高一数学必修导数的概念.ppt

1.1.2 导数的概念 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 问题2 高台跳水 在高台跳水运动中,运动员相对于水面的高 度h(单位：米)与起跳后的时间t（单位：秒） 存在函数关系 h(t)=-4.9t2+6.5t+10. 如何用运动员在某些时 间段内的平均速度粗略 地描述其运动状态? h t o Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 瞬时速度. 在高台跳水运动中,平均速度不能准确反 映他在这段时间里运动状态. 又如何求 瞬时速度呢? 我们把物体在某一时刻的速度称为瞬时速度 . Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 如何求（比如， t=2时的）瞬时速度？ 通过列表看出平均速度的变化趋势 ： 当t趋近于0时,平均 速度有什么变化趋势? Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 瞬时速度 我们用 表示 “当t=2, t趋近于0时,平均速度趋于确 定值-13.1”. 那么,运动员在某一时刻t0的瞬时速度? 局部以匀速代替变速，以平均速度代替瞬时速度，然后通过 取极限，从瞬时速度的近似值过渡到瞬时速度的精确值。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 导数的定义: 从函数y=f(x)在x=x0处的瞬时变化率是: Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 问题: 求函数y=3x2在x=1处的导数. 分析：先求f=y=f(x)-f() =6x+(x)2 再求 再求 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 应用： 例1 物体作自由落体运动,运动方程为： 其 中位 移单位是m,时间单位是s,g=10m/s2.求： (1) 物体在时间区间2,2.1上的平均速度； (2) 物体在时间区间2,2.01上的平均速度； (3) 物体在t=2(s)时的瞬时速度. 分析 : Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 解: (1)将 t=0.1代入上式，得: (2)将 t=0.01代入上式，得: Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 应用： 例2 将原油精练为汽油、柴油、塑胶等各种不 同产品，需要对原油进行冷却和加热。如果第 x(h)时，原油的温度（单位：0C）为 f(x)=x2-7x+15(0x8).计算第2（h) 和第6（ h）时，原由温度的瞬时变化率，并说明它们的 意义。 关键是求出： 它说明在第2（h)附近，原油 温度大约以3 0C/h的速度下降 ；在第6（h)附近，原油温度 大约以5 0C/H的速度上升。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 应用： 例3质量为kg的物体，按照 s(t)=3t2+t+4的规律做直线运动， （）求运动开始后s时物体的瞬时速度； （）求运动开始后s时物体的动能。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 小结： 1求物体运动的瞬时速度： （1）求位移增量s=s(t+t)-s(t) (2)求平均速度 （3）求极限 1由导数的定义可得求导数的一般步骤： （1）求函数的增量y=f(x0+t)-f(x0) (2)求平均变化率 （3）求极限 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 练习： (1)求函数y= 在x=1处的导数. (2)求函数y= 的导数. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 作业： 课本10页 A : 2，3。 Eva