1. 切线问题

考虑这样一个问题,一个抛物线(Parabola):

以及线上一点B(1,1),如何求线上B点的Tangent Line(切线)?

Tangent Line一个不严格的概念:

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that “just touches” the curve at that point.

如下图:

Fig.1 Parabola tangent line.
Fig.1 Parabola tangent line.

已知在平面上,一个点和斜率就可以唯一确定一条直线。B点已经有了,那么关键就在于如何求这条线的斜率?

一个思路是1

在抛物线上另找一点A,A, B就可以唯一确定一条直线l,且l的斜率为:

令A不断从右侧接近B,有

从左侧接近B,有

可以看出,当愈接近,即A点愈接近B点,的值愈接近2,如下图:

Fig.2 Point A goes to B.
Fig.2 Point A goes to B.

可以猜测,当无限趋近于的值为2,我们说函数即在处有极限:

即在抛物线上B点的斜率是2,从而也求得了相应的切线。

这个猜测是非常不严格的,只是一种直观的理解。可以把这种无限接近理解为一种approximation,并且在精度上可以无限增加。但是如何从无限接近到尽准值2这个跨越的,目前还不明白。

2. 极限

这里引出了极限的概念1

Suppose f(x) is defined when x is near the number a. (This means x that is defined on some open interval that contains a, except possibly at a itself.) Then we write:

and say “the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by
taking x to be sufficiently close to a (on either side of ) but not equal to a.

Roughly speaking, this says that the values of f(x) approach L as x approaches a. In other words, the values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of ) but x != a.

同样,这是一个直观的定义。

注意到极限是无限接近,并不需要在目标处有定义,如下图1a, b, c三种情况在a点都是有极限的。

Fig.3 different limit scenario.
Fig.3 different limit scenario.

一个没有极限的例子,如下图1,在t=0处没有极限:

Fig.3 no limit at 0.
Fig.3 no limit at 0.

左、右极限的概念类似,不再赘述。

  • 1st Rev. 11/06/19
  • 2nd Rev. 11/13/19
  1. Calculus, James Stewart, 7ed  2 3 4