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1、第 1 页 共 5 页 1、(1) sin 2limxxx0 . (2)d(arctan)x21d1+xx(3) 21dsinxx-cot+Cxx(4).2( )()xne22nxe.(5)4012 dx x26/3 2、(6) The right proposition in the following propositions is _A_. A. Iflim( )xaf xexists andlim( )xag xdoes not exist thenlim( )( )xaf xg xdoes not exist. B. Iflim( )xaf x,lim( )xag xdo both
2、not exist thenlim( )( )xaf xg xdoes not exist. C. Iflim( )xaf xexists andlim( )xag xdoes not exist thenlim( )( )xaf x g xdoes not exist. D. Iflim( )xaf xexists andlim( )xag xdoes not exist then( )lim( )xaf xg xdoes not exist. (7) The right proposition in the following propositions is _B_. A. Iflim(
3、)( )xaf xf athen( )faexists. B. Iflim( )( )xaf xf athen( )fadoes not exist. C. If( )fadoes not exist thenlim( )( )xaf xf a. D. If( )fadoes not exist then the cure( )yf xdoes not have tangent at( ,( )a f a. (8) The right statement in the following statements is _D_. A. sinlim1xxxB. 1lim(1)xxxeC. 11d1
4、xxxCD. 5511dd11bbaaxyxy(9) For continuous function( )f x, the erroneous expression in the following expressions is _D_.A.d( )d )( )dbafxxf bbB. d( )d )( )dbaf xxf aaC. d( )d )0dbaf xxxD. d( )d )( )( )dbaf xxf bf ax(10) The right proposition in the following propositions is _B_. A. If( )f xis discont
5、inuous on , a bthen( )f xis unbounded on , a b. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 1 页，共 5 页第 2 页 共 5 页 B. If( )f xis unbounded on , a bthen( )f xis discontinuous on , a b. C. If( )f xis bounded on , a bthen( )f xis continuous on , a b. D. If( )f xhas absolute extreme values on , a bthen
6、( )f xis continuous on , a b. 3、Evaluate2011lim()xxexx201=lim()xxexx01= lim()2xxex01= lim=22xxe考点课本 4.4 节洛比达法则，每年都会有一道求极限的解答题，大多数都是用洛比达法则去求解，所以大家要注意 4.4节的内容。注意洛比达法则的适用范围。 4Find 0d |xyand(0)yif20()dxxxyytte . 20()d)xxxyytte（）12( )2( )1xxyx y xeyx y xe00(2 0(0)1)0 xdyyedxdx(2( )1)2 ( )2( )xxyx y xey x
7、xy xe200()d-(0)0-01xxyytte xye002 (0)2 0 (0)=3yyye（ ）考察微积分基本定理与微分，书上5.3 节5、Find22arctand(1)xxxx=22221)arctand(1)xxxxxx（22arctanarctan=dd(1)xxxxxx-12311=-arctan +darctan+2xxxxx x22-1221+1=-arctan +darctan1+2xxxxxxxx（）-12211=-arctan +ddarctan1+2xxxxxxxx（）-12211=-arctan +InIn 1+arctan22xxxxx-1221=-arct
8、an +Inarctan+C21+xxxxx凑微分求不定积分，积分是微积分的重点及难点，大家一定要掌握透彻。6、Given that22( )1xf xx. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 2 页，共 5 页第 3 页 共 5 页 (1) Find the intervals on which( )f xis increasing or decreasing. 2222212( )1x xxxfxx（）（）2221xx（）When( )00fxx( )00fxxTherefore, the increasing interval i
9、s0，, the decreasing interval is 0，(2) Find the local maximum and minimum values of( )f x( )00fxxThe function is increasing in interval0，, decreasing in interval0，, therefore, the function exist the local minimum value, it is( )0f x(3)Find the intervals of concavity and the inflection points. 2222422
10、2242422181642( )111xxxxxxfxxxx（）（）（）（）（）422464233( )0133xxfxxor xx（）422464233( )000133xxfxxorxx（）331()()=334ffTherefore, the concave upward interval are33，,33，, the concave downward interval are3-03，,303，and the inflection points are 3 1-34，,3 134，(4) Find the asymptote lines of the cure( )yf x2221l
11、im=1111+xxxxTherefore, the liney = 1is a horizontal asymptote考点： 4.3 节， 4.5、4.6 节。近几年经常会考一道作图题。这种题目应该在注意的点主要包括函数的定义域，对称性，增减区间，极值点，凹凸性，拐点，以及渐近线等。大家参照课本的4.5 节进行作图7、Let R be the region bounded by the curve 1yx, and the lineyxand2x. (a)Evaluate the area of the region R. R=211xdxx2211=In2xx2211=2In21In12
12、23=In22(b)Find the volume of the solid generated by revolving the R about the y-axis. 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 3 页，共 5 页第 4 页 共 5 页 V=212121214)?4)ydydyy（12311211443xyyy3311114 224 114 142331283考点：求面积以及体积，课本6.1、6.2 节。这类题目是常考题，较简单。望同学一定要做相应的题目加以稳固。 8、 Determine the production le
13、vel that will maximize the profit for a company with cost and demand functions 23( )1450360.580.001C xxxxand( )600.01p xx. Solution 2( )(600.01 )600.01R xx xxx223( )( )( )600.01(1450 360.580.001 )P xR xC xxxxxx320.0010.57241450 xxx2( )0.0031.142P xxxLet( )040020P xxor xsince x0, then x=400 ( )0.0061
14、.14P xxWhen x=400 ( )0.006 400 1.141.260Px32(400)0.001 4000.57 40024 400 145035350PTherefore, when the production level is 400 that will maximize the profit 35350 考点：经济函数，课本4.8 节。此题型为常考题，属于送分题，大家可以做相应的4.8节的练习加以稳固9、State the second derivative test theorem testing maximum and prove it. Supposefis cont
15、inuous near c If cf（ )=0and cf（ ) 0, then has a local maximum at c . Proof: Because near cf （ ) 0c and sofis concave downward near c. This means that the graph of lies below its horizontal tangent at c and so has a local maximum at c. 10、Show that the equation3150 xxchas at most one root in the inte
16、rval 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 4 页，共 5 页第 5 页 共 5 页 2,2. Suppose the equation 3150 xxchas two root1212,andx xxxin the interval12, 2,2x x. Let 3F( )15xxxcthen there exist 12F()F()=0 xxUsing the Rolle s Theorem we can know that there exist one number c can satisfy 12F(c)0(,)cx x2F( )3150 2,2xxwhen xBut2F( )31502,2xxwhen x, Therefore, the suppose is wrong. Then the equation3150 xxchas at most one root in the interval 2,2. 考点：罗尔地理。书上4.2节。中值定理的证明题是历年考试证明题的热点，大家一定要吃透该定理，做一定的题目加以稳固精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 5 页，共 5 页