Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-010 · ★☆☆ Basic
What does the first part of the fundamental theorem of calculus state?
Choose one option.
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- C. The derivative of \(\int_a^x f(t)\,dt\) with respect to \(x\) is \(f(x)\)
Explanation
The first part says that if \(f\) is continuous, the function \(\int_a^x f(t)\,dt\) is differentiable and its derivative with respect to \(x\) is \(f(x)\).
Review the concept: Fundamental theorem of calculus
It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
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