Exercise · ZW-ANTIDERIVATIVE-008 · ★★★ Advanced
Let \(f(x)=0\) on \(D=(-1,0)\cup(0,1)\). Which statement is true?
Choose one option.
Show answer
- B. \(F(x)=0\) if \(x<0\) and \(F(x)=1\) if \(x>0\) is an antiderivative of \(f\) on \(D\)
Explanation
Each piece has derivative 0, but \(F\) equals 0 on one piece and 1 on the other, so it is not constant on \(D\). The constant rule needs an interval, not two pieces.
Review the concept: Antiderivative
An antiderivative of \(f\) is a function \(F\) such that \(F^{\prime}=f\). If \(F\) is an antiderivative of \(f\), all the others on an interval differ from \(F\) by a constant.
\[F'(x)=f(x)\]
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