微积分微分及其计算

2018/11/1
Edited by Lin Guojian
11
例(参数方程求导数法则) :
设参数方程
:
x y
= =
x(t) y(t)
t ∈[α , β ].
其中x(t), y(t)关于t可导且x′(t) ≠ 0,求 dy . dx
解 :由于dy = dy(t) = y′(t)dt, dx = dx(t) = x′(t)dt ⇒ dy = y′(t)dt = y′(t) ,t ∈[α , β ]. dx x′(t)dt x′(t)
2018/11/1
Edited by Lin Guojian
7
注 : 微分函数df (x) = f ′(x)dx是指它作为自变量x的函数, 而且习惯上也称f (x)为微分函数df (x)的原函数.
注 : 对可微函数y = f (x),由公式dy = df (x) = f ′(x)dx知 :
微分dy既与x有关又与dx有关, 但x与dx相互独立.
证 : ∆y = f (x0 + ∆x) − f (x0 ) = (x0 + ∆x)3 − x03
= x03 + 3x02∆x + 3x0∆x2 + ∆x3 − x03 = 3x02∆x + 3x0∆x2 + ∆x3
= 3x02∆x + o(∆x).
故y = f (x) = x3在x = x0处可微, 且微分dy = df (x) = 3x02∆x. 而且当x0 ≠ 0时, 微分df = df (x) = 3x02∆x是∆y的主部.
2018/11/1
Edited by Lin Guojian
4
由性质3.7知 :当f (x)在点x0可微时, f (x0 + ∆x) − f (x0 ) = f ′(x0 )∆x + o(∆x), (∆x → 0). 故当f ′(x0 ) ≠ 0且当∆x充分小时, o(∆x)可以忽略不计.
因此f (x0 + ∆x) − f (x0 ) ≈ f ′(x0 )∆x, 即f (x0 + ∆x) ≈ f (x0 ) + f ′(x0 )∆x,这是近似计算f (x0 + ∆x)的公式.
2018/11/1
Edited by Lin Guojian
10
由性质3.7知 : 对于可微函数y = f (x),有dy = df (x) = f ′(x)dx. 由性质3.9知 : 对于可微函数y = f (u)与u = g(x)有dy = df (u) = f ′(u)du. 因此无论v为自变量, 还是v为中间变量, 都有dy = df (v) = f ′(v)dv, 其形式是不变的.我们称这种性质为一阶微分形式不变性.
解二 : dy = d (x x ) = d (exln x ) = exln xd (x ln x) = exln x[(ln x)dx + xd (ln x)]
= exln x[(ln x)dx + x 1 dx] = exln x[(ln x) +1]dx. x
dx
=
g(x)
f
′(x)dx − f (x)g′(x)dx [ g ( x)]2
=
g(x)d[
f
(
x)] − f ( [ g ( x)]2
x)d
[
g
(
x)]
.
2018/11/1
Edited by Lin Guojian
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性质3.9 : 设y = f [g(x)]是由可微函数y = f (u)与u = g(x)复合而成,则y = f [g(x)]关于x可微.
而且求微分本质上是求导数.
例: 设y = f (x) = 3x2 ln x,求dy及dy . x=1
解 : dy = d (3x2 ln x) = (3x2 ln x)′dx = (6x ln x + 3x)dx.
dy = (6x ln x + 3x)dx = 3dx.
x=1
x=1
2018/11/1
x= x0
x= x0
若f (x)在(a,b)内处处可微,则称f (x)在(a,b)内可微, 且称f (x)是(a,b)内的可微
函数.这时微分dy = df (x) = A(x)∆x是两个独立变量x与∆x的函数.
2018/11/1
Edited by Lin Guojian
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例 : 用定义3.3证明函数y = f (x) = x3在x = x0点处可微.
5
2018/11/1
Edited by Lin Guojian
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例 : 求 sin 59的近似值.
解 : sin 59 = sin(60 − 1 ) = sin(π − π )
3 180 设y = f(x) = sin x.
由于f(x)在x =
π 点可导,故f(x)在x =
π 点可微且f ′(π ) = cos π
注 : ∆x是自变量x的微分. 设f (x) = x,则f (x)可微且f ′(x) = 1.故df (x) = dx = f ′(x)∆x = ∆x,即dx = ∆x.
因此任何一个可微函数y = f (x)的微分可以用自变量的微分表示为:
dy
ห้องสมุดไป่ตู้x= x0
= df (x) x= x0
=
f ′(x0 )∆x =
故 lim ∆x→0
∆y ∆x
=
lim
∆x→0
f
( x0
+ ∆x) − ∆x
f
(x0 )
=
lim
∆x→0
A∆x + o(∆x) ∆x
=
lim
∆x→0
A
+
o(∆x) ∆x
=
A.
因此f (x)在点x0可导且f ′(x0 ) = A.
从而dy
x= x0
= df
(x) x= x0
=
A∆x
=
f ′(x0 )∆x.
3.4、微分及其计算
例: 测量边长为x0正方形的面积. 由于测量时, 对真实的值x0总有误差∆x,这时测得正方形边长为x0 + ∆x. 故其面积为S (x0 + ∆x) = (x0 + ∆x)2 = x02 + 2x0∆x + (∆x)2.
而S(x0 ) = x02才是边长为x0正方形面积的真实值. 因此测量得到的面积与真实面积有误差 : ∆S = S (x0 + ∆x) − S (x0 ) = (x0 + ∆x)2 − x02 = x02 + 2x0∆x + (∆x)2 − x02 = 2x0∆x + (∆x)2. 由于∆x → 0时, (∆x)2是∆x的高阶无穷小量. 故∆S = 2x0∆x + o(∆x).
