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1、高等数学基础归类复习考试小抄一、单项选择题1-1下列各函数对中，（ C ）中的两个函数相等 A. ， B. ， C.， D. ，1-设函数的定义域为，则函数的图形关于（C ）对称A. 坐标原点 B. 轴 C. 轴 D. 设函数的定义域为，则函数的图形关于（D ）对称A. B. 轴 C. 轴 D. 坐标原点.函数的图形关于（ A ）对称(A) 坐标原点 (B) 轴 (C) 轴 (D) 1-下列函数中为奇函数是（ B ）A. B. C. D. 下列函数中为奇函数是（A ）A. B. C. D. 下列函数中为偶函数的是（ D ）A B C D 2-1 下列极限存计算不正确的是（ D ） A. B.
2、C. D. 2-2当时，变量（ C ）是无穷小量A. B. C. D. 当时，变量（ C ）是无穷小量A B C D .当时，变量（D ）是无穷小量A B C D 下列变量中，是无穷小量的为（ B ） A B C D.3-1设在点x=1处可导，则（ D ）A. B. C. D. 设在可导，则（ D ）A B C D 设在可导，则（ D ）A. B. C. D. 设，则（ A ） A B. C. D. 3-2. 下列等式不成立的是（D ）A. B C. D.下列等式中正确的是（B ）A. B. C. D.4-1函数的单调增加区间是（ D ） A. B. C. D. 函数在区间内满足（A ） A.
3、 先单调下降再单调上升 B. 单调下降 C. 先单调上升再单调下降 D. 单调上升.函数在区间（5，5）内满足（ A ）A 先单调下降再单调上升 B 单调下降 C先单调上升再单调下降 D 单调上升. 函数在区间内满足（D ）A. 先单调下降再单调上升 B. 单调下降 C. 先单调上升再单调下降 D. 单调上升5-1若的一个原函数是，则（D ） A. B. C. D. .若是 的一个原函数，则下列等式成立的是（ A ）。 A B C D5-2若，则（ B ） A. B. C. D. 下列等式成立的是（D ） A. B. C. D. （ B ） A. B. C. D. （ D ） A B C D
4、-3若，则（ B ） A. B. C. D. 补充： , 无穷积分收敛的是 函数的图形关于 y 轴 对称。二、填空题函数的定义域是（3，+） 函数的定义域是 （2，3） （3，4 函数的定义域是（5，2）若函数，则 1 2若函数，在处连续，则e .函数在处连续，则 2 函数的间断点是x=0 函数的间断点是 x=3 。函数的间断点是 x=0 3-曲线在处的切线斜率是1/2 曲线在处的切线斜率是 1/4 曲线在（0，2）处的切线斜率是 1 .曲线在处的切线斜率是 3 3-2 曲线在处的切线方程是y = 1 切线斜率是 0 曲线y = sinx 在点 (0,0)处的切线方程为 y = x 切线斜率是
5、 1 4.函数的单调减少区间是（，0 ） 函数的单调增加区间是（0，+） .函数的单调减少区间是 （，1 ） .函数的单调增加区间是 （0，+） 函数的单调减少区间是 （0，+） 5-1 . tan x +C 若，则9 sin 3x 5-2 3 0 0 下列积分计算正确的是（ B ）A B C D 三、计算题（一）、计算极限（1小题，11分）（1）利用极限的四则运算法则，主要是因式分解，消去零因子。（2）利用连续函数性质：有定义，则极限类型1： 利用重要极限 ， ， 计算1-1求 解： 1-2 求 解： 1-3 求 解：=类型2： 因式分解并利用重要极限 ， 化简计算。2-1求 解： =2-2
6、 解： 2-3 解： 类型3：因式分解并消去零因子，再计算极限3-1 解： =3-2 3-3 解 其他： ， ， （0807考题）计算 解： =（0801考题. ）计算 解 （0707考题.）= （二） 求函数的导数和微分（1小题，11分）（1）利用导数的四则运算法则 （2）利用导数基本公式和复合函数求导公式 类型1：加减法与乘法混合运算的求导，先加减求导，后乘法求导；括号求导最后计算。1-1 解：1-2 解：1-3 设，求解： 类型2：加减法与复合函数混合运算的求导，先加减求导，后复合求导2-1 ，求 解：2-2 ，求 解：2-3 ，求， 解：类型3： 乘积与复合函数混合运算的求导，先乘积求
7、导，后复合求导，求 。 解：其他：，求。 解：0807.设，求 解：0801.设，求 解：0707.设，求 解：0701.设，求 解：（三）积分计算：（2小题，共22分）凑微分类型1： 计算 解：0707.计算 解： 0701计算 解： 凑微分类型2：.计算 解： 0807.计算 解：0801.计算 解： 凑微分类型3：， 计算 解：.计算 解： 5 定积分计算题，分部积分法类型1：计算 解： ， 计算 解： ， 计算 解：，=0807 0707 类型2 （0801考题） 类型3： 四、应用题（1题，16分）类型1： 圆柱体上底的中心到下底的边沿的距离为l，问当底半径与高分别为多少时，圆柱体的
8、体积最大？l解：如图所示，圆柱体高与底半径满足 圆柱体的体积公式为 求导并令 得，并由此解出即当底半径，高时，圆柱体的体积最大类型2：已知体积或容积，求表面积最小时的尺寸。2-1（0801考题） 某制罐厂要生产一种体积为V的有盖圆柱形容器，问容器的底半径与高各为多少时用料最省？解：设容器的底半径为，高为，则其容积表面积为， 由得，此时。由实际问题可知，当底半径与高 时可使用料最省。一体积为V的圆柱体，问底半径与高各为多少时表面积最小？ 解： 本题的解法和结果与2-1完全相同。生产一种体积为V的无盖圆柱形容器，问容器的底半径与高各为多少时用料最省？ 解：设容器的底半径为，高为，则无盖圆柱形容器表
9、面积为 ，令 ， 得 ，由实际问题可知，当底半径与高 时可使用料最省。2-2欲做一个底为正方形，容积为32立方米的长方体开口容器，怎样做法用料最省？（0707考题）解： 设底边的边长为，高为，用材料为，由已知，表面积 ，令，得， 此时=2由实际问题可知，是函数的极小值点，所以当，时用料最省。欲做一个底为正方形，容积为62.5立方米的长方体开口容器，怎样做法用料最省？ 解： 本题的解法与2-2同，只需把V=62.5 代入即可。类型3 求求曲线上的点，使其到点的距离最短曲线上的点到点的距离平方为 ， 3-1在抛物线上求一点，使其与轴上的点的距离最短 解：设所求点P（x，y），则满足 ，点P 到点A
10、 的距离之平方为令，解得是唯一驻点，易知是函数的极小值点，当时，或，所以满足条件的有两个点（1，2）和（1，2）3-2求曲线上的点，使其到点的距离最短解：曲线上的点到点A（2，0） 的距离之平方为令，得， 由此， 即曲线上的点（1，）和（1，）到点A（2，0）的距离最短。08074 求曲线上的点，使其到点A（0，2）的距离最短。解： 曲线上的点到点A（0，2）的距离公式为 与在同一点取到最大值，为计算方便求的最大值点， 令 得，并由此解出，即曲线上的点（）和点（）到点A（0，2）的距离最短Emloyment tribunals sort out disagreements between em
