Yahoo Answers: Answers and Comments for 微積分基本定理和roll定理~ [數學]
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From 隨風飄飄天地任逍遙
zhHantTW
Tue, 18 Jul 2006 16:17:11 +0000
3
Yahoo Answers: Answers and Comments for 微積分基本定理和roll定理~ [數學]
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https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
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From Eric: （１）我想應該是：
Let f and f' be continuous and diffe...
https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
Wed, 19 Jul 2006 16:01:31 +0000
（１）我想應該是：
Let f and f' be continuous and differentiable functions.
Suppose the curve y = f(x) passes through the points (0,0) and (1, 1).
Find ∫(0,1) f'(x) dx.
We know that f(0) = 0, f(1) = 1.
∫(0,1) f'(x) dx = f(1)  f(0) (by the fundamental theorem of calculus)
= 1  0 = 1
（２）
Prove that x^33x^2+a=0 has at most one real solution in [0,1] regardless of the value of a.
Proof: Let a be any real number. Let f(x) = x^33x^2+a, which has derivative
f'(x) = 3x^26x = 3x(x2).
Clearly, f'(x) = 0 only at two points: x = 0 and x = 2.
Hence f'(x) ≠ 0 on the open interval (0, 1). (*).
Suppose there exist two solutions u, v with 0 ≤ u < v ≤ 1.
Then f(u) = f(v) = 0, and thus by Rolle's theorem, there exists a number c, u < c < v, such that f'(c) = 0  contradiction.
Hence, f(x) = 0 has at most one real solution in [0,1].
::

From 我的日子只有混: 我誤會了，不論a為何值，至多一個根在[0,1]之間
https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
Tue, 18 Jul 2006 17:53:36 +0000
我誤會了，不論a為何值，至多一個根在[0,1]之間

From 隨風飄飄天地任逍遙: 第二題:他說不管a是什麼…是什麼意思?
https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
https://tw.answers.yahoo.com/question/index?qid=20060718000014KK09359
Tue, 18 Jul 2006 17:33:11 +0000
第二題:他說不管a是什麼…是什麼意思?