Find the curve y = f(x) in the xy-plane that has the given properties.
-$f ( x )$ has a slope at each point given by $- \frac { 1 } { x ^ { 2 } }$ and passes through the point $\left( \frac { 1 } { 4 } , 7 \right)$
A) $y = \frac { 3 } { x ^ { 3 } } + 3$
B) $y = \frac { 1 } { x } + 3$
C) $y = \frac { 2 } { x } + 3$
D) $y = - \frac { 1 } { x } + 5$

Question

Quizplus · Accepted Answer

The slope of the curve is given as $- \frac { 1 } { x ^ { 2 } }$. Integrating this with respect to x gives us the formula for $f(x)$.
\begin{align*}
f'(x) & = - \frac { 1 } { x ^ { 2 } } \
\Rightarrow \quad f(x) & = \int - \frac { 1 } { x ^ { 2 } } dx \
& = \frac { 1 } { x } + C
\end{align*}
We use the point given to find the value of the constant C.
\begin{align*}
f \left( \frac { 1 } { 4 } \right) & = 7 \
\Rightarrow \quad 7 & = \frac { 1 } { \frac { 1 } { 4 } } + C \
\Rightarrow \quad C & = 3
\end{align*}
So the equation of the curve is
$$y = \frac { 1 } { x } + 3$$
Therefore, the correct answer is B.

Find the Curve Y = F(x) in the Xy-Plane That f(x)f ( x )f(x)

Find the Curve Y = F(x) in the Xy-Plane That $f ( x )$