文档介绍：一、微分的定义(Definition of Differential )
二、微分的几何意义(The Geometric Meaning of Differential)
四、微分在近似计算中的应用
Application of Differential in Approximation
第四节微分及其计算
Differential of a Function and the Rules for Differentiation
三、基本初等函数的微分公式与微分运算法则
The Differential Formulas of the Basic Elementary Funtiond and the Rules for Differentiation
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一、微分的定义(Definition of Differential )
问题的提出
一块正方形金属薄片受温度变化的影响,其边长
由变到(如图),问此薄片的面积
改变了多少?
一般地,如果函数y=f(x)满足一定条件,则函数的增量可表示为
其中A是不依赖于的常数,因此是的线性函数,且它与
之差
是比高阶的无穷小,所以,当,且很小时,我们就可
以近似地用来代替
定义设函数y=f(x)在某区间内有定义, 及在这区间内,如果函数的增量
可表示为
其中A是不依赖于的常数,而是比
高阶的无穷小,那么称y=f(x)在点是可微的,
而叫做函数y=f(x)在点相应于自变量增
量的微分,记作dy,即
Definition Let y=f(x) be a function defined on some interval I, ∈I, ∈I ,if the increment of the dependent variable y=f( )-f( ) can be expressed as △y=A△x,where A is a constant which is independent of △x ,then we say that f(x) is differentiable at ,and A△x is called the differential of y=f(x) at corresponding to the increment A△x of the independent varable,denoted by dy,= A△x .
由定义知:
定理:y=f(x)在可微的充分必要条件是f(x)在处
可导,且当f(x)在点可微时,其微分一定是
(1) 必要性
证明
Theorem: A function is derivable at x0 if and only if it is differentiable
at x0.
(2) 充分性
例1
解
例2
解
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