Answer
$f(x) = (8-x^2)^{\frac{3}{2}} -1$
Work Step by Step
For this problem, begin by finding $f(x)$ from its derivative, $f'(x)$. To do this, integrate $f'(x)$ with respect to x:
$\int -2x\sqrt {8-x^2}dx$
use $u$ substitution to evaluate this indefinite integral:
$u = 8-x^2$
$du = -2xdx$
Now do the substitution on the integral:
$\int \sqrt u du$
Now integrate:
$\int u^{\frac{1}{2}} du$
$u^{\frac{3}{2}} +c$
Now substitute back:
$f(x) = (8-x^2)^{\frac{3}{2}} +c$
Now, evaluate the point $(2,7)$ to find the value of $c$:
$f(2) = (8-(2)^2)^\frac{3}{2} + c = 7$
$8 + c = 7$
$c = -1$
Now set up the equation:
$f(x) = (8-x^2)^{\frac{3}{2}} -1$
