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1.3.1 函数的单调性与导数_图文


边城高级中学 张秀洲

1、理解导数与函数的单调性的关系.(易混点)

2、掌握利用导数判断函数单调性的方法.(重点)
3、会用导数求函数的单调区间.

自学教材 P23—P26 解决下列问题
一、掌握利用导数判断函数单调性的方法.
二、会用导数求函数的单调区间. 三、教材P26 练习.

y

y ? f ? x?

? x , f ? x ??
1 1

? x , f ? x ??
0 0

O

x

如图, 导数f ?( x0 )表示函数f ( x )在点( x0 , f ( x0 ))处的切线的斜率. 在 x ? x0 处, f ?( x0 ) ? 0, 切线是 "左下右上 "式的, 这时,函数f ( x )在 x0 附近单调递增; 在x ? x1处, f ' ( x1 ) ? 0, 切线是 "左上右下 "式的, 这时,函数f ( x )在x1附 近单调递减.

一般地,函数的单调性与导数的正负有如下关系 :
在某个区间(a , b)内, 如果f ( x ) ? 0, 那么函数y ? f ( x )在这个区间内单调递增;
' '

如果f ( x ) ? 0, 那么函数 y ? f ( x )在这个区间内单调递减.
如果在某个区间内恒有f ' ( x ) ? 0, 那么函数y ? f ( x )有什么特征?

思考 : 请同学们回顾一下函数单调性的定义, 并思考 某个区间上函数 y ? f ( x )的平均变化率的几何意义 与其导数正负的关系 .

类型1:导数与单调性的关系
已知导数f ' ( x )的下列信息 : 当1 ? x ? 4时, f ' ( x ) ? 0; 当x ? 4, 或x ? 1时, f ' ( x ) ? 0; 当x ? 4, 或x ? 1时, f ' ( x ) ? 0. 试画出函数f ( x )图象的大致形状.

y

O

1

4

x

解 当1 ? x ? 4时, f ' ( x ) ? 0, 可知f ( x )在此区间内单调递增;
当x ? 4, 或x ? 1时, f ' ( x ) ? 0, 可知f ( x )在这两个区间内 单调递减 ;

当x ? 4, 或x ? 1时, f ' ( x ) ? 0, 这两点比较特殊, 我们称 它们为"临界点". 综上所述,函数f ( x )图象的大致形状如图所示.

如图, 水以恒速(即单位时间内注入水的体积相同 )注入 下面四种底面积相同的容器中, 请分别找出与各容器对应的 高度h与时间t的函数关系图象.

? 1?
h h

? 2?
h

? 3?
h

? 4?

o

? A?

t

o

? B?

t

o

?C ?

t

o

? D?

t

分析 以容器 ? 2 ? 为例 ,由于容器上细下粗, 所以水 以恒速注入时, 开始阶段高度增加得慢,以后高度 增加得越来越快.反映在图象上, ? A? 符合上述变 化情况. 同理可知其他三种容器的情况.

h

?2?

? 1? ? ? B ? , ? 2 ? ? ? A ? , ? 3? ? ? D ? , ? 4? ? ? C ? .

o

? A?

t

思考 例 3 表明, 通过函数图象 , 不仅可以看出函数的增 与减 , 还可以看出其增减的快慢.结合图象, 你能从导数的 角度解释增减快慢的情况吗 ?

1.利用导数判断函数的单调性比利用函数单调性的定义简 单的多,只需判断导数在该区间内的正负即可.

2.通过图象研究函数单调性的方法:
(1)观察原函数的图象重在找出 “上升”“下降”产生变化 的点,分析函数值的变化趋势; (2)观察导函数的图象重在找出导函数图象与 x 轴的交点, 分析导数的正负.

设 f′(x)是函数 f(x)的导函数,y=f′(x)的图象如图所示, 则 y=f(x)的图象最有可能的是( C )

【解析】 由 f′(x)的图象知, 当 x∈(-∞, 0)∪ (2, +∞)时, f′(x)>0, x∈ (0,2)时, f′(x)<0,则函数 f(x)在区间(-∞,0)与 (2,+∞)上 是增函数,在区间(0,2)上是减函数,对照选项知应选 C.

