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

The equation of the given curve is $y=x^3$.

$\therefore \frac{dy}{dx}=3x^2$

When the slope of the tangent is equal to the $y$-coordinate of the point,

then $\frac{dy}{dx}=y=3x^2$ .

Also, we have $y=x^3$ .

$3x^2=x^3\Rightarrow x^2(x-3)=0\Rightarrow x=0,x=3$

When $x=0$, then $y=0$ and when $x=3$, then $y=3(3{)}^2=27$.

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