23-24高二下·全国·课前预习
1 . 知识点三 函数图象的变化趋势与导数的绝对值的大小的关系
一般地,设函数
,在区间
上:
一般地,设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c7028a5fa4d781d382ca3b73b74796e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5296b39d62215c77a923ed5674b10b01.png)
导数的绝对值 | 函数值变化 | 函数的图象 |
越大 | 比较“ | |
越小 | 比较“ |
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23-24高二下·全国·课前预习
2 . 知识点一 函数的单调性与其导数的正负之间的关系
定义在区间
内的函数
:
定义在区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![]() | ![]() |
![]() | 单调递 |
![]() | 单调递 |
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23-24高二下·全国·课前预习
3 . 全排列
(1)把n个不同的元素全部取出的一个排列,叫做n个元素的一个全排列,此时
,即有
,叫做n的阶乘,用______ 表示.
(2)性质:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/358164c251a0c14bea09aa6584fefc3c.png)
______ ,规定![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cfd8cf58a5571be8398ad43428ffa3a.png)
______ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6160f054fa94356b64157f16c7289037.png)
______ .
(1)把n个不同的元素全部取出的一个排列,叫做n个元素的一个全排列,此时
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0a4480988244a9d04ec293975db2cc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d390b17b67696d546e5426b4736b4ac5.png)
(2)性质:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/358164c251a0c14bea09aa6584fefc3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cfd8cf58a5571be8398ad43428ffa3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6160f054fa94356b64157f16c7289037.png)
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23-24高二下·全国·课前预习
4 . 知识点五 导函数的定义
从求函数
在
处导数的过程可以看出,当
时,
是一个唯一确定的数.这样,当x变化时,
就是x的函数,我们称它为
的________ (简称导数).
的导函数记作________ 或________ ,即![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9be1c518c1663eee81b4d13c8db456a.png)
.
从求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16515ca05229fe341868d8c23d9f2642.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d31f9ce464f2ce3b24833b70595941c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9be1c518c1663eee81b4d13c8db456a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4dcfdf0e8025c212c98808075d8afad.png)
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23-24高二下·全国·课前预习
5 . 导数的几何意义
函数
在点
处的导数的几何意义是曲线
在点
处的切线的斜率.也就是说,曲线
在点
处的切线的斜率是________ 相应地,切线方程为________ .
函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25be20e3724274132cb83b16deaeecfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25be20e3724274132cb83b16deaeecfc.png)
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23-24高二下·全国·课前预习
6 . 割线斜率与切线斜率
设函数
的图象如图所示,直线AB是过点
与点
的一条割线,此割线的斜率是![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85d193fe41eabf97168e5f5016324b7a.png)
________ .于是,当Δx→0时,割线AB的斜率无限趋近于过点A的切线AD的斜率k,即k=________ =
设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c2cec274d055a711936934698b13997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/905ac6407cb466d2434da38a7c5bc058.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85d193fe41eabf97168e5f5016324b7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8c1639ce321a0e100bdc793f61df3c3.png)
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23-24高二下·全国·课前预习
7 . 知识点三 函数在某点处的导数
如果当Δx→0时,平均变化率
无限趋近于一个确定的值,即
有极限,则称
在
处可导,并把这个确定的值叫做
在
处的导数(也称为瞬时变化率),记作________ ,即
=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6964df77cd765effefff5e6405c90098.png)
=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6964df77cd765effefff5e6405c90098.png)
.
如果当Δx→0时,平均变化率
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7688109e1a422042e8ce925007582a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7688109e1a422042e8ce925007582a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c7028a5fa4d781d382ca3b73b74796e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a12609054bd6ea75c0ab3214bb2e0b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c7028a5fa4d781d382ca3b73b74796e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a12609054bd6ea75c0ab3214bb2e0b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16515ca05229fe341868d8c23d9f2642.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6964df77cd765effefff5e6405c90098.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7688109e1a422042e8ce925007582a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6964df77cd765effefff5e6405c90098.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/504633259a36f33bb8e323620998b675.png)
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23-24高二下·全国·课前预习
8 . 知识点二 函数的平均变化率
对于函数
,设自变量x从
变化到
,相应地,函数值y就从
变化到
.这时,x的变化量为
,y的变化量为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03d9fb03d75c63e426c3c169d9844b7f.png)
________ 我们把比值
,即![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6989eef65f4476379f1a16f30cf5b1.png)
叫做函数
从
到
的平均变化率.
对于函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47db6b322f19cefcd70d0f8433917f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7417645b760b0e03cfe0bcdaa6a1d93e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d807ae26a784ed501d17c49adb96750.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1268e217016ff7e12b9bc51341c4cde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03d9fb03d75c63e426c3c169d9844b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1fde86bac61c7b0b0698580f7675a1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6989eef65f4476379f1a16f30cf5b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f29cd82a9de670075d0ac5727e52b3e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47db6b322f19cefcd70d0f8433917f0.png)
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23-24高二下·全国·课前预习
9 . 知识点一 瞬时速度
瞬时速度的定义
(1)物体在________ 的速度称为瞬时速度.
(2)一般地,设物体的运动规律是
,则物体在
到
这段时间内的平均速度为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cdb783174b72ceae5100dc051c4d925.png)
.如果
无限趋近于0时,
无限趋近于某个常数v,我们就说当
无限趋近于0时,
的________ 是v,这时v就是物体在时刻
时的瞬时速度,即瞬时速度![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8354e2efafc38e2f2e4f0f92dd89d618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cdb783174b72ceae5100dc051c4d925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/793345c6e43ebc1e9e7037421d4dc9d8.png)
.
瞬时速度的定义
(1)物体在
(2)一般地,设物体的运动规律是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e418a26507298f6829feec0d07def04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d9fd58e71dcae6cafaf9037d20ebd76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f19c2d30ba888aa60e5b05534cf32014.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cdb783174b72ceae5100dc051c4d925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af222fdff32ce2bcb3992a60ef694a9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ec106b92bc77e6716692a61a15a0d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9f58827347f7739452efeff88902307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ec106b92bc77e6716692a61a15a0d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9f58827347f7739452efeff88902307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82120861c1c4f7cc1a7a3f169f082a81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8354e2efafc38e2f2e4f0f92dd89d618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cdb783174b72ceae5100dc051c4d925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/793345c6e43ebc1e9e7037421d4dc9d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af222fdff32ce2bcb3992a60ef694a9f.png)
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