推 fix927 :已解決 感謝 01/21 19:04
這不就只是單純的微積分基本定理而已嗎?
f(x)€C[a,b] , c€(a,b)
x
Define F(x) = S f(x) dx
c
we have F(x)€C[a,b] and F€C^1(a,b) and F'(x) = f(x)
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√(1+x^2)€C(R) , 2€R (R is the set of real numbers)
x
Define F(x) = S √(1+x^2)
2
we have F'(x) = √(1+x^2) on R
so F'(2) = √5
2+h
S √(1+x^2) - 0
F(2+h) - F(2) 2
where F'(2) = lim ────── = lim ───────── = 題目所求
h→0 h h→0 h
P.S. 這個微積分基本定理可能跟一般書上不太一樣
一般是：f€C[a,b]
x
Define F(x) = S f(x) dx
a
we have F€C^1[a,b] and F'(x) = f(x) on [a,b]
但是微分的先決條件要該點在"附近"均有定義
所以F'(x) = f(x) on [a,b]這句話不太對
因為他在a點只有右微分，b點只有左微分
所以精確來講是：F€C^1(a,b) , €C[a,b] and F'(x) = f(x) on (a,b)
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