17) 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.

Answers (1)

Given equation of curve is  y = x ^3
Slope of tangent = \frac{dy}{dx} = 3x^2
it is given that  the slope of the tangent is equal to the y-coordinate of the point
3x^2 = y
We have  y = x ^3
3x^2 = x^3\\ 3x^2 - x^3=0\\ x^2(3-x)=0\\ x= 0 \ \ \ \ \ \ \ \ and \ \ \ \ \ \ \ \ \ \ x = 3
So, when x = 0 , y = 0
and when x = 3 , y = x^3 = 3^3 = 27

Hence, the coordinates are (3,27) and (0,0)

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