Question

A curve is such that .

    i.       A point P moves along the curve in such a way that the x-coordinate is increasing at a  constant rate of 0.3 units per second. Find the rate of change of the y-coordinate as P crosses the  y-axis.

The curve intersects the y-axis where  .

   ii.       Find the equation of the curve.

Solution

i.
 

We are given that;

We are required to find the rate of change of y-coordinates as P crosses the y-axis i.e. at y- intercept;

We know that;

We are given;

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;

Therefore;

Therefore, at  we have .

ii.
 

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

We are given that;

Therefore;

Rule for integration of  is:

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 curve crosses y-axis at . It is evident that at this point . 

Substituting these values in above equation;

Hence equation of the curve can be written as;