从而当误差∆x充分小时, (∆x)2可以忽略不计. 即S (x0 + ∆x) − S (x0 ) ≈ 2x0∆x.
2018/11/1
Edited by Lin Guojian
1
从类似的近似计算中可以抽象出一种数学概念 − 微分.
定义3.3 : 设y = f (x)在x0的某一领域内有定义,若在其中给x0一个改变量 ∆x,相应的函数值的改变量∆y可表示为如下 :
∆y = f (x0 + ∆x) − f (x0 ) = A∆x + o(∆x), (∆x → 0).
其中A与∆x无关,则称y = f (x)在点x0可微,且称A∆x为f (x)在点x0处的微分,
记为dy = df (x) = A∆x.
x= x0
x= x0
因此当A ≠ 0时,微分dy 是函数值改变量∆y 的主部.
令x = x0 + ∆x,则x → x0时, ∆x → 0. 故∆y = f (x0 + ∆x) − f (x0 ) = f ′(x0 )∆x + o(∆x), (∆x → 0). ⇒ f (x)在点x0可微.
"⇒"若f (x)在点x0可微,则∆y = f (x0 + ∆x) − f (x0 ) = A∆x + o(∆x), (∆x → 0).
dt,
故 dy = 6at − 3at4 = 2t − t4 .
dx 3a − 6at3 1 − 2t3
dy
=
6at(1+ t3 ) − 3at 2 (3t 2 ) (1+ t3 )2
dt
=
6at − 3at 4 (1+ t3 )2
dt,
⇒
dy dx
t =1
=
2at − at 4 a − 2at3
t =1
证 :由于y = f (u)可微与u = g(x)可微,故y = f (u)可导与u = g(x)可导, 由性质3.6知 : 复合函数y = f [g(x)]可导 ⇒ y = f [g(x)]可微.
那么dy = df [g(x)] = ( f [g(x)])′dx = f ′[g(x)]g′(x)dx = f ′[g(x)]d[g(x)] = f ′(u)du.
2018/11/1
Edited by Lin Guojian
3
性质3.7 : y = f (x)在点x0可微 ⇔ f (x)在点x0可导. 而且当f (x)在点x0可导时, df x=x0 = f ′(x0 )∆x.
证 :"⇐"若f (x)在点x0可导,则由有限增量公式知 : f (x) − f (x0 ) = f ′(x0 )(x − x0 ) + o(x − x0 ), (x → x0 ).
即dy = f ′(u)du = df (u) du = df (u) du dx ⇒ dy = df (u) du .
du
du dx
dx du dx
因此复合函数求导的链式法则 : dy = df (u) du 不仅具有(3 − 6)式中的含义, dx du dx
而且还具有导数可以作为微分的商进行运算.
例: 求5 0.99的近似值. 解 : 设y = f (x) = 5 x. 由于f (x)在x = 1点可导,故f (x)在x = 1点可微且f ′(1) = 1 .
5 那么有5 0.99 = f (1− 0.01) ≈ f (1) + f ′(1)(−0.01) = 1+ 1 (−0.01) = 0.998.
2018/11/1
Edited by Lin Guojian
12
例 : 设参数方程x
y
= =
3at
1 + t3 3at 2
1 + t3
,求
dy dx
及在t
=
1处的切线方程与法线方程.
解
:
dx
=
3a(1+ t3) − 3at(3t 2 ) (1+ t3 )2
dt
=
3a − 6at3 (1+ t3 )2
=
−1.
且t = 1时, x = 3 a, y = 3 a.
2
2
故参数方程在t = 1处的切线方程为: y − 3 a = −(x − 3 a) ⇒ y = −x + 3a.
2
2
故参数方程在t = 1处的法线方程为: y − 3 a = (x − 3 a) ⇒ y = x.
2
2
2018/11/1
Edited by Lin Guojian
f ′(x0 )dx.
或dy = df (x) = f ′(x)∆x = f ′(x)dx. ⇒ f ′(x) = dy = df (x) . dx dx
注 : 记号 dy 有了一个新的含义 : dx
在3.1节中 dy 是作为一个整体用来表示导数的. dx
而这里 dy 是作为dy为dx的商,因此导数也叫微商. dx
=
1 .
3
3
3
32
那么有 sin(π − π ) ≈ sin π + f ′(π )(− π ) = 3 + 1 (− π ).
3 180
3
3 180 2 2 180
2018/11/1
Edited by Lin Guojian
6
由性质3.7知 : f (x)可导 ⇔ f (x)可微,且df (x) = f ′(x)∆x,其中∆x是与x无关的变量.
13
例: 求y = xx的微分.
[ ] 解一 : dy = d (xx ) = d (exln x ) = exln x ′ dx = [exln x (x ln x)′]dx
= exln x[(ln x) +1]dx.
或者 : y′ = (x x )′ = (exln x )′ = exln x (x ln x)′ = exln x[(ln x) +1]. 故dy = exln x[(ln x) +1]dx.
Edited by Lin Guojian
8
由导数与微分的关系式dy = df (x) = f ′(x)dx容易得到微分的运算法则 :
性质3.8 : (1)d (C) = 0, (C为常数);
(2)d[Cf (x)] = Cd[ f (x)];
(3)d[ f (x) ± g(x)] = d[ f (x)] ± d[g(x)];
(4)d[ f (x)g(x)] = g(x)d[ f (x)] + f (x)d[g(x)];
(5)d
f (x)
g(x)
=
g(x)d[
f
(
x)] − f ( [ g ( x)]2
x)d
[
g
(
x)]
.
′
证 : (5)d
f (x)
g
(
x)
=
f (x)
g
(
x)
dx
=
g(x)
f
′(x) − f (x)g′(x) [ g ( x)]2