9) If F ( x) is an antiderivative of f ( x) , C is any constant, then _B___ is correct. A. F ( x) = C ∫ f ( x)dx C. F '( x) = f ( x) +C B. F ( x) = ∫ f ( x)dx
If f ( x) = e x , then
∫
f '(ln x) dx = _|x|+C___________. x
d2 f 1 −2sin x = − + f ( x) ln(cos x) + tan x , then 7) = 2 2 dx cos x cos3 x
If F ( x), f ( x), g ( x), h( x) are continuous in (−∞, ∞) . g ( x) ≤ f ( x) ≤ h( x) with
Increasing intervals: ( 5)
3 + 33 −3 + 33 , 0 ), ( , +∞) 4 4
= we have: a)
dy dt
(3 marks) Write out the concave up and concave down intervals of f ( x)
∫ f ( x)dx∫ g ( x)dx < 0
∫
b
a
f ( x)dx ∫ g ( x)dx < 0
a
b
2008-2009 学年第一学期《高等数学 D（英语） 》期末考试试卷（A 卷）--2
2. Fill in the blanks (10 marks)
1)
3 The domain of the function log
2) If a, b are in the domain of a decreasing function f ( x) , and a < b , A.
f (a ) ≤ f (b)
B. f (a ) ≥ f (b) C. f (a ) = f (b)
D. f (a ) ≈ f (b)
3) If f ( x) is a bounded function defined on [a,b], then f ( x) must be _C__ A. continuous 4) B. differentiable C. i ntegrable D. i ncreasing

2x is __ {x > 1} {x < 0} _____ and the x − 1
4)
region of this function is __ (−∞, log 3 2) (log 3 2, +∞) _______________. 2) The discontinuous point of
x →c
B. f '(c)= b − a D. 1 b f ( x)dx= b − a f (c) ∫a
此卷选为：期中考试( )、期终考试( √ )、重考( )试卷
年级 题号 得分
专业 一
二
学号 三
姓名 四 五
任课教师 总分
7) If lim = f ( x) lim = f '( x) 0, lim f ''( x) ≠ 0 but exists, then __A______.
10)
f (0) 0, = f (1) 2, , then If =
(
2 ∫ f '( x)e f ( x ) dx = 4( e 2 − 1) 2
0
1
)
2
3)
x →0
lim (1 − ln(1 − x) ) +
sin x
=1
2008-2009 学年第一学期《高等数学 D（英语） 》期末考试试卷（A 卷）--3
1. Choose a right answer of four to the following questions (10 marks)
C. lim
x→a
D. lim
x→a
1)
For the following concepts of a function, __D___ is not relative to a limitation A. continuity B. d erivative C. i ntegration D. va riable then _B__ 8) If f ( x) is a continuous on interval [a,b], then in [a,b], f ( x) at least have_ C__ A. a critical point. C. an absolute maximum point. B. a stationary point. D. an inflection point.
1)
(8 marks) Calculate the area of the region which is enclosed by functions
y = y = cos x and
2
π
π
| x | −1 .
2)
(3 marks) Write out all relative extreme points of f ( x) if there exist;
3. Calculations (30 marks)
1)
π x → 2
lim −
cos x =0 | x|
9)
∫
240 x 2 ( x − 1) x + 1dx = − −1 945
0
2)
3 x8 + sin x + 100 =0 x →+∞ 0.1e x + 7 ln x − 1 lim
3 + 33 −3 + 33 x= − ,x = 0, x = 4 4
AREA = 2 ∫ 2 cos x −
0

2x
π + 1 dx = 2( + 1) π 4
3)
(3 marks) Write out all inflection points of f ( x) if there exist;
x→a x→a
D.
none is A. B. C..
10) a and b are in the domains of f ( x) and g ( x) , then _A__ is correct. A. lim ( f ( x) g ( x) ) = lim f ( x) lim g ( x)
x→a x→a
lim = g ( x) lim = h( x) L , F ( x) is decreasing, then lim F ( f ( x)) = ___F(L)______.
8)
∫ e (e
−t
3t
− 4e −2t + 5cos(e − t ) ) dt =
1 2t 4 −3t e + e − 5sin e − t + C 2 3
x→a x→a x→a
B.
b
( f ( x) g ( x) ) ' =
b a
f '( x) g '( x)
b a
5) If f ( x), g ( x) are differentiable in [a,b], where f ( x) g ( x) < 0 , then __C_____ A. C.
x→a x →a x→a
A. lim
x→a
f ( x) = 0, f '( x) f ( x) = ∞, f '( x)
B. lim
x→a
f ( x) ≠ 0 but exists, f '( x) f ( x) ≠ ∞ but does not exist. f '( x)
（注意：本试卷共 5 大题，3 大张，满分 100 分．考试时间为 120 分钟。要求写出解题过程，否则不予计分）
C x
x→a
lim+ f ( x) exists, then __D_________
x→a
A. lim f ( x) = f (a ) ,
f ( x) = f (a + ) B. lim +
x→a
D. F ( x) = lim
h →∞
fห้องสมุดไป่ตู้( x + h) − f ( x ) h
f ( x) = lim f (a ) C. lim +
4. Graph Analysis
Analysis function f ( x) =x 4 + 2 x 3 − 3 x 2 : 1) (3 marks) Write out all roots of f ( x) if there exist;
x= −3, x = 0, x = 1
5. Applications
4)
(3 marks) Write out the increase and decrease intervals of f ( x) ;
3 + 33 −3 + 33 Decreasing intervals: (−∞, − ), (0, ) 4 4
y 2 + h2 = 3m , if
dh = 0.3m / s , dt
e x−y dy 2 x + ln y , then = 1 dx x− y
1
− f (− x) , then f (0) = __0____, and for any constant a, the definite If f ( x) =