类型2:利用导数研究函数的单调性
判断下列函数的单调性, 并求出单调区间 : (1)( f x) ? x 3 ? 3 x; (2)( f x) ? x2 ? 2x ? 3 ; (3)( f x) ? sin x ? x , x ? (0, ?) ; (4)( f x) ? 2 x 3 ? 3 x 2 ? 24 x ? 1.

y

( f x) ? x3 ? 3 x
o

解 (1)因为f ( x ) ? x 3 ? 3 x , 所以f ' ( x ) ? 3 x 2 ? 3 ? 3( x 2 ? 1) ? 0.

x

因此,函数 f ( x ) ? x 3 ? 3 x 在 x ? R 上单调递增 , 如图.

2 ' 2 因为 f ( x ) ? x ? 2 x ? 3, 所以 f ( x ) ? 2 x ? 2 ? 2( x ? 1). ? ?

当f ' ( x ) ? 0,即x ? 1时,函数f ( x) ? x 2 ? 2 x ? 3单调递增; 当f ' ( x ) ? 0,即x ? 1时,函数f ( x ) ? x 2 ? 2 x ? 3单调递减 . 函数f ( x ) ? x 2 ? 2 x ? 3的图象如图所示. y
f ? x ? ? x2 ? 2x ? 3
o 1

x

y

o

?

x
f ? x ? ? sin x ? x

' 3 因为 f ( x ) ? sin x ? x , x ? (0, ? ), 所以 f ( x) ? cos x ? 1 . ? ?

因此,函数f ? x ? ? sin x ? x, x ? ? 0, ? ? 内 单调递减 . 如图所示.

(4)因为f ( x) ? 2 x3 ? 3 x2 ? 24 x ? 1, 所以f ' ( x) ? 6 x 2 ? 6 x ? 24 .

当f ( x) ? 0,即
' '

时,函数f ( x) 时,函数f ( x)

; .

当f ( x) ? 0,即
y

f ( x) ? 2 x 3 ? 3 x 2 ? 24 x ? 1的图象如图所示.
如果不用导数的方法,
f ( x ) ? 2 x 3 ? 3 x 2 ? 24 x ? 1

直接运用单调性的定 义, 你如何求解本题 ?
x

5
O

1

运算过程 麻烦吗 ? 你 有什么体会 ?

利用导数研究函数单调性的方法:
第一步:求定义域,对函数求导; 第二步:解导数等于 0 时的方程; 第三步: 导数大于 0 的区间与定义域求交集为增区间, 小于 0 的区间与定义域求交集为减区间,即“正增负减”.

三、教材P26 练习.

2018年5月31日星期四

1 2 讨论函数 f(x)= ax +x-(a+1)ln x(a≥0)的单调性. 2

【解答】 函数 f(x)的定义域为(0,+∞), a+1 ax2+x-?a+1? f′(x)=ax+1- = , x x x-1 (1)当 a=0 时,f′(x)= , x 由 f′(x)>0,得 x>1;由 f′(x)<0,得 0<x<1. 所以,f(x)在(0,1)内为减函数,在(1,+∞)内为增函数;

(2)当 a>0 时,f′(x)=

? a+1? ? ? a?x+ ??x-1? a ? ?

x



a+1 因为 a>0,所以- <0, a 由 f′(x)>0,得 x>1;由 f′(x)<0,得 0<x<1. 所以,f(x)在 (0,1)内为减函数,在(1,+∞)内为增函数. 综上所述,a≥0 时 f(x)在 (0,1)内为减函数; 在(1,+∞)内为增函数.

本题易忽略a=0的情况而致错,同时,求函数的单调 性一定要注意函数的定义域.

设函数 f(x)=ln x-2ax. (1)若函数 y=f(x)的图象在点(1,f(1))处的切线为直线 l,且 直线 l 与圆(x+1)2+y2=1 相切,求 a 的值; (2)当 a>0 时,求函数 f(x)的单调区间.

