Question

A curve has equation  and it is given that . The point A is the only point  on the curve at which the gradient is −1.

     i.       Find the x-coordinate of A.

   ii.       Given that the curve also passes through the point (4,10), find the y-coordinate of A, giving  your answer as a fraction.

Solution

     i.
 

We are given that;

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is:

Therefore;

We are also given that point A is the only point on the curve at which the gradient is −1. 

Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point.

Gradient (slope)  of the curve  at a particular point  can be found by substituting x- coordinates of that point in the expression for gradient of the curve;

Therefore;

Hence;

Now we have two options.

Two possible values of  imply that there are two points on the curve, one at each value of , where  curve has gradient -1.

But we are given that point A is the only point on the curve at which the gradient is −1. 

We can see that substitution of both values of  gives;

For	For
	Hence,  is not valid option.

Therefore;

   ii.
 

We are required to find the y-coordinate of point A.

To find y-coordinate of a point  on the curve, we substitute value of x-coordinate of the point     on the curve in the equation of the curve.  

We have already found x-coordinate of point A in (i).

Therefore, we need equation of the curve.

We are given that;

We can find equation of the curve from its derivative through integration;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

If a point   lies on the curve , we can find out value of . We substitute values of  and    in the equation obtained from integration of the derivative of the curve i.e. .

We are given that the curve passes through the point (4,10).

Therefore, substituting  and  in the above derived equation of the curve;

Hence, equation of the curve is;

Now we can find y-coordinate of point A by substituting x-coordinate of point A  in equation of  the curve;