浙江大学数学分析[试卷+答案]2007年考研真题考研试题

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David Zhang
Department of Mathematics
Sun Yat-Sen Univsersity
QQ: 745657812
E-mail: zhangzujin360732@163.com
November 13, 2007
1 yyy²²²
1.1
ex
sin x − x(1 + x) = O(x3
)(x → 0)
y²:
lim
x→0
ex
sin x − x(1 + x)
x3
= lim
x→0
1 + x + x2
/2 + o(x2
) x − x3
/6 + o(x4
) − x − x2
3
= lim
x→0
x3
/3 + o(x3
)
x3
=
1
3
1.2
cos x + sin x > 1 + x − x2
, x ∈ (0, +∞)
y²:
f(x) = cos x + sin x − 1 − x + x2
u
f (x) = − sin x + cos x − 1 + 2x f (x) = − cos x − sin x + 2 > 0
l
x > 0 ⇒ f (x) > f (0) = −0 + 1 − 1 + 0 = 0
x > 0 ⇒ f(x) > f(0) = 1 + 0 − 1 − 0 + 0 = 0
1