Question

The equation of a curve passing through \[\left( {2,\dfrac{7}{2}} \right)\], and having gradient \[1 - \dfrac{1}{{{x^2}}}\]at (x,y) is:

Answer 1

Hint: Here the given question need to write the equation of the curve in which we are provided with the gradient of the curve with a point which satisfy the curve, in order to write the equation of the curve we need to use the gradient and point given.

Complete step-by-step solution:
Here we need to write the equation for the curve, where we have the gradient, here we will first solve the gradient, on solving we have:
\[\Rightarrow \dfrac{{dy}}{{dx}} = 1 - \dfrac{1}{{{x^2}}} \\
   \Rightarrow dy = \left( {1 - \dfrac{1}{{{x^2}}}} \right)dx \]
To solve the above formed equation here we need to integrate both side of the equations, on solving we get:
\[ \Rightarrow \int {dy} = \int {\left( {1 - \dfrac{1}{{{x^2}}}} \right)dx} \\
   \Rightarrow y = \int {dx - \int {\dfrac{1}{{{x^2}}}} } \\
   \Rightarrow y = x + \left( {\dfrac{1}{x}} \right) + c \]
Now here we know that the equation passes through \[\left( {2,\dfrac{7}{2}} \right)\]
\[
   \Rightarrow y = x + \left( {\dfrac{1}{x}} \right) + c \\
   \Rightarrow \dfrac{7}{2} = 2 + \dfrac{1}{2} + c \\
   \Rightarrow c = 1 \]
Here we get the value of the integration constant “c”.
Now putting the value of the constant in the equation obtained we get:
\[ \Rightarrow y = x + \left( {\dfrac{1}{x}} \right) + 1 \\
   \Rightarrow xy = {x^2} + x + 1 \]
This is the required equation of the curve.

Note: Here in the given question we are provided with the gradient and thus needed to solve for the equation of the curve, and accordingly we first solve the gradient given and then by putting values we get the final equation of the curve.