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“ |
您最近一年使用:0次
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 |
您最近一年使用:0次
23-24高二下·全国·课前预习
3 . 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)
您最近一年使用:0次
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)
您最近一年使用:0次
23-24高二下·全国·课前预习
5 . 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)
您最近一年使用:0次
23-24高二下·全国·课前预习
6 . 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)
您最近一年使用:0次
23-24高二下·全国·课前预习
7 . 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)
您最近一年使用:0次
23-24高二下·全国·课前预习
8 . 知识点三 导数的运算法则
已知
,
为可导函数,且
.
(1)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102f5fa5fc56cc91b36b256653dbd307.png)
______ .
(2)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3788a9fea4a60d4e107eaa0a41eae60a.png)
______ ,特别地,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f742e4258a9b3ae1ba4967d7f34e6fcb.png)
______ .
(3)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bae54ea79b90e1710336975ad63f55cd.png)
______ .
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3f1a17ec04e606e9159f7a22b144008.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102f5fa5fc56cc91b36b256653dbd307.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3788a9fea4a60d4e107eaa0a41eae60a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f742e4258a9b3ae1ba4967d7f34e6fcb.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bae54ea79b90e1710336975ad63f55cd.png)
您最近一年使用:0次
23-24高二下·全国·课前预习
9 . 知识点01等比数列的概念
1、等比数列的定义
如果一个数列从第2项起,_______ 等于同一常数,那么这个数列叫做等比数列,这个常数叫做等比数列的_______ ,通常用字母_______ 表示
.
2、对等比数列概念的理解
(1)“从第2项起”,是因为首项没有“前一项”,同时注意公比是每一项与前一项的比,前后次序不能颠倒,另外等比数列中至少含有三项;
(2)定义中的“同一常数”是定义的核心之一,一定不能把“同”字省略,这是因为如果一个数列从第2项起,每一项与它的前一项的比都是一个与
无关的常数,但是如果这些常数不相同,那么此数列也不是等比数列,当且仅当这些常数相同时,数列才是等比数列;
(3)若一个数列不是从第2项起,而是从第3项起或第
项起,每一项与它的前一项的比等于同一常数,则此数列不是等比数列;
(4)由定义可知,等比数列的任一项都不为0,且公比
;
(5)不为0的常数列是特殊的等比数列,其公比为1.
1、等比数列的定义
如果一个数列从第2项起,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455931a43e525371730e61177c48aec6.png)
2、对等比数列概念的理解
(1)“从第2项起”,是因为首项没有“前一项”,同时注意公比是每一项与前一项的比,前后次序不能颠倒,另外等比数列中至少含有三项;
(2)定义中的“同一常数”是定义的核心之一,一定不能把“同”字省略,这是因为如果一个数列从第2项起,每一项与它的前一项的比都是一个与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若一个数列不是从第2项起,而是从第3项起或第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bccd5c35461ea19c93e24f80e8538f2d.png)
(4)由定义可知,等比数列的任一项都不为0,且公比
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3ac83c571110d41a396d12d8eea1c6.png)
(5)不为0的常数列是特殊的等比数列,其公比为1.
您最近一年使用:0次
23-24高二下·全国·课前预习
10 . 知识点04等比中项
1、等比中项定义:如果在
与
中间插入一个数
,使
成等比数列,那么
叫做
与
的_______ ,即
是
与
的等比中项
成等比数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
_______
2、对等比中项概念的理解
(1)
是
与
的等比中项,则
与
的符号相同,符号相反的两个实数不存在等比中项.此时,
,即等比中项有两个,且互为相反数.
(2)
时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
_______ 是
与
的等比中项.例如
,但
不是等比数列;
(3)在等比数列
中,从第2项起,每一项是它相邻两项的等比中项;
(4)与等比数列中的任一项“等距离”的两项之积等于该项的平方,即在等比数列
中,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6169cb2b9f2a373abc0cadecadd4f2.png)
3、等差中项与等比中项区别
(1)任意两数都存在等差中项,但并不是任意两数都存在等比中项,当且仅当两数同号且均不为0时才存在等比中项;
(2)任意两数的等差中项是______ 的,而若两数有等比中项,则等比中项______ .
1、等比中项定义:如果在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6703df340de9d28c32832badbd30f22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83ab5539817e40ffaf20a517e0978b80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
2、对等比中项概念的理解
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede7e2c31ca68ce700cffa87764dc484.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1c8379fe535e68721fd84be969d257f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce438ef49c36ad7b8a27e918137e9ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419abd403ca442c5aadd04165fc9a528.png)
(3)在等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(4)与等比数列中的任一项“等距离”的两项之积等于该项的平方,即在等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6169cb2b9f2a373abc0cadecadd4f2.png)
3、等差中项与等比中项区别
(1)任意两数都存在等差中项,但并不是任意两数都存在等比中项,当且仅当两数同号且均不为0时才存在等比中项;
(2)任意两数的等差中项是
您最近一年使用:0次