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.

Solution

The equation of curve is $straight y space equals space straight x cubed$
$therefore space space space space dy over dx space equals space 3 straight x squared$
$Let space space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space be space the space point.$
$At space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space dy over dx space equals space 3 straight x subscript 1 squared comma space space space which space is space straight a space slope space of space tangent. space$
From given condition,
$3 straight x subscript 1 squared space equals space straight y subscript 1$ ...(1)
Also, $left parenthesis straight x subscript 1 comma space straight y subscript 1 right parenthesis comma space lies space on space straight y space equals space straight x cubed$
$therefore space space space space space space space straight y subscript 1 space equals space straight x subscript 1 cubed$
From (1) and (2), $straight x subscript 1 cubed space equals space 3 straight x subscript 1 squared space space space space rightwards double arrow space space space space straight x subscript 1 squared left parenthesis straight x subscript 1 minus 3 right parenthesis space equals space 0 space space space space rightwards double arrow space space space space straight x subscript 1 space equals space 0 comma space space 3$
From (1), $straight y subscript 1 space equals 0 comma space 27$
$therefore space space space space space space points space are space left parenthesis 0 comma space 0 right parenthesis comma space space space left parenthesis 3 comma space 27 right parenthesis.$

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Application Of Derivatives

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

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

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

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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