解题方法
1 . 1903年前苏联(俄罗斯)航天之父齐奥尔科夫斯基推导出火箭的理想速度公式为:
,其中
为火箭的初始质量,
为火箭燃烧完毕熄火后的剩余质量,
称为火箭的质量比,
为火箭的发动机的喷气速度.100多年来,所有的大小火箭都遵循齐奥尔科夫斯基公式的基本规律.已知某型号火箭的发动机的喷气速度为第一宇宙速度7900m/s.
(1)当该型号火箭的质量比为10时,求该型号火箭的理想速度;
(2)经过材料更新和技术改进后,某型号火箭的发动机的喷气速度提高到了原来的2倍,质量比缩小为原来的
,若要使火箭的理想速度至少增加3950m/s.求在材料更新和技术改进前质量比的最小整数值,参考数据:
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1076848dada14ff109fb834e3dfb8d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5e86a882ef57f44f0ad22836079afe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be5fa4b8d343adfa41504daeacd80d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
(1)当该型号火箭的质量比为10时,求该型号火箭的理想速度;
(2)经过材料更新和技术改进后,某型号火箭的发动机的喷气速度提高到了原来的2倍,质量比缩小为原来的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a348b6d1ee170a4150eb059a7433cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ec4f8b1590946700c7ad33415460705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb92b2f1b067084b3eb3103bb1353520.png)
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名校
2 . 对数对大数据运算具有独特优势,法国著名天文学家拉普拉斯曾说:“对数,可以缩短计算时间使天文学家的寿命翻倍,所有天文学家都应该感谢对数的发现”.现有一大数据
,用科学记数法可表示为
,其中
,已知
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb7a8804c5ce9bbf8d9eed77e0bb2352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d43ce7c852ffd312d4c87c17e6c4845.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ff4830bcb25f9c9fb380e0a6302c840.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b979a5218bb3d73e38dd12229f48025.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3cb8d72bb2e281b943b3b430138ef7.png)
A.953 | B.954 | C.955 | D.956 |
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3 . 数学中有许多美丽的错误,法国数学家费马通过观察计算曾提出猜想:形如![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c3365590c38e541c3b9200e6bf0fe0c.png)
的数都是质数,这就是费马素数猜想.半个世纪后善于发现的欧拉算出第5个费马数不是质数,从而否定了这一种猜想.现设:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcba8b34ba2649cfa3d9392dd7fb1617.png)
,
为常数,
表示数列
的前
项和,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5940dcec52d61d8c33ab9e2c1e6b47.png)
_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c3365590c38e541c3b9200e6bf0fe0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65446775b23351f5973a89dbf7973e98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcba8b34ba2649cfa3d9392dd7fb1617.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8fce31fc8cef06de4036231dbf11e84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab4717e4827480f0f6f4ded85e52eab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ce82db19a6dcb76aedb107203cf425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5940dcec52d61d8c33ab9e2c1e6b47.png)
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4 . 若函数f(x)的导数
存在导数,记
的导数为
.如果f(x)对任意x∈(a,b),都有
成立,则f(x)有如下性质:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b64b2549a849fb225f1e7b7592a74424.png)
.其中n∈N*,x1,x2,…,xn∈(a,b).若f(x)=lnx,则
=___________ ;根据上述性质推断:当x1+x2+x3=3e且x1,x2,x3∈(0,+∞)时,根据上述性质推断:
的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2495c66dd202153fcdad0e2a34abf50c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2495c66dd202153fcdad0e2a34abf50c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aac282e92da3691942a6ba8511de2303.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfceb27b2073133b327f7d2c28645b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b64b2549a849fb225f1e7b7592a74424.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e8574bb168aee51d7a9773c1b12fd21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aac282e92da3691942a6ba8511de2303.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9519a8872ccbd9029425c8e6f981fe24.png)
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