Question

If \(f\) is continuous on \([a,b]\) and \[A(x)=\int_a^x f(t)\,dt,\] what is \(A'(x)\) for every \(x\) in \((a,b)\)?

Options

• \[A'(x)=f(x)\]

• \[A'(x)=F(b)-F(a)\]

• \[A'(x)=\int_a^b f(x)\,dx\]

• \[A'(x)=f(t)\]

Answer 1

The First Fundamental Theorem states that \(A'(x)=f(x)\) for every \(x\in(a,b)\).
Thus, the derivative of the accumulated area function is the original function itself.