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用導(dǎo)數(shù)的定義求極限(limit)_AP 微積分

2022-10-15 23:03 作者:第一性原理  | 我要投稿

DEFINITION OF DERIVATIVE ( 導(dǎo)數(shù)的定義)


f'(a)%3D%5Clim_%7Bh%5Cto0%7D%20%5Cfrac%7Bf(a%2Bh)-f(a)%7D%7Bh%7D%20

The fraction??%20%5Cfrac%7Bf(a%2Bh)-f(a)%7D%7Bh%7D%20 is called the difference quotient for %5C%20f(x)at %5C%20x%3Da%20?and represents the average rate of change (ARC) of%5C%20f(x)? from a to a + h.?


Geometrically, it is the slope of the secant PQ to the curve y = f(x) through the points P(a, f(a)) and Q(a + h, f(a + h)). The limit, f′(a), of the difference quotient is the (instantaneous) rate of change of f at point a. Geometrically (see Figure 3–1a), the derivative f′(a) is the limit of the slope of secant PQ as Q approaches P—that is, as h approaches zero. This limit is the slope of the curve at P. The tangent to the curve at P is the line through P with this slope.

在幾何學(xué)上,它是通過點P(a, f(a))和Q(a+h, f(a+h))的曲線y=f(x)的割線PQ的斜率。? ? ? ? ? ? ? ? ?差商的極限f′(a)是f(x)在a點的(瞬時)變化率。這個極限就是曲線在P點的斜率,曲線在P點的切線就是通過P點的斜率的直線。


key

f'(a)%3D%5Clim_%7Bh%5Cto0%7D%20%5Cfrac%7Bf(a%2Bh)-f(a)%7D%7Bh%7D%20

35.?f(x)%3Dx%5E6%20?,%5C%20x%3D1?,?f(1)%3D1;key?f'(1)%3D6%5Ctimes%201%5E5%3D1%20

36.?f(x)%3D%5Csqrt%5B3%5D%7Bx%7D%20?,%5C%20x%3D8,?f(8)%3D2; key?f'(8)%3D%5Cfrac%7B1%7D%7B3%5Ctimes8%5E%5Cfrac%7B2%7D%7B3%7D%20%7D%3D%5Cfrac%7B1%7D%7B3%5Ctimes4%20%7D%20%3D%5Cfrac%7B1%7D%7B12%7D%20

37.?f(x)%3D%5Cln%20x%2C%20x%3De%2C%20f(e)%3D1;key?f'(e)%3D%5Cfrac%7B1%7D%7Be%7D%20

38.?f(x)%3D%5Ccos%20x%20?,%5C%20x%3D0,?f(0)%3D1; key?f'(0)%3D-%5Csin%200%3D0%20


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