中科院微积分期末试卷

cos3(x) sin4(x)
dx;
a) e−2x sin(5x)dx;
c)
2x3
+x2+2x−1 x4−1
dx.
7. (15©¤OŽiùÈ©:
b) 01(1 − x2)2015dx;
a)
π+1 1
sin2(x)dx;
c)
π/2 0
sin(x)
sin(2x)
sin(3x)dx.
8. (10©) ¼êf (x)34«m[a, b]þëYŒ‡ …f (a) = 0. y²Ø
‡È©A-1•(Ï"•Á, 2015.1.20)
1. £5©¤|^
− δŠóy² lim n→∞
2015×2n+n2+20×sin(n) n!
=
0.
2. £10©¤¦4•µ
a)
lim
x→+∞
; x2015
ex
b) lim xxx−1.
x→0+
3. £10©¤ ëY¼êf (x)÷v
sup |f (x + y) − f (x) − f (y)| < ∞
x,y∈
… lim n→∞
f (n) n
=
2015.
žy²sup |f (x)
x∈
−
2015x|
<
∞.
4. £10©¤3[0, 1]þ E˜‡¢¼êf (x), ¦ §3[0, 1]þüN§3¤ k kn:þÑØëY …÷vf (0) = 0 Úf (1) = 1"
5. (10©) ¼êf (x) Vúª£.‚KF{‘¤•
f (x)
=
f (x0)+f
(x0)(x−x0)+·
·
·+
f
(n)(x0 n!
)
(x−x0)n+
f (n+1) (n +
(ξn) 1)!
(x−x0)n+1,
ξn0ux0Úxƒm"XJf (n+2)(x0)
=
0,
žy lim x→x0
ξn−x0 x−x0
=
1 n+2
.
6. £15©¤OŽØ½È©:
wenku.baidu.com
b)
ªM 2 ≤ (b − a)
b a
(f
(x))2dx,
Ù¥M
=
sup
|f (x)|.
a≤x≤b
n
∞
9.
£10©¤-An
=
k=1
ln(k
+
1),
y²?ê
n=1
1 An
uÑ"
10.
(5©¤OŽ‡~È©
+∞ 0
ln(x) (1+xln(x) )x
dx