【解】

1 (1)依题意有,f′(x)= -2a. x

因此 y=f(x)在点(1,f(1))处的切线的斜率为 1-2a, 又 f(1)=-2a, 所以,y=f(x)在点(1,f(1))处的切线方程为 y+2a=(1-2a)(x-1), 即(2a-1)x+y+1=0. 又已知圆的圆心为(-1,0),半径为 1, |1-2a+1| 依题意, =1, 2 ?2a-1? +1 1 解得 a= . 2

(2)由题意知 f(x)=ln x-2ax 的定义域为(0,+∞), 1 又 f′(x)= -2a, x 1 因为 a>0,x>0,令 -2a>0,则 1-2ax>0. x 所以在 在
? 1? ? x∈?0, ? ?时,f(x)=ln 2 a ? ?

x-2ax 是增函数;

?1 ? ? x∈? ,+∞? ?时,f(x)=ln 2 a ? ?

x-2ax 是减函数.

所以当 a>0 时,函数 函数

? 1? ? f(x)的单调增区间是?0, ? ?, 2 a ? ?

?1 ? ? f(x)的单调减区间是? ,+∞? ?. 2 a ? ?

类型3:利用函数单调性证明不等式

证明:e ? 1 ? x, x ? 0
x

【证明】设 f ( x ) ? e x ? 1 ? x , x ? 0
f ?( x ) ? e x ? 1, x ? 0
?当 x ? 0 时, f ?( x ) ? e x ? 1 ? 0 , f ( x )单调递增,

f ( x ) ? e x ? 1 ? x ? f (0) ? 0

当 x ? 0 时, f ?( x ) ? e x ? 1 ? 0 , f ( x )单调递减,
f ( x ) ? e x ? 1 ? x ? f (0) ? 0

综上所述, e x ? 1 ? x ? 0,即 e x ? 1 ? x

你学会了吗?

※对自己说,你有什么收获?
※对同学说,你有什么提示? ※对老师说,你有什么疑惑?

2018年5月31日星期

1.求函数 f(x)的单调区间时,先确定函数的定义域,在定义 域内通过解 f′(x)>0 或 f′(x)<0 得到,两个单调性相同的区间, 不能用并集符号连接. 2. 已知函数 f(x)在某个区间上的单调性求参数的取值范围时, 可转化为 f′(x)≥0 或 f′(x)≤0 恒成立问题,并注意验证等号成 立时是否符合题意.

2018年5月31日 1次
必做题:《教材》 P31 A组第1(2)、2(2)(4)题

选做题:《基础训练》 变式训练3
《教材》 P32 B组第1(1)题 【预习】课本P27-P29《函数的极值与导数》

类型4:已知函数的单调性求参数的范围
a 已知函数 f(x)=x + (x≠0,常数 a∈R),若函数 f(x) x
2

在 x∈[2,+∞)上是单调递增的,求 a 的取值范围.

【解答】

3 a 2x - a f′(x)=2x- 2= 2 . x x

要使 f(x)在[2, +∞)上是单调递增的, 则 f′(x)≥0 在 x∈[2, +∞)时恒成立, 2x3-a 即 2 ≥0 在 x∈[2,+∞)时恒成立. x ∵x2>0,∴2x3-a≥0, ∴a≤2x3 在 x∈[2,+∞)上恒成立. ∴a≤(2x3)min.

∵x∈[2,+∞),y=2x3 是单调递增的, ∴(2x3)min=16,∴a≤16. 2x3-16 当 a=16 时, f′(x)= ≥0(x∈[2, +∞))有且只有 f′(2)=0, 2 x ∴a 的取值范围是(-∞,16].

已知 f(x)在区间(a,b)上的单调性,求参数范围的方法: (1)利用集合的包含关系处理:f(x)在(a,b)上单调递增(减),则 区间(a,b)是相应单调区间的子集; (2)利用不等式的恒成立处理:f(x)在(a,b)上单调递增(减),则 f′(x)≥0(f′(x)≤0)在(a,b)内恒成立,注意验证等号是否成立.

若存在正数 x 使 2x(x-a)<1 成立, 则 a 的取值范围是( D ) A.(-∞,+∞) B.(-2,+∞) C.(0,+∞) D.(-1,+∞)

1 【解析】 ∵2 (x-a)<1,∴a>x- x. 2
x

1 令 f(x)=x- x,∴f′(x)=1+2-xln 2>0. 2 ∴f(x)在(0,+∞)上单调递增, ∴f(x)>f(0)=0-1=-1, ∴a 的取值范围为(-1,+∞),故选 D.


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