AP Calculus AB / Integration and Accumulation of Change / Question No. 42

42. If \( f'(x) = \cos(x^2 - 1) \) and \( f(-1) = 1.5 \), then \( f(5) = \)
(calculator)

1.554

2.841

3.024

3.456

Answer is: option3

3.024

Solution:

Since \( f'(x) = \cos(x^2 - 1) \), then

\( f(5) = f(-1) + \int_{-1}^{5} \cos(x^2 - 1) \, dx \)

Given \( f(-1) = 1.5 \), we need to evaluate:

\( f(5) = 1.5 + \int_{-1}^{5} \cos(x^2 - 1) \, dx \)

Using a graphing calculator, we get:

\( \int_{-1}^{5} \cos(x^2 - 1) \, dx = 1.524 \)

Substituting values:

\( f(5) = 1.5 + 1.524 = \boxed{3.024} \)

Hence option C

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