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高等数学笔记(22)

2014-02-06 20:15 浏览: 1418772 次 我要评论(0 条) 字号:

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【前言】
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【正文】

第3章 导数与微分

(1)由于自变量(x)的变化引起函数(y = f(x))变化的“快慢”问题——函数的变化率/导数。
(2)由于自变量的微小改变(增量(Delta x)很小时)引起(y = f(x))的改变量(Delta y)的近似值问题——微分问题。
(3)求导数或微分——微分法。

(xi )1 导数概念

一、两个实例
1.直线运动的瞬时速度问题
设质点沿直线作非匀速运动,其走过的路程(s)与时间(t)的函数关系(s = s(t)),求某一时刻({t_0})时的瞬时速度。
设从时刻({t_0})到({t_0} + Delta t)这段时间内质点走过的路程为(Delta s = s({t_0} + Delta t) – s({t_0}))

从({t_0})到({t_0} + Delta t)这段时间内,平均速度(overline v = frac{{Delta s}}{{Delta t}} = frac{{s({t_0} + Delta t) – s({t_0})}}{{Delta t}})
对非匀速运动的质点,平均速度(overline v )可以作为({t_0})时刻瞬时速度的近似值(({Delta t})很小时):
({left. v right|_{t = {t_0}}} approx overline v )
(Delta t)越小,(overline v )与({left. v right|_{t = {t_0}}})越接近。
如果当(Delta t to 0)时,(overline v )的极限存在,即:
(mathop {lim }limits_{Delta t to 0} overline v = mathop {lim }limits_{Delta t to 0} frac{{Delta s}}{{Delta t}} = mathop {lim }limits_{Delta t to 0} frac{{s({t_0} + Delta t) – s({t_0})}}{{Delta t}} = {v_0})
则有({left. v right|_{t = {t_0}}} = {v_0})
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2.曲线在一点处的切线斜率

切线:当(P' to {P_0})时,割线({P_0}P')的极限位置({P_0}T)称为曲线的切线

割线:({P_0}({x_0},f({x_0})),P'({x_0} + Delta x,f({x_0} + Delta x)))
割线斜率(bar k = tan {alpha _1} = frac{{Delta y}}{{Delta x}} = frac{{f({x_0} + Delta x) – f({x_0})}}{{Delta x}})
当(P' to {P_0})时,(Delta x to 0)
(mathop {lim }limits_{Delta x to 0} bar k = mathop {lim }limits_{Delta x to 0} frac{{Delta y}}{{Delta x}} = mathop {lim }limits_{Delta x to 0} frac{{f({x_0} + Delta x) – f({x_0})}}{{Delta x}})
切线({P_0}T)的斜率(k = tan alpha = mathop {lim }limits_{Delta x to 0} bar k = mathop {lim }limits_{Delta x to 0} frac{{f({x_0} + Delta x) – f({x_0})}}{{Delta x}})
(注:(alpha )为切线的倾斜角)
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二、导数定义
<定义1> 设(y = f(x))在(N({x_0},delta ),delta > 0)内有定义,当自变量(x)在({x_0})点有增量({Delta x})(({x_0} + Delta x in N({x_0},delta ))),函数(y = f(x))相应的增量为(Delta y = f({x_0} + Delta x) – f({x_0})),如果极限(mathop {lim }limits_{Delta x to 0} frac{{Delta y}}{{Delta x}} = mathop {lim }limits_{Delta x to 0} frac{{f({x_0} + Delta x) – f({x_0})}}{{Delta x}})存在,则称(y = f(x))在({x_0})点可导,并称此极限值为(y = f(x))在({x_0})点的导数。
记为:
(y'{|_{x = {x_0}}},;f'({x_0}),;{left. {frac{{dy}}{{dx}}} right|_{x = {x_0}}},;{left. {frac{{df(x)}}{{dx}}} right|_{x = {x_0}}})
即(f'({x_0}) = mathop {lim }limits_{x to {x_0}} frac{{f({x_0} + Delta x) – f({x_0})}}{{Delta x}})
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直线运动的瞬时速度(v{|_{t = {t_0}}} = s'(t){|_{t = {t_0}}})
曲线在(({x_0},f({x_0})))的切线斜率(k{|_{x = {x_0}}} = f'({x_0}))

导数定义的另一种极限形式——(y = f(x))在({x_0})点的导数可以定义为:
若记(x = {x_0} + Delta x)(即(Delta x = x – {x_0}))
当(Delta x to 0)时,(x to {x_0})
(Delta y = f({x_0} + Delta x) – f({x_0}) = f(x) – f({x_0}))
(f'(x) = mathop {lim }limits_{Delta x to 0} frac{{f({x_0} + Delta x) – f({x_0})}}{{Delta x}} = mathop {lim }limits_{x to {x_0}} frac{{f(x) – f({x_0})}}{{x – {x_0}}})
文章来源:http://www.codelast.com/
(y = f(x))在({{x_0}})点可导,记为(f(x) in D({x_0}))
(y = f(x))在((a,b))内每一点处都可导,则称(y = f(x))在((a,b))内可导,记为(f(x) in D(a,b))
(y = f(x))在区间(I)上可导,记为(f(x) in D(I))
若(f(x))在((a,b))内可导,(forall x in (a,b)),就有(f'(x))与(x)对应,由函数定义,可知(f'(x))是定义在((a,b))上的函数,(f'(x))称为导函数,一般还称为导数
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例1. 求函数(y = frac{1}{{{x^2}}})的导函数((x ne 0))
解:
(y = frac{1}{{{x^2}}})定义域为(( – infty ,0) cup (0, + infty ))
(forall x in ( – infty ,0) cup (0, + infty )),自变量有增量(Delta x),且(x + Delta x in ( – infty ,0) cup (0, + infty ))
函数(y = frac{1}{{{x^2}}})对应的增量(Delta y = frac{1}{{{{(x + Delta x)}^2}}} – frac{1}{{{x^2}}} = frac{{ – 2x cdot Delta x – {{(Delta x)}^2}}}{{{x^2}{{(x + Delta x)}^2}}})
作比值:
(frac{{Delta y}}{{Delta x}} = frac{{ – 2x – Delta x}}{{{x^2}{{(x + Delta x)}^2}}})
求极限:
(mathop {lim }limits_{Delta x to 0} frac{{Delta y}}{{Delta x}} = mathop {lim }limits_{Delta x to 0} frac{{ – 2x – Delta x}}{{{x^2}{{(x + Delta x)}^2}}} = – frac{2}{{{x^3}}})
即({left( {frac{1}{{{x^2}}}} right)^prime } = – frac{2}{{{x^3}}})
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(第22课完)



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