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wwwxpj33050com:2015人教版高中数学选修4-5学案:1.2.1绝对值不等式

选修 4-5 学案 1.2.1 绝对值不等式 ☆学习目标: 1.对深化绝对值的定义及其几何意义的理解和掌握; 2. 理解关于绝对值三角不等式并会简单应用 ? 知识情景: 定理 3 如果 a, b, c ? R ? , 那么 定理 3 的国语表述: 推论:对于 n 个正数 a1 , a2 ,?, an , 它们的 即 当且仅当 a ? b ? c 时,等号成立.

a?b?c 3 ? abc , 当且仅当 a ? b ? c 时, 等号成立. 3

☆探究:许多不等关系都涉及到距离的长短、面积或体积的大小、重量,等等,它们都要通过 非负数来表示.因此,研究含有绝对值的不等式具有重要大的意义. ? 建构新知:
? 1.绝对值的定义: ?a ? R , | a |? ? ? ? ?

2. 绝对值的几何意义: ⑴实数 a 的绝对值 | a | ,表示数轴上坐标为 a 的点 A

⑵ ? 两个实数 a , b ,它们在数轴上对应的点分别为 A, B , 那么 | a ? b | 的几何意义是 例 1 设函数 f ( x) ? x ?1 ? x ? 4 .

?1? 解不等式 f ( x) ? 2 ; ? 2 ? 求函数 y ?

f ( x) 的最值.

2. 绝对值三角不等式:探究 | a | , | b | , | a ? b | 之间的关系. ① a ? b ? 0 时,如下图, 容易得: | a ? b |

| a| ?|b|.

-1-

② a ? b ? 0 时,如图, 容易得: | a ? b |

| a| ?|b|.

③ a ? b ? 0 时,显然有: | a ? b | 综上,得 定理 1 如果 a, b ? R , 那么 | a ? b |

| a| ?|b|.

| a | ? | b | . 当且仅当

时, 等号成立.

在上面不等式中,用向量 a, b 分别替换实数 a , b , 则当 a, b 不共线时, 由向量加法三角形法则: 向量 a, b , a ? b 构成三角形, 因此有 | a ? b | 它的几何意义就是:

? ?

? ?

? ? ? ?

| a| ?|b|

定理 2 如果 a, b, c ? R , 那么

| a ?c |

| a ? b | ? | b ? c | . 当且仅当

时, 等号成立.

☆案例学习: 例 2、 ⑴已知 x ? a ? 2 , y ? b ? 2 ,求证 ( x ? y) ? (a ? b) ? c. ,
c c

⑵已知 ? ? 0, x ? a ? ? , y ? b ? ? ,求证: 2x ? 3 y ? 2a ? 3b ? 5? 。

-2-


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