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高中数学第三章导数及其应用3.3导数的应用3.3.3导数的实际应用课前引导素材新人教B版选修1_1

3.3.3 导数的实际应用 课前导引 问题导入 已知一扇形的周长为 c(c>0),当扇形的弧长为何值时,它有最大面积?并求出面积的最大 值. 思路分析:设扇形的半径为 R,弧长为 x,面积为 S,则 c?x ,(x<c). 2 1 1 c?x 于是 S= Rx= ? ·x 2 2 2 1 2 = (cx-x ). 4 1 c c c 由于 S′= (c-2x)=0 只有一根 x= ,又当 x< 时,S′>0.当 x> 时,S′<0.这样当 2 2 4 2 c=2R+x,∴R= x= c2 c 时,面积取得最大值 . 2 16 知识预览 1.生活中经常遇到求________、________、________等问题,这些问题通常称为优化问题. 答案:利润最大 用料最省 效率最高 2. 利 用 导 数 解 决 优 化 问 题 的 实 质 是 _______________________________________________________________________________ _________. 答案:将实际问题转化为函数模型,进而运用导数求得函数的最值 1

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