file_5174fb699d910同济大学高等数学英语版

同济大学课程考核试卷（A 卷） 20102011 学年第一学期 命题教师签名：梁进梁进 审核教师签名： 课号：122008122008 课名：高等数学高等数学 D D（英语）（英语） 考试考查：考试考试 此卷选为：期中考试( )、期终考试( )、重考( )试卷 年级 专业 学号 姓名 任课教师 题号 一 二 三 四 五 总分 得分 （注意：本试卷共（注意：本试卷共 5 5 大题，大题，3 3 大张，满分大张，满分 100100 分考试时间为分考试时间为 120120 分钟分钟。要。要求写出解题过程，否则不予计分求写出解题过程，否则不予计分） 1. Choose a right answer of four to the following questions (10 marks) 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 2) If , a b are in the domain of a decreasing function( )f x, andab, then _B_ A. ( )( )f af b B. ( )( )f af b C. ( )( )f af b= D. ( )( )f af b 3) If ( )f x is a bounded function defined on a,b, then( )f x must be _C_ A. continuous B. differentiable C. i ntegrable D. i ncreasing 4) lim( ) xa f x + exists, then _D_ A. lim( )( ) xa f xf a =, B. lim( )() xa f xf a + + = C. lim( )lim( ) xa xa f xf a + = D. none is A. B. C. 5) If ( ),( )f xg x are differentiable in a,b, where( )( )0f x g x , then _C_ A. () ()( )( )( )( )0f bf ag bg a B. ( )( )0fx g x C. ( )( )0f x dxg x dx D. ( )( )0 bb aa f x dxg x dx _ and the region of this function is _ 33 (,log2)(log2,)+_. 2) The discontinuous point of sin 1 x x e is _x=0_. 3) The inverse function of 215yx= + is _ 2 (5)1 2 x+ _. 4) Suppose ( )f x is differentiable, then the value of( )f xat xa= is_( )f a_, and the slope of the tangent line of ( )f x at this point is_( )fa_. 5) If ( )()f xfx= , then (0)f=_0_, and for any constant a, the definite integration ( ) a a f x dx =_0_. 6) If ( ) x f xe=, then (ln )fx dx x = _|x|+C_. 7) If ( ),( ),( ), ( )F xf xg xh x are continuous in (,) . ( )( )( )g xf xh x with lim ( )lim ( ) xaxa g xh xL =, ( )F x is decreasing, then lim( ( ) xa F f x = _F(L)_. 3. Calculations (30 marks) 1) 2 cos lim | x x x =0 2) 8 3sin100 lim 0.17ln1 x x xx ex + + + =0 3) () sin 0 lim 1 ln(1) x x x + =1 4) ()tan x d ex dx = 2 1 (tan) cos x ex x + 5) 2 0 1 1 x dx dxx = + =1 6) Suppose ln x xyey=+, then dy dx = 1 2 1 x ey x x y 7) ( )ln(cos )tanf xxx=+, then 2 223 12sin coscos d fx dxxx = + 8) () 3223 14 45cos()5sin 23 ttttttt eeeedteeeC +=+ 9) 0 2 1 240 (1)1 945 xxxdx += 10) If (0)0,(1)2,ff=, then ( ) 2 1 ( )22 0 2( )4(1) f x fx edxe= 4. Graph Analysis Analysis function 432 ( )23f xxxx=+ : 1) (3 marks) Write out all roots of ( )f x if there exist; 3,0,1xxx= = 2) (3 marks) Write out all relative extreme points of ( )f x if there exist; 333333 ,0, 44 xxx + + = = 3) (3 marks) Write out all inflection points of ( )f x if there exist; 1313 , 44 xx + + = = 4) (3 marks) Write out the increase and decrease intervals of( )f x; Decreasing intervals: 333333 (,),(0,) 44 + + Increasing intervals: 333333 (,0),(,) 44 + + + 5) (3 marks) Write out the concave up and concave down intervals of ( )f x Concave up 1313 (,),(,) 44 + + +, concave down 1313 (,) 44 + + 6) (3 marks) Write out the infinite behaviors of ( )f x lim( ) x f x = + 7) (5 markes) Sketch the figure of the function 5. Applications 1) (8 marks) Calculate the area of the region which is enclosed by functions cosyx= and 2 | 1yx =. AREA 2 0 2 2cos12(1) 4 x xdx =+=+ 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 22 3yhm+=, if 0.3/ dh m s dt =, we have: a) 22 0.3 99 dyhdhh dtdt hh = b) 1 2 0.30.3 |0.106 9 1 9 h dyh dt h = = 3) (8 marks) A closed rectangular container with a square base is to have a volume of 1000 cm3. It costs twice as much per square centimeter for the top as it does for the sides and bottom. Find the dimensions of the container of least cost. Square side =