Find the function with the given derivative whose graph passes through the point P.
-$g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 2,2 )$
A) $g ( x ) = - \frac { 1 } { x } - x ^ { 2 } - \frac { 5 } { 2 }$
B) $g ( x ) = \frac { 1 } { x } + x ^ { 2 } + \frac { 5 } { 2 }$
C) $g ( x ) = \frac { 1 } { x } + x ^ { 2 } - \frac { 5 } { 2 }$
D) $g ( x ) = - \frac { 1 } { x } + x ^ { 2 } - \frac { 5 } { 2 }$

Question

Quizplus · Accepted Answer

To find the function $g(x)$ given its derivative $g'(x) = \frac{1}{x^2} + 2x$, we integrate $g'(x)$ with respect to $x$. The integral of $\frac{1}{x^2}$ is $-\frac{1}{x}$, and the integral of $2x$ is $x^2$. Thus, the general form of $g(x)$ is $-\frac{1}{x} + x^2 + C$, where $C$ is the constant of integration. To find $C$, we use the point $P(-2, 2)$ which lies on the graph of $g(x)$. Substituting $x = -2$ and $g(x) = 2$ into the equation gives $2 = -\frac{1}{-2} + (-2)^2 + C$, which simplifies to $2 = \frac{1}{2} + 4 + C$. Solving for $C$ gives $C = 2 - \frac{1}{2} - 4 = -\frac{5}{2}$. Therefore, the correct function is $g(x) = -\frac{1}{x} + x^2 - \frac{5}{2}$.

Find the Function with the Given Derivative Whose Graph Passes g′(x)=1x2+2x,P(−2,2)g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 2,2 )g′(x)=x21​+2x,P(−2,2)

Find the Function with the Given Derivative Whose Graph Passes $g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 2,2 )$