Find the equation of the function f whose graph passes through the point $\left( 0 , \frac { 7 } { 3 } \right)$ and whose derivative is $f ^ { \prime } ( x ) = x \sqrt { 1 - x ^ { 2 } }$ .
A) $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$
B) $f ( x ) = \frac { 1 } { 3 } \left[ 8 - \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } \right]$
C) $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$
D) $f ( x ) = \frac { 1 } { 3 } \left[ 10 + \left( 1 - x ^ { 2 } \right) ^ { 1 / 2 } \right]$
E) $f ( x ) = \frac { 1 } { 3 } \left[ 10 - \left( 1 - x ^ { 2 } \right) ^ { 2 / 3 } \right]$

Question

Quizplus · Accepted Answer

To find the equation of $f(x)$, we integrate $f'(x) = x\sqrt{1-x^2}$. The antiderivative of $x\sqrt{1-x^2}$ is $-\frac{1}{3}(1-x^2)^{3/2} + C$. Using the point $\left(0, \frac{7}{3}\right)$, we find $C = \frac{8}{3}$. Thus, $f(x) = \frac{1}{3}\left[8 - (1-x^2)^{3/2}\right]$.

Find the Equation of the Function F Whose Graph Passes (0,73)\left( 0 , \frac { 7 } { 3 } \right)(0,37​)

Find the Equation of the Function F Whose Graph Passes $\left( 0 , \frac { 7 } { 3 } \right)$