I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if someone could explain how to solve this problem, and why the answer is 7.982. It's a calculator problem.
If $f(2) = 8$ and $f '(x) = \frac{\cos(1-x^2)}{(x^2 + \sqrt{x})}$ , what is $f(7) = ?$
Answer
From the fundamental theorem, (since $f'$ is continuous on [2,7])
$$\int_2 ^7 f'(x) dx = f(7) - f(2) = f(7) - 8$$
If memory serves correctly, your calculator should be able to compute definite integrals numerically.
$\int_2 ^7 f'(x) dx \approx -0.0182 $ and thus
$f(7) \approx 8 - 0.0182 = 7.982$.
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