Question

Find the points on the curve y = x 3 at which the slope of the tangent is equal to the y -coordinate of the point.

Answer 1

Equation of the curve is given as,

y= x 3 (1)

The slope of the tangent to the curve at a given point x= x 0 is given as,

slope = ( dy dx ) x= x 0

The slope of the given curve is given as,

dy dx = d( x 3 ) dx =3 x 2 (2)

It is given that the tangent is equal to the y coordinate of the point, hence from equation (1) and (2)

y=3 x 2 x 3 =3 x 2 x 3 −3 x 2 =0 x 2 ( x−3 )=0 x=0, 0 and 3

Coordinate of y when x=0 is,

y=3 ( 0 ) 2 =0

Coordinate of y when x=3 is,

y=3 ( 3 ) 2 =27

Thus, the points at which the slope of the tangent is equal to y-coordinate are ( 0,0 ) and ( 3,27 ).