AP Calculus AB / Integration and Accumulation of Change / Question No. 37

37. If \( F(x) = \int_0^{\sqrt{x}} \cos(t^2)\,dt \), then \( F'(4) = \)

\( \cos 2 \)

\( \frac{\cos 4}{4} \)

\( \frac{\cos 4}{\sqrt{2}} \)

\( \sqrt{2} \cos 4 \)

Answer is: option2

\( \frac{\cos 4}{4} \)

Solution:

Let: \[ F(x) = \int_0^{u(x)} f(t)\,dt, \quad \text{then} \quad F'(x) = f(u(x)) \cdot u'(x) \]

Here:

So: \[ F'(x) = \cos((\sqrt{x})^2) \cdot \frac{1}{2\sqrt{x}} = \cos(x) \cdot \frac{1}{2\sqrt{x}} \]

\[ F'(4) = \frac{1}{2\sqrt{4}} \cdot \cos(4) = \frac{1}{2 \cdot 2} \cdot \cos(4) = \frac{\cos 4}{4} \]

Hence option B.

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