Question

CALCULUS EXPERT WANTED (ignore the stuff I was writing)

Answer 1

Given how complicated the derivative,

$f'(x)=\dfrac{\cos(1-x^2)}{x^2+\sqrt x}$

looks, integrating it to find $f(x)$ seems futile, and actually not what needs to be done. You're asked to approximate $f(7)$ using a given point and the value of the derivative at any point. To do that, you can use a linear approximation.

$f(7)\approx f(2)+f'(2)(7-2)$

which comes from exploiting the mean value theorem (solve for $f'(2)$ above to see how)

Then

$f(7)\approx8+\dfrac{5\cos(1-2^2)}{2^2+\sqrt2}\approx7.086$

The closest answer would be B. The reason the listed answer is so far off is because 7 is not so close to 2.