11、ployers and employees. You may need to make a claim to an employment tribunal if: you dont agree with the disciplinary action your employer has taken against you your employer dismisses you and you think that you have been dismissed unfairly. For more informu, take advice from one of the organisatio
12、ns listed underFur ther help. Employment tribunals are less formal than some other courts, but it is still a legal process and you will need to give evidence under an oath or affirmation. Most people find making a claim to an employment tribunal challenging. If you are thinking about making a claim
13、to an employment tribunal, you should get help straight away from one of the organisations listed underFurther help. ation about dismissal and unfair dismissal, seeDismissal. You can make a claim to an employment tribunal, even if you haventappealedagainst the disciplinary action your employer has t
14、aken against you. However, if you win your case, the tribunal may reduce any compensation awarded to you as a result of your failure to appeal. Remember that in most cases you must make an application to an employment tribunal within three months of the date when the event you are complaining about
15、happened. If your application is received after this time limit, the tribunal will not usually accept i. If you are worried about how the time limits apply to you If you are being represented by a solicitor at the tribunal, they may ask you to sign an agreement where you pay their fee out of your co
16、mpensation if you win the case. This is known as adamages-based agreement. In England and Wales, your solicitor cant charge you more than 35% of your compensationif you win the case. You are clear about the terms of the agreement. It might be best to get advice from an experienced adviser, for examp
17、le, at a Citizens Advice Bureau. To find your nearest CAB, including those that give advice by e-mail, click onnearest CAB. For more information about making a claim to an employment tribunal, seeEmployment tribunals. The (lack of) air up there Watch m Cay man Islands-based Webb, the head of Fifas a
18、nti-racism taskforce, is in London for the Football Associations 150th anniversary celebrations and will attend Citys Premier League match at Chelsea on Sunday. I am going to be at the match tomorrow and I have asked to meet Ya ya Toure, he told BBC Sport. For me its about how he felt and I would li
19、ke to speak to him first to find out what his experience was. Uefa hasopened disciplinary proceedings against CSKAfor the racist behaviour of their fans duringCitys 2-1 win. Michel Platini, president of European footballs governing body, has also ordered an immediate investigation into the referees
20、actions. CSKA said they were surprised and disappointed by Toures complaint. In a statement the Russian side added: We found no racist insults from fans of CSKA. Age has reached the end of the beginning of a word. May be guilty in his seems to passing a lot of different life became the appearance of
