数学专业英语(课堂PPT)
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2.8 函数的导数和它的几何意义 The Derivative of a Function and Its
Geometric interpretation
New Words & Expressions:
acceleration 加速度
interval 区间
altitude 高度
numerator 分子
速度v(t)是位置函数f的导数。这通常说成速度是位移 关于时间的变化率。
In general, the limit process which produces f'(x) from f(x) gives us a way of obtaining a new function f' from a given function f. This process is called differentiation, and f' is called the first derivative of f.
上一节描述的例子指出了介绍导数概念的方法。
We begin with a function f defined at least on some open interval (a,b) on the x-axis. 我们从一个至少定义在x轴的开区间(a,b)上的函数入 手。
Then we choose a fixed point x in this interval and introduce the difference quotient
f (xh) f (x) h
where the number h, which may be positive or negative (but not zero), is such that x+h also lie in (a,b).
接下来在这个区间中选择一个固定点，并且引进差商， 这里数h可正可负(但是不能为0)，且使得x+h也在(a,b) 中。
通过对比(8.2)和上节的(7.3)式，可以看出瞬时速度的 概念只是导数概念的一个例子。
The velocity v(t) is equal to the derivative f'(t) , where f is the function which measures position. This is often described by saying that velocity is the rate of change of position with respect to time.
aห้องสมุดไป่ตู้proach 趋于
rectilinear motion 直线运动
bound 界，限
slope 斜率
derivative 导数
tangent 正切，切线
fraction 分数，分式
velocity 速度
Key points:
the definition of the derivative of a function
The numerator of this quotient measures the change in the function when x changes from x to x+h. The quotient itself is referred to as the average rate of the change of f in the interval joining x to x + h.
Difficult points:
some relevant terms
8-A The derivative of a function
The example described in the foregoing section points the way to the introduction of the concept of derivative.
如果差商以某个确定的值为极限(这蕴含着不论 h 取 正的值趋于 0 还是取负的值趋于 0，其极限一样)，那 么这个极限称为 f 在 x点 的导数，记作f' (x) (读成“f 一撇x”)。
Thus, the formal definition of f'(x) may be stated as follows:
called the rate of change of f at x.
如果这个极限存在，则这个等式定义了导数f'(x) ，它 也称为f在x处的变化率。
By comparing (8.2) with (7.3) in the foregoing section, we see that the concept of instantaneous velocity is merely an example of the concept of derivative.
这个商的分子度量了当x由x变到x+h时函数的改变量。 差商本身表示函数 f 在连接 x 与 x + h 的区间上的平 均变化率。
Now we let h approach zero and see what happens to this quotient.
现在令h趋于0，看这个商如何变化。
※If the quotient approaches some definite value as a limit (which implies that the limit is the same whether h approaches zero through positive values or through negative values), then this limit is called the derivative of f at x and is denoted by the symbol f'(x) (read as “f prime of x” ).
这样， f'(x)的正式定义可以叙述如下：
DEFINITION. The derivative f'(x) is defined by the
equation
f'(x ) lim f(x h ) f(x )
h 0
h
(8 .2 )
provided the limit exists. The number f'(x) is also
Geometric interpretation
New Words & Expressions:
acceleration 加速度
interval 区间
altitude 高度
numerator 分子
速度v(t)是位置函数f的导数。这通常说成速度是位移 关于时间的变化率。
In general, the limit process which produces f'(x) from f(x) gives us a way of obtaining a new function f' from a given function f. This process is called differentiation, and f' is called the first derivative of f.
上一节描述的例子指出了介绍导数概念的方法。
We begin with a function f defined at least on some open interval (a,b) on the x-axis. 我们从一个至少定义在x轴的开区间(a,b)上的函数入 手。
Then we choose a fixed point x in this interval and introduce the difference quotient
f (xh) f (x) h
where the number h, which may be positive or negative (but not zero), is such that x+h also lie in (a,b).
接下来在这个区间中选择一个固定点，并且引进差商， 这里数h可正可负(但是不能为0)，且使得x+h也在(a,b) 中。
通过对比(8.2)和上节的(7.3)式，可以看出瞬时速度的 概念只是导数概念的一个例子。
The velocity v(t) is equal to the derivative f'(t) , where f is the function which measures position. This is often described by saying that velocity is the rate of change of position with respect to time.
aห้องสมุดไป่ตู้proach 趋于
rectilinear motion 直线运动
bound 界，限
slope 斜率
derivative 导数
tangent 正切，切线
fraction 分数，分式
velocity 速度
Key points:
the definition of the derivative of a function
The numerator of this quotient measures the change in the function when x changes from x to x+h. The quotient itself is referred to as the average rate of the change of f in the interval joining x to x + h.
Difficult points:
some relevant terms
8-A The derivative of a function
The example described in the foregoing section points the way to the introduction of the concept of derivative.
如果差商以某个确定的值为极限(这蕴含着不论 h 取 正的值趋于 0 还是取负的值趋于 0，其极限一样)，那 么这个极限称为 f 在 x点 的导数，记作f' (x) (读成“f 一撇x”)。
Thus, the formal definition of f'(x) may be stated as follows:
called the rate of change of f at x.
如果这个极限存在，则这个等式定义了导数f'(x) ，它 也称为f在x处的变化率。
By comparing (8.2) with (7.3) in the foregoing section, we see that the concept of instantaneous velocity is merely an example of the concept of derivative.
这个商的分子度量了当x由x变到x+h时函数的改变量。 差商本身表示函数 f 在连接 x 与 x + h 的区间上的平 均变化率。
Now we let h approach zero and see what happens to this quotient.
现在令h趋于0，看这个商如何变化。
※If the quotient approaches some definite value as a limit (which implies that the limit is the same whether h approaches zero through positive values or through negative values), then this limit is called the derivative of f at x and is denoted by the symbol f'(x) (read as “f prime of x” ).
这样， f'(x)的正式定义可以叙述如下：
DEFINITION. The derivative f'(x) is defined by the
equation
f'(x ) lim f(x h ) f(x )
h 0
h
(8 .2 )
provided the limit exists. The number f'(x) is also