＊急＊微積分antiderivative問題！！

Question

是關於Ｔ/Ｆ的問題
想請大家幫忙一下～＞＜
感謝

１　If F is an antiderivative of f and if G is an antiderivative of g then F·G is an antiderivative of f·g.

２　If x=F(t) is an antiderivative of x=f(t), then x=F(t) is a solution of the differential equation dx/dt=f(t).

３　An equation that involves the derivatives of a function is called a differential equation.

就這三題我非常不確定答案～
麻煩大家了
再次感謝！！

Answer 1

1. False
(F·G)'=f·G+F·g<>f·g
2. 題目敘述有問題:應該是
If F(t) is an antiderivative of f(t), then F(t) is a solution of the differential equation dx/dt=f(t).
Ans: True
因F(t)代入dx/dt=f(t)中之未知函數x,得dx/dt=F'(t)=f(t),滿足Diff Eq.故x=F(t)為dx/dt=f(t)之一解
3. True
微分方程式,即含未知函數及其導函數的方程式

Answer 2

F·G 不是 f·g 的 antiderivative, 這只要微分驗證, 或舉個例子就知道了!

"x = F(t) = antiderivative of f(t)" 當然滿足 "dx/dt = f(t)", 也就是該微方之解.

微方的定義, 查書不就知道了?