Chapter 3

导数 Derivative 导数 可导的 Derivable 可导的 可导性 Derivability 可导性 单侧导数 One-sided derivative 单侧导数 左导数 Left-hand derivative 左导数 右导数 Right-hand derivative 右导数 割线 切线 Secant line 割线 Tangent line 切线 瞬时变化率 Instantaneous rate of change 瞬时变化率 Vocabulary

微积分的创立 微积分 (Calculus) 是微分学 (Differential calculus) 和积分学 (Integral calculus) 的总称， 它是由牛顿与 莱布尼兹在研究物理和几何的过程中总结前人的经 验，于十七世纪后期建立起来的。 牛顿 [ 英国 ] (Isaac Newton) 1642—1727 莱布尼兹 [ 德国 ] (G.W.Leibniz) 1646—1716

关于牛顿 牛顿的三大成就：  流数术（微积分）  万有引力定律  光学分析的基本思想 我不知道在别人看来，我是什么样的人；但在我 自己看来，我不过就象是一个在海滨玩耍的小孩，为 不时发现比寻常更为光滑的一块卵石或比寻常更为美 丽的一片贝壳而沾沾自喜，而对于展现在我面前的浩 瀚的的真理的海洋，却全然没有发现。 —— 牛顿

1. Introduction One of central ideas of calculus is the notion of derivative. The derivative originated from a problem in geometry, that is, the problem of finding the tangent line at a point of a curve. It was soon found that it also provides a way to calculate velocity and, more generally, the rate of change of a function. These problems contain the essential features of the derivative concept and may help to motivate the general definition of derivative which is given in this section.

(1) The problem of finding the tangent line at a point of a curve See figure Solution

(2) The problem involving velocity To find speed of an object moving on a line whose distant at time t is given by f(t)? Solution

2. The Definition of Derivatives Definition 1

Notes

(2) The derivative of a function defined on an open interval I Definition 2 Note:

(3) One-sided derivative Left-hand derivative Right-hand derivative Remarks:

Example 1

Solution:According to the definition of derivative

The above example illustrate the concept of the derivative. 3. Examples for Finding Derivatives

Solution Using definition of derivative, we have at any point x Solution By definition of the derivative,

Note

SolutionBy definition of the derivative,

SolutionBy definition of the derivative,

SolutionBy definition of the derivative,

Solution

4. Geometric Interpretation of the Derivative Then equation of tangent line is and equation of normal line is

Remark: Example 7 Solution By definition of the derivative, we have

The slope of the secant line PQ is

5. The Derivative and Continuity Theorem 1 Proof

The converse of theorem 1 is not true, that is, a continuous function needs not always be derivable. Note:

Solution Example 8

But we have

Example 9 Solution

Example 10

Solution

Example 11 Solution By definition of the derivative,

Example 12 Solution