Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-018 · ★★☆ Intermediate
Compute \(\displaystyle\lim_{x\to0}\frac1x\int_0^x\cos(t^2)\,dt\).
Choose one option.
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- C. \(1\)
Explanation
It is the derivative at \(0\) of \(F(x)=\int_0^x\cos(t^2)\,dt\) (with \(F(0)=0\)), which equals \(\cos 0=1\).
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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