file_5174fb699d910同济大学高等数学英语版

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) =
integration 6) 7)
x→a
∫
a
−a
f ( x)dx =___0_____.
2008-2009 学年第一学期《高等数学 D（英语） 》期末考试试卷（A 卷）--1
同济大学课程考核试卷（A 卷） 2010—2011 学年第一学期
命题教师签名：梁进 课号：122008 审核教师签名： 课名：高等数学 D（英语） 考试考查：考试
6) If f ( x) is an integrable function, then there exists c ∈ (a, b) , such that__ A. f (c)= b − a C. lim f ( x)= b − a
3) 4)
y The inverse function of =
Suppose f ( x) is differentiable, then the value of f ( x) at x = a is__ f (a ) ____,
6)
xy e Suppose =
x
and the slope of the tangent line of f ( x) at this point is_______ f '(a ) __________. 5)
( f (b) − f (a) )( g (b) − g (a) ) < 0
C. ∫ f ( x) g ( x)dx = ∫ f ( x)dx ∫ g ( x)dx
D. ∫ f ( x) g ( x)dx = ∫ f ( x)dx ∫ g ( x)dx
a
B. f '( x) g '( x) < 0 D.
d x 1 e tan x ) = e x (tan x + ) ( dx cos 2 x
sin x is __x=0______________________. ex −1
2 x − 1 + 5 is _____
( x − 5) 2 + 1 ____________. 2
5)
d x −1 =1 dx x 2 + 1 x =0
1+ 3 −1 + 3 x= − ,x = 4 4
2) (11 marks) A 3-m ladder is leaning against a wall. If the top of the ladder slips down the wall at a constant rate of 0.3m/s, a) What is the rate function of the foot moving away from the wall, where the independent variable of the function is the distance between the bottom and the wall? b) What is the exact figure of the rate when the top is 1 m above the ground. The distance to the wall is y, the top of the ladder to the bottom of the wall is h， then