cx^{2}+dx^{3}

p(x) - tan x \sim x^{3}, \text { when } x \to 0

Find a, b, c, d ?

[Solution] : 

1. Don’t use l’Hôpital Rule for  \displaystyle \lim \frac {f}{g}

2. Apply Taylor expansion :

\tan x = x + \frac {1}{3}x^{3} + o (x^{3})
p (x) - \tan x = a + (b -1)x + (c - \frac {1}{3})x^{3} + o (x^{3})

p(x) - tan x \sim x^{3}, \text { when } x \to 0

\iff \boxed {a=0, b=1, c=\frac {4}{3}}

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