# 幫我翻譯微積分數學 20點

One fundamental interpretation of the derivative of a function is that it is the slope of the tangent line to the graph of the function. (Still, it is important to realize that this is not the definition of the thing, and that there are other possible and important interpretations as well).

The precise statement of this fundamental idea is as follows. Let f be a function. For each fixed value X0 of the input to f, the value f’(x0) of the derivative f’ of f evaluated at X0 is the slope of the tangent line to the graph of f at the particular point (x0, f(x0)) on the graph.

Recall the point-slope form of a line with slope m through a point (x0, y0):

y - y0 = m(x - x0)

In the present context, the slope is f’(x0) and the point is (x0, f(x0)), so the equation of the tangent line to the graph of f at (x0, f(x0)) is

y - f(x0) = f’(x0)(x - x0)

The normal line to a curve at a particular point is the line through that point and perpendicular to the tangent. A person might remember from analytic geometry that the slope of any line perpendicular to a line with slope m is the negative reciprocal -1/m. Thus, just changing this aspect of the equation for the tangent line, we can say generally that the equation of the normal line to the graph of f at (x0, f(x0)) is

y - f(x0) = (-1/f’(x0))*(x - x0)

### 1 個解答

• Leslie
Lv 7
1 0 年前
最佳解答

函數之導數的一個基本解釋就是此函數的圖形的切線.

(然而, 很重要的是, 您要了解這並非導數的定義. 並且,

導數還有其他可能的及重要的解釋)

此基本觀念的精確敘述如下.

設 f 為一函數.

對各給定的函數輸入值 x0,

f 在 x0 的導數值, f'(x0),

就是 f 的圖形在點 (x0, f(x0)) 的切線之斜率.

回想一下所謂的 "點-斜率" 直線公式:

一條通過點 (x0, y0), 且斜率為 m 的直線, 其公式為

y - y0 = m(x - x0)

而我們現在講的是, 點是 (x0, f(x0)), 斜率為 f'(x0),

因此, f 的圖形在點 (x0, f(x0)) 的切線之公式為

y - f(x0) = f'(x0)(x - x0)

另外,

所謂一條曲線上的某個點的 "法線", 指的是

通過此點, 並與此點的切線垂直的直線.

您若記得解析幾何中說:

和斜率為 m 的直線垂直的任何直線, 其斜率為 m 的負的倒數,

即 -1/m.

因此, 利用到我們的切線(及法線)上, 我們便可以說,

f 的圖形在點 (x0, f(x0)) 的法線之公式為

y - f(x0) = (-1/f'(x0))*(x - x0)

參考資料： D I F F E R E N C I A L C A L C U L U S