21、 the same day; May be back in the past, to oneself the paranoid weird belief disillusionment, these days, my mind has been very messy, in my mind constantly. Always feel oneself should go to do something, or write something. Twenty years of life trajectory deeply shallow, suddenly feel something, do
22、 it.一字开头的年龄已经到了尾声。或许是愧疚于自己似乎把转瞬即逝的很多个不同的日子过成了同一天的样子；或许是追溯过去，对自己那些近乎偏执的怪异信念的醒悟，这些天以来，思绪一直很凌乱，在脑海中不断纠缠。总觉得自己似乎应该去做点什么，或者写点什么。二十年的人生轨迹深深浅浅，突然就感觉到有些事情，非做不可了。The end of our life, and can meet many things really do?而穷尽我们的一生，又能遇到多少事情是真正地非做不可？ During my childhood, think lucky money and new clothes are neces
23、sary for New Year, but as the advance of the age, will be more and more found that those things are optional; Junior high school, thought to have a crush on just means that the real growth, but over the past three years later, his writing of alumni in peace, suddenly found that isnt really grow up,
24、it seems is not so important; Then in high school, think dont want to give vent to out your inner voice can be in the high school children of the feelings in a period, but was eventually infarction when graduation party in the throat, later again stood on the pitch he has sweat profusely, looked at
25、his thrown a basketball hoops, suddenly found himself has already cant remember his appearance. Baumgartner the disappointing news: Mission aborted. r plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or hu
26、midity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitud
27、e of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence. We often close ourselves off when traumatic events happen in our lives; instead of letting the world soften us, we let it drive
28、us deeper into ourselves. We try to deflect the hurt and pain by pretending it doesnt exist, but although we can try this all we want, in the end, we cant hide from ourselves. We need to learn to open our hearts to the potentials of life and let the world soften us.生活发生不幸时，我们常常会关上心门；世界不仅没能慰藉我们，反倒使我们
29、更加消沉。我们假装一切仿佛都不曾发生，以此试图忘却伤痛，可就算隐藏得再好，最终也还是骗不了自己。既然如此，何不尝试打开心门，拥抱生活中的各种可能，让世界感化我们呢？ Whenever we start to let our fears and seriousness get the best of us, we should take a step back and re-evaluate our behavior. The items listed below are six ways you can open your heart more fully and completely.当恐惧
30、与焦虑来袭时，我们应该退后一步，重新反思自己的言行。下面六个方法有助于你更完满透彻地敞开心扉。Whenever a painful situation arises in your life, try to embrace it instead of running away or trying to mask the hurt. When the sadness strikes, take a deep breath and lean into it. When we run away from sadness thats unfolding in our lives, it gets st
31、ronger and more real. We take an emotion thats fleeting and make it a solid event, instead of something that passes through us.当生活中出现痛苦的事情时，别再逃跑或隐藏痛苦，试着拥抱它吧；当悲伤来袭时，试着深呼吸，然后直面它。如果我们一味逃避生活中的悲伤，悲伤只会变得更强烈更真实悲伤原本只是稍纵即逝的情绪，我们却固执地耿耿于怀By utilizing our breath we soften our experiences. If we dam them up, our lives will stagnate, but when we keep them flowing, we allow more newness and greater experiences to blossom.深呼吸能减缓我们的感受。屏住呼吸，生活停滞；呼出呼吸，更多新奇与经历又将拉开序幕。
