Question

Find the points on the curve y = x³ at which the slope of the tangent is equal to the Y-coordinate of the point.

Answer 1

The equation of the given curve is y = x³

\(\frac{dy}{dx}\)=3x²

The slope of the tangent to a curve at (x, y) is given by,

Therefore, the slope of the tangent at the point where x = 2 is given by,

dy/dx]_{x,y) =3x}²

When the slope of the tangent is equal to the y-coordinate of the point, then y = 3x² . Also, we have y = x³ .

∴3x² = x³

∴ x² (x − 3) = 0

∴ x = 0, x = 3

When x = 0, then y = 0 and when x = 3, then y = 3(3) ² = 27.

Hence, the required points are (0, 0) and (3, 27).