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www.191msc.com:高中数学选修1-1(北师版)第四章导数的应用4.2(与最新教材完全匹配)知识点总结含同步练习题及答案

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高中数学选修1-1(北师版)知识点总结含同步练习题及答案
第四章 导数应用 4.2 导数在实际问题中的应用

一、知识清单
利用导数处理生活中的优化问 题

二、知识讲解
1.利用导数处理生活中的优化问题 描述: 生活中经常遇到求利润最大、用料最省、效率最高等问题,这些问题通常称为优化问题.

例题: 货车欲以 x km/h 的速度行驶到 130 km 远的某地.按照交通法则,车辆行驶速度的允许范围 是 50 km ? 100 km.假设汽油的价格为 2 元/L ,而汽车耗油的速度是 (2 + 机的工资是 14元/h ,试问最经济的车速是多少?这次行车的总费用最低是多少?

x2 ) L/h ,司 360

130 130 x2 h ,耗油量为 ? (2 + ) L ,耗油费用为 x x 360 130 130 x2 元. 2? ? (2 + ) 元,司机的工资为 14 ? x 360 x
解:汽车行驶的时间为 故这次行车的总费用为

y =2?
所以

130 130 x 18 x2 ? (2 + ) + 14 ? = 130( + )元, x 360 x 180 x 1 18 ? ). 180 x2

y ′ = 130(


y ′ = 0,


? ? ? x = 18√? 10 或x = ?18√10 (舍去). ? ≈ 57 km/h 时,y 取得最小值.故最经济的车速约为 因为 50 ? x ? 100,所以当 x = 18√? 10 57 km/h,最低费用为 130 × ( ? 18√? 18 10 + ? ) ≈ 82.2 元. 180 18√? 10

某单位用木料制作如图所示的框架,框架的下部是边长分别为 x、y (单位:m )的矩形,上部 是等腰直角三角形,要求框架围成的总面积为 8 m 2 ,问 x、y 分别为多少时用料最省(精确到 0.001 m)?

解:依题意,有

xy +
所以

1 x x ? = 8, 2 2

y=
于是框架用料总长度为

8 x ? (0 < x < 4√2 ), x 4

3 16 √2 x = ( + √2 )x + . 2 2 x 3 16 L ′ = + √2 ? 2 , 2 x L = 2x + 2y + 2 ?
令 L ′ = 0 ,即

3 16 + √2 ? = 0, 2 x2
解得

x1 = 8 ? 4√2 ,x2 = 4√2 ? 8(舍去),
当 0 < x < 8 ? 4√2 时,

L ′ < 0;
当 8 ? 4√2 < x < 4√2 时,

L ′ > 0.
所以当 x = 8 ? 4√2 时,L 取得最小值,此时

x = 8 ? 4√2 ≈ 2.343 m,y = 2.828 m.
即当 x 为 2.343 m ,y 为 2.828 m 时,用料最省. 如图,有一块半椭圆形的钢板,其长半轴长为 2r ,短半轴长为 r ,计划将此钢板切割成等腰梯 形的形状,下底 AB 是半椭圆的短轴,上底 CD 的端点在椭圆上.记 CD = 2x,梯形面积为 S. (1)求面积 S 以 x 为自变量的函数式,并写出定义域; (2)求面积 S 的最大值.

解:(1)依题意,以 AB 的中点 O 为原点建立直角坐标系 O ? xy(如图),

则点 C 的横坐标为 x .点 C 的纵坐标 y 满足方程

解得 y = 2√r2 ? x 2 (0 < x < r) .

? ? ? ? ? ?

y2 x2 + = 1(y ? 0), 4r2 r2 ? ? ? ? ? ? ? ? ? ? ? ? 1 (2x + 2r) ? 2√r2 ? x2 = 2(x + r) ? √r2 ? x2 , 2

S=

其定义域为 {x | 0 < x < r}. (2)记

f (x) = 4(x + r)2 (r2 ? x2 ),0 < x < r,
则 f ′ (x) = 8(x + r)2 (r ? 2x).令

f ′ (x) = 0,
得 x=

1 r 时, r.因为当 0 < x < 2 2 f ′ (x) > 0;



r < x < r 时, 2 f ′ (x) < 0,

r r 时,S 也取得最大值,最大值为 ) 是 f (x) 的最大值.因为,当 x = 2 2 ? ? ? ? ? 3√3 2 3√3 2 r r ,即梯形面积 S 的最大值为 r . √f ( ) = 2 2 2
所以 f (

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